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979fa03571
Now rounding functionality comes first, and then exception functionality. Before, it was a back-and-forth between them.
8161 lines
240 KiB
D
8161 lines
240 KiB
D
// Written in the D programming language.
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/**
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* Contains the elementary mathematical functions (powers, roots,
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* and trigonometric functions), and low-level floating-point operations.
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* Mathematical special functions are available in $(D std.mathspecial).
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*
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$(SCRIPT inhibitQuickIndex = 1;)
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||
|
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$(DIVC quickindex,
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$(BOOKTABLE ,
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$(TR $(TH Category) $(TH Members) )
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||
$(TR $(TDNW Constants) $(TD
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||
$(MYREF E) $(MYREF PI) $(MYREF PI_2) $(MYREF PI_4) $(MYREF M_1_PI)
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$(MYREF M_2_PI) $(MYREF M_2_SQRTPI) $(MYREF LN10) $(MYREF LN2)
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$(MYREF LOG2) $(MYREF LOG2E) $(MYREF LOG2T) $(MYREF LOG10E)
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$(MYREF SQRT2) $(MYREF SQRT1_2)
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))
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$(TR $(TDNW Classics) $(TD
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||
$(MYREF abs) $(MYREF fabs) $(MYREF sqrt) $(MYREF cbrt) $(MYREF hypot)
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$(MYREF poly) $(MYREF nextPow2) $(MYREF truncPow2)
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))
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$(TR $(TDNW Trigonometry) $(TD
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$(MYREF sin) $(MYREF cos) $(MYREF tan) $(MYREF asin) $(MYREF acos)
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$(MYREF atan) $(MYREF atan2) $(MYREF sinh) $(MYREF cosh) $(MYREF tanh)
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$(MYREF asinh) $(MYREF acosh) $(MYREF atanh) $(MYREF expi)
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))
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$(TR $(TDNW Rounding) $(TD
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$(MYREF ceil) $(MYREF floor) $(MYREF round) $(MYREF lround)
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$(MYREF trunc) $(MYREF rint) $(MYREF lrint) $(MYREF nearbyint)
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$(MYREF rndtol) $(MYREF quantize)
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))
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$(TR $(TDNW Exponentiation & Logarithms) $(TD
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$(MYREF pow) $(MYREF exp) $(MYREF exp2) $(MYREF expm1) $(MYREF ldexp)
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$(MYREF frexp) $(MYREF log) $(MYREF log2) $(MYREF log10) $(MYREF logb)
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$(MYREF ilogb) $(MYREF log1p) $(MYREF scalbn)
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))
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$(TR $(TDNW Modulus) $(TD
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$(MYREF fmod) $(MYREF modf) $(MYREF remainder)
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))
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$(TR $(TDNW Floating-point operations) $(TD
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||
$(MYREF approxEqual) $(MYREF feqrel) $(MYREF fdim) $(MYREF fmax)
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$(MYREF fmin) $(MYREF fma) $(MYREF nextDown) $(MYREF nextUp)
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$(MYREF nextafter) $(MYREF NaN) $(MYREF getNaNPayload)
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$(MYREF cmp)
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))
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$(TR $(TDNW Introspection) $(TD
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$(MYREF isFinite) $(MYREF isIdentical) $(MYREF isInfinity) $(MYREF isNaN)
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$(MYREF isNormal) $(MYREF isSubnormal) $(MYREF signbit) $(MYREF sgn)
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$(MYREF copysign) $(MYREF isPowerOf2)
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))
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$(TR $(TDNW Complex Numbers) $(TD
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||
$(MYREF abs) $(MYREF conj) $(MYREF sin) $(MYREF cos) $(MYREF expi)
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))
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$(TR $(TDNW Hardware Control) $(TD
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$(MYREF IeeeFlags) $(MYREF FloatingPointControl)
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))
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)
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)
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* The functionality closely follows the IEEE754-2008 standard for
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* floating-point arithmetic, including the use of camelCase names rather
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* than C99-style lower case names. All of these functions behave correctly
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* when presented with an infinity or NaN.
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*
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* The following IEEE 'real' formats are currently supported:
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* $(UL
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* $(LI 64 bit Big-endian 'double' (eg PowerPC))
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* $(LI 128 bit Big-endian 'quadruple' (eg SPARC))
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* $(LI 64 bit Little-endian 'double' (eg x86-SSE2))
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* $(LI 80 bit Little-endian, with implied bit 'real80' (eg x87, Itanium))
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* $(LI 128 bit Little-endian 'quadruple' (not implemented on any known processor!))
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* $(LI Non-IEEE 128 bit Big-endian 'doubledouble' (eg PowerPC) has partial support)
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* )
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* Unlike C, there is no global 'errno' variable. Consequently, almost all of
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* these functions are pure nothrow.
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*
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* Status:
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* The semantics and names of feqrel and approxEqual will be revised.
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*
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* Macros:
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* TABLE_SV = <table border="1" cellpadding="4" cellspacing="0">
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* <caption>Special Values</caption>
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* $0</table>
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* SVH = $(TR $(TH $1) $(TH $2))
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* SV = $(TR $(TD $1) $(TD $2))
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* TH3 = $(TR $(TH $1) $(TH $2) $(TH $3))
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* TD3 = $(TR $(TD $1) $(TD $2) $(TD $3))
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* TABLE_DOMRG = <table border="1" cellpadding="4" cellspacing="0">
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* $(SVH Domain X, Range Y)
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$(SV $1, $2)
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* </table>
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* DOMAIN=$1
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* RANGE=$1
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|
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* NAN = $(RED NAN)
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* SUP = <span style="vertical-align:super;font-size:smaller">$0</span>
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* GAMMA = Γ
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* THETA = θ
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* INTEGRAL = ∫
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* INTEGRATE = $(BIG ∫<sub>$(SMALL $1)</sub><sup>$2</sup>)
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* POWER = $1<sup>$2</sup>
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* SUB = $1<sub>$2</sub>
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* BIGSUM = $(BIG Σ <sup>$2</sup><sub>$(SMALL $1)</sub>)
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* CHOOSE = $(BIG () <sup>$(SMALL $1)</sup><sub>$(SMALL $2)</sub> $(BIG ))
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* PLUSMN = ±
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* INFIN = ∞
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* PLUSMNINF = ±∞
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* PI = π
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* LT = <
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* GT = >
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* SQRT = √
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* HALF = ½
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*
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* Copyright: Copyright Digital Mars 2000 - 2011.
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* D implementations of tan, atan, atan2, exp, expm1, exp2, log, log10, log1p,
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* log2, floor, ceil and lrint functions are based on the CEPHES math library,
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* which is Copyright (C) 2001 Stephen L. Moshier $(LT)steve@moshier.net$(GT)
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* and are incorporated herein by permission of the author. The author
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* reserves the right to distribute this material elsewhere under different
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* copying permissions. These modifications are distributed here under
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* the following terms:
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* License: $(HTTP www.boost.org/LICENSE_1_0.txt, Boost License 1.0).
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* Authors: $(HTTP digitalmars.com, Walter Bright), Don Clugston,
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* Conversion of CEPHES math library to D by Iain Buclaw and David Nadlinger
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* Source: $(PHOBOSSRC std/_math.d)
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*/
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module std.math;
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version (Win64)
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{
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version (D_InlineAsm_X86_64)
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version = Win64_DMD_InlineAsm;
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}
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static import core.math;
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static import core.stdc.math;
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import std.traits; // CommonType, isFloatingPoint, isIntegral, isSigned, isUnsigned, Largest, Unqual
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version(LDC)
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{
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import ldc.intrinsics;
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}
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version(DigitalMars)
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{
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version = INLINE_YL2X; // x87 has opcodes for these
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}
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version (X86) version = X86_Any;
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version (X86_64) version = X86_Any;
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version (PPC) version = PPC_Any;
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version (PPC64) version = PPC_Any;
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version(D_InlineAsm_X86)
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{
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version = InlineAsm_X86_Any;
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}
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else version(D_InlineAsm_X86_64)
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{
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version = InlineAsm_X86_Any;
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}
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version (X86_64) version = StaticallyHaveSSE;
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version (X86) version (OSX) version = StaticallyHaveSSE;
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version (StaticallyHaveSSE)
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{
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private enum bool haveSSE = true;
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}
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else
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{
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static import core.cpuid;
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private alias haveSSE = core.cpuid.sse;
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}
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version(unittest)
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{
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import core.stdc.stdio; // : sprintf;
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static if (real.sizeof > double.sizeof)
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enum uint useDigits = 16;
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else
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enum uint useDigits = 15;
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/******************************************
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* Compare floating point numbers to n decimal digits of precision.
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* Returns:
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* 1 match
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* 0 nomatch
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*/
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private bool equalsDigit(real x, real y, uint ndigits)
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{
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if (signbit(x) != signbit(y))
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return 0;
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if (isInfinity(x) && isInfinity(y))
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return 1;
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if (isInfinity(x) || isInfinity(y))
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return 0;
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if (isNaN(x) && isNaN(y))
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return 1;
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if (isNaN(x) || isNaN(y))
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return 0;
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char[30] bufx;
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char[30] bufy;
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assert(ndigits < bufx.length);
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int ix;
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int iy;
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version(CRuntime_Microsoft)
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alias real_t = double;
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else
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alias real_t = real;
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ix = sprintf(bufx.ptr, "%.*Lg", ndigits, cast(real_t) x);
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iy = sprintf(bufy.ptr, "%.*Lg", ndigits, cast(real_t) y);
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assert(ix < bufx.length && ix > 0);
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assert(ix < bufy.length && ix > 0);
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return bufx[0 .. ix] == bufy[0 .. iy];
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}
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}
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package:
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// The following IEEE 'real' formats are currently supported.
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version(LittleEndian)
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{
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static assert(real.mant_dig == 53 || real.mant_dig == 64
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|| real.mant_dig == 113,
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"Only 64-bit, 80-bit, and 128-bit reals"~
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" are supported for LittleEndian CPUs");
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}
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else
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{
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static assert(real.mant_dig == 53 || real.mant_dig == 106
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|| real.mant_dig == 113,
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"Only 64-bit and 128-bit reals are supported for BigEndian CPUs."~
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" double-double reals have partial support");
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}
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// Underlying format exposed through floatTraits
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enum RealFormat
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{
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ieeeHalf,
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ieeeSingle,
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ieeeDouble,
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ieeeExtended, // x87 80-bit real
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ieeeExtended53, // x87 real rounded to precision of double.
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ibmExtended, // IBM 128-bit extended
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ieeeQuadruple,
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}
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// Constants used for extracting the components of the representation.
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// They supplement the built-in floating point properties.
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template floatTraits(T)
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{
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// EXPMASK is a ushort mask to select the exponent portion (without sign)
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// EXPSHIFT is the number of bits the exponent is left-shifted by in its ushort
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// EXPBIAS is the exponent bias - 1 (exp == EXPBIAS yields ×2^-1).
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// EXPPOS_SHORT is the index of the exponent when represented as a ushort array.
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// SIGNPOS_BYTE is the index of the sign when represented as a ubyte array.
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// RECIP_EPSILON is the value such that (smallest_subnormal) * RECIP_EPSILON == T.min_normal
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enum T RECIP_EPSILON = (1/T.epsilon);
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static if (T.mant_dig == 24)
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{
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// Single precision float
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enum ushort EXPMASK = 0x7F80;
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enum ushort EXPSHIFT = 7;
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enum ushort EXPBIAS = 0x3F00;
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enum uint EXPMASK_INT = 0x7F80_0000;
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enum uint MANTISSAMASK_INT = 0x007F_FFFF;
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enum realFormat = RealFormat.ieeeSingle;
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version(LittleEndian)
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{
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enum EXPPOS_SHORT = 1;
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enum SIGNPOS_BYTE = 3;
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}
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else
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{
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enum EXPPOS_SHORT = 0;
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enum SIGNPOS_BYTE = 0;
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}
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}
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else static if (T.mant_dig == 53)
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{
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static if (T.sizeof == 8)
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{
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// Double precision float, or real == double
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enum ushort EXPMASK = 0x7FF0;
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enum ushort EXPSHIFT = 4;
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enum ushort EXPBIAS = 0x3FE0;
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enum uint EXPMASK_INT = 0x7FF0_0000;
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enum uint MANTISSAMASK_INT = 0x000F_FFFF; // for the MSB only
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enum realFormat = RealFormat.ieeeDouble;
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version(LittleEndian)
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{
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enum EXPPOS_SHORT = 3;
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enum SIGNPOS_BYTE = 7;
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}
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else
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{
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enum EXPPOS_SHORT = 0;
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enum SIGNPOS_BYTE = 0;
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}
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}
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else static if (T.sizeof == 12)
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{
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// Intel extended real80 rounded to double
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enum ushort EXPMASK = 0x7FFF;
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enum ushort EXPSHIFT = 0;
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enum ushort EXPBIAS = 0x3FFE;
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enum realFormat = RealFormat.ieeeExtended53;
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version(LittleEndian)
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{
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enum EXPPOS_SHORT = 4;
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enum SIGNPOS_BYTE = 9;
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}
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else
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{
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enum EXPPOS_SHORT = 0;
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enum SIGNPOS_BYTE = 0;
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}
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}
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else
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static assert(false, "No traits support for " ~ T.stringof);
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}
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else static if (T.mant_dig == 64)
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{
|
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// Intel extended real80
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enum ushort EXPMASK = 0x7FFF;
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enum ushort EXPSHIFT = 0;
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enum ushort EXPBIAS = 0x3FFE;
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enum realFormat = RealFormat.ieeeExtended;
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version(LittleEndian)
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{
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enum EXPPOS_SHORT = 4;
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enum SIGNPOS_BYTE = 9;
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||
}
|
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else
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{
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enum EXPPOS_SHORT = 0;
|
||
enum SIGNPOS_BYTE = 0;
|
||
}
|
||
}
|
||
else static if (T.mant_dig == 113)
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{
|
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// Quadruple precision float
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enum ushort EXPMASK = 0x7FFF;
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enum ushort EXPSHIFT = 0;
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enum ushort EXPBIAS = 0x3FFE;
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enum realFormat = RealFormat.ieeeQuadruple;
|
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version(LittleEndian)
|
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{
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enum EXPPOS_SHORT = 7;
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enum SIGNPOS_BYTE = 15;
|
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}
|
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else
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{
|
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enum EXPPOS_SHORT = 0;
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enum SIGNPOS_BYTE = 0;
|
||
}
|
||
}
|
||
else static if (T.mant_dig == 106)
|
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{
|
||
// IBM Extended doubledouble
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enum ushort EXPMASK = 0x7FF0;
|
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enum ushort EXPSHIFT = 4;
|
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enum realFormat = RealFormat.ibmExtended;
|
||
// the exponent byte is not unique
|
||
version(LittleEndian)
|
||
{
|
||
enum EXPPOS_SHORT = 7; // [3] is also an exp short
|
||
enum SIGNPOS_BYTE = 15;
|
||
}
|
||
else
|
||
{
|
||
enum EXPPOS_SHORT = 0; // [4] is also an exp short
|
||
enum SIGNPOS_BYTE = 0;
|
||
}
|
||
}
|
||
else
|
||
static assert(false, "No traits support for " ~ T.stringof);
|
||
}
|
||
|
||
// These apply to all floating-point types
|
||
version(LittleEndian)
|
||
{
|
||
enum MANTISSA_LSB = 0;
|
||
enum MANTISSA_MSB = 1;
|
||
}
|
||
else
|
||
{
|
||
enum MANTISSA_LSB = 1;
|
||
enum MANTISSA_MSB = 0;
|
||
}
|
||
|
||
// Common code for math implementations.
|
||
|
||
// Helper for floor/ceil
|
||
T floorImpl(T)(const T x) @trusted pure nothrow @nogc
|
||
{
|
||
alias F = floatTraits!(T);
|
||
// Take care not to trigger library calls from the compiler,
|
||
// while ensuring that we don't get defeated by some optimizers.
|
||
union floatBits
|
||
{
|
||
T rv;
|
||
ushort[T.sizeof/2] vu;
|
||
}
|
||
floatBits y = void;
|
||
y.rv = x;
|
||
|
||
// Find the exponent (power of 2)
|
||
static if (F.realFormat == RealFormat.ieeeSingle)
|
||
{
|
||
int exp = ((y.vu[F.EXPPOS_SHORT] >> 7) & 0xff) - 0x7f;
|
||
|
||
version (LittleEndian)
|
||
int pos = 0;
|
||
else
|
||
int pos = 3;
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeDouble)
|
||
{
|
||
int exp = ((y.vu[F.EXPPOS_SHORT] >> 4) & 0x7ff) - 0x3ff;
|
||
|
||
version (LittleEndian)
|
||
int pos = 0;
|
||
else
|
||
int pos = 3;
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeExtended)
|
||
{
|
||
int exp = (y.vu[F.EXPPOS_SHORT] & 0x7fff) - 0x3fff;
|
||
|
||
version (LittleEndian)
|
||
int pos = 0;
|
||
else
|
||
int pos = 4;
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeQuadruple)
|
||
{
|
||
int exp = (y.vu[F.EXPPOS_SHORT] & 0x7fff) - 0x3fff;
|
||
|
||
version (LittleEndian)
|
||
int pos = 0;
|
||
else
|
||
int pos = 7;
|
||
}
|
||
else
|
||
static assert(false, "Not implemented for this architecture");
|
||
|
||
if (exp < 0)
|
||
{
|
||
if (x < 0.0)
|
||
return -1.0;
|
||
else
|
||
return 0.0;
|
||
}
|
||
|
||
exp = (T.mant_dig - 1) - exp;
|
||
|
||
// Zero 16 bits at a time.
|
||
while (exp >= 16)
|
||
{
|
||
version (LittleEndian)
|
||
y.vu[pos++] = 0;
|
||
else
|
||
y.vu[pos--] = 0;
|
||
exp -= 16;
|
||
}
|
||
|
||
// Clear the remaining bits.
|
||
if (exp > 0)
|
||
y.vu[pos] &= 0xffff ^ ((1 << exp) - 1);
|
||
|
||
if ((x < 0.0) && (x != y.rv))
|
||
y.rv -= 1.0;
|
||
|
||
return y.rv;
|
||
}
|
||
|
||
public:
|
||
|
||
// Values obtained from Wolfram Alpha. 116 bits ought to be enough for anybody.
|
||
// Wolfram Alpha LLC. 2011. Wolfram|Alpha. http://www.wolframalpha.com/input/?i=e+in+base+16 (access July 6, 2011).
|
||
enum real E = 0x1.5bf0a8b1457695355fb8ac404e7a8p+1L; /** e = 2.718281... */
|
||
enum real LOG2T = 0x1.a934f0979a3715fc9257edfe9b5fbp+1L; /** $(SUB log, 2)10 = 3.321928... */
|
||
enum real LOG2E = 0x1.71547652b82fe1777d0ffda0d23a8p+0L; /** $(SUB log, 2)e = 1.442695... */
|
||
enum real LOG2 = 0x1.34413509f79fef311f12b35816f92p-2L; /** $(SUB log, 10)2 = 0.301029... */
|
||
enum real LOG10E = 0x1.bcb7b1526e50e32a6ab7555f5a67cp-2L; /** $(SUB log, 10)e = 0.434294... */
|
||
enum real LN2 = 0x1.62e42fefa39ef35793c7673007e5fp-1L; /** ln 2 = 0.693147... */
|
||
enum real LN10 = 0x1.26bb1bbb5551582dd4adac5705a61p+1L; /** ln 10 = 2.302585... */
|
||
enum real PI = 0x1.921fb54442d18469898cc51701b84p+1L; /** $(_PI) = 3.141592... */
|
||
enum real PI_2 = PI/2; /** $(PI) / 2 = 1.570796... */
|
||
enum real PI_4 = PI/4; /** $(PI) / 4 = 0.785398... */
|
||
enum real M_1_PI = 0x1.45f306dc9c882a53f84eafa3ea69cp-2L; /** 1 / $(PI) = 0.318309... */
|
||
enum real M_2_PI = 2*M_1_PI; /** 2 / $(PI) = 0.636619... */
|
||
enum real M_2_SQRTPI = 0x1.20dd750429b6d11ae3a914fed7fd8p+0L; /** 2 / $(SQRT)$(PI) = 1.128379... */
|
||
enum real SQRT2 = 0x1.6a09e667f3bcc908b2fb1366ea958p+0L; /** $(SQRT)2 = 1.414213... */
|
||
enum real SQRT1_2 = SQRT2/2; /** $(SQRT)$(HALF) = 0.707106... */
|
||
// Note: Make sure the magic numbers in compiler backend for x87 match these.
|
||
|
||
|
||
/***********************************
|
||
* Calculates the absolute value of a number
|
||
*
|
||
* Params:
|
||
* Num = (template parameter) type of number
|
||
* x = real number value
|
||
* z = complex number value
|
||
* y = imaginary number value
|
||
*
|
||
* Returns:
|
||
* The absolute value of the number. If floating-point or integral,
|
||
* the return type will be the same as the input; if complex or
|
||
* imaginary, the returned value will be the corresponding floating
|
||
* point type.
|
||
*
|
||
* For complex numbers, abs(z) = sqrt( $(POWER z.re, 2) + $(POWER z.im, 2) )
|
||
* = hypot(z.re, z.im).
|
||
*/
|
||
Num abs(Num)(Num x) @safe pure nothrow
|
||
if (is(typeof(Num.init >= 0)) && is(typeof(-Num.init)) &&
|
||
!(is(Num* : const(ifloat*)) || is(Num* : const(idouble*))
|
||
|| is(Num* : const(ireal*))))
|
||
{
|
||
static if (isFloatingPoint!(Num))
|
||
return fabs(x);
|
||
else
|
||
return x >= 0 ? x : -x;
|
||
}
|
||
|
||
/// ditto
|
||
auto abs(Num)(Num z) @safe pure nothrow @nogc
|
||
if (is(Num* : const(cfloat*)) || is(Num* : const(cdouble*))
|
||
|| is(Num* : const(creal*)))
|
||
{
|
||
return hypot(z.re, z.im);
|
||
}
|
||
|
||
/// ditto
|
||
auto abs(Num)(Num y) @safe pure nothrow @nogc
|
||
if (is(Num* : const(ifloat*)) || is(Num* : const(idouble*))
|
||
|| is(Num* : const(ireal*)))
|
||
{
|
||
return fabs(y.im);
|
||
}
|
||
|
||
/// ditto
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
assert(isIdentical(abs(-0.0L), 0.0L));
|
||
assert(isNaN(abs(real.nan)));
|
||
assert(abs(-real.infinity) == real.infinity);
|
||
assert(abs(-3.2Li) == 3.2L);
|
||
assert(abs(71.6Li) == 71.6L);
|
||
assert(abs(-56) == 56);
|
||
assert(abs(2321312L) == 2321312L);
|
||
assert(abs(-1L+1i) == sqrt(2.0L));
|
||
}
|
||
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
import std.meta : AliasSeq;
|
||
foreach (T; AliasSeq!(float, double, real))
|
||
{
|
||
T f = 3;
|
||
assert(abs(f) == f);
|
||
assert(abs(-f) == f);
|
||
}
|
||
foreach (T; AliasSeq!(cfloat, cdouble, creal))
|
||
{
|
||
T f = -12+3i;
|
||
assert(abs(f) == hypot(f.re, f.im));
|
||
assert(abs(-f) == hypot(f.re, f.im));
|
||
}
|
||
}
|
||
|
||
/***********************************
|
||
* Complex conjugate
|
||
*
|
||
* conj(x + iy) = x - iy
|
||
*
|
||
* Note that z * conj(z) = $(POWER z.re, 2) - $(POWER z.im, 2)
|
||
* is always a real number
|
||
*/
|
||
auto conj(Num)(Num z) @safe pure nothrow @nogc
|
||
if (is(Num* : const(cfloat*)) || is(Num* : const(cdouble*))
|
||
|| is(Num* : const(creal*)))
|
||
{
|
||
//FIXME
|
||
//Issue 14206
|
||
static if (is(Num* : const(cdouble*)))
|
||
return cast(cdouble) conj(cast(creal) z);
|
||
else
|
||
return z.re - z.im*1fi;
|
||
}
|
||
|
||
/** ditto */
|
||
auto conj(Num)(Num y) @safe pure nothrow @nogc
|
||
if (is(Num* : const(ifloat*)) || is(Num* : const(idouble*))
|
||
|| is(Num* : const(ireal*)))
|
||
{
|
||
return -y;
|
||
}
|
||
|
||
///
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
creal c = 7 + 3Li;
|
||
assert(conj(c) == 7-3Li);
|
||
ireal z = -3.2Li;
|
||
assert(conj(z) == -z);
|
||
}
|
||
//Issue 14206
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
cdouble c = 7 + 3i;
|
||
assert(conj(c) == 7-3i);
|
||
idouble z = -3.2i;
|
||
assert(conj(z) == -z);
|
||
}
|
||
//Issue 14206
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
cfloat c = 7f + 3fi;
|
||
assert(conj(c) == 7f-3fi);
|
||
ifloat z = -3.2fi;
|
||
assert(conj(z) == -z);
|
||
}
|
||
|
||
/***********************************
|
||
* Returns cosine of x. x is in radians.
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH x) $(TH cos(x)) $(TH invalid?))
|
||
* $(TR $(TD $(NAN)) $(TD $(NAN)) $(TD yes) )
|
||
* $(TR $(TD $(PLUSMN)$(INFIN)) $(TD $(NAN)) $(TD yes) )
|
||
* )
|
||
* Bugs:
|
||
* Results are undefined if |x| >= $(POWER 2,64).
|
||
*/
|
||
|
||
real cos(real x) @safe pure nothrow @nogc { pragma(inline, true); return core.math.cos(x); }
|
||
//FIXME
|
||
///ditto
|
||
double cos(double x) @safe pure nothrow @nogc { return cos(cast(real) x); }
|
||
//FIXME
|
||
///ditto
|
||
float cos(float x) @safe pure nothrow @nogc { return cos(cast(real) x); }
|
||
|
||
@safe unittest
|
||
{
|
||
real function(real) pcos = &cos;
|
||
assert(pcos != null);
|
||
}
|
||
|
||
/***********************************
|
||
* Returns $(HTTP en.wikipedia.org/wiki/Sine, sine) of x. x is in $(HTTP en.wikipedia.org/wiki/Radian, radians).
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TH3 x , sin(x) , invalid?)
|
||
* $(TD3 $(NAN) , $(NAN) , yes )
|
||
* $(TD3 $(PLUSMN)0.0, $(PLUSMN)0.0, no )
|
||
* $(TD3 $(PLUSMNINF), $(NAN) , yes )
|
||
* )
|
||
*
|
||
* Params:
|
||
* x = angle in radians (not degrees)
|
||
* Returns:
|
||
* sine of x
|
||
* See_Also:
|
||
* $(MYREF cos), $(MYREF tan), $(MYREF asin)
|
||
* Bugs:
|
||
* Results are undefined if |x| >= $(POWER 2,64).
|
||
*/
|
||
|
||
real sin(real x) @safe pure nothrow @nogc { pragma(inline, true); return core.math.sin(x); }
|
||
//FIXME
|
||
///ditto
|
||
double sin(double x) @safe pure nothrow @nogc { return sin(cast(real) x); }
|
||
//FIXME
|
||
///ditto
|
||
float sin(float x) @safe pure nothrow @nogc { return sin(cast(real) x); }
|
||
|
||
///
|
||
@safe unittest
|
||
{
|
||
import std.math : sin, PI;
|
||
import std.stdio : writefln;
|
||
|
||
void someFunc()
|
||
{
|
||
real x = 30.0;
|
||
auto result = sin(x * (PI / 180)); // convert degrees to radians
|
||
writefln("The sine of %s degrees is %s", x, result);
|
||
}
|
||
}
|
||
|
||
@safe unittest
|
||
{
|
||
real function(real) psin = &sin;
|
||
assert(psin != null);
|
||
}
|
||
|
||
/***********************************
|
||
* Returns sine for complex and imaginary arguments.
|
||
*
|
||
* sin(z) = sin(z.re)*cosh(z.im) + cos(z.re)*sinh(z.im)i
|
||
*
|
||
* If both sin($(THETA)) and cos($(THETA)) are required,
|
||
* it is most efficient to use expi($(THETA)).
|
||
*/
|
||
creal sin(creal z) @safe pure nothrow @nogc
|
||
{
|
||
const creal cs = expi(z.re);
|
||
const creal csh = coshisinh(z.im);
|
||
return cs.im * csh.re + cs.re * csh.im * 1i;
|
||
}
|
||
|
||
/** ditto */
|
||
ireal sin(ireal y) @safe pure nothrow @nogc
|
||
{
|
||
return cosh(y.im)*1i;
|
||
}
|
||
|
||
///
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
assert(sin(0.0+0.0i) == 0.0);
|
||
assert(sin(2.0+0.0i) == sin(2.0L) );
|
||
}
|
||
|
||
/***********************************
|
||
* cosine, complex and imaginary
|
||
*
|
||
* cos(z) = cos(z.re)*cosh(z.im) - sin(z.re)*sinh(z.im)i
|
||
*/
|
||
creal cos(creal z) @safe pure nothrow @nogc
|
||
{
|
||
const creal cs = expi(z.re);
|
||
const creal csh = coshisinh(z.im);
|
||
return cs.re * csh.re - cs.im * csh.im * 1i;
|
||
}
|
||
|
||
/** ditto */
|
||
real cos(ireal y) @safe pure nothrow @nogc
|
||
{
|
||
return cosh(y.im);
|
||
}
|
||
|
||
///
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
assert(cos(0.0+0.0i)==1.0);
|
||
assert(cos(1.3L+0.0i)==cos(1.3L));
|
||
assert(cos(5.2Li)== cosh(5.2L));
|
||
}
|
||
|
||
/****************************************************************************
|
||
* Returns tangent of x. x is in radians.
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH x) $(TH tan(x)) $(TH invalid?))
|
||
* $(TR $(TD $(NAN)) $(TD $(NAN)) $(TD yes))
|
||
* $(TR $(TD $(PLUSMN)0.0) $(TD $(PLUSMN)0.0) $(TD no))
|
||
* $(TR $(TD $(PLUSMNINF)) $(TD $(NAN)) $(TD yes))
|
||
* )
|
||
*/
|
||
|
||
real tan(real x) @trusted pure nothrow @nogc
|
||
{
|
||
version(D_InlineAsm_X86)
|
||
{
|
||
asm pure nothrow @nogc
|
||
{
|
||
fld x[EBP] ; // load theta
|
||
fxam ; // test for oddball values
|
||
fstsw AX ;
|
||
sahf ;
|
||
jc trigerr ; // x is NAN, infinity, or empty
|
||
// 387's can handle subnormals
|
||
SC18: fptan ;
|
||
fstsw AX ;
|
||
sahf ;
|
||
jnp Clear1 ; // C2 = 1 (x is out of range)
|
||
|
||
// Do argument reduction to bring x into range
|
||
fldpi ;
|
||
fxch ;
|
||
SC17: fprem1 ;
|
||
fstsw AX ;
|
||
sahf ;
|
||
jp SC17 ;
|
||
fstp ST(1) ; // remove pi from stack
|
||
jmp SC18 ;
|
||
|
||
trigerr:
|
||
jnp Lret ; // if theta is NAN, return theta
|
||
fstp ST(0) ; // dump theta
|
||
}
|
||
return real.nan;
|
||
|
||
Clear1: asm pure nothrow @nogc{
|
||
fstp ST(0) ; // dump X, which is always 1
|
||
}
|
||
|
||
Lret: {}
|
||
}
|
||
else version(D_InlineAsm_X86_64)
|
||
{
|
||
version (Win64)
|
||
{
|
||
asm pure nothrow @nogc
|
||
{
|
||
fld real ptr [RCX] ; // load theta
|
||
}
|
||
}
|
||
else
|
||
{
|
||
asm pure nothrow @nogc
|
||
{
|
||
fld x[RBP] ; // load theta
|
||
}
|
||
}
|
||
asm pure nothrow @nogc
|
||
{
|
||
fxam ; // test for oddball values
|
||
fstsw AX ;
|
||
test AH,1 ;
|
||
jnz trigerr ; // x is NAN, infinity, or empty
|
||
// 387's can handle subnormals
|
||
SC18: fptan ;
|
||
fstsw AX ;
|
||
test AH,4 ;
|
||
jz Clear1 ; // C2 = 1 (x is out of range)
|
||
|
||
// Do argument reduction to bring x into range
|
||
fldpi ;
|
||
fxch ;
|
||
SC17: fprem1 ;
|
||
fstsw AX ;
|
||
test AH,4 ;
|
||
jnz SC17 ;
|
||
fstp ST(1) ; // remove pi from stack
|
||
jmp SC18 ;
|
||
|
||
trigerr:
|
||
test AH,4 ;
|
||
jz Lret ; // if theta is NAN, return theta
|
||
fstp ST(0) ; // dump theta
|
||
}
|
||
return real.nan;
|
||
|
||
Clear1: asm pure nothrow @nogc{
|
||
fstp ST(0) ; // dump X, which is always 1
|
||
}
|
||
|
||
Lret: {}
|
||
}
|
||
else
|
||
{
|
||
// Coefficients for tan(x) and PI/4 split into three parts.
|
||
static if (floatTraits!real.realFormat == RealFormat.ieeeQuadruple)
|
||
{
|
||
static immutable real[6] P = [
|
||
2.883414728874239697964612246732416606301E10L,
|
||
-2.307030822693734879744223131873392503321E9L,
|
||
5.160188250214037865511600561074819366815E7L,
|
||
-4.249691853501233575668486667664718192660E5L,
|
||
1.272297782199996882828849455156962260810E3L,
|
||
-9.889929415807650724957118893791829849557E-1L
|
||
];
|
||
static immutable real[7] Q = [
|
||
8.650244186622719093893836740197250197602E10L
|
||
-4.152206921457208101480801635640958361612E10L,
|
||
2.758476078803232151774723646710890525496E9L,
|
||
-5.733709132766856723608447733926138506824E7L,
|
||
4.529422062441341616231663543669583527923E5L,
|
||
-1.317243702830553658702531997959756728291E3L,
|
||
1.0
|
||
];
|
||
|
||
enum real DP1 =
|
||
7.853981633974483067550664827649598009884357452392578125E-1L;
|
||
enum real DP2 =
|
||
2.8605943630549158983813312792950660807511260829685741796657E-18L;
|
||
enum real DP3 =
|
||
2.1679525325309452561992610065108379921905808E-35L;
|
||
}
|
||
else
|
||
{
|
||
static immutable real[3] P = [
|
||
-1.7956525197648487798769E7L,
|
||
1.1535166483858741613983E6L,
|
||
-1.3093693918138377764608E4L,
|
||
];
|
||
static immutable real[5] Q = [
|
||
-5.3869575592945462988123E7L,
|
||
2.5008380182335791583922E7L,
|
||
-1.3208923444021096744731E6L,
|
||
1.3681296347069295467845E4L,
|
||
1.0000000000000000000000E0L,
|
||
];
|
||
|
||
enum real P1 = 7.853981554508209228515625E-1L;
|
||
enum real P2 = 7.946627356147928367136046290398E-9L;
|
||
enum real P3 = 3.061616997868382943065164830688E-17L;
|
||
}
|
||
|
||
// Special cases.
|
||
if (x == 0.0 || isNaN(x))
|
||
return x;
|
||
if (isInfinity(x))
|
||
return real.nan;
|
||
|
||
// Make argument positive but save the sign.
|
||
bool sign = false;
|
||
if (signbit(x))
|
||
{
|
||
sign = true;
|
||
x = -x;
|
||
}
|
||
|
||
// Compute x mod PI/4.
|
||
real y = floor(x / PI_4);
|
||
// Strip high bits of integer part.
|
||
real z = ldexp(y, -4);
|
||
// Compute y - 16 * (y / 16).
|
||
z = y - ldexp(floor(z), 4);
|
||
|
||
// Integer and fraction part modulo one octant.
|
||
int j = cast(int)(z);
|
||
|
||
// Map zeros and singularities to origin.
|
||
if (j & 1)
|
||
{
|
||
j += 1;
|
||
y += 1.0;
|
||
}
|
||
|
||
z = ((x - y * P1) - y * P2) - y * P3;
|
||
const real zz = z * z;
|
||
|
||
if (zz > 1.0e-20L)
|
||
y = z + z * (zz * poly(zz, P) / poly(zz, Q));
|
||
else
|
||
y = z;
|
||
|
||
if (j & 2)
|
||
y = -1.0 / y;
|
||
|
||
return (sign) ? -y : y;
|
||
}
|
||
}
|
||
|
||
@safe nothrow @nogc unittest
|
||
{
|
||
static real[2][] vals = // angle,tan
|
||
[
|
||
[ 0, 0],
|
||
[ .5, .5463024898],
|
||
[ 1, 1.557407725],
|
||
[ 1.5, 14.10141995],
|
||
[ 2, -2.185039863],
|
||
[ 2.5,-.7470222972],
|
||
[ 3, -.1425465431],
|
||
[ 3.5, .3745856402],
|
||
[ 4, 1.157821282],
|
||
[ 4.5, 4.637332055],
|
||
[ 5, -3.380515006],
|
||
[ 5.5,-.9955840522],
|
||
[ 6, -.2910061914],
|
||
[ 6.5, .2202772003],
|
||
[ 10, .6483608275],
|
||
|
||
// special angles
|
||
[ PI_4, 1],
|
||
//[ PI_2, real.infinity], // PI_2 is not _exactly_ pi/2.
|
||
[ 3*PI_4, -1],
|
||
[ PI, 0],
|
||
[ 5*PI_4, 1],
|
||
//[ 3*PI_2, -real.infinity],
|
||
[ 7*PI_4, -1],
|
||
[ 2*PI, 0],
|
||
];
|
||
int i;
|
||
|
||
for (i = 0; i < vals.length; i++)
|
||
{
|
||
real x = vals[i][0];
|
||
real r = vals[i][1];
|
||
real t = tan(x);
|
||
|
||
//printf("tan(%Lg) = %Lg, should be %Lg\n", x, t, r);
|
||
if (!isIdentical(r, t)) assert(fabs(r-t) <= .0000001);
|
||
|
||
x = -x;
|
||
r = -r;
|
||
t = tan(x);
|
||
//printf("tan(%Lg) = %Lg, should be %Lg\n", x, t, r);
|
||
if (!isIdentical(r, t) && !(r != r && t != t)) assert(fabs(r-t) <= .0000001);
|
||
}
|
||
// overflow
|
||
assert(isNaN(tan(real.infinity)));
|
||
assert(isNaN(tan(-real.infinity)));
|
||
// NaN propagation
|
||
assert(isIdentical( tan(NaN(0x0123L)), NaN(0x0123L) ));
|
||
}
|
||
|
||
@system unittest
|
||
{
|
||
assert(equalsDigit(tan(PI / 3), std.math.sqrt(3.0), useDigits));
|
||
}
|
||
|
||
/***************
|
||
* Calculates the arc cosine of x,
|
||
* returning a value ranging from 0 to $(PI).
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH x) $(TH acos(x)) $(TH invalid?))
|
||
* $(TR $(TD $(GT)1.0) $(TD $(NAN)) $(TD yes))
|
||
* $(TR $(TD $(LT)-1.0) $(TD $(NAN)) $(TD yes))
|
||
* $(TR $(TD $(NAN)) $(TD $(NAN)) $(TD yes))
|
||
* )
|
||
*/
|
||
real acos(real x) @safe pure nothrow @nogc
|
||
{
|
||
return atan2(sqrt(1-x*x), x);
|
||
}
|
||
|
||
/// ditto
|
||
double acos(double x) @safe pure nothrow @nogc { return acos(cast(real) x); }
|
||
|
||
/// ditto
|
||
float acos(float x) @safe pure nothrow @nogc { return acos(cast(real) x); }
|
||
|
||
@system unittest
|
||
{
|
||
assert(equalsDigit(acos(0.5), std.math.PI / 3, useDigits));
|
||
}
|
||
|
||
/***************
|
||
* Calculates the arc sine of x,
|
||
* returning a value ranging from -$(PI)/2 to $(PI)/2.
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH x) $(TH asin(x)) $(TH invalid?))
|
||
* $(TR $(TD $(PLUSMN)0.0) $(TD $(PLUSMN)0.0) $(TD no))
|
||
* $(TR $(TD $(GT)1.0) $(TD $(NAN)) $(TD yes))
|
||
* $(TR $(TD $(LT)-1.0) $(TD $(NAN)) $(TD yes))
|
||
* )
|
||
*/
|
||
real asin(real x) @safe pure nothrow @nogc
|
||
{
|
||
return atan2(x, sqrt(1-x*x));
|
||
}
|
||
|
||
/// ditto
|
||
double asin(double x) @safe pure nothrow @nogc { return asin(cast(real) x); }
|
||
|
||
/// ditto
|
||
float asin(float x) @safe pure nothrow @nogc { return asin(cast(real) x); }
|
||
|
||
@system unittest
|
||
{
|
||
assert(equalsDigit(asin(0.5), PI / 6, useDigits));
|
||
}
|
||
|
||
/***************
|
||
* Calculates the arc tangent of x,
|
||
* returning a value ranging from -$(PI)/2 to $(PI)/2.
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH x) $(TH atan(x)) $(TH invalid?))
|
||
* $(TR $(TD $(PLUSMN)0.0) $(TD $(PLUSMN)0.0) $(TD no))
|
||
* $(TR $(TD $(PLUSMN)$(INFIN)) $(TD $(NAN)) $(TD yes))
|
||
* )
|
||
*/
|
||
real atan(real x) @safe pure nothrow @nogc
|
||
{
|
||
version(InlineAsm_X86_Any)
|
||
{
|
||
return atan2(x, 1.0L);
|
||
}
|
||
else
|
||
{
|
||
// Coefficients for atan(x)
|
||
static if (floatTraits!real.realFormat == RealFormat.ieeeQuadruple)
|
||
{
|
||
static immutable real[9] P = [
|
||
-6.880597774405940432145577545328795037141E2L,
|
||
-2.514829758941713674909996882101723647996E3L,
|
||
-3.696264445691821235400930243493001671932E3L,
|
||
-2.792272753241044941703278827346430350236E3L,
|
||
-1.148164399808514330375280133523543970854E3L,
|
||
-2.497759878476618348858065206895055957104E2L,
|
||
-2.548067867495502632615671450650071218995E1L,
|
||
-8.768423468036849091777415076702113400070E-1L,
|
||
-6.635810778635296712545011270011752799963E-4L
|
||
];
|
||
static immutable real[9] Q = [
|
||
2.064179332321782129643673263598686441900E3L,
|
||
8.782996876218210302516194604424986107121E3L,
|
||
1.547394317752562611786521896296215170819E4L,
|
||
1.458510242529987155225086911411015961174E4L,
|
||
7.928572347062145288093560392463784743935E3L,
|
||
2.494680540950601626662048893678584497900E3L,
|
||
4.308348370818927353321556740027020068897E2L,
|
||
3.566239794444800849656497338030115886153E1L,
|
||
1.0
|
||
];
|
||
}
|
||
else
|
||
{
|
||
static immutable real[5] P = [
|
||
-5.0894116899623603312185E1L,
|
||
-9.9988763777265819915721E1L,
|
||
-6.3976888655834347413154E1L,
|
||
-1.4683508633175792446076E1L,
|
||
-8.6863818178092187535440E-1L,
|
||
];
|
||
static immutable real[6] Q = [
|
||
1.5268235069887081006606E2L,
|
||
3.9157570175111990631099E2L,
|
||
3.6144079386152023162701E2L,
|
||
1.4399096122250781605352E2L,
|
||
2.2981886733594175366172E1L,
|
||
1.0000000000000000000000E0L,
|
||
];
|
||
}
|
||
|
||
// tan(PI/8)
|
||
enum real TAN_PI_8 = 0.414213562373095048801688724209698078569672L;
|
||
// tan(3 * PI/8)
|
||
enum real TAN3_PI_8 = 2.414213562373095048801688724209698078569672L;
|
||
|
||
// Special cases.
|
||
if (x == 0.0)
|
||
return x;
|
||
if (isInfinity(x))
|
||
return copysign(PI_2, x);
|
||
|
||
// Make argument positive but save the sign.
|
||
bool sign = false;
|
||
if (signbit(x))
|
||
{
|
||
sign = true;
|
||
x = -x;
|
||
}
|
||
|
||
// Range reduction.
|
||
real y;
|
||
if (x > TAN3_PI_8)
|
||
{
|
||
y = PI_2;
|
||
x = -(1.0 / x);
|
||
}
|
||
else if (x > TAN_PI_8)
|
||
{
|
||
y = PI_4;
|
||
x = (x - 1.0)/(x + 1.0);
|
||
}
|
||
else
|
||
y = 0.0;
|
||
|
||
// Rational form in x^^2.
|
||
const real z = x * x;
|
||
y = y + (poly(z, P) / poly(z, Q)) * z * x + x;
|
||
|
||
return (sign) ? -y : y;
|
||
}
|
||
}
|
||
|
||
/// ditto
|
||
double atan(double x) @safe pure nothrow @nogc { return atan(cast(real) x); }
|
||
|
||
/// ditto
|
||
float atan(float x) @safe pure nothrow @nogc { return atan(cast(real) x); }
|
||
|
||
@system unittest
|
||
{
|
||
assert(equalsDigit(atan(std.math.sqrt(3.0)), PI / 3, useDigits));
|
||
}
|
||
|
||
/***************
|
||
* Calculates the arc tangent of y / x,
|
||
* returning a value ranging from -$(PI) to $(PI).
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH y) $(TH x) $(TH atan(y, x)))
|
||
* $(TR $(TD $(NAN)) $(TD anything) $(TD $(NAN)) )
|
||
* $(TR $(TD anything) $(TD $(NAN)) $(TD $(NAN)) )
|
||
* $(TR $(TD $(PLUSMN)0.0) $(TD $(GT)0.0) $(TD $(PLUSMN)0.0) )
|
||
* $(TR $(TD $(PLUSMN)0.0) $(TD +0.0) $(TD $(PLUSMN)0.0) )
|
||
* $(TR $(TD $(PLUSMN)0.0) $(TD $(LT)0.0) $(TD $(PLUSMN)$(PI)))
|
||
* $(TR $(TD $(PLUSMN)0.0) $(TD -0.0) $(TD $(PLUSMN)$(PI)))
|
||
* $(TR $(TD $(GT)0.0) $(TD $(PLUSMN)0.0) $(TD $(PI)/2) )
|
||
* $(TR $(TD $(LT)0.0) $(TD $(PLUSMN)0.0) $(TD -$(PI)/2) )
|
||
* $(TR $(TD $(GT)0.0) $(TD $(INFIN)) $(TD $(PLUSMN)0.0) )
|
||
* $(TR $(TD $(PLUSMN)$(INFIN)) $(TD anything) $(TD $(PLUSMN)$(PI)/2))
|
||
* $(TR $(TD $(GT)0.0) $(TD -$(INFIN)) $(TD $(PLUSMN)$(PI)) )
|
||
* $(TR $(TD $(PLUSMN)$(INFIN)) $(TD $(INFIN)) $(TD $(PLUSMN)$(PI)/4))
|
||
* $(TR $(TD $(PLUSMN)$(INFIN)) $(TD -$(INFIN)) $(TD $(PLUSMN)3$(PI)/4))
|
||
* )
|
||
*/
|
||
real atan2(real y, real x) @trusted pure nothrow @nogc
|
||
{
|
||
version(InlineAsm_X86_Any)
|
||
{
|
||
version (Win64)
|
||
{
|
||
asm pure nothrow @nogc {
|
||
naked;
|
||
fld real ptr [RDX]; // y
|
||
fld real ptr [RCX]; // x
|
||
fpatan;
|
||
ret;
|
||
}
|
||
}
|
||
else
|
||
{
|
||
asm pure nothrow @nogc {
|
||
fld y;
|
||
fld x;
|
||
fpatan;
|
||
}
|
||
}
|
||
}
|
||
else
|
||
{
|
||
// Special cases.
|
||
if (isNaN(x) || isNaN(y))
|
||
return real.nan;
|
||
if (y == 0.0)
|
||
{
|
||
if (x >= 0 && !signbit(x))
|
||
return copysign(0, y);
|
||
else
|
||
return copysign(PI, y);
|
||
}
|
||
if (x == 0.0)
|
||
return copysign(PI_2, y);
|
||
if (isInfinity(x))
|
||
{
|
||
if (signbit(x))
|
||
{
|
||
if (isInfinity(y))
|
||
return copysign(3*PI_4, y);
|
||
else
|
||
return copysign(PI, y);
|
||
}
|
||
else
|
||
{
|
||
if (isInfinity(y))
|
||
return copysign(PI_4, y);
|
||
else
|
||
return copysign(0.0, y);
|
||
}
|
||
}
|
||
if (isInfinity(y))
|
||
return copysign(PI_2, y);
|
||
|
||
// Call atan and determine the quadrant.
|
||
real z = atan(y / x);
|
||
|
||
if (signbit(x))
|
||
{
|
||
if (signbit(y))
|
||
z = z - PI;
|
||
else
|
||
z = z + PI;
|
||
}
|
||
|
||
if (z == 0.0)
|
||
return copysign(z, y);
|
||
|
||
return z;
|
||
}
|
||
}
|
||
|
||
/// ditto
|
||
double atan2(double y, double x) @safe pure nothrow @nogc
|
||
{
|
||
return atan2(cast(real) y, cast(real) x);
|
||
}
|
||
|
||
/// ditto
|
||
float atan2(float y, float x) @safe pure nothrow @nogc
|
||
{
|
||
return atan2(cast(real) y, cast(real) x);
|
||
}
|
||
|
||
@system unittest
|
||
{
|
||
assert(equalsDigit(atan2(1.0L, std.math.sqrt(3.0L)), PI / 6, useDigits));
|
||
}
|
||
|
||
/***********************************
|
||
* Calculates the hyperbolic cosine of x.
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH x) $(TH cosh(x)) $(TH invalid?))
|
||
* $(TR $(TD $(PLUSMN)$(INFIN)) $(TD $(PLUSMN)0.0) $(TD no) )
|
||
* )
|
||
*/
|
||
real cosh(real x) @safe pure nothrow @nogc
|
||
{
|
||
// cosh = (exp(x)+exp(-x))/2.
|
||
// The naive implementation works correctly.
|
||
const real y = exp(x);
|
||
return (y + 1.0/y) * 0.5;
|
||
}
|
||
|
||
/// ditto
|
||
double cosh(double x) @safe pure nothrow @nogc { return cosh(cast(real) x); }
|
||
|
||
/// ditto
|
||
float cosh(float x) @safe pure nothrow @nogc { return cosh(cast(real) x); }
|
||
|
||
@system unittest
|
||
{
|
||
assert(equalsDigit(cosh(1.0), (E + 1.0 / E) / 2, useDigits));
|
||
}
|
||
|
||
/***********************************
|
||
* Calculates the hyperbolic sine of x.
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH x) $(TH sinh(x)) $(TH invalid?))
|
||
* $(TR $(TD $(PLUSMN)0.0) $(TD $(PLUSMN)0.0) $(TD no))
|
||
* $(TR $(TD $(PLUSMN)$(INFIN)) $(TD $(PLUSMN)$(INFIN)) $(TD no))
|
||
* )
|
||
*/
|
||
real sinh(real x) @safe pure nothrow @nogc
|
||
{
|
||
// sinh(x) = (exp(x)-exp(-x))/2;
|
||
// Very large arguments could cause an overflow, but
|
||
// the maximum value of x for which exp(x) + exp(-x)) != exp(x)
|
||
// is x = 0.5 * (real.mant_dig) * LN2. // = 22.1807 for real80.
|
||
if (fabs(x) > real.mant_dig * LN2)
|
||
{
|
||
return copysign(0.5 * exp(fabs(x)), x);
|
||
}
|
||
|
||
const real y = expm1(x);
|
||
return 0.5 * y / (y+1) * (y+2);
|
||
}
|
||
|
||
/// ditto
|
||
double sinh(double x) @safe pure nothrow @nogc { return sinh(cast(real) x); }
|
||
|
||
/// ditto
|
||
float sinh(float x) @safe pure nothrow @nogc { return sinh(cast(real) x); }
|
||
|
||
@system unittest
|
||
{
|
||
assert(equalsDigit(sinh(1.0), (E - 1.0 / E) / 2, useDigits));
|
||
}
|
||
|
||
/***********************************
|
||
* Calculates the hyperbolic tangent of x.
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH x) $(TH tanh(x)) $(TH invalid?))
|
||
* $(TR $(TD $(PLUSMN)0.0) $(TD $(PLUSMN)0.0) $(TD no) )
|
||
* $(TR $(TD $(PLUSMN)$(INFIN)) $(TD $(PLUSMN)1.0) $(TD no))
|
||
* )
|
||
*/
|
||
real tanh(real x) @safe pure nothrow @nogc
|
||
{
|
||
// tanh(x) = (exp(x) - exp(-x))/(exp(x)+exp(-x))
|
||
if (fabs(x) > real.mant_dig * LN2)
|
||
{
|
||
return copysign(1, x);
|
||
}
|
||
|
||
const real y = expm1(2*x);
|
||
return y / (y + 2);
|
||
}
|
||
|
||
/// ditto
|
||
double tanh(double x) @safe pure nothrow @nogc { return tanh(cast(real) x); }
|
||
|
||
/// ditto
|
||
float tanh(float x) @safe pure nothrow @nogc { return tanh(cast(real) x); }
|
||
|
||
@system unittest
|
||
{
|
||
assert(equalsDigit(tanh(1.0), sinh(1.0) / cosh(1.0), 15));
|
||
}
|
||
|
||
package:
|
||
|
||
/* Returns cosh(x) + I * sinh(x)
|
||
* Only one call to exp() is performed.
|
||
*/
|
||
creal coshisinh(real x) @safe pure nothrow @nogc
|
||
{
|
||
// See comments for cosh, sinh.
|
||
if (fabs(x) > real.mant_dig * LN2)
|
||
{
|
||
const real y = exp(fabs(x));
|
||
return y * 0.5 + 0.5i * copysign(y, x);
|
||
}
|
||
else
|
||
{
|
||
const real y = expm1(x);
|
||
return (y + 1.0 + 1.0/(y + 1.0)) * 0.5 + 0.5i * y / (y+1) * (y+2);
|
||
}
|
||
}
|
||
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
creal c = coshisinh(3.0L);
|
||
assert(c.re == cosh(3.0L));
|
||
assert(c.im == sinh(3.0L));
|
||
}
|
||
|
||
public:
|
||
|
||
/***********************************
|
||
* Calculates the inverse hyperbolic cosine of x.
|
||
*
|
||
* Mathematically, acosh(x) = log(x + sqrt( x*x - 1))
|
||
*
|
||
* $(TABLE_DOMRG
|
||
* $(DOMAIN 1..$(INFIN)),
|
||
* $(RANGE 0..$(INFIN))
|
||
* )
|
||
*
|
||
* $(TABLE_SV
|
||
* $(SVH x, acosh(x) )
|
||
* $(SV $(NAN), $(NAN) )
|
||
* $(SV $(LT)1, $(NAN) )
|
||
* $(SV 1, 0 )
|
||
* $(SV +$(INFIN),+$(INFIN))
|
||
* )
|
||
*/
|
||
real acosh(real x) @safe pure nothrow @nogc
|
||
{
|
||
if (x > 1/real.epsilon)
|
||
return LN2 + log(x);
|
||
else
|
||
return log(x + sqrt(x*x - 1));
|
||
}
|
||
|
||
/// ditto
|
||
double acosh(double x) @safe pure nothrow @nogc { return acosh(cast(real) x); }
|
||
|
||
/// ditto
|
||
float acosh(float x) @safe pure nothrow @nogc { return acosh(cast(real) x); }
|
||
|
||
|
||
@system unittest
|
||
{
|
||
assert(isNaN(acosh(0.9)));
|
||
assert(isNaN(acosh(real.nan)));
|
||
assert(acosh(1.0)==0.0);
|
||
assert(acosh(real.infinity) == real.infinity);
|
||
assert(isNaN(acosh(0.5)));
|
||
assert(equalsDigit(acosh(cosh(3.0)), 3, useDigits));
|
||
}
|
||
|
||
/***********************************
|
||
* Calculates the inverse hyperbolic sine of x.
|
||
*
|
||
* Mathematically,
|
||
* ---------------
|
||
* asinh(x) = log( x + sqrt( x*x + 1 )) // if x >= +0
|
||
* asinh(x) = -log(-x + sqrt( x*x + 1 )) // if x <= -0
|
||
* -------------
|
||
*
|
||
* $(TABLE_SV
|
||
* $(SVH x, asinh(x) )
|
||
* $(SV $(NAN), $(NAN) )
|
||
* $(SV $(PLUSMN)0, $(PLUSMN)0 )
|
||
* $(SV $(PLUSMN)$(INFIN),$(PLUSMN)$(INFIN))
|
||
* )
|
||
*/
|
||
real asinh(real x) @safe pure nothrow @nogc
|
||
{
|
||
return (fabs(x) > 1 / real.epsilon)
|
||
// beyond this point, x*x + 1 == x*x
|
||
? copysign(LN2 + log(fabs(x)), x)
|
||
// sqrt(x*x + 1) == 1 + x * x / ( 1 + sqrt(x*x + 1) )
|
||
: copysign(log1p(fabs(x) + x*x / (1 + sqrt(x*x + 1)) ), x);
|
||
}
|
||
|
||
/// ditto
|
||
double asinh(double x) @safe pure nothrow @nogc { return asinh(cast(real) x); }
|
||
|
||
/// ditto
|
||
float asinh(float x) @safe pure nothrow @nogc { return asinh(cast(real) x); }
|
||
|
||
@system unittest
|
||
{
|
||
assert(isIdentical(asinh(0.0), 0.0));
|
||
assert(isIdentical(asinh(-0.0), -0.0));
|
||
assert(asinh(real.infinity) == real.infinity);
|
||
assert(asinh(-real.infinity) == -real.infinity);
|
||
assert(isNaN(asinh(real.nan)));
|
||
assert(equalsDigit(asinh(sinh(3.0)), 3, useDigits));
|
||
}
|
||
|
||
/***********************************
|
||
* Calculates the inverse hyperbolic tangent of x,
|
||
* returning a value from ranging from -1 to 1.
|
||
*
|
||
* Mathematically, atanh(x) = log( (1+x)/(1-x) ) / 2
|
||
*
|
||
* $(TABLE_DOMRG
|
||
* $(DOMAIN -$(INFIN)..$(INFIN)),
|
||
* $(RANGE -1 .. 1)
|
||
* )
|
||
* $(BR)
|
||
* $(TABLE_SV
|
||
* $(SVH x, acosh(x) )
|
||
* $(SV $(NAN), $(NAN) )
|
||
* $(SV $(PLUSMN)0, $(PLUSMN)0)
|
||
* $(SV -$(INFIN), -0)
|
||
* )
|
||
*/
|
||
real atanh(real x) @safe pure nothrow @nogc
|
||
{
|
||
// log( (1+x)/(1-x) ) == log ( 1 + (2*x)/(1-x) )
|
||
return 0.5 * log1p( 2 * x / (1 - x) );
|
||
}
|
||
|
||
/// ditto
|
||
double atanh(double x) @safe pure nothrow @nogc { return atanh(cast(real) x); }
|
||
|
||
/// ditto
|
||
float atanh(float x) @safe pure nothrow @nogc { return atanh(cast(real) x); }
|
||
|
||
|
||
@system unittest
|
||
{
|
||
assert(isIdentical(atanh(0.0), 0.0));
|
||
assert(isIdentical(atanh(-0.0),-0.0));
|
||
assert(isNaN(atanh(real.nan)));
|
||
assert(isNaN(atanh(-real.infinity)));
|
||
assert(atanh(0.0) == 0);
|
||
assert(equalsDigit(atanh(tanh(0.5L)), 0.5, useDigits));
|
||
}
|
||
|
||
/*****************************************
|
||
* Returns x rounded to a long value using the current rounding mode.
|
||
* If the integer value of x is
|
||
* greater than long.max, the result is
|
||
* indeterminate.
|
||
*/
|
||
long rndtol(real x) @nogc @safe pure nothrow { pragma(inline, true); return core.math.rndtol(x); }
|
||
//FIXME
|
||
///ditto
|
||
long rndtol(double x) @safe pure nothrow @nogc { return rndtol(cast(real) x); }
|
||
//FIXME
|
||
///ditto
|
||
long rndtol(float x) @safe pure nothrow @nogc { return rndtol(cast(real) x); }
|
||
|
||
@safe unittest
|
||
{
|
||
long function(real) prndtol = &rndtol;
|
||
assert(prndtol != null);
|
||
}
|
||
|
||
/*****************************************
|
||
* Returns x rounded to a long value using the FE_TONEAREST rounding mode.
|
||
* If the integer value of x is
|
||
* greater than long.max, the result is
|
||
* indeterminate.
|
||
*/
|
||
extern (C) real rndtonl(real x);
|
||
|
||
/***************************************
|
||
* Compute square root of x.
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH x) $(TH sqrt(x)) $(TH invalid?))
|
||
* $(TR $(TD -0.0) $(TD -0.0) $(TD no))
|
||
* $(TR $(TD $(LT)0.0) $(TD $(NAN)) $(TD yes))
|
||
* $(TR $(TD +$(INFIN)) $(TD +$(INFIN)) $(TD no))
|
||
* )
|
||
*/
|
||
float sqrt(float x) @nogc @safe pure nothrow { pragma(inline, true); return core.math.sqrt(x); }
|
||
|
||
/// ditto
|
||
double sqrt(double x) @nogc @safe pure nothrow { pragma(inline, true); return core.math.sqrt(x); }
|
||
|
||
/// ditto
|
||
real sqrt(real x) @nogc @safe pure nothrow { pragma(inline, true); return core.math.sqrt(x); }
|
||
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
//ctfe
|
||
enum ZX80 = sqrt(7.0f);
|
||
enum ZX81 = sqrt(7.0);
|
||
enum ZX82 = sqrt(7.0L);
|
||
|
||
assert(isNaN(sqrt(-1.0f)));
|
||
assert(isNaN(sqrt(-1.0)));
|
||
assert(isNaN(sqrt(-1.0L)));
|
||
}
|
||
|
||
@safe unittest
|
||
{
|
||
float function(float) psqrtf = &sqrt;
|
||
assert(psqrtf != null);
|
||
double function(double) psqrtd = &sqrt;
|
||
assert(psqrtd != null);
|
||
real function(real) psqrtr = &sqrt;
|
||
assert(psqrtr != null);
|
||
}
|
||
|
||
creal sqrt(creal z) @nogc @safe pure nothrow
|
||
{
|
||
creal c;
|
||
real x,y,w,r;
|
||
|
||
if (z == 0)
|
||
{
|
||
c = 0 + 0i;
|
||
}
|
||
else
|
||
{
|
||
const real z_re = z.re;
|
||
const real z_im = z.im;
|
||
|
||
x = fabs(z_re);
|
||
y = fabs(z_im);
|
||
if (x >= y)
|
||
{
|
||
r = y / x;
|
||
w = sqrt(x) * sqrt(0.5 * (1 + sqrt(1 + r * r)));
|
||
}
|
||
else
|
||
{
|
||
r = x / y;
|
||
w = sqrt(y) * sqrt(0.5 * (r + sqrt(1 + r * r)));
|
||
}
|
||
|
||
if (z_re >= 0)
|
||
{
|
||
c = w + (z_im / (w + w)) * 1.0i;
|
||
}
|
||
else
|
||
{
|
||
if (z_im < 0)
|
||
w = -w;
|
||
c = z_im / (w + w) + w * 1.0i;
|
||
}
|
||
}
|
||
return c;
|
||
}
|
||
|
||
/**
|
||
* Calculates e$(SUPERSCRIPT x).
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH x) $(TH e$(SUPERSCRIPT x)) )
|
||
* $(TR $(TD +$(INFIN)) $(TD +$(INFIN)) )
|
||
* $(TR $(TD -$(INFIN)) $(TD +0.0) )
|
||
* $(TR $(TD $(NAN)) $(TD $(NAN)) )
|
||
* )
|
||
*/
|
||
real exp(real x) @trusted pure nothrow @nogc
|
||
{
|
||
version(D_InlineAsm_X86)
|
||
{
|
||
// e^^x = 2^^(LOG2E*x)
|
||
// (This is valid because the overflow & underflow limits for exp
|
||
// and exp2 are so similar).
|
||
return exp2(LOG2E*x);
|
||
}
|
||
else version(D_InlineAsm_X86_64)
|
||
{
|
||
// e^^x = 2^^(LOG2E*x)
|
||
// (This is valid because the overflow & underflow limits for exp
|
||
// and exp2 are so similar).
|
||
return exp2(LOG2E*x);
|
||
}
|
||
else
|
||
{
|
||
alias F = floatTraits!real;
|
||
static if (F.realFormat == RealFormat.ieeeDouble)
|
||
{
|
||
// Coefficients for exp(x)
|
||
static immutable real[3] P = [
|
||
9.99999999999999999910E-1L,
|
||
3.02994407707441961300E-2L,
|
||
1.26177193074810590878E-4L,
|
||
];
|
||
static immutable real[4] Q = [
|
||
2.00000000000000000009E0L,
|
||
2.27265548208155028766E-1L,
|
||
2.52448340349684104192E-3L,
|
||
3.00198505138664455042E-6L,
|
||
];
|
||
|
||
// C1 + C2 = LN2.
|
||
enum real C1 = 6.93145751953125E-1;
|
||
enum real C2 = 1.42860682030941723212E-6;
|
||
|
||
// Overflow and Underflow limits.
|
||
enum real OF = 7.09782712893383996732E2; // ln((1-2^-53) * 2^1024)
|
||
enum real UF = -7.451332191019412076235E2; // ln(2^-1075)
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeExtended)
|
||
{
|
||
// Coefficients for exp(x)
|
||
static immutable real[3] P = [
|
||
9.9999999999999999991025E-1L,
|
||
3.0299440770744196129956E-2L,
|
||
1.2617719307481059087798E-4L,
|
||
];
|
||
static immutable real[4] Q = [
|
||
2.0000000000000000000897E0L,
|
||
2.2726554820815502876593E-1L,
|
||
2.5244834034968410419224E-3L,
|
||
3.0019850513866445504159E-6L,
|
||
];
|
||
|
||
// C1 + C2 = LN2.
|
||
enum real C1 = 6.9314575195312500000000E-1L;
|
||
enum real C2 = 1.4286068203094172321215E-6L;
|
||
|
||
// Overflow and Underflow limits.
|
||
enum real OF = 1.1356523406294143949492E4L; // ln((1-2^-64) * 2^16384)
|
||
enum real UF = -1.13994985314888605586758E4L; // ln(2^-16446)
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeQuadruple)
|
||
{
|
||
// Coefficients for exp(x) - 1
|
||
static immutable real[5] P = [
|
||
9.999999999999999999999999999999999998502E-1L,
|
||
3.508710990737834361215404761139478627390E-2L,
|
||
2.708775201978218837374512615596512792224E-4L,
|
||
6.141506007208645008909088812338454698548E-7L,
|
||
3.279723985560247033712687707263393506266E-10L
|
||
];
|
||
static immutable real[6] Q = [
|
||
2.000000000000000000000000000000000000150E0,
|
||
2.368408864814233538909747618894558968880E-1L,
|
||
3.611828913847589925056132680618007270344E-3L,
|
||
1.504792651814944826817779302637284053660E-5L,
|
||
1.771372078166251484503904874657985291164E-8L,
|
||
2.980756652081995192255342779918052538681E-12L
|
||
];
|
||
|
||
// C1 + C2 = LN2.
|
||
enum real C1 = 6.93145751953125E-1L;
|
||
enum real C2 = 1.428606820309417232121458176568075500134E-6L;
|
||
|
||
// Overflow and Underflow limits.
|
||
enum real OF = 1.135583025911358400418251384584930671458833e4L;
|
||
enum real UF = -1.143276959615573793352782661133116431383730e4L;
|
||
}
|
||
else
|
||
static assert(0, "Not implemented for this architecture");
|
||
|
||
// Special cases. Raises an overflow or underflow flag accordingly,
|
||
// except in the case for CTFE, where there are no hardware controls.
|
||
if (isNaN(x))
|
||
return x;
|
||
if (x > OF)
|
||
{
|
||
if (__ctfe)
|
||
return real.infinity;
|
||
else
|
||
return real.max * copysign(real.max, real.infinity);
|
||
}
|
||
if (x < UF)
|
||
{
|
||
if (__ctfe)
|
||
return 0.0;
|
||
else
|
||
return real.min_normal * copysign(real.min_normal, 0.0);
|
||
}
|
||
|
||
// Express: e^^x = e^^g * 2^^n
|
||
// = e^^g * e^^(n * LOG2E)
|
||
// = e^^(g + n * LOG2E)
|
||
int n = cast(int) floor(LOG2E * x + 0.5);
|
||
x -= n * C1;
|
||
x -= n * C2;
|
||
|
||
// Rational approximation for exponential of the fractional part:
|
||
// e^^x = 1 + 2x P(x^^2) / (Q(x^^2) - P(x^^2))
|
||
const real xx = x * x;
|
||
const real px = x * poly(xx, P);
|
||
x = px / (poly(xx, Q) - px);
|
||
x = 1.0 + ldexp(x, 1);
|
||
|
||
// Scale by power of 2.
|
||
x = ldexp(x, n);
|
||
|
||
return x;
|
||
}
|
||
}
|
||
|
||
/// ditto
|
||
double exp(double x) @safe pure nothrow @nogc { return exp(cast(real) x); }
|
||
|
||
/// ditto
|
||
float exp(float x) @safe pure nothrow @nogc { return exp(cast(real) x); }
|
||
|
||
@system unittest
|
||
{
|
||
assert(equalsDigit(exp(3.0L), E * E * E, useDigits));
|
||
}
|
||
|
||
/**
|
||
* Calculates the value of the natural logarithm base (e)
|
||
* raised to the power of x, minus 1.
|
||
*
|
||
* For very small x, expm1(x) is more accurate
|
||
* than exp(x)-1.
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH x) $(TH e$(SUPERSCRIPT x)-1) )
|
||
* $(TR $(TD $(PLUSMN)0.0) $(TD $(PLUSMN)0.0) )
|
||
* $(TR $(TD +$(INFIN)) $(TD +$(INFIN)) )
|
||
* $(TR $(TD -$(INFIN)) $(TD -1.0) )
|
||
* $(TR $(TD $(NAN)) $(TD $(NAN)) )
|
||
* )
|
||
*/
|
||
real expm1(real x) @trusted pure nothrow @nogc
|
||
{
|
||
version(D_InlineAsm_X86)
|
||
{
|
||
enum PARAMSIZE = (real.sizeof+3)&(0xFFFF_FFFC); // always a multiple of 4
|
||
asm pure nothrow @nogc
|
||
{
|
||
/* expm1() for x87 80-bit reals, IEEE754-2008 conformant.
|
||
* Author: Don Clugston.
|
||
*
|
||
* expm1(x) = 2^^(rndint(y))* 2^^(y-rndint(y)) - 1 where y = LN2*x.
|
||
* = 2rndy * 2ym1 + 2rndy - 1, where 2rndy = 2^^(rndint(y))
|
||
* and 2ym1 = (2^^(y-rndint(y))-1).
|
||
* If 2rndy < 0.5*real.epsilon, result is -1.
|
||
* Implementation is otherwise the same as for exp2()
|
||
*/
|
||
naked;
|
||
fld real ptr [ESP+4] ; // x
|
||
mov AX, [ESP+4+8]; // AX = exponent and sign
|
||
sub ESP, 12+8; // Create scratch space on the stack
|
||
// [ESP,ESP+2] = scratchint
|
||
// [ESP+4..+6, +8..+10, +10] = scratchreal
|
||
// set scratchreal mantissa = 1.0
|
||
mov dword ptr [ESP+8], 0;
|
||
mov dword ptr [ESP+8+4], 0x80000000;
|
||
and AX, 0x7FFF; // drop sign bit
|
||
cmp AX, 0x401D; // avoid InvalidException in fist
|
||
jae L_extreme;
|
||
fldl2e;
|
||
fmulp ST(1), ST; // y = x*log2(e)
|
||
fist dword ptr [ESP]; // scratchint = rndint(y)
|
||
fisub dword ptr [ESP]; // y - rndint(y)
|
||
// and now set scratchreal exponent
|
||
mov EAX, [ESP];
|
||
add EAX, 0x3fff;
|
||
jle short L_largenegative;
|
||
cmp EAX,0x8000;
|
||
jge short L_largepositive;
|
||
mov [ESP+8+8],AX;
|
||
f2xm1; // 2ym1 = 2^^(y-rndint(y)) -1
|
||
fld real ptr [ESP+8] ; // 2rndy = 2^^rndint(y)
|
||
fmul ST(1), ST; // ST=2rndy, ST(1)=2rndy*2ym1
|
||
fld1;
|
||
fsubp ST(1), ST; // ST = 2rndy-1, ST(1) = 2rndy * 2ym1 - 1
|
||
faddp ST(1), ST; // ST = 2rndy * 2ym1 + 2rndy - 1
|
||
add ESP,12+8;
|
||
ret PARAMSIZE;
|
||
|
||
L_extreme: // Extreme exponent. X is very large positive, very
|
||
// large negative, infinity, or NaN.
|
||
fxam;
|
||
fstsw AX;
|
||
test AX, 0x0400; // NaN_or_zero, but we already know x != 0
|
||
jz L_was_nan; // if x is NaN, returns x
|
||
test AX, 0x0200;
|
||
jnz L_largenegative;
|
||
L_largepositive:
|
||
// Set scratchreal = real.max.
|
||
// squaring it will create infinity, and set overflow flag.
|
||
mov word ptr [ESP+8+8], 0x7FFE;
|
||
fstp ST(0);
|
||
fld real ptr [ESP+8]; // load scratchreal
|
||
fmul ST(0), ST; // square it, to create havoc!
|
||
L_was_nan:
|
||
add ESP,12+8;
|
||
ret PARAMSIZE;
|
||
L_largenegative:
|
||
fstp ST(0);
|
||
fld1;
|
||
fchs; // return -1. Underflow flag is not set.
|
||
add ESP,12+8;
|
||
ret PARAMSIZE;
|
||
}
|
||
}
|
||
else version(D_InlineAsm_X86_64)
|
||
{
|
||
asm pure nothrow @nogc
|
||
{
|
||
naked;
|
||
}
|
||
version (Win64)
|
||
{
|
||
asm pure nothrow @nogc
|
||
{
|
||
fld real ptr [RCX]; // x
|
||
mov AX,[RCX+8]; // AX = exponent and sign
|
||
}
|
||
}
|
||
else
|
||
{
|
||
asm pure nothrow @nogc
|
||
{
|
||
fld real ptr [RSP+8]; // x
|
||
mov AX,[RSP+8+8]; // AX = exponent and sign
|
||
}
|
||
}
|
||
asm pure nothrow @nogc
|
||
{
|
||
/* expm1() for x87 80-bit reals, IEEE754-2008 conformant.
|
||
* Author: Don Clugston.
|
||
*
|
||
* expm1(x) = 2^(rndint(y))* 2^(y-rndint(y)) - 1 where y = LN2*x.
|
||
* = 2rndy * 2ym1 + 2rndy - 1, where 2rndy = 2^(rndint(y))
|
||
* and 2ym1 = (2^(y-rndint(y))-1).
|
||
* If 2rndy < 0.5*real.epsilon, result is -1.
|
||
* Implementation is otherwise the same as for exp2()
|
||
*/
|
||
sub RSP, 24; // Create scratch space on the stack
|
||
// [RSP,RSP+2] = scratchint
|
||
// [RSP+4..+6, +8..+10, +10] = scratchreal
|
||
// set scratchreal mantissa = 1.0
|
||
mov dword ptr [RSP+8], 0;
|
||
mov dword ptr [RSP+8+4], 0x80000000;
|
||
and AX, 0x7FFF; // drop sign bit
|
||
cmp AX, 0x401D; // avoid InvalidException in fist
|
||
jae L_extreme;
|
||
fldl2e;
|
||
fmul ; // y = x*log2(e)
|
||
fist dword ptr [RSP]; // scratchint = rndint(y)
|
||
fisub dword ptr [RSP]; // y - rndint(y)
|
||
// and now set scratchreal exponent
|
||
mov EAX, [RSP];
|
||
add EAX, 0x3fff;
|
||
jle short L_largenegative;
|
||
cmp EAX,0x8000;
|
||
jge short L_largepositive;
|
||
mov [RSP+8+8],AX;
|
||
f2xm1; // 2^(y-rndint(y)) -1
|
||
fld real ptr [RSP+8] ; // 2^rndint(y)
|
||
fmul ST(1), ST;
|
||
fld1;
|
||
fsubp ST(1), ST;
|
||
fadd;
|
||
add RSP,24;
|
||
ret;
|
||
|
||
L_extreme: // Extreme exponent. X is very large positive, very
|
||
// large negative, infinity, or NaN.
|
||
fxam;
|
||
fstsw AX;
|
||
test AX, 0x0400; // NaN_or_zero, but we already know x != 0
|
||
jz L_was_nan; // if x is NaN, returns x
|
||
test AX, 0x0200;
|
||
jnz L_largenegative;
|
||
L_largepositive:
|
||
// Set scratchreal = real.max.
|
||
// squaring it will create infinity, and set overflow flag.
|
||
mov word ptr [RSP+8+8], 0x7FFE;
|
||
fstp ST(0);
|
||
fld real ptr [RSP+8]; // load scratchreal
|
||
fmul ST(0), ST; // square it, to create havoc!
|
||
L_was_nan:
|
||
add RSP,24;
|
||
ret;
|
||
|
||
L_largenegative:
|
||
fstp ST(0);
|
||
fld1;
|
||
fchs; // return -1. Underflow flag is not set.
|
||
add RSP,24;
|
||
ret;
|
||
}
|
||
}
|
||
else
|
||
{
|
||
// Coefficients for exp(x) - 1 and overflow/underflow limits.
|
||
static if (floatTraits!real.realFormat == RealFormat.ieeeQuadruple)
|
||
{
|
||
static immutable real[8] P = [
|
||
2.943520915569954073888921213330863757240E8L,
|
||
-5.722847283900608941516165725053359168840E7L,
|
||
8.944630806357575461578107295909719817253E6L,
|
||
-7.212432713558031519943281748462837065308E5L,
|
||
4.578962475841642634225390068461943438441E4L,
|
||
-1.716772506388927649032068540558788106762E3L,
|
||
4.401308817383362136048032038528753151144E1L,
|
||
-4.888737542888633647784737721812546636240E-1L
|
||
];
|
||
|
||
static immutable real[9] Q = [
|
||
1.766112549341972444333352727998584753865E9L,
|
||
-7.848989743695296475743081255027098295771E8L,
|
||
1.615869009634292424463780387327037251069E8L,
|
||
-2.019684072836541751428967854947019415698E7L,
|
||
1.682912729190313538934190635536631941751E6L,
|
||
-9.615511549171441430850103489315371768998E4L,
|
||
3.697714952261803935521187272204485251835E3L,
|
||
-8.802340681794263968892934703309274564037E1L,
|
||
1.0
|
||
];
|
||
|
||
enum real OF = 1.1356523406294143949491931077970764891253E4L;
|
||
enum real UF = -1.143276959615573793352782661133116431383730e4L;
|
||
}
|
||
else
|
||
{
|
||
static immutable real[5] P = [
|
||
-1.586135578666346600772998894928250240826E4L,
|
||
2.642771505685952966904660652518429479531E3L,
|
||
-3.423199068835684263987132888286791620673E2L,
|
||
1.800826371455042224581246202420972737840E1L,
|
||
-5.238523121205561042771939008061958820811E-1L,
|
||
];
|
||
static immutable real[6] Q = [
|
||
-9.516813471998079611319047060563358064497E4L,
|
||
3.964866271411091674556850458227710004570E4L,
|
||
-7.207678383830091850230366618190187434796E3L,
|
||
7.206038318724600171970199625081491823079E2L,
|
||
-4.002027679107076077238836622982900945173E1L,
|
||
1.0
|
||
];
|
||
|
||
enum real OF = 1.1356523406294143949492E4L;
|
||
enum real UF = -4.5054566736396445112120088E1L;
|
||
}
|
||
|
||
|
||
// C1 + C2 = LN2.
|
||
enum real C1 = 6.9314575195312500000000E-1L;
|
||
enum real C2 = 1.428606820309417232121458176568075500134E-6L;
|
||
|
||
// Special cases. Raises an overflow flag, except in the case
|
||
// for CTFE, where there are no hardware controls.
|
||
if (x > OF)
|
||
{
|
||
if (__ctfe)
|
||
return real.infinity;
|
||
else
|
||
return real.max * copysign(real.max, real.infinity);
|
||
}
|
||
if (x == 0.0)
|
||
return x;
|
||
if (x < UF)
|
||
return -1.0;
|
||
|
||
// Express x = LN2 (n + remainder), remainder not exceeding 1/2.
|
||
int n = cast(int) floor(0.5 + x / LN2);
|
||
x -= n * C1;
|
||
x -= n * C2;
|
||
|
||
// Rational approximation:
|
||
// exp(x) - 1 = x + 0.5 x^^2 + x^^3 P(x) / Q(x)
|
||
real px = x * poly(x, P);
|
||
real qx = poly(x, Q);
|
||
const real xx = x * x;
|
||
qx = x + (0.5 * xx + xx * px / qx);
|
||
|
||
// We have qx = exp(remainder LN2) - 1, so:
|
||
// exp(x) - 1 = 2^^n (qx + 1) - 1 = 2^^n qx + 2^^n - 1.
|
||
px = ldexp(1.0, n);
|
||
x = px * qx + (px - 1.0);
|
||
|
||
return x;
|
||
}
|
||
}
|
||
|
||
|
||
|
||
/**
|
||
* Calculates 2$(SUPERSCRIPT x).
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH x) $(TH exp2(x)) )
|
||
* $(TR $(TD +$(INFIN)) $(TD +$(INFIN)) )
|
||
* $(TR $(TD -$(INFIN)) $(TD +0.0) )
|
||
* $(TR $(TD $(NAN)) $(TD $(NAN)) )
|
||
* )
|
||
*/
|
||
real exp2(real x) @nogc @trusted pure nothrow
|
||
{
|
||
version(D_InlineAsm_X86)
|
||
{
|
||
enum PARAMSIZE = (real.sizeof+3)&(0xFFFF_FFFC); // always a multiple of 4
|
||
|
||
asm pure nothrow @nogc
|
||
{
|
||
/* exp2() for x87 80-bit reals, IEEE754-2008 conformant.
|
||
* Author: Don Clugston.
|
||
*
|
||
* exp2(x) = 2^^(rndint(x))* 2^^(y-rndint(x))
|
||
* The trick for high performance is to avoid the fscale(28cycles on core2),
|
||
* frndint(19 cycles), leaving f2xm1(19 cycles) as the only slow instruction.
|
||
*
|
||
* We can do frndint by using fist. BUT we can't use it for huge numbers,
|
||
* because it will set the Invalid Operation flag if overflow or NaN occurs.
|
||
* Fortunately, whenever this happens the result would be zero or infinity.
|
||
*
|
||
* We can perform fscale by directly poking into the exponent. BUT this doesn't
|
||
* work for the (very rare) cases where the result is subnormal. So we fall back
|
||
* to the slow method in that case.
|
||
*/
|
||
naked;
|
||
fld real ptr [ESP+4] ; // x
|
||
mov AX, [ESP+4+8]; // AX = exponent and sign
|
||
sub ESP, 12+8; // Create scratch space on the stack
|
||
// [ESP,ESP+2] = scratchint
|
||
// [ESP+4..+6, +8..+10, +10] = scratchreal
|
||
// set scratchreal mantissa = 1.0
|
||
mov dword ptr [ESP+8], 0;
|
||
mov dword ptr [ESP+8+4], 0x80000000;
|
||
and AX, 0x7FFF; // drop sign bit
|
||
cmp AX, 0x401D; // avoid InvalidException in fist
|
||
jae L_extreme;
|
||
fist dword ptr [ESP]; // scratchint = rndint(x)
|
||
fisub dword ptr [ESP]; // x - rndint(x)
|
||
// and now set scratchreal exponent
|
||
mov EAX, [ESP];
|
||
add EAX, 0x3fff;
|
||
jle short L_subnormal;
|
||
cmp EAX,0x8000;
|
||
jge short L_overflow;
|
||
mov [ESP+8+8],AX;
|
||
L_normal:
|
||
f2xm1;
|
||
fld1;
|
||
faddp ST(1), ST; // 2^^(x-rndint(x))
|
||
fld real ptr [ESP+8] ; // 2^^rndint(x)
|
||
add ESP,12+8;
|
||
fmulp ST(1), ST;
|
||
ret PARAMSIZE;
|
||
|
||
L_subnormal:
|
||
// Result will be subnormal.
|
||
// In this rare case, the simple poking method doesn't work.
|
||
// The speed doesn't matter, so use the slow fscale method.
|
||
fild dword ptr [ESP]; // scratchint
|
||
fld1;
|
||
fscale;
|
||
fstp real ptr [ESP+8]; // scratchreal = 2^^scratchint
|
||
fstp ST(0); // drop scratchint
|
||
jmp L_normal;
|
||
|
||
L_extreme: // Extreme exponent. X is very large positive, very
|
||
// large negative, infinity, or NaN.
|
||
fxam;
|
||
fstsw AX;
|
||
test AX, 0x0400; // NaN_or_zero, but we already know x != 0
|
||
jz L_was_nan; // if x is NaN, returns x
|
||
// set scratchreal = real.min_normal
|
||
// squaring it will return 0, setting underflow flag
|
||
mov word ptr [ESP+8+8], 1;
|
||
test AX, 0x0200;
|
||
jnz L_waslargenegative;
|
||
L_overflow:
|
||
// Set scratchreal = real.max.
|
||
// squaring it will create infinity, and set overflow flag.
|
||
mov word ptr [ESP+8+8], 0x7FFE;
|
||
L_waslargenegative:
|
||
fstp ST(0);
|
||
fld real ptr [ESP+8]; // load scratchreal
|
||
fmul ST(0), ST; // square it, to create havoc!
|
||
L_was_nan:
|
||
add ESP,12+8;
|
||
ret PARAMSIZE;
|
||
}
|
||
}
|
||
else version(D_InlineAsm_X86_64)
|
||
{
|
||
asm pure nothrow @nogc
|
||
{
|
||
naked;
|
||
}
|
||
version (Win64)
|
||
{
|
||
asm pure nothrow @nogc
|
||
{
|
||
fld real ptr [RCX]; // x
|
||
mov AX,[RCX+8]; // AX = exponent and sign
|
||
}
|
||
}
|
||
else
|
||
{
|
||
asm pure nothrow @nogc
|
||
{
|
||
fld real ptr [RSP+8]; // x
|
||
mov AX,[RSP+8+8]; // AX = exponent and sign
|
||
}
|
||
}
|
||
asm pure nothrow @nogc
|
||
{
|
||
/* exp2() for x87 80-bit reals, IEEE754-2008 conformant.
|
||
* Author: Don Clugston.
|
||
*
|
||
* exp2(x) = 2^(rndint(x))* 2^(y-rndint(x))
|
||
* The trick for high performance is to avoid the fscale(28cycles on core2),
|
||
* frndint(19 cycles), leaving f2xm1(19 cycles) as the only slow instruction.
|
||
*
|
||
* We can do frndint by using fist. BUT we can't use it for huge numbers,
|
||
* because it will set the Invalid Operation flag is overflow or NaN occurs.
|
||
* Fortunately, whenever this happens the result would be zero or infinity.
|
||
*
|
||
* We can perform fscale by directly poking into the exponent. BUT this doesn't
|
||
* work for the (very rare) cases where the result is subnormal. So we fall back
|
||
* to the slow method in that case.
|
||
*/
|
||
sub RSP, 24; // Create scratch space on the stack
|
||
// [RSP,RSP+2] = scratchint
|
||
// [RSP+4..+6, +8..+10, +10] = scratchreal
|
||
// set scratchreal mantissa = 1.0
|
||
mov dword ptr [RSP+8], 0;
|
||
mov dword ptr [RSP+8+4], 0x80000000;
|
||
and AX, 0x7FFF; // drop sign bit
|
||
cmp AX, 0x401D; // avoid InvalidException in fist
|
||
jae L_extreme;
|
||
fist dword ptr [RSP]; // scratchint = rndint(x)
|
||
fisub dword ptr [RSP]; // x - rndint(x)
|
||
// and now set scratchreal exponent
|
||
mov EAX, [RSP];
|
||
add EAX, 0x3fff;
|
||
jle short L_subnormal;
|
||
cmp EAX,0x8000;
|
||
jge short L_overflow;
|
||
mov [RSP+8+8],AX;
|
||
L_normal:
|
||
f2xm1;
|
||
fld1;
|
||
fadd; // 2^(x-rndint(x))
|
||
fld real ptr [RSP+8] ; // 2^rndint(x)
|
||
add RSP,24;
|
||
fmulp ST(1), ST;
|
||
ret;
|
||
|
||
L_subnormal:
|
||
// Result will be subnormal.
|
||
// In this rare case, the simple poking method doesn't work.
|
||
// The speed doesn't matter, so use the slow fscale method.
|
||
fild dword ptr [RSP]; // scratchint
|
||
fld1;
|
||
fscale;
|
||
fstp real ptr [RSP+8]; // scratchreal = 2^scratchint
|
||
fstp ST(0); // drop scratchint
|
||
jmp L_normal;
|
||
|
||
L_extreme: // Extreme exponent. X is very large positive, very
|
||
// large negative, infinity, or NaN.
|
||
fxam;
|
||
fstsw AX;
|
||
test AX, 0x0400; // NaN_or_zero, but we already know x != 0
|
||
jz L_was_nan; // if x is NaN, returns x
|
||
// set scratchreal = real.min
|
||
// squaring it will return 0, setting underflow flag
|
||
mov word ptr [RSP+8+8], 1;
|
||
test AX, 0x0200;
|
||
jnz L_waslargenegative;
|
||
L_overflow:
|
||
// Set scratchreal = real.max.
|
||
// squaring it will create infinity, and set overflow flag.
|
||
mov word ptr [RSP+8+8], 0x7FFE;
|
||
L_waslargenegative:
|
||
fstp ST(0);
|
||
fld real ptr [RSP+8]; // load scratchreal
|
||
fmul ST(0), ST; // square it, to create havoc!
|
||
L_was_nan:
|
||
add RSP,24;
|
||
ret;
|
||
}
|
||
}
|
||
else
|
||
{
|
||
// Coefficients for exp2(x)
|
||
static if (floatTraits!real.realFormat == RealFormat.ieeeQuadruple)
|
||
{
|
||
static immutable real[5] P = [
|
||
9.079594442980146270952372234833529694788E12L,
|
||
1.530625323728429161131811299626419117557E11L,
|
||
5.677513871931844661829755443994214173883E8L,
|
||
6.185032670011643762127954396427045467506E5L,
|
||
1.587171580015525194694938306936721666031E2L
|
||
];
|
||
|
||
static immutable real[6] Q = [
|
||
2.619817175234089411411070339065679229869E13L,
|
||
1.490560994263653042761789432690793026977E12L,
|
||
1.092141473886177435056423606755843616331E10L,
|
||
2.186249607051644894762167991800811827835E7L,
|
||
1.236602014442099053716561665053645270207E4L,
|
||
1.0
|
||
];
|
||
}
|
||
else
|
||
{
|
||
static immutable real[3] P = [
|
||
2.0803843631901852422887E6L,
|
||
3.0286971917562792508623E4L,
|
||
6.0614853552242266094567E1L,
|
||
];
|
||
static immutable real[4] Q = [
|
||
6.0027204078348487957118E6L,
|
||
3.2772515434906797273099E5L,
|
||
1.7492876999891839021063E3L,
|
||
1.0000000000000000000000E0L,
|
||
];
|
||
}
|
||
|
||
// Overflow and Underflow limits.
|
||
enum real OF = 16_384.0L;
|
||
enum real UF = -16_382.0L;
|
||
|
||
// Special cases. Raises an overflow or underflow flag accordingly,
|
||
// except in the case for CTFE, where there are no hardware controls.
|
||
if (isNaN(x))
|
||
return x;
|
||
if (x > OF)
|
||
{
|
||
if (__ctfe)
|
||
return real.infinity;
|
||
else
|
||
return real.max * copysign(real.max, real.infinity);
|
||
}
|
||
if (x < UF)
|
||
{
|
||
if (__ctfe)
|
||
return 0.0;
|
||
else
|
||
return real.min_normal * copysign(real.min_normal, 0.0);
|
||
}
|
||
|
||
// Separate into integer and fractional parts.
|
||
int n = cast(int) floor(x + 0.5);
|
||
x -= n;
|
||
|
||
// Rational approximation:
|
||
// exp2(x) = 1.0 + 2x P(x^^2) / (Q(x^^2) - P(x^^2))
|
||
const real xx = x * x;
|
||
const real px = x * poly(xx, P);
|
||
x = px / (poly(xx, Q) - px);
|
||
x = 1.0 + ldexp(x, 1);
|
||
|
||
// Scale by power of 2.
|
||
x = ldexp(x, n);
|
||
|
||
return x;
|
||
}
|
||
}
|
||
|
||
///
|
||
@safe unittest
|
||
{
|
||
assert(feqrel(exp2(0.5L), SQRT2) >= real.mant_dig -1);
|
||
assert(exp2(8.0L) == 256.0);
|
||
assert(exp2(-9.0L)== 1.0L/512.0);
|
||
}
|
||
|
||
@safe unittest
|
||
{
|
||
version(CRuntime_Microsoft) {} else // aexp2/exp2f/exp2l not implemented
|
||
{
|
||
assert( core.stdc.math.exp2f(0.0f) == 1 );
|
||
assert( core.stdc.math.exp2 (0.0) == 1 );
|
||
assert( core.stdc.math.exp2l(0.0L) == 1 );
|
||
}
|
||
}
|
||
|
||
@system unittest
|
||
{
|
||
FloatingPointControl ctrl;
|
||
if (FloatingPointControl.hasExceptionTraps)
|
||
ctrl.disableExceptions(FloatingPointControl.allExceptions);
|
||
ctrl.rounding = FloatingPointControl.roundToNearest;
|
||
|
||
static if (real.mant_dig == 113)
|
||
{
|
||
static immutable real[2][] exptestpoints =
|
||
[ // x exp(x)
|
||
[ 1.0L, E ],
|
||
[ 0.5L, 0x1.a61298e1e069bc972dfefab6df34p+0L ],
|
||
[ 3.0L, E*E*E ],
|
||
[ 0x1.6p+13L, 0x1.6e509d45728655cdb4840542acb5p+16250L ], // near overflow
|
||
[ 0x1.7p+13L, real.infinity ], // close overflow
|
||
[ 0x1p+80L, real.infinity ], // far overflow
|
||
[ real.infinity, real.infinity ],
|
||
[-0x1.18p+13L, 0x1.5e4bf54b4807034ea97fef0059a6p-12927L ], // near underflow
|
||
[-0x1.625p+13L, 0x1.a6bd68a39d11fec3a250cd97f524p-16358L ], // ditto
|
||
[-0x1.62dafp+13L, 0x0.cb629e9813b80ed4d639e875be6cp-16382L ], // near underflow - subnormal
|
||
[-0x1.6549p+13L, 0x0.0000000000000000000000000001p-16382L ], // ditto
|
||
[-0x1.655p+13L, 0 ], // close underflow
|
||
[-0x1p+30L, 0 ], // far underflow
|
||
];
|
||
}
|
||
else static if (real.mant_dig == 64) // 80-bit reals
|
||
{
|
||
static immutable real[2][] exptestpoints =
|
||
[ // x exp(x)
|
||
[ 1.0L, E ],
|
||
[ 0.5L, 0x1.a61298e1e069bc97p+0L ],
|
||
[ 3.0L, E*E*E ],
|
||
[ 0x1.1p+13L, 0x1.29aeffefc8ec645p+12557L ], // near overflow
|
||
[ 0x1.7p+13L, real.infinity ], // close overflow
|
||
[ 0x1p+80L, real.infinity ], // far overflow
|
||
[ real.infinity, real.infinity ],
|
||
[-0x1.18p+13L, 0x1.5e4bf54b4806db9p-12927L ], // near underflow
|
||
[-0x1.625p+13L, 0x1.a6bd68a39d11f35cp-16358L ], // ditto
|
||
[-0x1.62dafp+13L, 0x1.96c53d30277021dp-16383L ], // near underflow - subnormal
|
||
[-0x1.643p+13L, 0x1p-16444L ], // ditto
|
||
[-0x1.645p+13L, 0 ], // close underflow
|
||
[-0x1p+30L, 0 ], // far underflow
|
||
];
|
||
}
|
||
else static if (real.mant_dig == 53) // 64-bit reals
|
||
{
|
||
static immutable real[2][] exptestpoints =
|
||
[ // x, exp(x)
|
||
[ 1.0L, E ],
|
||
[ 0.5L, 0x1.a61298e1e069cp+0L ],
|
||
[ 3.0L, E*E*E ],
|
||
[ 0x1.6p+9L, 0x1.93bf4ec282efbp+1015L ], // near overflow
|
||
[ 0x1.7p+9L, real.infinity ], // close overflow
|
||
[ 0x1p+80L, real.infinity ], // far overflow
|
||
[ real.infinity, real.infinity ],
|
||
[-0x1.6p+9L, 0x1.44a3824e5285fp-1016L ], // near underflow
|
||
[-0x1.64p+9L, 0x0.06f84920bb2d4p-1022L ], // near underflow - subnormal
|
||
[-0x1.743p+9L, 0x0.0000000000001p-1022L ], // ditto
|
||
[-0x1.8p+9L, 0 ], // close underflow
|
||
[-0x1p30L, 0 ], // far underflow
|
||
];
|
||
}
|
||
else
|
||
static assert(0, "No exp() tests for real type!");
|
||
|
||
const minEqualMantissaBits = real.mant_dig - 2;
|
||
real x;
|
||
IeeeFlags f;
|
||
foreach (ref pair; exptestpoints)
|
||
{
|
||
resetIeeeFlags();
|
||
x = exp(pair[0]);
|
||
f = ieeeFlags;
|
||
assert(feqrel(x, pair[1]) >= minEqualMantissaBits);
|
||
|
||
version (IeeeFlagsSupport)
|
||
{
|
||
// Check the overflow bit
|
||
if (x == real.infinity)
|
||
{
|
||
// don't care about the overflow bit if input was inf
|
||
// (e.g., the LLVM intrinsic doesn't set it on Linux x86_64)
|
||
assert(pair[0] == real.infinity || f.overflow);
|
||
}
|
||
else
|
||
assert(!f.overflow);
|
||
// Check the underflow bit
|
||
assert(f.underflow == (fabs(x) < real.min_normal));
|
||
// Invalid and div by zero shouldn't be affected.
|
||
assert(!f.invalid);
|
||
assert(!f.divByZero);
|
||
}
|
||
}
|
||
// Ideally, exp(0) would not set the inexact flag.
|
||
// Unfortunately, fldl2e sets it!
|
||
// So it's not realistic to avoid setting it.
|
||
assert(exp(0.0L) == 1.0);
|
||
|
||
// NaN propagation. Doesn't set flags, bcos was already NaN.
|
||
resetIeeeFlags();
|
||
x = exp(real.nan);
|
||
f = ieeeFlags;
|
||
assert(isIdentical(abs(x), real.nan));
|
||
assert(f.flags == 0);
|
||
|
||
resetIeeeFlags();
|
||
x = exp(-real.nan);
|
||
f = ieeeFlags;
|
||
assert(isIdentical(abs(x), real.nan));
|
||
assert(f.flags == 0);
|
||
|
||
x = exp(NaN(0x123));
|
||
assert(isIdentical(x, NaN(0x123)));
|
||
|
||
// High resolution test (verified against GNU MPFR/Mathematica).
|
||
assert(exp(0.5L) == 0x1.A612_98E1_E069_BC97_2DFE_FAB6_DF34p+0L);
|
||
}
|
||
|
||
|
||
/**
|
||
* Calculate cos(y) + i sin(y).
|
||
*
|
||
* On many CPUs (such as x86), this is a very efficient operation;
|
||
* almost twice as fast as calculating sin(y) and cos(y) separately,
|
||
* and is the preferred method when both are required.
|
||
*/
|
||
creal expi(real y) @trusted pure nothrow @nogc
|
||
{
|
||
version(InlineAsm_X86_Any)
|
||
{
|
||
version (Win64)
|
||
{
|
||
asm pure nothrow @nogc
|
||
{
|
||
naked;
|
||
fld real ptr [ECX];
|
||
fsincos;
|
||
fxch ST(1), ST(0);
|
||
ret;
|
||
}
|
||
}
|
||
else
|
||
{
|
||
asm pure nothrow @nogc
|
||
{
|
||
fld y;
|
||
fsincos;
|
||
fxch ST(1), ST(0);
|
||
}
|
||
}
|
||
}
|
||
else
|
||
{
|
||
return cos(y) + sin(y)*1i;
|
||
}
|
||
}
|
||
|
||
///
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
assert(expi(1.3e5L) == cos(1.3e5L) + sin(1.3e5L) * 1i);
|
||
assert(expi(0.0L) == 1L + 0.0Li);
|
||
}
|
||
|
||
/*********************************************************************
|
||
* Separate floating point value into significand and exponent.
|
||
*
|
||
* Returns:
|
||
* Calculate and return $(I x) and $(I exp) such that
|
||
* value =$(I x)*2$(SUPERSCRIPT exp) and
|
||
* .5 $(LT)= |$(I x)| $(LT) 1.0
|
||
*
|
||
* $(I x) has same sign as value.
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH value) $(TH returns) $(TH exp))
|
||
* $(TR $(TD $(PLUSMN)0.0) $(TD $(PLUSMN)0.0) $(TD 0))
|
||
* $(TR $(TD +$(INFIN)) $(TD +$(INFIN)) $(TD int.max))
|
||
* $(TR $(TD -$(INFIN)) $(TD -$(INFIN)) $(TD int.min))
|
||
* $(TR $(TD $(PLUSMN)$(NAN)) $(TD $(PLUSMN)$(NAN)) $(TD int.min))
|
||
* )
|
||
*/
|
||
T frexp(T)(const T value, out int exp) @trusted pure nothrow @nogc
|
||
if (isFloatingPoint!T)
|
||
{
|
||
Unqual!T vf = value;
|
||
ushort* vu = cast(ushort*)&vf;
|
||
static if (is(Unqual!T == float))
|
||
int* vi = cast(int*)&vf;
|
||
else
|
||
long* vl = cast(long*)&vf;
|
||
int ex;
|
||
alias F = floatTraits!T;
|
||
|
||
ex = vu[F.EXPPOS_SHORT] & F.EXPMASK;
|
||
static if (F.realFormat == RealFormat.ieeeExtended)
|
||
{
|
||
if (ex)
|
||
{ // If exponent is non-zero
|
||
if (ex == F.EXPMASK) // infinity or NaN
|
||
{
|
||
if (*vl & 0x7FFF_FFFF_FFFF_FFFF) // NaN
|
||
{
|
||
*vl |= 0xC000_0000_0000_0000; // convert NaNS to NaNQ
|
||
exp = int.min;
|
||
}
|
||
else if (vu[F.EXPPOS_SHORT] & 0x8000) // negative infinity
|
||
exp = int.min;
|
||
else // positive infinity
|
||
exp = int.max;
|
||
|
||
}
|
||
else
|
||
{
|
||
exp = ex - F.EXPBIAS;
|
||
vu[F.EXPPOS_SHORT] = (0x8000 & vu[F.EXPPOS_SHORT]) | 0x3FFE;
|
||
}
|
||
}
|
||
else if (!*vl)
|
||
{
|
||
// vf is +-0.0
|
||
exp = 0;
|
||
}
|
||
else
|
||
{
|
||
// subnormal
|
||
|
||
vf *= F.RECIP_EPSILON;
|
||
ex = vu[F.EXPPOS_SHORT] & F.EXPMASK;
|
||
exp = ex - F.EXPBIAS - T.mant_dig + 1;
|
||
vu[F.EXPPOS_SHORT] = (~F.EXPMASK & vu[F.EXPPOS_SHORT]) | 0x3FFE;
|
||
}
|
||
return vf;
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeQuadruple)
|
||
{
|
||
if (ex) // If exponent is non-zero
|
||
{
|
||
if (ex == F.EXPMASK)
|
||
{
|
||
// infinity or NaN
|
||
if (vl[MANTISSA_LSB] |
|
||
(vl[MANTISSA_MSB] & 0x0000_FFFF_FFFF_FFFF)) // NaN
|
||
{
|
||
// convert NaNS to NaNQ
|
||
vl[MANTISSA_MSB] |= 0x0000_8000_0000_0000;
|
||
exp = int.min;
|
||
}
|
||
else if (vu[F.EXPPOS_SHORT] & 0x8000) // negative infinity
|
||
exp = int.min;
|
||
else // positive infinity
|
||
exp = int.max;
|
||
}
|
||
else
|
||
{
|
||
exp = ex - F.EXPBIAS;
|
||
vu[F.EXPPOS_SHORT] = F.EXPBIAS | (0x8000 & vu[F.EXPPOS_SHORT]);
|
||
}
|
||
}
|
||
else if ((vl[MANTISSA_LSB] |
|
||
(vl[MANTISSA_MSB] & 0x0000_FFFF_FFFF_FFFF)) == 0)
|
||
{
|
||
// vf is +-0.0
|
||
exp = 0;
|
||
}
|
||
else
|
||
{
|
||
// subnormal
|
||
vf *= F.RECIP_EPSILON;
|
||
ex = vu[F.EXPPOS_SHORT] & F.EXPMASK;
|
||
exp = ex - F.EXPBIAS - T.mant_dig + 1;
|
||
vu[F.EXPPOS_SHORT] = F.EXPBIAS | (0x8000 & vu[F.EXPPOS_SHORT]);
|
||
}
|
||
return vf;
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeDouble)
|
||
{
|
||
if (ex) // If exponent is non-zero
|
||
{
|
||
if (ex == F.EXPMASK) // infinity or NaN
|
||
{
|
||
if (*vl == 0x7FF0_0000_0000_0000) // positive infinity
|
||
{
|
||
exp = int.max;
|
||
}
|
||
else if (*vl == 0xFFF0_0000_0000_0000) // negative infinity
|
||
exp = int.min;
|
||
else
|
||
{ // NaN
|
||
*vl |= 0x0008_0000_0000_0000; // convert NaNS to NaNQ
|
||
exp = int.min;
|
||
}
|
||
}
|
||
else
|
||
{
|
||
exp = (ex - F.EXPBIAS) >> 4;
|
||
vu[F.EXPPOS_SHORT] = cast(ushort)((0x800F & vu[F.EXPPOS_SHORT]) | 0x3FE0);
|
||
}
|
||
}
|
||
else if (!(*vl & 0x7FFF_FFFF_FFFF_FFFF))
|
||
{
|
||
// vf is +-0.0
|
||
exp = 0;
|
||
}
|
||
else
|
||
{
|
||
// subnormal
|
||
vf *= F.RECIP_EPSILON;
|
||
ex = vu[F.EXPPOS_SHORT] & F.EXPMASK;
|
||
exp = ((ex - F.EXPBIAS) >> 4) - T.mant_dig + 1;
|
||
vu[F.EXPPOS_SHORT] =
|
||
cast(ushort)((~F.EXPMASK & vu[F.EXPPOS_SHORT]) | 0x3FE0);
|
||
}
|
||
return vf;
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeSingle)
|
||
{
|
||
if (ex) // If exponent is non-zero
|
||
{
|
||
if (ex == F.EXPMASK) // infinity or NaN
|
||
{
|
||
if (*vi == 0x7F80_0000) // positive infinity
|
||
{
|
||
exp = int.max;
|
||
}
|
||
else if (*vi == 0xFF80_0000) // negative infinity
|
||
exp = int.min;
|
||
else
|
||
{ // NaN
|
||
*vi |= 0x0040_0000; // convert NaNS to NaNQ
|
||
exp = int.min;
|
||
}
|
||
}
|
||
else
|
||
{
|
||
exp = (ex - F.EXPBIAS) >> 7;
|
||
vu[F.EXPPOS_SHORT] = cast(ushort)((0x807F & vu[F.EXPPOS_SHORT]) | 0x3F00);
|
||
}
|
||
}
|
||
else if (!(*vi & 0x7FFF_FFFF))
|
||
{
|
||
// vf is +-0.0
|
||
exp = 0;
|
||
}
|
||
else
|
||
{
|
||
// subnormal
|
||
vf *= F.RECIP_EPSILON;
|
||
ex = vu[F.EXPPOS_SHORT] & F.EXPMASK;
|
||
exp = ((ex - F.EXPBIAS) >> 7) - T.mant_dig + 1;
|
||
vu[F.EXPPOS_SHORT] =
|
||
cast(ushort)((~F.EXPMASK & vu[F.EXPPOS_SHORT]) | 0x3F00);
|
||
}
|
||
return vf;
|
||
}
|
||
else // static if (F.realFormat == RealFormat.ibmExtended)
|
||
{
|
||
assert(0, "frexp not implemented");
|
||
}
|
||
}
|
||
|
||
///
|
||
@system unittest
|
||
{
|
||
int exp;
|
||
real mantissa = frexp(123.456L, exp);
|
||
|
||
// check if values are equal to 19 decimal digits of precision
|
||
assert(equalsDigit(mantissa * pow(2.0L, cast(real) exp), 123.456L, 19));
|
||
|
||
assert(frexp(-real.nan, exp) && exp == int.min);
|
||
assert(frexp(real.nan, exp) && exp == int.min);
|
||
assert(frexp(-real.infinity, exp) == -real.infinity && exp == int.min);
|
||
assert(frexp(real.infinity, exp) == real.infinity && exp == int.max);
|
||
assert(frexp(-0.0, exp) == -0.0 && exp == 0);
|
||
assert(frexp(0.0, exp) == 0.0 && exp == 0);
|
||
}
|
||
|
||
@safe unittest
|
||
{
|
||
import std.meta : AliasSeq;
|
||
import std.typecons : tuple, Tuple;
|
||
|
||
foreach (T; AliasSeq!(real, double, float))
|
||
{
|
||
Tuple!(T, T, int)[] vals = // x,frexp,exp
|
||
[
|
||
tuple(T(0.0), T( 0.0 ), 0),
|
||
tuple(T(-0.0), T( -0.0), 0),
|
||
tuple(T(1.0), T( .5 ), 1),
|
||
tuple(T(-1.0), T( -.5 ), 1),
|
||
tuple(T(2.0), T( .5 ), 2),
|
||
tuple(T(float.min_normal/2.0f), T(.5), -126),
|
||
tuple(T.infinity, T.infinity, int.max),
|
||
tuple(-T.infinity, -T.infinity, int.min),
|
||
tuple(T.nan, T.nan, int.min),
|
||
tuple(-T.nan, -T.nan, int.min),
|
||
|
||
// Phobos issue #16026:
|
||
tuple(3 * (T.min_normal * T.epsilon), T( .75), (T.min_exp - T.mant_dig) + 2)
|
||
];
|
||
|
||
foreach (elem; vals)
|
||
{
|
||
T x = elem[0];
|
||
T e = elem[1];
|
||
int exp = elem[2];
|
||
int eptr;
|
||
T v = frexp(x, eptr);
|
||
assert(isIdentical(e, v));
|
||
assert(exp == eptr);
|
||
|
||
}
|
||
|
||
static if (floatTraits!(T).realFormat == RealFormat.ieeeExtended)
|
||
{
|
||
static T[3][] extendedvals = [ // x,frexp,exp
|
||
[0x1.a5f1c2eb3fe4efp+73L, 0x1.A5F1C2EB3FE4EFp-1L, 74], // normal
|
||
[0x1.fa01712e8f0471ap-1064L, 0x1.fa01712e8f0471ap-1L, -1063],
|
||
[T.min_normal, .5, -16381],
|
||
[T.min_normal/2.0L, .5, -16382] // subnormal
|
||
];
|
||
foreach (elem; extendedvals)
|
||
{
|
||
T x = elem[0];
|
||
T e = elem[1];
|
||
int exp = cast(int) elem[2];
|
||
int eptr;
|
||
T v = frexp(x, eptr);
|
||
assert(isIdentical(e, v));
|
||
assert(exp == eptr);
|
||
|
||
}
|
||
}
|
||
}
|
||
}
|
||
|
||
@safe unittest
|
||
{
|
||
import std.meta : AliasSeq;
|
||
void foo() {
|
||
foreach (T; AliasSeq!(real, double, float))
|
||
{
|
||
int exp;
|
||
const T a = 1;
|
||
immutable T b = 2;
|
||
auto c = frexp(a, exp);
|
||
auto d = frexp(b, exp);
|
||
}
|
||
}
|
||
}
|
||
|
||
/******************************************
|
||
* Extracts the exponent of x as a signed integral value.
|
||
*
|
||
* If x is not a special value, the result is the same as
|
||
* $(D cast(int) logb(x)).
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH x) $(TH ilogb(x)) $(TH Range error?))
|
||
* $(TR $(TD 0) $(TD FP_ILOGB0) $(TD yes))
|
||
* $(TR $(TD $(PLUSMN)$(INFIN)) $(TD int.max) $(TD no))
|
||
* $(TR $(TD $(NAN)) $(TD FP_ILOGBNAN) $(TD no))
|
||
* )
|
||
*/
|
||
int ilogb(T)(const T x) @trusted pure nothrow @nogc
|
||
if (isFloatingPoint!T)
|
||
{
|
||
import core.bitop : bsr;
|
||
alias F = floatTraits!T;
|
||
|
||
union floatBits
|
||
{
|
||
T rv;
|
||
ushort[T.sizeof/2] vu;
|
||
uint[T.sizeof/4] vui;
|
||
static if (T.sizeof >= 8)
|
||
ulong[T.sizeof/8] vul;
|
||
}
|
||
floatBits y = void;
|
||
y.rv = x;
|
||
|
||
int ex = y.vu[F.EXPPOS_SHORT] & F.EXPMASK;
|
||
static if (F.realFormat == RealFormat.ieeeExtended)
|
||
{
|
||
if (ex)
|
||
{
|
||
// If exponent is non-zero
|
||
if (ex == F.EXPMASK) // infinity or NaN
|
||
{
|
||
if (y.vul[0] & 0x7FFF_FFFF_FFFF_FFFF) // NaN
|
||
return FP_ILOGBNAN;
|
||
else // +-infinity
|
||
return int.max;
|
||
}
|
||
else
|
||
{
|
||
return ex - F.EXPBIAS - 1;
|
||
}
|
||
}
|
||
else if (!y.vul[0])
|
||
{
|
||
// vf is +-0.0
|
||
return FP_ILOGB0;
|
||
}
|
||
else
|
||
{
|
||
// subnormal
|
||
return ex - F.EXPBIAS - T.mant_dig + 1 + bsr(y.vul[0]);
|
||
}
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeQuadruple)
|
||
{
|
||
if (ex) // If exponent is non-zero
|
||
{
|
||
if (ex == F.EXPMASK)
|
||
{
|
||
// infinity or NaN
|
||
if (y.vul[MANTISSA_LSB] | ( y.vul[MANTISSA_MSB] & 0x0000_FFFF_FFFF_FFFF)) // NaN
|
||
return FP_ILOGBNAN;
|
||
else // +- infinity
|
||
return int.max;
|
||
}
|
||
else
|
||
{
|
||
return ex - F.EXPBIAS - 1;
|
||
}
|
||
}
|
||
else if ((y.vul[MANTISSA_LSB] | (y.vul[MANTISSA_MSB] & 0x0000_FFFF_FFFF_FFFF)) == 0)
|
||
{
|
||
// vf is +-0.0
|
||
return FP_ILOGB0;
|
||
}
|
||
else
|
||
{
|
||
// subnormal
|
||
const ulong msb = y.vul[MANTISSA_MSB] & 0x0000_FFFF_FFFF_FFFF;
|
||
const ulong lsb = y.vul[MANTISSA_LSB];
|
||
if (msb)
|
||
return ex - F.EXPBIAS - T.mant_dig + 1 + bsr(msb) + 64;
|
||
else
|
||
return ex - F.EXPBIAS - T.mant_dig + 1 + bsr(lsb);
|
||
}
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeDouble)
|
||
{
|
||
if (ex) // If exponent is non-zero
|
||
{
|
||
if (ex == F.EXPMASK) // infinity or NaN
|
||
{
|
||
if ((y.vul[0] & 0x7FFF_FFFF_FFFF_FFFF) == 0x7FF0_0000_0000_0000) // +- infinity
|
||
return int.max;
|
||
else // NaN
|
||
return FP_ILOGBNAN;
|
||
}
|
||
else
|
||
{
|
||
return ((ex - F.EXPBIAS) >> 4) - 1;
|
||
}
|
||
}
|
||
else if (!(y.vul[0] & 0x7FFF_FFFF_FFFF_FFFF))
|
||
{
|
||
// vf is +-0.0
|
||
return FP_ILOGB0;
|
||
}
|
||
else
|
||
{
|
||
// subnormal
|
||
enum MANTISSAMASK_64 = ((cast(ulong) F.MANTISSAMASK_INT) << 32) | 0xFFFF_FFFF;
|
||
return ((ex - F.EXPBIAS) >> 4) - T.mant_dig + 1 + bsr(y.vul[0] & MANTISSAMASK_64);
|
||
}
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeSingle)
|
||
{
|
||
if (ex) // If exponent is non-zero
|
||
{
|
||
if (ex == F.EXPMASK) // infinity or NaN
|
||
{
|
||
if ((y.vui[0] & 0x7FFF_FFFF) == 0x7F80_0000) // +- infinity
|
||
return int.max;
|
||
else // NaN
|
||
return FP_ILOGBNAN;
|
||
}
|
||
else
|
||
{
|
||
return ((ex - F.EXPBIAS) >> 7) - 1;
|
||
}
|
||
}
|
||
else if (!(y.vui[0] & 0x7FFF_FFFF))
|
||
{
|
||
// vf is +-0.0
|
||
return FP_ILOGB0;
|
||
}
|
||
else
|
||
{
|
||
// subnormal
|
||
const uint mantissa = y.vui[0] & F.MANTISSAMASK_INT;
|
||
return ((ex - F.EXPBIAS) >> 7) - T.mant_dig + 1 + bsr(mantissa);
|
||
}
|
||
}
|
||
else // static if (F.realFormat == RealFormat.ibmExtended)
|
||
{
|
||
core.stdc.math.ilogbl(x);
|
||
}
|
||
}
|
||
/// ditto
|
||
int ilogb(T)(const T x) @safe pure nothrow @nogc
|
||
if (isIntegral!T && isUnsigned!T)
|
||
{
|
||
import core.bitop : bsr;
|
||
if (x == 0)
|
||
return FP_ILOGB0;
|
||
else
|
||
{
|
||
static assert(T.sizeof <= ulong.sizeof, "integer size too large for the current ilogb implementation");
|
||
return bsr(x);
|
||
}
|
||
}
|
||
/// ditto
|
||
int ilogb(T)(const T x) @safe pure nothrow @nogc
|
||
if (isIntegral!T && isSigned!T)
|
||
{
|
||
import std.traits : Unsigned;
|
||
// Note: abs(x) can not be used because the return type is not Unsigned and
|
||
// the return value would be wrong for x == int.min
|
||
Unsigned!T absx = x >= 0 ? x : -x;
|
||
return ilogb(absx);
|
||
}
|
||
|
||
alias FP_ILOGB0 = core.stdc.math.FP_ILOGB0;
|
||
alias FP_ILOGBNAN = core.stdc.math.FP_ILOGBNAN;
|
||
|
||
@system nothrow @nogc unittest
|
||
{
|
||
import std.meta : AliasSeq;
|
||
import std.typecons : Tuple;
|
||
foreach (F; AliasSeq!(float, double, real))
|
||
{
|
||
alias T = Tuple!(F, int);
|
||
T[13] vals = // x, ilogb(x)
|
||
[
|
||
T( F.nan , FP_ILOGBNAN ),
|
||
T( -F.nan , FP_ILOGBNAN ),
|
||
T( F.infinity, int.max ),
|
||
T( -F.infinity, int.max ),
|
||
T( 0.0 , FP_ILOGB0 ),
|
||
T( -0.0 , FP_ILOGB0 ),
|
||
T( 2.0 , 1 ),
|
||
T( 2.0001 , 1 ),
|
||
T( 1.9999 , 0 ),
|
||
T( 0.5 , -1 ),
|
||
T( 123.123 , 6 ),
|
||
T( -123.123 , 6 ),
|
||
T( 0.123 , -4 ),
|
||
];
|
||
|
||
foreach (elem; vals)
|
||
{
|
||
assert(ilogb(elem[0]) == elem[1]);
|
||
}
|
||
}
|
||
|
||
// min_normal and subnormals
|
||
assert(ilogb(-float.min_normal) == -126);
|
||
assert(ilogb(nextUp(-float.min_normal)) == -127);
|
||
assert(ilogb(nextUp(-float(0.0))) == -149);
|
||
assert(ilogb(-double.min_normal) == -1022);
|
||
assert(ilogb(nextUp(-double.min_normal)) == -1023);
|
||
assert(ilogb(nextUp(-double(0.0))) == -1074);
|
||
static if (floatTraits!(real).realFormat == RealFormat.ieeeExtended)
|
||
{
|
||
assert(ilogb(-real.min_normal) == -16382);
|
||
assert(ilogb(nextUp(-real.min_normal)) == -16383);
|
||
assert(ilogb(nextUp(-real(0.0))) == -16445);
|
||
}
|
||
else static if (floatTraits!(real).realFormat == RealFormat.ieeeDouble)
|
||
{
|
||
assert(ilogb(-real.min_normal) == -1022);
|
||
assert(ilogb(nextUp(-real.min_normal)) == -1023);
|
||
assert(ilogb(nextUp(-real(0.0))) == -1074);
|
||
}
|
||
|
||
// test integer types
|
||
assert(ilogb(0) == FP_ILOGB0);
|
||
assert(ilogb(int.max) == 30);
|
||
assert(ilogb(int.min) == 31);
|
||
assert(ilogb(uint.max) == 31);
|
||
assert(ilogb(long.max) == 62);
|
||
assert(ilogb(long.min) == 63);
|
||
assert(ilogb(ulong.max) == 63);
|
||
}
|
||
|
||
/*******************************************
|
||
* Compute n * 2$(SUPERSCRIPT exp)
|
||
* References: frexp
|
||
*/
|
||
|
||
real ldexp(real n, int exp) @nogc @safe pure nothrow { pragma(inline, true); return core.math.ldexp(n, exp); }
|
||
//FIXME
|
||
///ditto
|
||
double ldexp(double n, int exp) @safe pure nothrow @nogc { return ldexp(cast(real) n, exp); }
|
||
//FIXME
|
||
///ditto
|
||
float ldexp(float n, int exp) @safe pure nothrow @nogc { return ldexp(cast(real) n, exp); }
|
||
|
||
///
|
||
@nogc @safe pure nothrow unittest
|
||
{
|
||
import std.meta : AliasSeq;
|
||
foreach (T; AliasSeq!(float, double, real))
|
||
{
|
||
T r;
|
||
|
||
r = ldexp(3.0L, 3);
|
||
assert(r == 24);
|
||
|
||
r = ldexp(cast(T) 3.0, cast(int) 3);
|
||
assert(r == 24);
|
||
|
||
T n = 3.0;
|
||
int exp = 3;
|
||
r = ldexp(n, exp);
|
||
assert(r == 24);
|
||
}
|
||
}
|
||
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
static if (floatTraits!(real).realFormat == RealFormat.ieeeExtended)
|
||
{
|
||
assert(ldexp(1.0L, -16384) == 0x1p-16384L);
|
||
assert(ldexp(1.0L, -16382) == 0x1p-16382L);
|
||
int x;
|
||
real n = frexp(0x1p-16384L, x);
|
||
assert(n == 0.5L);
|
||
assert(x==-16383);
|
||
assert(ldexp(n, x)==0x1p-16384L);
|
||
}
|
||
else static if (floatTraits!(real).realFormat == RealFormat.ieeeDouble)
|
||
{
|
||
assert(ldexp(1.0L, -1024) == 0x1p-1024L);
|
||
assert(ldexp(1.0L, -1022) == 0x1p-1022L);
|
||
int x;
|
||
real n = frexp(0x1p-1024L, x);
|
||
assert(n == 0.5L);
|
||
assert(x==-1023);
|
||
assert(ldexp(n, x)==0x1p-1024L);
|
||
}
|
||
else static assert(false, "Floating point type real not supported");
|
||
}
|
||
|
||
/* workaround Issue 14718, float parsing depends on platform strtold
|
||
typed_allocator.d
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
assert(ldexp(1.0, -1024) == 0x1p-1024);
|
||
assert(ldexp(1.0, -1022) == 0x1p-1022);
|
||
int x;
|
||
double n = frexp(0x1p-1024, x);
|
||
assert(n == 0.5);
|
||
assert(x==-1023);
|
||
assert(ldexp(n, x)==0x1p-1024);
|
||
}
|
||
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
assert(ldexp(1.0f, -128) == 0x1p-128f);
|
||
assert(ldexp(1.0f, -126) == 0x1p-126f);
|
||
int x;
|
||
float n = frexp(0x1p-128f, x);
|
||
assert(n == 0.5f);
|
||
assert(x==-127);
|
||
assert(ldexp(n, x)==0x1p-128f);
|
||
}
|
||
*/
|
||
|
||
@system unittest
|
||
{
|
||
static real[3][] vals = // value,exp,ldexp
|
||
[
|
||
[ 0, 0, 0],
|
||
[ 1, 0, 1],
|
||
[ -1, 0, -1],
|
||
[ 1, 1, 2],
|
||
[ 123, 10, 125952],
|
||
[ real.max, int.max, real.infinity],
|
||
[ real.max, -int.max, 0],
|
||
[ real.min_normal, -int.max, 0],
|
||
];
|
||
int i;
|
||
|
||
for (i = 0; i < vals.length; i++)
|
||
{
|
||
real x = vals[i][0];
|
||
int exp = cast(int) vals[i][1];
|
||
real z = vals[i][2];
|
||
real l = ldexp(x, exp);
|
||
|
||
assert(equalsDigit(z, l, 7));
|
||
}
|
||
|
||
real function(real, int) pldexp = &ldexp;
|
||
assert(pldexp != null);
|
||
}
|
||
|
||
private
|
||
{
|
||
version (INLINE_YL2X) {} else
|
||
{
|
||
static if (floatTraits!real.realFormat == RealFormat.ieeeQuadruple)
|
||
{
|
||
// Coefficients for log(1 + x) = x - x**2/2 + x**3 P(x)/Q(x)
|
||
static immutable real[13] logCoeffsP = [
|
||
1.313572404063446165910279910527789794488E4L,
|
||
7.771154681358524243729929227226708890930E4L,
|
||
2.014652742082537582487669938141683759923E5L,
|
||
3.007007295140399532324943111654767187848E5L,
|
||
2.854829159639697837788887080758954924001E5L,
|
||
1.797628303815655343403735250238293741397E5L,
|
||
7.594356839258970405033155585486712125861E4L,
|
||
2.128857716871515081352991964243375186031E4L,
|
||
3.824952356185897735160588078446136783779E3L,
|
||
4.114517881637811823002128927449878962058E2L,
|
||
2.321125933898420063925789532045674660756E1L,
|
||
4.998469661968096229986658302195402690910E-1L,
|
||
1.538612243596254322971797716843006400388E-6L
|
||
];
|
||
static immutable real[13] logCoeffsQ = [
|
||
3.940717212190338497730839731583397586124E4L,
|
||
2.626900195321832660448791748036714883242E5L,
|
||
7.777690340007566932935753241556479363645E5L,
|
||
1.347518538384329112529391120390701166528E6L,
|
||
1.514882452993549494932585972882995548426E6L,
|
||
1.158019977462989115839826904108208787040E6L,
|
||
6.132189329546557743179177159925690841200E5L,
|
||
2.248234257620569139969141618556349415120E5L,
|
||
5.605842085972455027590989944010492125825E4L,
|
||
9.147150349299596453976674231612674085381E3L,
|
||
9.104928120962988414618126155557301584078E2L,
|
||
4.839208193348159620282142911143429644326E1L,
|
||
1.0
|
||
];
|
||
|
||
// Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2)
|
||
// where z = 2(x-1)/(x+1)
|
||
static immutable real[6] logCoeffsR = [
|
||
-8.828896441624934385266096344596648080902E-1L,
|
||
8.057002716646055371965756206836056074715E1L,
|
||
-2.024301798136027039250415126250455056397E3L,
|
||
2.048819892795278657810231591630928516206E4L,
|
||
-8.977257995689735303686582344659576526998E4L,
|
||
1.418134209872192732479751274970992665513E5L
|
||
];
|
||
static immutable real[6] logCoeffsS = [
|
||
1.701761051846631278975701529965589676574E6L
|
||
-1.332535117259762928288745111081235577029E6L,
|
||
4.001557694070773974936904547424676279307E5L,
|
||
-5.748542087379434595104154610899551484314E4L,
|
||
3.998526750980007367835804959888064681098E3L,
|
||
-1.186359407982897997337150403816839480438E2L,
|
||
1.0
|
||
];
|
||
}
|
||
else
|
||
{
|
||
// Coefficients for log(1 + x) = x - x**2/2 + x**3 P(x)/Q(x)
|
||
static immutable real[7] logCoeffsP = [
|
||
2.0039553499201281259648E1L,
|
||
5.7112963590585538103336E1L,
|
||
6.0949667980987787057556E1L,
|
||
2.9911919328553073277375E1L,
|
||
6.5787325942061044846969E0L,
|
||
4.9854102823193375972212E-1L,
|
||
4.5270000862445199635215E-5L,
|
||
];
|
||
static immutable real[7] logCoeffsQ = [
|
||
6.0118660497603843919306E1L,
|
||
2.1642788614495947685003E2L,
|
||
3.0909872225312059774938E2L,
|
||
2.2176239823732856465394E2L,
|
||
8.3047565967967209469434E1L,
|
||
1.5062909083469192043167E1L,
|
||
1.0000000000000000000000E0L,
|
||
];
|
||
|
||
// Coefficients for log(x) = z + z^3 P(z^2)/Q(z^2)
|
||
// where z = 2(x-1)/(x+1)
|
||
static immutable real[4] logCoeffsR = [
|
||
-3.5717684488096787370998E1L,
|
||
1.0777257190312272158094E1L,
|
||
-7.1990767473014147232598E-1L,
|
||
1.9757429581415468984296E-3L,
|
||
];
|
||
static immutable real[4] logCoeffsS = [
|
||
-4.2861221385716144629696E2L,
|
||
1.9361891836232102174846E2L,
|
||
-2.6201045551331104417768E1L,
|
||
1.0000000000000000000000E0L,
|
||
];
|
||
}
|
||
}
|
||
}
|
||
|
||
/**************************************
|
||
* Calculate the natural logarithm of x.
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH x) $(TH log(x)) $(TH divide by 0?) $(TH invalid?))
|
||
* $(TR $(TD $(PLUSMN)0.0) $(TD -$(INFIN)) $(TD yes) $(TD no))
|
||
* $(TR $(TD $(LT)0.0) $(TD $(NAN)) $(TD no) $(TD yes))
|
||
* $(TR $(TD +$(INFIN)) $(TD +$(INFIN)) $(TD no) $(TD no))
|
||
* )
|
||
*/
|
||
real log(real x) @safe pure nothrow @nogc
|
||
{
|
||
version (INLINE_YL2X)
|
||
return core.math.yl2x(x, LN2);
|
||
else
|
||
{
|
||
// C1 + C2 = LN2.
|
||
enum real C1 = 6.93145751953125E-1L;
|
||
enum real C2 = 1.428606820309417232121458176568075500134E-6L;
|
||
|
||
// Special cases.
|
||
if (isNaN(x))
|
||
return x;
|
||
if (isInfinity(x) && !signbit(x))
|
||
return x;
|
||
if (x == 0.0)
|
||
return -real.infinity;
|
||
if (x < 0.0)
|
||
return real.nan;
|
||
|
||
// Separate mantissa from exponent.
|
||
// Note, frexp is used so that denormal numbers will be handled properly.
|
||
real y, z;
|
||
int exp;
|
||
|
||
x = frexp(x, exp);
|
||
|
||
// Logarithm using log(x) = z + z^^3 R(z) / S(z),
|
||
// where z = 2(x - 1)/(x + 1)
|
||
if ((exp > 2) || (exp < -2))
|
||
{
|
||
if (x < SQRT1_2)
|
||
{ // 2(2x - 1)/(2x + 1)
|
||
exp -= 1;
|
||
z = x - 0.5;
|
||
y = 0.5 * z + 0.5;
|
||
}
|
||
else
|
||
{ // 2(x - 1)/(x + 1)
|
||
z = x - 0.5;
|
||
z -= 0.5;
|
||
y = 0.5 * x + 0.5;
|
||
}
|
||
x = z / y;
|
||
z = x * x;
|
||
z = x * (z * poly(z, logCoeffsR) / poly(z, logCoeffsS));
|
||
z += exp * C2;
|
||
z += x;
|
||
z += exp * C1;
|
||
|
||
return z;
|
||
}
|
||
|
||
// Logarithm using log(1 + x) = x - .5x^^2 + x^^3 P(x) / Q(x)
|
||
if (x < SQRT1_2)
|
||
{ // 2x - 1
|
||
exp -= 1;
|
||
x = ldexp(x, 1) - 1.0;
|
||
}
|
||
else
|
||
{
|
||
x = x - 1.0;
|
||
}
|
||
z = x * x;
|
||
y = x * (z * poly(x, logCoeffsP) / poly(x, logCoeffsQ));
|
||
y += exp * C2;
|
||
z = y - ldexp(z, -1);
|
||
|
||
// Note, the sum of above terms does not exceed x/4,
|
||
// so it contributes at most about 1/4 lsb to the error.
|
||
z += x;
|
||
z += exp * C1;
|
||
|
||
return z;
|
||
}
|
||
}
|
||
|
||
///
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
assert(log(E) == 1);
|
||
}
|
||
|
||
/**************************************
|
||
* Calculate the base-10 logarithm of x.
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH x) $(TH log10(x)) $(TH divide by 0?) $(TH invalid?))
|
||
* $(TR $(TD $(PLUSMN)0.0) $(TD -$(INFIN)) $(TD yes) $(TD no))
|
||
* $(TR $(TD $(LT)0.0) $(TD $(NAN)) $(TD no) $(TD yes))
|
||
* $(TR $(TD +$(INFIN)) $(TD +$(INFIN)) $(TD no) $(TD no))
|
||
* )
|
||
*/
|
||
real log10(real x) @safe pure nothrow @nogc
|
||
{
|
||
version (INLINE_YL2X)
|
||
return core.math.yl2x(x, LOG2);
|
||
else
|
||
{
|
||
// log10(2) split into two parts.
|
||
enum real L102A = 0.3125L;
|
||
enum real L102B = -1.14700043360188047862611052755069732318101185E-2L;
|
||
|
||
// log10(e) split into two parts.
|
||
enum real L10EA = 0.5L;
|
||
enum real L10EB = -6.570551809674817234887108108339491770560299E-2L;
|
||
|
||
// Special cases are the same as for log.
|
||
if (isNaN(x))
|
||
return x;
|
||
if (isInfinity(x) && !signbit(x))
|
||
return x;
|
||
if (x == 0.0)
|
||
return -real.infinity;
|
||
if (x < 0.0)
|
||
return real.nan;
|
||
|
||
// Separate mantissa from exponent.
|
||
// Note, frexp is used so that denormal numbers will be handled properly.
|
||
real y, z;
|
||
int exp;
|
||
|
||
x = frexp(x, exp);
|
||
|
||
// Logarithm using log(x) = z + z^^3 R(z) / S(z),
|
||
// where z = 2(x - 1)/(x + 1)
|
||
if ((exp > 2) || (exp < -2))
|
||
{
|
||
if (x < SQRT1_2)
|
||
{ // 2(2x - 1)/(2x + 1)
|
||
exp -= 1;
|
||
z = x - 0.5;
|
||
y = 0.5 * z + 0.5;
|
||
}
|
||
else
|
||
{ // 2(x - 1)/(x + 1)
|
||
z = x - 0.5;
|
||
z -= 0.5;
|
||
y = 0.5 * x + 0.5;
|
||
}
|
||
x = z / y;
|
||
z = x * x;
|
||
y = x * (z * poly(z, logCoeffsR) / poly(z, logCoeffsS));
|
||
goto Ldone;
|
||
}
|
||
|
||
// Logarithm using log(1 + x) = x - .5x^^2 + x^^3 P(x) / Q(x)
|
||
if (x < SQRT1_2)
|
||
{ // 2x - 1
|
||
exp -= 1;
|
||
x = ldexp(x, 1) - 1.0;
|
||
}
|
||
else
|
||
x = x - 1.0;
|
||
|
||
z = x * x;
|
||
y = x * (z * poly(x, logCoeffsP) / poly(x, logCoeffsQ));
|
||
y = y - ldexp(z, -1);
|
||
|
||
// Multiply log of fraction by log10(e) and base 2 exponent by log10(2).
|
||
// This sequence of operations is critical and it may be horribly
|
||
// defeated by some compiler optimizers.
|
||
Ldone:
|
||
z = y * L10EB;
|
||
z += x * L10EB;
|
||
z += exp * L102B;
|
||
z += y * L10EA;
|
||
z += x * L10EA;
|
||
z += exp * L102A;
|
||
|
||
return z;
|
||
}
|
||
}
|
||
|
||
///
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
assert(fabs(log10(1000) - 3) < .000001);
|
||
}
|
||
|
||
/******************************************
|
||
* Calculates the natural logarithm of 1 + x.
|
||
*
|
||
* For very small x, log1p(x) will be more accurate than
|
||
* log(1 + x).
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH x) $(TH log1p(x)) $(TH divide by 0?) $(TH invalid?))
|
||
* $(TR $(TD $(PLUSMN)0.0) $(TD $(PLUSMN)0.0) $(TD no) $(TD no))
|
||
* $(TR $(TD -1.0) $(TD -$(INFIN)) $(TD yes) $(TD no))
|
||
* $(TR $(TD $(LT)-1.0) $(TD $(NAN)) $(TD no) $(TD yes))
|
||
* $(TR $(TD +$(INFIN)) $(TD -$(INFIN)) $(TD no) $(TD no))
|
||
* )
|
||
*/
|
||
real log1p(real x) @safe pure nothrow @nogc
|
||
{
|
||
version(INLINE_YL2X)
|
||
{
|
||
// On x87, yl2xp1 is valid if and only if -0.5 <= lg(x) <= 0.5,
|
||
// ie if -0.29 <= x <= 0.414
|
||
return (fabs(x) <= 0.25) ? core.math.yl2xp1(x, LN2) : core.math.yl2x(x+1, LN2);
|
||
}
|
||
else
|
||
{
|
||
// Special cases.
|
||
if (isNaN(x) || x == 0.0)
|
||
return x;
|
||
if (isInfinity(x) && !signbit(x))
|
||
return x;
|
||
if (x == -1.0)
|
||
return -real.infinity;
|
||
if (x < -1.0)
|
||
return real.nan;
|
||
|
||
return log(x + 1.0);
|
||
}
|
||
}
|
||
|
||
/***************************************
|
||
* Calculates the base-2 logarithm of x:
|
||
* $(SUB log, 2)x
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH x) $(TH log2(x)) $(TH divide by 0?) $(TH invalid?))
|
||
* $(TR $(TD $(PLUSMN)0.0) $(TD -$(INFIN)) $(TD yes) $(TD no) )
|
||
* $(TR $(TD $(LT)0.0) $(TD $(NAN)) $(TD no) $(TD yes) )
|
||
* $(TR $(TD +$(INFIN)) $(TD +$(INFIN)) $(TD no) $(TD no) )
|
||
* )
|
||
*/
|
||
real log2(real x) @safe pure nothrow @nogc
|
||
{
|
||
version (INLINE_YL2X)
|
||
return core.math.yl2x(x, 1);
|
||
else
|
||
{
|
||
// Special cases are the same as for log.
|
||
if (isNaN(x))
|
||
return x;
|
||
if (isInfinity(x) && !signbit(x))
|
||
return x;
|
||
if (x == 0.0)
|
||
return -real.infinity;
|
||
if (x < 0.0)
|
||
return real.nan;
|
||
|
||
// Separate mantissa from exponent.
|
||
// Note, frexp is used so that denormal numbers will be handled properly.
|
||
real y, z;
|
||
int exp;
|
||
|
||
x = frexp(x, exp);
|
||
|
||
// Logarithm using log(x) = z + z^^3 R(z) / S(z),
|
||
// where z = 2(x - 1)/(x + 1)
|
||
if ((exp > 2) || (exp < -2))
|
||
{
|
||
if (x < SQRT1_2)
|
||
{ // 2(2x - 1)/(2x + 1)
|
||
exp -= 1;
|
||
z = x - 0.5;
|
||
y = 0.5 * z + 0.5;
|
||
}
|
||
else
|
||
{ // 2(x - 1)/(x + 1)
|
||
z = x - 0.5;
|
||
z -= 0.5;
|
||
y = 0.5 * x + 0.5;
|
||
}
|
||
x = z / y;
|
||
z = x * x;
|
||
y = x * (z * poly(z, logCoeffsR) / poly(z, logCoeffsS));
|
||
goto Ldone;
|
||
}
|
||
|
||
// Logarithm using log(1 + x) = x - .5x^^2 + x^^3 P(x) / Q(x)
|
||
if (x < SQRT1_2)
|
||
{ // 2x - 1
|
||
exp -= 1;
|
||
x = ldexp(x, 1) - 1.0;
|
||
}
|
||
else
|
||
x = x - 1.0;
|
||
|
||
z = x * x;
|
||
y = x * (z * poly(x, logCoeffsP) / poly(x, logCoeffsQ));
|
||
y = y - ldexp(z, -1);
|
||
|
||
// Multiply log of fraction by log10(e) and base 2 exponent by log10(2).
|
||
// This sequence of operations is critical and it may be horribly
|
||
// defeated by some compiler optimizers.
|
||
Ldone:
|
||
z = y * (LOG2E - 1.0);
|
||
z += x * (LOG2E - 1.0);
|
||
z += y;
|
||
z += x;
|
||
z += exp;
|
||
|
||
return z;
|
||
}
|
||
}
|
||
|
||
///
|
||
@system unittest
|
||
{
|
||
// check if values are equal to 19 decimal digits of precision
|
||
assert(equalsDigit(log2(1024.0L), 10, 19));
|
||
}
|
||
|
||
/*****************************************
|
||
* Extracts the exponent of x as a signed integral value.
|
||
*
|
||
* If x is subnormal, it is treated as if it were normalized.
|
||
* For a positive, finite x:
|
||
*
|
||
* 1 $(LT)= $(I x) * FLT_RADIX$(SUPERSCRIPT -logb(x)) $(LT) FLT_RADIX
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH x) $(TH logb(x)) $(TH divide by 0?) )
|
||
* $(TR $(TD $(PLUSMN)$(INFIN)) $(TD +$(INFIN)) $(TD no))
|
||
* $(TR $(TD $(PLUSMN)0.0) $(TD -$(INFIN)) $(TD yes) )
|
||
* )
|
||
*/
|
||
real logb(real x) @trusted nothrow @nogc
|
||
{
|
||
version (Win64_DMD_InlineAsm)
|
||
{
|
||
asm pure nothrow @nogc
|
||
{
|
||
naked ;
|
||
fld real ptr [RCX] ;
|
||
fxtract ;
|
||
fstp ST(0) ;
|
||
ret ;
|
||
}
|
||
}
|
||
else version (CRuntime_Microsoft)
|
||
{
|
||
asm pure nothrow @nogc
|
||
{
|
||
fld x ;
|
||
fxtract ;
|
||
fstp ST(0) ;
|
||
}
|
||
}
|
||
else
|
||
return core.stdc.math.logbl(x);
|
||
}
|
||
|
||
/************************************
|
||
* Calculates the remainder from the calculation x/y.
|
||
* Returns:
|
||
* The value of x - i * y, where i is the number of times that y can
|
||
* be completely subtracted from x. The result has the same sign as x.
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH x) $(TH y) $(TH fmod(x, y)) $(TH invalid?))
|
||
* $(TR $(TD $(PLUSMN)0.0) $(TD not 0.0) $(TD $(PLUSMN)0.0) $(TD no))
|
||
* $(TR $(TD $(PLUSMNINF)) $(TD anything) $(TD $(NAN)) $(TD yes))
|
||
* $(TR $(TD anything) $(TD $(PLUSMN)0.0) $(TD $(NAN)) $(TD yes))
|
||
* $(TR $(TD !=$(PLUSMNINF)) $(TD $(PLUSMNINF)) $(TD x) $(TD no))
|
||
* )
|
||
*/
|
||
real fmod(real x, real y) @trusted nothrow @nogc
|
||
{
|
||
version (CRuntime_Microsoft)
|
||
{
|
||
return x % y;
|
||
}
|
||
else
|
||
return core.stdc.math.fmodl(x, y);
|
||
}
|
||
|
||
/************************************
|
||
* Breaks x into an integral part and a fractional part, each of which has
|
||
* the same sign as x. The integral part is stored in i.
|
||
* Returns:
|
||
* The fractional part of x.
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH x) $(TH i (on input)) $(TH modf(x, i)) $(TH i (on return)))
|
||
* $(TR $(TD $(PLUSMNINF)) $(TD anything) $(TD $(PLUSMN)0.0) $(TD $(PLUSMNINF)))
|
||
* )
|
||
*/
|
||
real modf(real x, ref real i) @trusted nothrow @nogc
|
||
{
|
||
version (CRuntime_Microsoft)
|
||
{
|
||
i = trunc(x);
|
||
return copysign(isInfinity(x) ? 0.0 : x - i, x);
|
||
}
|
||
else
|
||
return core.stdc.math.modfl(x,&i);
|
||
}
|
||
|
||
/*************************************
|
||
* Efficiently calculates x * 2$(SUPERSCRIPT n).
|
||
*
|
||
* scalbn handles underflow and overflow in
|
||
* the same fashion as the basic arithmetic operators.
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH x) $(TH scalb(x)))
|
||
* $(TR $(TD $(PLUSMNINF)) $(TD $(PLUSMNINF)) )
|
||
* $(TR $(TD $(PLUSMN)0.0) $(TD $(PLUSMN)0.0) )
|
||
* )
|
||
*/
|
||
real scalbn(real x, int n) @trusted nothrow @nogc
|
||
{
|
||
version(InlineAsm_X86_Any)
|
||
{
|
||
// scalbnl is not supported on DMD-Windows, so use asm pure nothrow @nogc.
|
||
version (Win64)
|
||
{
|
||
asm pure nothrow @nogc {
|
||
naked ;
|
||
mov 16[RSP],RCX ;
|
||
fild word ptr 16[RSP] ;
|
||
fld real ptr [RDX] ;
|
||
fscale ;
|
||
fstp ST(1) ;
|
||
ret ;
|
||
}
|
||
}
|
||
else
|
||
{
|
||
asm pure nothrow @nogc {
|
||
fild n;
|
||
fld x;
|
||
fscale;
|
||
fstp ST(1);
|
||
}
|
||
}
|
||
}
|
||
else
|
||
{
|
||
return core.stdc.math.scalbnl(x, n);
|
||
}
|
||
}
|
||
|
||
///
|
||
@safe nothrow @nogc unittest
|
||
{
|
||
assert(scalbn(-real.infinity, 5) == -real.infinity);
|
||
}
|
||
|
||
/***************
|
||
* Calculates the cube root of x.
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH $(I x)) $(TH cbrt(x)) $(TH invalid?))
|
||
* $(TR $(TD $(PLUSMN)0.0) $(TD $(PLUSMN)0.0) $(TD no) )
|
||
* $(TR $(TD $(NAN)) $(TD $(NAN)) $(TD yes) )
|
||
* $(TR $(TD $(PLUSMN)$(INFIN)) $(TD $(PLUSMN)$(INFIN)) $(TD no) )
|
||
* )
|
||
*/
|
||
real cbrt(real x) @trusted nothrow @nogc
|
||
{
|
||
version (CRuntime_Microsoft)
|
||
{
|
||
version (INLINE_YL2X)
|
||
return copysign(exp2(core.math.yl2x(fabs(x), 1.0L/3.0L)), x);
|
||
else
|
||
return core.stdc.math.cbrtl(x);
|
||
}
|
||
else
|
||
return core.stdc.math.cbrtl(x);
|
||
}
|
||
|
||
|
||
/*******************************
|
||
* Returns |x|
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH x) $(TH fabs(x)))
|
||
* $(TR $(TD $(PLUSMN)0.0) $(TD +0.0) )
|
||
* $(TR $(TD $(PLUSMN)$(INFIN)) $(TD +$(INFIN)) )
|
||
* )
|
||
*/
|
||
real fabs(real x) @safe pure nothrow @nogc { pragma(inline, true); return core.math.fabs(x); }
|
||
//FIXME
|
||
///ditto
|
||
double fabs(double x) @safe pure nothrow @nogc { return fabs(cast(real) x); }
|
||
//FIXME
|
||
///ditto
|
||
float fabs(float x) @safe pure nothrow @nogc { return fabs(cast(real) x); }
|
||
|
||
@safe unittest
|
||
{
|
||
real function(real) pfabs = &fabs;
|
||
assert(pfabs != null);
|
||
}
|
||
|
||
/***********************************************************************
|
||
* Calculates the length of the
|
||
* hypotenuse of a right-angled triangle with sides of length x and y.
|
||
* The hypotenuse is the value of the square root of
|
||
* the sums of the squares of x and y:
|
||
*
|
||
* sqrt($(POWER x, 2) + $(POWER y, 2))
|
||
*
|
||
* Note that hypot(x, y), hypot(y, x) and
|
||
* hypot(x, -y) are equivalent.
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH x) $(TH y) $(TH hypot(x, y)) $(TH invalid?))
|
||
* $(TR $(TD x) $(TD $(PLUSMN)0.0) $(TD |x|) $(TD no))
|
||
* $(TR $(TD $(PLUSMNINF)) $(TD y) $(TD +$(INFIN)) $(TD no))
|
||
* $(TR $(TD $(PLUSMNINF)) $(TD $(NAN)) $(TD +$(INFIN)) $(TD no))
|
||
* )
|
||
*/
|
||
|
||
real hypot(real x, real y) @safe pure nothrow @nogc
|
||
{
|
||
// Scale x and y to avoid underflow and overflow.
|
||
// If one is huge and the other tiny, return the larger.
|
||
// If both are huge, avoid overflow by scaling by 1/sqrt(real.max/2).
|
||
// If both are tiny, avoid underflow by scaling by sqrt(real.min_normal*real.epsilon).
|
||
|
||
enum real SQRTMIN = 0.5 * sqrt(real.min_normal); // This is a power of 2.
|
||
enum real SQRTMAX = 1.0L / SQRTMIN; // 2^^((max_exp)/2) = nextUp(sqrt(real.max))
|
||
|
||
static assert(2*(SQRTMAX/2)*(SQRTMAX/2) <= real.max);
|
||
|
||
// Proves that sqrt(real.max) ~~ 0.5/sqrt(real.min_normal)
|
||
static assert(real.min_normal*real.max > 2 && real.min_normal*real.max <= 4);
|
||
|
||
real u = fabs(x);
|
||
real v = fabs(y);
|
||
if (!(u >= v)) // check for NaN as well.
|
||
{
|
||
v = u;
|
||
u = fabs(y);
|
||
if (u == real.infinity) return u; // hypot(inf, nan) == inf
|
||
if (v == real.infinity) return v; // hypot(nan, inf) == inf
|
||
}
|
||
|
||
// Now u >= v, or else one is NaN.
|
||
if (v >= SQRTMAX*0.5)
|
||
{
|
||
// hypot(huge, huge) -- avoid overflow
|
||
u *= SQRTMIN*0.5;
|
||
v *= SQRTMIN*0.5;
|
||
return sqrt(u*u + v*v) * SQRTMAX * 2.0;
|
||
}
|
||
|
||
if (u <= SQRTMIN)
|
||
{
|
||
// hypot (tiny, tiny) -- avoid underflow
|
||
// This is only necessary to avoid setting the underflow
|
||
// flag.
|
||
u *= SQRTMAX / real.epsilon;
|
||
v *= SQRTMAX / real.epsilon;
|
||
return sqrt(u*u + v*v) * SQRTMIN * real.epsilon;
|
||
}
|
||
|
||
if (u * real.epsilon > v)
|
||
{
|
||
// hypot (huge, tiny) = huge
|
||
return u;
|
||
}
|
||
|
||
// both are in the normal range
|
||
return sqrt(u*u + v*v);
|
||
}
|
||
|
||
@safe unittest
|
||
{
|
||
static real[3][] vals = // x,y,hypot
|
||
[
|
||
[ 0.0, 0.0, 0.0],
|
||
[ 0.0, -0.0, 0.0],
|
||
[ -0.0, -0.0, 0.0],
|
||
[ 3.0, 4.0, 5.0],
|
||
[ -300, -400, 500],
|
||
[0.0, 7.0, 7.0],
|
||
[9.0, 9*real.epsilon, 9.0],
|
||
[88/(64*sqrt(real.min_normal)), 105/(64*sqrt(real.min_normal)), 137/(64*sqrt(real.min_normal))],
|
||
[88/(128*sqrt(real.min_normal)), 105/(128*sqrt(real.min_normal)), 137/(128*sqrt(real.min_normal))],
|
||
[3*real.min_normal*real.epsilon, 4*real.min_normal*real.epsilon, 5*real.min_normal*real.epsilon],
|
||
[ real.min_normal, real.min_normal, sqrt(2.0L)*real.min_normal],
|
||
[ real.max/sqrt(2.0L), real.max/sqrt(2.0L), real.max],
|
||
[ real.infinity, real.nan, real.infinity],
|
||
[ real.nan, real.infinity, real.infinity],
|
||
[ real.nan, real.nan, real.nan],
|
||
[ real.nan, real.max, real.nan],
|
||
[ real.max, real.nan, real.nan],
|
||
];
|
||
for (int i = 0; i < vals.length; i++)
|
||
{
|
||
real x = vals[i][0];
|
||
real y = vals[i][1];
|
||
real z = vals[i][2];
|
||
real h = hypot(x, y);
|
||
assert(isIdentical(z,h) || feqrel(z, h) >= real.mant_dig - 1);
|
||
}
|
||
}
|
||
|
||
/**************************************
|
||
* Returns the value of x rounded upward to the next integer
|
||
* (toward positive infinity).
|
||
*/
|
||
real ceil(real x) @trusted pure nothrow @nogc
|
||
{
|
||
version (Win64_DMD_InlineAsm)
|
||
{
|
||
asm pure nothrow @nogc
|
||
{
|
||
naked ;
|
||
fld real ptr [RCX] ;
|
||
fstcw 8[RSP] ;
|
||
mov AL,9[RSP] ;
|
||
mov DL,AL ;
|
||
and AL,0xC3 ;
|
||
or AL,0x08 ; // round to +infinity
|
||
mov 9[RSP],AL ;
|
||
fldcw 8[RSP] ;
|
||
frndint ;
|
||
mov 9[RSP],DL ;
|
||
fldcw 8[RSP] ;
|
||
ret ;
|
||
}
|
||
}
|
||
else version(CRuntime_Microsoft)
|
||
{
|
||
short cw;
|
||
asm pure nothrow @nogc
|
||
{
|
||
fld x ;
|
||
fstcw cw ;
|
||
mov AL,byte ptr cw+1 ;
|
||
mov DL,AL ;
|
||
and AL,0xC3 ;
|
||
or AL,0x08 ; // round to +infinity
|
||
mov byte ptr cw+1,AL ;
|
||
fldcw cw ;
|
||
frndint ;
|
||
mov byte ptr cw+1,DL ;
|
||
fldcw cw ;
|
||
}
|
||
}
|
||
else
|
||
{
|
||
// Special cases.
|
||
if (isNaN(x) || isInfinity(x))
|
||
return x;
|
||
|
||
real y = floorImpl(x);
|
||
if (y < x)
|
||
y += 1.0;
|
||
|
||
return y;
|
||
}
|
||
}
|
||
|
||
///
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
assert(ceil(+123.456L) == +124);
|
||
assert(ceil(-123.456L) == -123);
|
||
assert(ceil(-1.234L) == -1);
|
||
assert(ceil(-0.123L) == 0);
|
||
assert(ceil(0.0L) == 0);
|
||
assert(ceil(+0.123L) == 1);
|
||
assert(ceil(+1.234L) == 2);
|
||
assert(ceil(real.infinity) == real.infinity);
|
||
assert(isNaN(ceil(real.nan)));
|
||
assert(isNaN(ceil(real.init)));
|
||
}
|
||
|
||
// ditto
|
||
double ceil(double x) @trusted pure nothrow @nogc
|
||
{
|
||
// Special cases.
|
||
if (isNaN(x) || isInfinity(x))
|
||
return x;
|
||
|
||
double y = floorImpl(x);
|
||
if (y < x)
|
||
y += 1.0;
|
||
|
||
return y;
|
||
}
|
||
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
assert(ceil(+123.456) == +124);
|
||
assert(ceil(-123.456) == -123);
|
||
assert(ceil(-1.234) == -1);
|
||
assert(ceil(-0.123) == 0);
|
||
assert(ceil(0.0) == 0);
|
||
assert(ceil(+0.123) == 1);
|
||
assert(ceil(+1.234) == 2);
|
||
assert(ceil(double.infinity) == double.infinity);
|
||
assert(isNaN(ceil(double.nan)));
|
||
assert(isNaN(ceil(double.init)));
|
||
}
|
||
|
||
// ditto
|
||
float ceil(float x) @trusted pure nothrow @nogc
|
||
{
|
||
// Special cases.
|
||
if (isNaN(x) || isInfinity(x))
|
||
return x;
|
||
|
||
float y = floorImpl(x);
|
||
if (y < x)
|
||
y += 1.0;
|
||
|
||
return y;
|
||
}
|
||
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
assert(ceil(+123.456f) == +124);
|
||
assert(ceil(-123.456f) == -123);
|
||
assert(ceil(-1.234f) == -1);
|
||
assert(ceil(-0.123f) == 0);
|
||
assert(ceil(0.0f) == 0);
|
||
assert(ceil(+0.123f) == 1);
|
||
assert(ceil(+1.234f) == 2);
|
||
assert(ceil(float.infinity) == float.infinity);
|
||
assert(isNaN(ceil(float.nan)));
|
||
assert(isNaN(ceil(float.init)));
|
||
}
|
||
|
||
/**************************************
|
||
* Returns the value of x rounded downward to the next integer
|
||
* (toward negative infinity).
|
||
*/
|
||
real floor(real x) @trusted pure nothrow @nogc
|
||
{
|
||
version (Win64_DMD_InlineAsm)
|
||
{
|
||
asm pure nothrow @nogc
|
||
{
|
||
naked ;
|
||
fld real ptr [RCX] ;
|
||
fstcw 8[RSP] ;
|
||
mov AL,9[RSP] ;
|
||
mov DL,AL ;
|
||
and AL,0xC3 ;
|
||
or AL,0x04 ; // round to -infinity
|
||
mov 9[RSP],AL ;
|
||
fldcw 8[RSP] ;
|
||
frndint ;
|
||
mov 9[RSP],DL ;
|
||
fldcw 8[RSP] ;
|
||
ret ;
|
||
}
|
||
}
|
||
else version(CRuntime_Microsoft)
|
||
{
|
||
short cw;
|
||
asm pure nothrow @nogc
|
||
{
|
||
fld x ;
|
||
fstcw cw ;
|
||
mov AL,byte ptr cw+1 ;
|
||
mov DL,AL ;
|
||
and AL,0xC3 ;
|
||
or AL,0x04 ; // round to -infinity
|
||
mov byte ptr cw+1,AL ;
|
||
fldcw cw ;
|
||
frndint ;
|
||
mov byte ptr cw+1,DL ;
|
||
fldcw cw ;
|
||
}
|
||
}
|
||
else
|
||
{
|
||
// Special cases.
|
||
if (isNaN(x) || isInfinity(x) || x == 0.0)
|
||
return x;
|
||
|
||
return floorImpl(x);
|
||
}
|
||
}
|
||
|
||
///
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
assert(floor(+123.456L) == +123);
|
||
assert(floor(-123.456L) == -124);
|
||
assert(floor(-1.234L) == -2);
|
||
assert(floor(-0.123L) == -1);
|
||
assert(floor(0.0L) == 0);
|
||
assert(floor(+0.123L) == 0);
|
||
assert(floor(+1.234L) == 1);
|
||
assert(floor(real.infinity) == real.infinity);
|
||
assert(isNaN(floor(real.nan)));
|
||
assert(isNaN(floor(real.init)));
|
||
}
|
||
|
||
// ditto
|
||
double floor(double x) @trusted pure nothrow @nogc
|
||
{
|
||
// Special cases.
|
||
if (isNaN(x) || isInfinity(x) || x == 0.0)
|
||
return x;
|
||
|
||
return floorImpl(x);
|
||
}
|
||
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
assert(floor(+123.456) == +123);
|
||
assert(floor(-123.456) == -124);
|
||
assert(floor(-1.234) == -2);
|
||
assert(floor(-0.123) == -1);
|
||
assert(floor(0.0) == 0);
|
||
assert(floor(+0.123) == 0);
|
||
assert(floor(+1.234) == 1);
|
||
assert(floor(double.infinity) == double.infinity);
|
||
assert(isNaN(floor(double.nan)));
|
||
assert(isNaN(floor(double.init)));
|
||
}
|
||
|
||
// ditto
|
||
float floor(float x) @trusted pure nothrow @nogc
|
||
{
|
||
// Special cases.
|
||
if (isNaN(x) || isInfinity(x) || x == 0.0)
|
||
return x;
|
||
|
||
return floorImpl(x);
|
||
}
|
||
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
assert(floor(+123.456f) == +123);
|
||
assert(floor(-123.456f) == -124);
|
||
assert(floor(-1.234f) == -2);
|
||
assert(floor(-0.123f) == -1);
|
||
assert(floor(0.0f) == 0);
|
||
assert(floor(+0.123f) == 0);
|
||
assert(floor(+1.234f) == 1);
|
||
assert(floor(float.infinity) == float.infinity);
|
||
assert(isNaN(floor(float.nan)));
|
||
assert(isNaN(floor(float.init)));
|
||
}
|
||
|
||
/**
|
||
* Round `val` to a multiple of `unit`. `rfunc` specifies the rounding
|
||
* function to use; by default this is `rint`, which uses the current
|
||
* rounding mode.
|
||
*/
|
||
Unqual!F quantize(alias rfunc = rint, F)(const F val, const F unit)
|
||
if (is(typeof(rfunc(F.init)) : F) && isFloatingPoint!F)
|
||
{
|
||
typeof(return) ret = val;
|
||
if (unit != 0)
|
||
{
|
||
const scaled = val / unit;
|
||
if (!scaled.isInfinity)
|
||
ret = rfunc(scaled) * unit;
|
||
}
|
||
return ret;
|
||
}
|
||
|
||
///
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
assert(12345.6789L.quantize(0.01L) == 12345.68L);
|
||
assert(12345.6789L.quantize!floor(0.01L) == 12345.67L);
|
||
assert(12345.6789L.quantize(22.0L) == 12342.0L);
|
||
}
|
||
|
||
///
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
assert(12345.6789L.quantize(0) == 12345.6789L);
|
||
assert(12345.6789L.quantize(real.infinity).isNaN);
|
||
assert(12345.6789L.quantize(real.nan).isNaN);
|
||
assert(real.infinity.quantize(0.01L) == real.infinity);
|
||
assert(real.infinity.quantize(real.nan).isNaN);
|
||
assert(real.nan.quantize(0.01L).isNaN);
|
||
assert(real.nan.quantize(real.infinity).isNaN);
|
||
assert(real.nan.quantize(real.nan).isNaN);
|
||
}
|
||
|
||
/**
|
||
* Round `val` to a multiple of `pow(base, exp)`. `rfunc` specifies the
|
||
* rounding function to use; by default this is `rint`, which uses the
|
||
* current rounding mode.
|
||
*/
|
||
Unqual!F quantize(real base, alias rfunc = rint, F, E)(const F val, const E exp)
|
||
if (is(typeof(rfunc(F.init)) : F) && isFloatingPoint!F && isIntegral!E)
|
||
{
|
||
// TODO: Compile-time optimization for power-of-two bases?
|
||
return quantize!rfunc(val, pow(cast(F) base, exp));
|
||
}
|
||
|
||
/// ditto
|
||
Unqual!F quantize(real base, long exp = 1, alias rfunc = rint, F)(const F val)
|
||
if (is(typeof(rfunc(F.init)) : F) && isFloatingPoint!F)
|
||
{
|
||
enum unit = cast(F) pow(base, exp);
|
||
return quantize!rfunc(val, unit);
|
||
}
|
||
|
||
///
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
assert(12345.6789L.quantize!10(-2) == 12345.68L);
|
||
assert(12345.6789L.quantize!(10, -2) == 12345.68L);
|
||
assert(12345.6789L.quantize!(10, floor)(-2) == 12345.67L);
|
||
assert(12345.6789L.quantize!(10, -2, floor) == 12345.67L);
|
||
|
||
assert(12345.6789L.quantize!22(1) == 12342.0L);
|
||
assert(12345.6789L.quantize!22 == 12342.0L);
|
||
}
|
||
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
import std.meta : AliasSeq;
|
||
|
||
foreach (F; AliasSeq!(real, double, float))
|
||
{
|
||
const maxL10 = cast(int) F.max.log10.floor;
|
||
const maxR10 = pow(cast(F) 10, maxL10);
|
||
assert((cast(F) 0.9L * maxR10).quantize!10(maxL10) == maxR10);
|
||
assert((cast(F)-0.9L * maxR10).quantize!10(maxL10) == -maxR10);
|
||
|
||
assert(F.max.quantize(F.min_normal) == F.max);
|
||
assert((-F.max).quantize(F.min_normal) == -F.max);
|
||
assert(F.min_normal.quantize(F.max) == 0);
|
||
assert((-F.min_normal).quantize(F.max) == 0);
|
||
assert(F.min_normal.quantize(F.min_normal) == F.min_normal);
|
||
assert((-F.min_normal).quantize(F.min_normal) == -F.min_normal);
|
||
}
|
||
}
|
||
|
||
/******************************************
|
||
* Rounds x to the nearest integer value, using the current rounding
|
||
* mode.
|
||
*
|
||
* Unlike the rint functions, nearbyint does not raise the
|
||
* FE_INEXACT exception.
|
||
*/
|
||
real nearbyint(real x) @trusted nothrow @nogc
|
||
{
|
||
version (CRuntime_Microsoft)
|
||
{
|
||
assert(0); // not implemented in C library
|
||
}
|
||
else
|
||
return core.stdc.math.nearbyintl(x);
|
||
}
|
||
|
||
/**********************************
|
||
* Rounds x to the nearest integer value, using the current rounding
|
||
* mode.
|
||
* If the return value is not equal to x, the FE_INEXACT
|
||
* exception is raised.
|
||
* $(B nearbyint) performs
|
||
* the same operation, but does not set the FE_INEXACT exception.
|
||
*/
|
||
real rint(real x) @safe pure nothrow @nogc { pragma(inline, true); return core.math.rint(x); }
|
||
//FIXME
|
||
///ditto
|
||
double rint(double x) @safe pure nothrow @nogc { return rint(cast(real) x); }
|
||
//FIXME
|
||
///ditto
|
||
float rint(float x) @safe pure nothrow @nogc { return rint(cast(real) x); }
|
||
|
||
@safe unittest
|
||
{
|
||
real function(real) print = &rint;
|
||
assert(print != null);
|
||
}
|
||
|
||
/***************************************
|
||
* Rounds x to the nearest integer value, using the current rounding
|
||
* mode.
|
||
*
|
||
* This is generally the fastest method to convert a floating-point number
|
||
* to an integer. Note that the results from this function
|
||
* depend on the rounding mode, if the fractional part of x is exactly 0.5.
|
||
* If using the default rounding mode (ties round to even integers)
|
||
* lrint(4.5) == 4, lrint(5.5)==6.
|
||
*/
|
||
long lrint(real x) @trusted pure nothrow @nogc
|
||
{
|
||
version(InlineAsm_X86_Any)
|
||
{
|
||
version (Win64)
|
||
{
|
||
asm pure nothrow @nogc
|
||
{
|
||
naked;
|
||
fld real ptr [RCX];
|
||
fistp qword ptr 8[RSP];
|
||
mov RAX,8[RSP];
|
||
ret;
|
||
}
|
||
}
|
||
else
|
||
{
|
||
long n;
|
||
asm pure nothrow @nogc
|
||
{
|
||
fld x;
|
||
fistp n;
|
||
}
|
||
return n;
|
||
}
|
||
}
|
||
else
|
||
{
|
||
alias F = floatTraits!(real);
|
||
static if (F.realFormat == RealFormat.ieeeDouble)
|
||
{
|
||
long result;
|
||
|
||
// Rounding limit when casting from real(double) to ulong.
|
||
enum real OF = 4.50359962737049600000E15L;
|
||
|
||
uint* vi = cast(uint*)(&x);
|
||
|
||
// Find the exponent and sign
|
||
uint msb = vi[MANTISSA_MSB];
|
||
uint lsb = vi[MANTISSA_LSB];
|
||
int exp = ((msb >> 20) & 0x7ff) - 0x3ff;
|
||
const int sign = msb >> 31;
|
||
msb &= 0xfffff;
|
||
msb |= 0x100000;
|
||
|
||
if (exp < 63)
|
||
{
|
||
if (exp >= 52)
|
||
result = (cast(long) msb << (exp - 20)) | (lsb << (exp - 52));
|
||
else
|
||
{
|
||
// Adjust x and check result.
|
||
const real j = sign ? -OF : OF;
|
||
x = (j + x) - j;
|
||
msb = vi[MANTISSA_MSB];
|
||
lsb = vi[MANTISSA_LSB];
|
||
exp = ((msb >> 20) & 0x7ff) - 0x3ff;
|
||
msb &= 0xfffff;
|
||
msb |= 0x100000;
|
||
|
||
if (exp < 0)
|
||
result = 0;
|
||
else if (exp < 20)
|
||
result = cast(long) msb >> (20 - exp);
|
||
else if (exp == 20)
|
||
result = cast(long) msb;
|
||
else
|
||
result = (cast(long) msb << (exp - 20)) | (lsb >> (52 - exp));
|
||
}
|
||
}
|
||
else
|
||
{
|
||
// It is left implementation defined when the number is too large.
|
||
return cast(long) x;
|
||
}
|
||
|
||
return sign ? -result : result;
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeExtended)
|
||
{
|
||
long result;
|
||
|
||
// Rounding limit when casting from real(80-bit) to ulong.
|
||
enum real OF = 9.22337203685477580800E18L;
|
||
|
||
ushort* vu = cast(ushort*)(&x);
|
||
uint* vi = cast(uint*)(&x);
|
||
|
||
// Find the exponent and sign
|
||
int exp = (vu[F.EXPPOS_SHORT] & 0x7fff) - 0x3fff;
|
||
const int sign = (vu[F.EXPPOS_SHORT] >> 15) & 1;
|
||
|
||
if (exp < 63)
|
||
{
|
||
// Adjust x and check result.
|
||
const real j = sign ? -OF : OF;
|
||
x = (j + x) - j;
|
||
exp = (vu[F.EXPPOS_SHORT] & 0x7fff) - 0x3fff;
|
||
|
||
version (LittleEndian)
|
||
{
|
||
if (exp < 0)
|
||
result = 0;
|
||
else if (exp <= 31)
|
||
result = vi[1] >> (31 - exp);
|
||
else
|
||
result = (cast(long) vi[1] << (exp - 31)) | (vi[0] >> (63 - exp));
|
||
}
|
||
else
|
||
{
|
||
if (exp < 0)
|
||
result = 0;
|
||
else if (exp <= 31)
|
||
result = vi[1] >> (31 - exp);
|
||
else
|
||
result = (cast(long) vi[1] << (exp - 31)) | (vi[2] >> (63 - exp));
|
||
}
|
||
}
|
||
else
|
||
{
|
||
// It is left implementation defined when the number is too large
|
||
// to fit in a 64bit long.
|
||
return cast(long) x;
|
||
}
|
||
|
||
return sign ? -result : result;
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeQuadruple)
|
||
{
|
||
const vu = cast(ushort*)(&x);
|
||
|
||
// Find the exponent and sign
|
||
const sign = (vu[F.EXPPOS_SHORT] >> 15) & 1;
|
||
if ((vu[F.EXPPOS_SHORT] & F.EXPMASK) - (F.EXPBIAS + 1) > 63)
|
||
{
|
||
// The result is left implementation defined when the number is
|
||
// too large to fit in a 64 bit long.
|
||
return cast(long) x;
|
||
}
|
||
|
||
// Force rounding of lower bits according to current rounding
|
||
// mode by adding ±2^-112 and subtracting it again.
|
||
enum OF = 5.19229685853482762853049632922009600E33L;
|
||
const j = sign ? -OF : OF;
|
||
x = (j + x) - j;
|
||
|
||
const implicitOne = 1UL << 48;
|
||
auto vl = cast(ulong*)(&x);
|
||
vl[MANTISSA_MSB] &= implicitOne - 1;
|
||
vl[MANTISSA_MSB] |= implicitOne;
|
||
|
||
long result;
|
||
|
||
const exp = (vu[F.EXPPOS_SHORT] & F.EXPMASK) - (F.EXPBIAS + 1);
|
||
if (exp < 0)
|
||
result = 0;
|
||
else if (exp <= 48)
|
||
result = vl[MANTISSA_MSB] >> (48 - exp);
|
||
else
|
||
result = (vl[MANTISSA_MSB] << (exp - 48)) | (vl[MANTISSA_LSB] >> (112 - exp));
|
||
|
||
return sign ? -result : result;
|
||
}
|
||
else
|
||
{
|
||
static assert(false, "real type not supported by lrint()");
|
||
}
|
||
}
|
||
}
|
||
|
||
///
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
assert(lrint(4.5) == 4);
|
||
assert(lrint(5.5) == 6);
|
||
assert(lrint(-4.5) == -4);
|
||
assert(lrint(-5.5) == -6);
|
||
|
||
assert(lrint(int.max - 0.5) == 2147483646L);
|
||
assert(lrint(int.max + 0.5) == 2147483648L);
|
||
assert(lrint(int.min - 0.5) == -2147483648L);
|
||
assert(lrint(int.min + 0.5) == -2147483648L);
|
||
}
|
||
|
||
static if (real.mant_dig >= long.sizeof * 8)
|
||
{
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
assert(lrint(long.max - 1.5L) == long.max - 1);
|
||
assert(lrint(long.max - 0.5L) == long.max - 1);
|
||
assert(lrint(long.min + 0.5L) == long.min);
|
||
assert(lrint(long.min + 1.5L) == long.min + 2);
|
||
}
|
||
}
|
||
|
||
/*******************************************
|
||
* Return the value of x rounded to the nearest integer.
|
||
* If the fractional part of x is exactly 0.5, the return value is
|
||
* rounded away from zero.
|
||
*/
|
||
real round(real x) @trusted nothrow @nogc
|
||
{
|
||
version (CRuntime_Microsoft)
|
||
{
|
||
auto old = FloatingPointControl.getControlState();
|
||
FloatingPointControl.setControlState(
|
||
(old & ~FloatingPointControl.ROUNDING_MASK) | FloatingPointControl.roundToZero
|
||
);
|
||
x = rint((x >= 0) ? x + 0.5 : x - 0.5);
|
||
FloatingPointControl.setControlState(old);
|
||
return x;
|
||
}
|
||
else
|
||
return core.stdc.math.roundl(x);
|
||
}
|
||
|
||
/**********************************************
|
||
* Return the value of x rounded to the nearest integer.
|
||
*
|
||
* If the fractional part of x is exactly 0.5, the return value is rounded
|
||
* away from zero.
|
||
*/
|
||
long lround(real x) @trusted nothrow @nogc
|
||
{
|
||
version (Posix)
|
||
return core.stdc.math.llroundl(x);
|
||
else
|
||
assert(0, "lround not implemented");
|
||
}
|
||
|
||
version(Posix)
|
||
{
|
||
@safe nothrow @nogc unittest
|
||
{
|
||
assert(lround(0.49) == 0);
|
||
assert(lround(0.5) == 1);
|
||
assert(lround(1.5) == 2);
|
||
}
|
||
}
|
||
|
||
/****************************************************
|
||
* Returns the integer portion of x, dropping the fractional portion.
|
||
*
|
||
* This is also known as "chop" rounding.
|
||
*/
|
||
real trunc(real x) @trusted nothrow @nogc
|
||
{
|
||
version (Win64_DMD_InlineAsm)
|
||
{
|
||
asm pure nothrow @nogc
|
||
{
|
||
naked ;
|
||
fld real ptr [RCX] ;
|
||
fstcw 8[RSP] ;
|
||
mov AL,9[RSP] ;
|
||
mov DL,AL ;
|
||
and AL,0xC3 ;
|
||
or AL,0x0C ; // round to 0
|
||
mov 9[RSP],AL ;
|
||
fldcw 8[RSP] ;
|
||
frndint ;
|
||
mov 9[RSP],DL ;
|
||
fldcw 8[RSP] ;
|
||
ret ;
|
||
}
|
||
}
|
||
else version(CRuntime_Microsoft)
|
||
{
|
||
short cw;
|
||
asm pure nothrow @nogc
|
||
{
|
||
fld x ;
|
||
fstcw cw ;
|
||
mov AL,byte ptr cw+1 ;
|
||
mov DL,AL ;
|
||
and AL,0xC3 ;
|
||
or AL,0x0C ; // round to 0
|
||
mov byte ptr cw+1,AL ;
|
||
fldcw cw ;
|
||
frndint ;
|
||
mov byte ptr cw+1,DL ;
|
||
fldcw cw ;
|
||
}
|
||
}
|
||
else
|
||
return core.stdc.math.truncl(x);
|
||
}
|
||
|
||
/****************************************************
|
||
* Calculate the remainder x REM y, following IEC 60559.
|
||
*
|
||
* REM is the value of x - y * n, where n is the integer nearest the exact
|
||
* value of x / y.
|
||
* If |n - x / y| == 0.5, n is even.
|
||
* If the result is zero, it has the same sign as x.
|
||
* Otherwise, the sign of the result is the sign of x / y.
|
||
* Precision mode has no effect on the remainder functions.
|
||
*
|
||
* remquo returns n in the parameter n.
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH x) $(TH y) $(TH remainder(x, y)) $(TH n) $(TH invalid?))
|
||
* $(TR $(TD $(PLUSMN)0.0) $(TD not 0.0) $(TD $(PLUSMN)0.0) $(TD 0.0) $(TD no))
|
||
* $(TR $(TD $(PLUSMNINF)) $(TD anything) $(TD $(NAN)) $(TD ?) $(TD yes))
|
||
* $(TR $(TD anything) $(TD $(PLUSMN)0.0) $(TD $(NAN)) $(TD ?) $(TD yes))
|
||
* $(TR $(TD != $(PLUSMNINF)) $(TD $(PLUSMNINF)) $(TD x) $(TD ?) $(TD no))
|
||
* )
|
||
*
|
||
* Note: remquo not supported on windows
|
||
*/
|
||
real remainder(real x, real y) @trusted nothrow @nogc
|
||
{
|
||
version (CRuntime_Microsoft)
|
||
{
|
||
int n;
|
||
return remquo(x, y, n);
|
||
}
|
||
else
|
||
return core.stdc.math.remainderl(x, y);
|
||
}
|
||
|
||
real remquo(real x, real y, out int n) @trusted nothrow @nogc /// ditto
|
||
{
|
||
version (Posix)
|
||
return core.stdc.math.remquol(x, y, &n);
|
||
else
|
||
assert(0, "remquo not implemented");
|
||
}
|
||
|
||
/** IEEE exception status flags ('sticky bits')
|
||
|
||
These flags indicate that an exceptional floating-point condition has occurred.
|
||
They indicate that a NaN or an infinity has been generated, that a result
|
||
is inexact, or that a signalling NaN has been encountered. If floating-point
|
||
exceptions are enabled (unmasked), a hardware exception will be generated
|
||
instead of setting these flags.
|
||
*/
|
||
struct IeeeFlags
|
||
{
|
||
private:
|
||
// The x87 FPU status register is 16 bits.
|
||
// The Pentium SSE2 status register is 32 bits.
|
||
uint flags;
|
||
version (X86_Any)
|
||
{
|
||
// Applies to both x87 status word (16 bits) and SSE2 status word(32 bits).
|
||
enum : int
|
||
{
|
||
INEXACT_MASK = 0x20,
|
||
UNDERFLOW_MASK = 0x10,
|
||
OVERFLOW_MASK = 0x08,
|
||
DIVBYZERO_MASK = 0x04,
|
||
INVALID_MASK = 0x01,
|
||
|
||
EXCEPTIONS_MASK = 0b11_1111
|
||
}
|
||
// Don't bother about subnormals, they are not supported on most CPUs.
|
||
// SUBNORMAL_MASK = 0x02;
|
||
}
|
||
else version (PPC_Any)
|
||
{
|
||
// PowerPC FPSCR is a 32-bit register.
|
||
enum : int
|
||
{
|
||
INEXACT_MASK = 0x02000000,
|
||
DIVBYZERO_MASK = 0x04000000,
|
||
UNDERFLOW_MASK = 0x08000000,
|
||
OVERFLOW_MASK = 0x10000000,
|
||
INVALID_MASK = 0x20000000 // Summary as PowerPC has five types of invalid exceptions.
|
||
}
|
||
}
|
||
else version (ARM)
|
||
{
|
||
// ARM FPSCR is a 32bit register
|
||
enum : int
|
||
{
|
||
INEXACT_MASK = 0x10,
|
||
UNDERFLOW_MASK = 0x08,
|
||
OVERFLOW_MASK = 0x04,
|
||
DIVBYZERO_MASK = 0x02,
|
||
INVALID_MASK = 0x01
|
||
}
|
||
}
|
||
else version(SPARC)
|
||
{
|
||
// SPARC FSR is a 32bit register
|
||
//(64 bits for Sparc 7 & 8, but high 32 bits are uninteresting).
|
||
enum : int
|
||
{
|
||
INEXACT_MASK = 0x020,
|
||
UNDERFLOW_MASK = 0x080,
|
||
OVERFLOW_MASK = 0x100,
|
||
DIVBYZERO_MASK = 0x040,
|
||
INVALID_MASK = 0x200
|
||
}
|
||
}
|
||
else
|
||
static assert(0, "Not implemented");
|
||
private:
|
||
static uint getIeeeFlags()
|
||
{
|
||
version(InlineAsm_X86_Any)
|
||
{
|
||
ushort sw;
|
||
asm pure nothrow @nogc { fstsw sw; }
|
||
|
||
// OR the result with the SSE2 status register (MXCSR).
|
||
if (haveSSE)
|
||
{
|
||
uint mxcsr;
|
||
asm pure nothrow @nogc { stmxcsr mxcsr; }
|
||
return (sw | mxcsr) & EXCEPTIONS_MASK;
|
||
}
|
||
else return sw & EXCEPTIONS_MASK;
|
||
}
|
||
else version (SPARC)
|
||
{
|
||
/*
|
||
int retval;
|
||
asm pure nothrow @nogc { st %fsr, retval; }
|
||
return retval;
|
||
*/
|
||
assert(0, "Not yet supported");
|
||
}
|
||
else version (ARM)
|
||
{
|
||
assert(false, "Not yet supported.");
|
||
}
|
||
else
|
||
assert(0, "Not yet supported");
|
||
}
|
||
static void resetIeeeFlags() @nogc
|
||
{
|
||
version(InlineAsm_X86_Any)
|
||
{
|
||
asm pure nothrow @nogc
|
||
{
|
||
fnclex;
|
||
}
|
||
|
||
// Also clear exception flags in MXCSR, SSE's control register.
|
||
if (haveSSE)
|
||
{
|
||
uint mxcsr;
|
||
asm nothrow @nogc { stmxcsr mxcsr; }
|
||
mxcsr &= ~EXCEPTIONS_MASK;
|
||
asm nothrow @nogc { ldmxcsr mxcsr; }
|
||
}
|
||
}
|
||
else
|
||
{
|
||
/* SPARC:
|
||
int tmpval;
|
||
asm pure nothrow @nogc { st %fsr, tmpval; }
|
||
tmpval &=0xFFFF_FC00;
|
||
asm pure nothrow @nogc { ld tmpval, %fsr; }
|
||
*/
|
||
assert(0, "Not yet supported");
|
||
}
|
||
}
|
||
public:
|
||
version (IeeeFlagsSupport)
|
||
{
|
||
|
||
/**
|
||
* The result cannot be represented exactly, so rounding occurred.
|
||
* Example: `x = sin(0.1);`
|
||
*/
|
||
@property bool inexact() const { return (flags & INEXACT_MASK) != 0; }
|
||
|
||
/**
|
||
* A zero was generated by underflow
|
||
* Example: `x = real.min*real.epsilon/2;`
|
||
*/
|
||
@property bool underflow() const { return (flags & UNDERFLOW_MASK) != 0; }
|
||
|
||
/**
|
||
* An infinity was generated by overflow
|
||
* Example: `x = real.max*2;`
|
||
*/
|
||
@property bool overflow() const { return (flags & OVERFLOW_MASK) != 0; }
|
||
|
||
/**
|
||
* An infinity was generated by division by zero
|
||
* Example: `x = 3/0.0;`
|
||
*/
|
||
@property bool divByZero() const { return (flags & DIVBYZERO_MASK) != 0; }
|
||
|
||
/**
|
||
* A machine NaN was generated.
|
||
* Example: `x = real.infinity * 0.0;`
|
||
*/
|
||
@property bool invalid() const { return (flags & INVALID_MASK) != 0; }
|
||
|
||
}
|
||
}
|
||
|
||
///
|
||
@system unittest
|
||
{
|
||
static void func() {
|
||
int a = 10 * 10;
|
||
}
|
||
|
||
real a=3.5;
|
||
// Set all the flags to zero
|
||
resetIeeeFlags();
|
||
assert(!ieeeFlags.divByZero);
|
||
// Perform a division by zero.
|
||
a/=0.0L;
|
||
assert(a == real.infinity);
|
||
assert(ieeeFlags.divByZero);
|
||
// Create a NaN
|
||
a*=0.0L;
|
||
assert(ieeeFlags.invalid);
|
||
assert(isNaN(a));
|
||
|
||
// Check that calling func() has no effect on the
|
||
// status flags.
|
||
IeeeFlags f = ieeeFlags;
|
||
func();
|
||
assert(ieeeFlags == f);
|
||
}
|
||
|
||
@system unittest
|
||
{
|
||
import std.meta : AliasSeq;
|
||
|
||
static struct Test
|
||
{
|
||
void delegate() action;
|
||
bool function() ieeeCheck;
|
||
}
|
||
|
||
foreach (T; AliasSeq!(float, double, real))
|
||
{
|
||
T x; /* Needs to be here to trick -O. It would optimize away the
|
||
calculations if x were local to the function literals. */
|
||
auto tests = [
|
||
Test(
|
||
() { x = 1; x += 0.1; },
|
||
() => ieeeFlags.inexact
|
||
),
|
||
Test(
|
||
() { x = T.min_normal; x /= T.max; },
|
||
() => ieeeFlags.underflow
|
||
),
|
||
Test(
|
||
() { x = T.max; x += T.max; },
|
||
() => ieeeFlags.overflow
|
||
),
|
||
Test(
|
||
() { x = 1; x /= 0; },
|
||
() => ieeeFlags.divByZero
|
||
),
|
||
Test(
|
||
() { x = 0; x /= 0; },
|
||
() => ieeeFlags.invalid
|
||
)
|
||
];
|
||
foreach (test; tests)
|
||
{
|
||
resetIeeeFlags();
|
||
assert(!test.ieeeCheck());
|
||
test.action();
|
||
assert(test.ieeeCheck());
|
||
}
|
||
}
|
||
}
|
||
|
||
version(X86_Any)
|
||
{
|
||
version = IeeeFlagsSupport;
|
||
}
|
||
else version(ARM)
|
||
{
|
||
version = IeeeFlagsSupport;
|
||
}
|
||
|
||
/// Set all of the floating-point status flags to false.
|
||
void resetIeeeFlags() @nogc { IeeeFlags.resetIeeeFlags(); }
|
||
|
||
/// Returns: snapshot of the current state of the floating-point status flags
|
||
@property IeeeFlags ieeeFlags()
|
||
{
|
||
return IeeeFlags(IeeeFlags.getIeeeFlags());
|
||
}
|
||
|
||
/** Control the Floating point hardware
|
||
|
||
Change the IEEE754 floating-point rounding mode and the floating-point
|
||
hardware exceptions.
|
||
|
||
By default, the rounding mode is roundToNearest and all hardware exceptions
|
||
are disabled. For most applications, debugging is easier if the $(I division
|
||
by zero), $(I overflow), and $(I invalid operation) exceptions are enabled.
|
||
These three are combined into a $(I severeExceptions) value for convenience.
|
||
Note in particular that if $(I invalidException) is enabled, a hardware trap
|
||
will be generated whenever an uninitialized floating-point variable is used.
|
||
|
||
All changes are temporary. The previous state is restored at the
|
||
end of the scope.
|
||
|
||
|
||
Example:
|
||
----
|
||
{
|
||
FloatingPointControl fpctrl;
|
||
|
||
// Enable hardware exceptions for division by zero, overflow to infinity,
|
||
// invalid operations, and uninitialized floating-point variables.
|
||
fpctrl.enableExceptions(FloatingPointControl.severeExceptions);
|
||
|
||
// This will generate a hardware exception, if x is a
|
||
// default-initialized floating point variable:
|
||
real x; // Add `= 0` or even `= real.nan` to not throw the exception.
|
||
real y = x * 3.0;
|
||
|
||
// The exception is only thrown for default-uninitialized NaN-s.
|
||
// NaN-s with other payload are valid:
|
||
real z = y * real.nan; // ok
|
||
|
||
// Changing the rounding mode:
|
||
fpctrl.rounding = FloatingPointControl.roundUp;
|
||
assert(rint(1.1) == 2);
|
||
|
||
// The set hardware exceptions will be disabled when leaving this scope.
|
||
// The original rounding mode will also be restored.
|
||
}
|
||
|
||
// Ensure previous values are returned:
|
||
assert(!FloatingPointControl.enabledExceptions);
|
||
assert(FloatingPointControl.rounding == FloatingPointControl.roundToNearest);
|
||
assert(rint(1.1) == 1);
|
||
----
|
||
|
||
*/
|
||
struct FloatingPointControl
|
||
{
|
||
alias RoundingMode = uint; ///
|
||
|
||
version(StdDdoc)
|
||
{
|
||
enum : RoundingMode
|
||
{
|
||
/** IEEE rounding modes.
|
||
* The default mode is roundToNearest.
|
||
*/
|
||
roundToNearest,
|
||
roundDown, /// ditto
|
||
roundUp, /// ditto
|
||
roundToZero /// ditto
|
||
}
|
||
}
|
||
else version(ARM)
|
||
{
|
||
enum : RoundingMode
|
||
{
|
||
roundToNearest = 0x000000,
|
||
roundDown = 0x800000,
|
||
roundUp = 0x400000,
|
||
roundToZero = 0xC00000
|
||
}
|
||
}
|
||
else version(PPC_Any)
|
||
{
|
||
enum : RoundingMode
|
||
{
|
||
roundToNearest = 0x00000000,
|
||
roundDown = 0x00000003,
|
||
roundUp = 0x00000002,
|
||
roundToZero = 0x00000001
|
||
}
|
||
}
|
||
else
|
||
{
|
||
enum : RoundingMode
|
||
{
|
||
roundToNearest = 0x0000,
|
||
roundDown = 0x0400,
|
||
roundUp = 0x0800,
|
||
roundToZero = 0x0C00
|
||
}
|
||
}
|
||
|
||
//// Change the floating-point hardware rounding mode
|
||
@property void rounding(RoundingMode newMode) @nogc
|
||
{
|
||
initialize();
|
||
setControlState((getControlState() & ~ROUNDING_MASK) | (newMode & ROUNDING_MASK));
|
||
}
|
||
|
||
/// Returns: the currently active rounding mode
|
||
@property static RoundingMode rounding() @nogc
|
||
{
|
||
return cast(RoundingMode)(getControlState() & ROUNDING_MASK);
|
||
}
|
||
|
||
version(StdDdoc)
|
||
{
|
||
enum : uint
|
||
{
|
||
/** IEEE hardware exceptions.
|
||
* By default, all exceptions are masked (disabled).
|
||
*
|
||
* severeExceptions = The overflow, division by zero, and invalid
|
||
* exceptions.
|
||
*/
|
||
subnormalException,
|
||
inexactException, /// ditto
|
||
underflowException, /// ditto
|
||
overflowException, /// ditto
|
||
divByZeroException, /// ditto
|
||
invalidException, /// ditto
|
||
severeExceptions, /// ditto
|
||
allExceptions, /// ditto
|
||
}
|
||
}
|
||
else version(ARM)
|
||
{
|
||
enum : uint
|
||
{
|
||
subnormalException = 0x8000,
|
||
inexactException = 0x1000,
|
||
underflowException = 0x0800,
|
||
overflowException = 0x0400,
|
||
divByZeroException = 0x0200,
|
||
invalidException = 0x0100,
|
||
severeExceptions = overflowException | divByZeroException
|
||
| invalidException,
|
||
allExceptions = severeExceptions | underflowException
|
||
| inexactException | subnormalException,
|
||
}
|
||
}
|
||
else version(PPC_Any)
|
||
{
|
||
enum : uint
|
||
{
|
||
inexactException = 0x0008,
|
||
divByZeroException = 0x0010,
|
||
underflowException = 0x0020,
|
||
overflowException = 0x0040,
|
||
invalidException = 0x0080,
|
||
severeExceptions = overflowException | divByZeroException
|
||
| invalidException,
|
||
allExceptions = severeExceptions | underflowException
|
||
| inexactException,
|
||
}
|
||
}
|
||
else
|
||
{
|
||
enum : uint
|
||
{
|
||
inexactException = 0x20,
|
||
underflowException = 0x10,
|
||
overflowException = 0x08,
|
||
divByZeroException = 0x04,
|
||
subnormalException = 0x02,
|
||
invalidException = 0x01,
|
||
severeExceptions = overflowException | divByZeroException
|
||
| invalidException,
|
||
allExceptions = severeExceptions | underflowException
|
||
| inexactException | subnormalException,
|
||
}
|
||
}
|
||
|
||
private:
|
||
version(ARM)
|
||
{
|
||
enum uint EXCEPTION_MASK = 0x9F00;
|
||
enum uint ROUNDING_MASK = 0xC00000;
|
||
}
|
||
else version(PPC_Any)
|
||
{
|
||
enum uint EXCEPTION_MASK = 0x00F8;
|
||
enum uint ROUNDING_MASK = 0x0003;
|
||
}
|
||
else version(X86)
|
||
{
|
||
enum ushort EXCEPTION_MASK = 0x3F;
|
||
enum ushort ROUNDING_MASK = 0xC00;
|
||
}
|
||
else version(X86_64)
|
||
{
|
||
enum ushort EXCEPTION_MASK = 0x3F;
|
||
enum ushort ROUNDING_MASK = 0xC00;
|
||
}
|
||
else
|
||
static assert(false, "Architecture not supported");
|
||
|
||
public:
|
||
/// Returns: true if the current FPU supports exception trapping
|
||
@property static bool hasExceptionTraps() @safe nothrow @nogc
|
||
{
|
||
version(X86_Any)
|
||
return true;
|
||
else version(PPC_Any)
|
||
return true;
|
||
else version(ARM)
|
||
{
|
||
auto oldState = getControlState();
|
||
// If exceptions are not supported, we set the bit but read it back as zero
|
||
// https://sourceware.org/ml/libc-ports/2012-06/msg00091.html
|
||
setControlState(oldState | (divByZeroException & EXCEPTION_MASK));
|
||
immutable result = (getControlState() & EXCEPTION_MASK) != 0;
|
||
setControlState(oldState);
|
||
return result;
|
||
}
|
||
else
|
||
static assert(false, "Not implemented for this architecture");
|
||
}
|
||
|
||
/// Enable (unmask) specific hardware exceptions. Multiple exceptions may be ORed together.
|
||
void enableExceptions(uint exceptions) @nogc
|
||
{
|
||
assert(hasExceptionTraps);
|
||
initialize();
|
||
version(X86_Any)
|
||
setControlState(getControlState() & ~(exceptions & EXCEPTION_MASK));
|
||
else
|
||
setControlState(getControlState() | (exceptions & EXCEPTION_MASK));
|
||
}
|
||
|
||
/// Disable (mask) specific hardware exceptions. Multiple exceptions may be ORed together.
|
||
void disableExceptions(uint exceptions) @nogc
|
||
{
|
||
assert(hasExceptionTraps);
|
||
initialize();
|
||
version(X86_Any)
|
||
setControlState(getControlState() | (exceptions & EXCEPTION_MASK));
|
||
else
|
||
setControlState(getControlState() & ~(exceptions & EXCEPTION_MASK));
|
||
}
|
||
|
||
/// Returns: the exceptions which are currently enabled (unmasked)
|
||
@property static uint enabledExceptions() @nogc
|
||
{
|
||
assert(hasExceptionTraps);
|
||
version(X86_Any)
|
||
return (getControlState() & EXCEPTION_MASK) ^ EXCEPTION_MASK;
|
||
else
|
||
return (getControlState() & EXCEPTION_MASK);
|
||
}
|
||
|
||
/// Clear all pending exceptions, then restore the original exception state and rounding mode.
|
||
~this() @nogc
|
||
{
|
||
clearExceptions();
|
||
if (initialized)
|
||
setControlState(savedState);
|
||
}
|
||
|
||
private:
|
||
ControlState savedState;
|
||
|
||
bool initialized = false;
|
||
|
||
version(ARM)
|
||
{
|
||
alias ControlState = uint;
|
||
}
|
||
else version(PPC_Any)
|
||
{
|
||
alias ControlState = uint;
|
||
}
|
||
else
|
||
{
|
||
alias ControlState = ushort;
|
||
}
|
||
|
||
void initialize() @nogc
|
||
{
|
||
// BUG: This works around the absence of this() constructors.
|
||
if (initialized) return;
|
||
clearExceptions();
|
||
savedState = getControlState();
|
||
initialized = true;
|
||
}
|
||
|
||
// Clear all pending exceptions
|
||
static void clearExceptions() @nogc
|
||
{
|
||
resetIeeeFlags();
|
||
}
|
||
|
||
// Read from the control register
|
||
static ControlState getControlState() @trusted nothrow @nogc
|
||
{
|
||
version (D_InlineAsm_X86)
|
||
{
|
||
short cont;
|
||
asm nothrow @nogc
|
||
{
|
||
xor EAX, EAX;
|
||
fstcw cont;
|
||
}
|
||
return cont;
|
||
}
|
||
else
|
||
version (D_InlineAsm_X86_64)
|
||
{
|
||
short cont;
|
||
asm nothrow @nogc
|
||
{
|
||
xor RAX, RAX;
|
||
fstcw cont;
|
||
}
|
||
return cont;
|
||
}
|
||
else
|
||
assert(0, "Not yet supported");
|
||
}
|
||
|
||
// Set the control register
|
||
static void setControlState(ControlState newState) @trusted nothrow @nogc
|
||
{
|
||
version (InlineAsm_X86_Any)
|
||
{
|
||
asm nothrow @nogc
|
||
{
|
||
fclex;
|
||
fldcw newState;
|
||
}
|
||
|
||
// Also update MXCSR, SSE's control register.
|
||
if (haveSSE)
|
||
{
|
||
uint mxcsr;
|
||
asm nothrow @nogc { stmxcsr mxcsr; }
|
||
|
||
/* In the FPU control register, rounding mode is in bits 10 and
|
||
11. In MXCSR it's in bits 13 and 14. */
|
||
enum ROUNDING_MASK_SSE = ROUNDING_MASK << 3;
|
||
immutable newRoundingModeSSE = (newState & ROUNDING_MASK) << 3;
|
||
mxcsr &= ~ROUNDING_MASK_SSE; // delete old rounding mode
|
||
mxcsr |= newRoundingModeSSE; // write new rounding mode
|
||
|
||
/* In the FPU control register, masks are bits 0 through 5.
|
||
In MXCSR they're 7 through 12. */
|
||
enum EXCEPTION_MASK_SSE = EXCEPTION_MASK << 7;
|
||
immutable newExceptionMasks = (newState & EXCEPTION_MASK) << 7;
|
||
mxcsr &= ~EXCEPTION_MASK_SSE; // delete old masks
|
||
mxcsr |= newExceptionMasks; // write new exception masks
|
||
|
||
asm nothrow @nogc { ldmxcsr mxcsr; }
|
||
}
|
||
}
|
||
else
|
||
assert(0, "Not yet supported");
|
||
}
|
||
}
|
||
|
||
@system unittest
|
||
{
|
||
void ensureDefaults()
|
||
{
|
||
assert(FloatingPointControl.rounding
|
||
== FloatingPointControl.roundToNearest);
|
||
if (FloatingPointControl.hasExceptionTraps)
|
||
assert(FloatingPointControl.enabledExceptions == 0);
|
||
}
|
||
|
||
{
|
||
FloatingPointControl ctrl;
|
||
}
|
||
ensureDefaults();
|
||
|
||
version(D_HardFloat)
|
||
{
|
||
{
|
||
FloatingPointControl ctrl;
|
||
ctrl.rounding = FloatingPointControl.roundDown;
|
||
assert(FloatingPointControl.rounding == FloatingPointControl.roundDown);
|
||
}
|
||
ensureDefaults();
|
||
}
|
||
|
||
if (FloatingPointControl.hasExceptionTraps)
|
||
{
|
||
FloatingPointControl ctrl;
|
||
ctrl.enableExceptions(FloatingPointControl.divByZeroException
|
||
| FloatingPointControl.overflowException);
|
||
assert(ctrl.enabledExceptions ==
|
||
(FloatingPointControl.divByZeroException
|
||
| FloatingPointControl.overflowException));
|
||
|
||
ctrl.rounding = FloatingPointControl.roundUp;
|
||
assert(FloatingPointControl.rounding == FloatingPointControl.roundUp);
|
||
}
|
||
ensureDefaults();
|
||
}
|
||
|
||
@system unittest // rounding
|
||
{
|
||
import std.meta : AliasSeq;
|
||
|
||
foreach (T; AliasSeq!(float, double, real))
|
||
{
|
||
FloatingPointControl fpctrl;
|
||
|
||
fpctrl.rounding = FloatingPointControl.roundUp;
|
||
T u = 1;
|
||
u += 0.1;
|
||
|
||
fpctrl.rounding = FloatingPointControl.roundDown;
|
||
T d = 1;
|
||
d += 0.1;
|
||
|
||
fpctrl.rounding = FloatingPointControl.roundToZero;
|
||
T z = 1;
|
||
z += 0.1;
|
||
|
||
assert(u > d);
|
||
assert(z == d);
|
||
|
||
fpctrl.rounding = FloatingPointControl.roundUp;
|
||
u = -1;
|
||
u -= 0.1;
|
||
|
||
fpctrl.rounding = FloatingPointControl.roundDown;
|
||
d = -1;
|
||
d -= 0.1;
|
||
|
||
fpctrl.rounding = FloatingPointControl.roundToZero;
|
||
z = -1;
|
||
z -= 0.1;
|
||
|
||
assert(u > d);
|
||
assert(z == u);
|
||
}
|
||
}
|
||
|
||
|
||
/*********************************
|
||
* Determines if $(D_PARAM x) is NaN.
|
||
* params:
|
||
* x = a floating point number.
|
||
* returns:
|
||
* $(D true) if $(D_PARAM x) is Nan.
|
||
*/
|
||
bool isNaN(X)(X x) @nogc @trusted pure nothrow
|
||
if (isFloatingPoint!(X))
|
||
{
|
||
alias F = floatTraits!(X);
|
||
static if (F.realFormat == RealFormat.ieeeSingle)
|
||
{
|
||
const uint p = *cast(uint *)&x;
|
||
return ((p & 0x7F80_0000) == 0x7F80_0000)
|
||
&& p & 0x007F_FFFF; // not infinity
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeDouble)
|
||
{
|
||
const ulong p = *cast(ulong *)&x;
|
||
return ((p & 0x7FF0_0000_0000_0000) == 0x7FF0_0000_0000_0000)
|
||
&& p & 0x000F_FFFF_FFFF_FFFF; // not infinity
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeExtended)
|
||
{
|
||
const ushort e = F.EXPMASK & (cast(ushort *)&x)[F.EXPPOS_SHORT];
|
||
const ulong ps = *cast(ulong *)&x;
|
||
return e == F.EXPMASK &&
|
||
ps & 0x7FFF_FFFF_FFFF_FFFF; // not infinity
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeQuadruple)
|
||
{
|
||
const ushort e = F.EXPMASK & (cast(ushort *)&x)[F.EXPPOS_SHORT];
|
||
const ulong psLsb = (cast(ulong *)&x)[MANTISSA_LSB];
|
||
const ulong psMsb = (cast(ulong *)&x)[MANTISSA_MSB];
|
||
return e == F.EXPMASK &&
|
||
(psLsb | (psMsb& 0x0000_FFFF_FFFF_FFFF)) != 0;
|
||
}
|
||
else
|
||
{
|
||
return x != x;
|
||
}
|
||
}
|
||
|
||
///
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
assert( isNaN(float.init));
|
||
assert( isNaN(-double.init));
|
||
assert( isNaN(real.nan));
|
||
assert( isNaN(-real.nan));
|
||
assert(!isNaN(cast(float) 53.6));
|
||
assert(!isNaN(cast(real)-53.6));
|
||
}
|
||
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
import std.meta : AliasSeq;
|
||
|
||
foreach (T; AliasSeq!(float, double, real))
|
||
{
|
||
// CTFE-able tests
|
||
assert(isNaN(T.init));
|
||
assert(isNaN(-T.init));
|
||
assert(isNaN(T.nan));
|
||
assert(isNaN(-T.nan));
|
||
assert(!isNaN(T.infinity));
|
||
assert(!isNaN(-T.infinity));
|
||
assert(!isNaN(cast(T) 53.6));
|
||
assert(!isNaN(cast(T)-53.6));
|
||
|
||
// Runtime tests
|
||
shared T f;
|
||
f = T.init;
|
||
assert(isNaN(f));
|
||
assert(isNaN(-f));
|
||
f = T.nan;
|
||
assert(isNaN(f));
|
||
assert(isNaN(-f));
|
||
f = T.infinity;
|
||
assert(!isNaN(f));
|
||
assert(!isNaN(-f));
|
||
f = cast(T) 53.6;
|
||
assert(!isNaN(f));
|
||
assert(!isNaN(-f));
|
||
}
|
||
}
|
||
|
||
/*********************************
|
||
* Determines if $(D_PARAM x) is finite.
|
||
* params:
|
||
* x = a floating point number.
|
||
* returns:
|
||
* $(D true) if $(D_PARAM x) is finite.
|
||
*/
|
||
bool isFinite(X)(X x) @trusted pure nothrow @nogc
|
||
{
|
||
alias F = floatTraits!(X);
|
||
ushort* pe = cast(ushort *)&x;
|
||
return (pe[F.EXPPOS_SHORT] & F.EXPMASK) != F.EXPMASK;
|
||
}
|
||
|
||
///
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
assert( isFinite(1.23f));
|
||
assert( isFinite(float.max));
|
||
assert( isFinite(float.min_normal));
|
||
assert(!isFinite(float.nan));
|
||
assert(!isFinite(float.infinity));
|
||
}
|
||
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
assert(isFinite(1.23));
|
||
assert(isFinite(double.max));
|
||
assert(isFinite(double.min_normal));
|
||
assert(!isFinite(double.nan));
|
||
assert(!isFinite(double.infinity));
|
||
|
||
assert(isFinite(1.23L));
|
||
assert(isFinite(real.max));
|
||
assert(isFinite(real.min_normal));
|
||
assert(!isFinite(real.nan));
|
||
assert(!isFinite(real.infinity));
|
||
}
|
||
|
||
|
||
/*********************************
|
||
* Determines if $(D_PARAM x) is normalized.
|
||
*
|
||
* A normalized number must not be zero, subnormal, infinite nor $(NAN).
|
||
*
|
||
* params:
|
||
* x = a floating point number.
|
||
* returns:
|
||
* $(D true) if $(D_PARAM x) is normalized.
|
||
*/
|
||
|
||
/* Need one for each format because subnormal floats might
|
||
* be converted to normal reals.
|
||
*/
|
||
bool isNormal(X)(X x) @trusted pure nothrow @nogc
|
||
{
|
||
alias F = floatTraits!(X);
|
||
static if (F.realFormat == RealFormat.ibmExtended)
|
||
{
|
||
// doubledouble is normal if the least significant part is normal.
|
||
return isNormal((cast(double*)&x)[MANTISSA_LSB]);
|
||
}
|
||
else
|
||
{
|
||
ushort e = F.EXPMASK & (cast(ushort *)&x)[F.EXPPOS_SHORT];
|
||
return (e != F.EXPMASK && e != 0);
|
||
}
|
||
}
|
||
|
||
///
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
float f = 3;
|
||
double d = 500;
|
||
real e = 10e+48;
|
||
|
||
assert(isNormal(f));
|
||
assert(isNormal(d));
|
||
assert(isNormal(e));
|
||
f = d = e = 0;
|
||
assert(!isNormal(f));
|
||
assert(!isNormal(d));
|
||
assert(!isNormal(e));
|
||
assert(!isNormal(real.infinity));
|
||
assert(isNormal(-real.max));
|
||
assert(!isNormal(real.min_normal/4));
|
||
|
||
}
|
||
|
||
/*********************************
|
||
* Determines if $(D_PARAM x) is subnormal.
|
||
*
|
||
* Subnormals (also known as "denormal number"), have a 0 exponent
|
||
* and a 0 most significant mantissa bit.
|
||
*
|
||
* params:
|
||
* x = a floating point number.
|
||
* returns:
|
||
* $(D true) if $(D_PARAM x) is a denormal number.
|
||
*/
|
||
bool isSubnormal(X)(X x) @trusted pure nothrow @nogc
|
||
{
|
||
/*
|
||
Need one for each format because subnormal floats might
|
||
be converted to normal reals.
|
||
*/
|
||
alias F = floatTraits!(X);
|
||
static if (F.realFormat == RealFormat.ieeeSingle)
|
||
{
|
||
uint *p = cast(uint *)&x;
|
||
return (*p & F.EXPMASK_INT) == 0 && *p & F.MANTISSAMASK_INT;
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeDouble)
|
||
{
|
||
uint *p = cast(uint *)&x;
|
||
return (p[MANTISSA_MSB] & F.EXPMASK_INT) == 0
|
||
&& (p[MANTISSA_LSB] || p[MANTISSA_MSB] & F.MANTISSAMASK_INT);
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeQuadruple)
|
||
{
|
||
ushort e = F.EXPMASK & (cast(ushort *)&x)[F.EXPPOS_SHORT];
|
||
long* ps = cast(long *)&x;
|
||
return (e == 0 &&
|
||
((ps[MANTISSA_LSB]|(ps[MANTISSA_MSB]& 0x0000_FFFF_FFFF_FFFF)) != 0));
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeExtended)
|
||
{
|
||
ushort* pe = cast(ushort *)&x;
|
||
long* ps = cast(long *)&x;
|
||
|
||
return (pe[F.EXPPOS_SHORT] & F.EXPMASK) == 0 && *ps > 0;
|
||
}
|
||
else static if (F.realFormat == RealFormat.ibmExtended)
|
||
{
|
||
return isSubnormal((cast(double*)&x)[MANTISSA_MSB]);
|
||
}
|
||
else
|
||
{
|
||
static assert(false, "Not implemented for this architecture");
|
||
}
|
||
}
|
||
|
||
///
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
import std.meta : AliasSeq;
|
||
|
||
foreach (T; AliasSeq!(float, double, real))
|
||
{
|
||
T f;
|
||
for (f = 1.0; !isSubnormal(f); f /= 2)
|
||
assert(f != 0);
|
||
}
|
||
}
|
||
|
||
/*********************************
|
||
* Determines if $(D_PARAM x) is $(PLUSMN)$(INFIN).
|
||
* params:
|
||
* x = a floating point number.
|
||
* returns:
|
||
* $(D true) if $(D_PARAM x) is $(PLUSMN)$(INFIN).
|
||
*/
|
||
bool isInfinity(X)(X x) @nogc @trusted pure nothrow
|
||
if (isFloatingPoint!(X))
|
||
{
|
||
alias F = floatTraits!(X);
|
||
static if (F.realFormat == RealFormat.ieeeSingle)
|
||
{
|
||
return ((*cast(uint *)&x) & 0x7FFF_FFFF) == 0x7F80_0000;
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeDouble)
|
||
{
|
||
return ((*cast(ulong *)&x) & 0x7FFF_FFFF_FFFF_FFFF)
|
||
== 0x7FF0_0000_0000_0000;
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeExtended)
|
||
{
|
||
const ushort e = cast(ushort)(F.EXPMASK & (cast(ushort *)&x)[F.EXPPOS_SHORT]);
|
||
const ulong ps = *cast(ulong *)&x;
|
||
|
||
// On Motorola 68K, infinity can have hidden bit = 1 or 0. On x86, it is always 1.
|
||
return e == F.EXPMASK && (ps & 0x7FFF_FFFF_FFFF_FFFF) == 0;
|
||
}
|
||
else static if (F.realFormat == RealFormat.ibmExtended)
|
||
{
|
||
return (((cast(ulong *)&x)[MANTISSA_MSB]) & 0x7FFF_FFFF_FFFF_FFFF)
|
||
== 0x7FF8_0000_0000_0000;
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeQuadruple)
|
||
{
|
||
const long psLsb = (cast(long *)&x)[MANTISSA_LSB];
|
||
const long psMsb = (cast(long *)&x)[MANTISSA_MSB];
|
||
return (psLsb == 0)
|
||
&& (psMsb & 0x7FFF_FFFF_FFFF_FFFF) == 0x7FFF_0000_0000_0000;
|
||
}
|
||
else
|
||
{
|
||
return (x < -X.max) || (X.max < x);
|
||
}
|
||
}
|
||
|
||
///
|
||
@nogc @safe pure nothrow unittest
|
||
{
|
||
assert(!isInfinity(float.init));
|
||
assert(!isInfinity(-float.init));
|
||
assert(!isInfinity(float.nan));
|
||
assert(!isInfinity(-float.nan));
|
||
assert(isInfinity(float.infinity));
|
||
assert(isInfinity(-float.infinity));
|
||
assert(isInfinity(-1.0f / 0.0f));
|
||
}
|
||
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
// CTFE-able tests
|
||
assert(!isInfinity(double.init));
|
||
assert(!isInfinity(-double.init));
|
||
assert(!isInfinity(double.nan));
|
||
assert(!isInfinity(-double.nan));
|
||
assert(isInfinity(double.infinity));
|
||
assert(isInfinity(-double.infinity));
|
||
assert(isInfinity(-1.0 / 0.0));
|
||
|
||
assert(!isInfinity(real.init));
|
||
assert(!isInfinity(-real.init));
|
||
assert(!isInfinity(real.nan));
|
||
assert(!isInfinity(-real.nan));
|
||
assert(isInfinity(real.infinity));
|
||
assert(isInfinity(-real.infinity));
|
||
assert(isInfinity(-1.0L / 0.0L));
|
||
|
||
// Runtime tests
|
||
shared float f;
|
||
f = float.init;
|
||
assert(!isInfinity(f));
|
||
assert(!isInfinity(-f));
|
||
f = float.nan;
|
||
assert(!isInfinity(f));
|
||
assert(!isInfinity(-f));
|
||
f = float.infinity;
|
||
assert(isInfinity(f));
|
||
assert(isInfinity(-f));
|
||
f = (-1.0f / 0.0f);
|
||
assert(isInfinity(f));
|
||
|
||
shared double d;
|
||
d = double.init;
|
||
assert(!isInfinity(d));
|
||
assert(!isInfinity(-d));
|
||
d = double.nan;
|
||
assert(!isInfinity(d));
|
||
assert(!isInfinity(-d));
|
||
d = double.infinity;
|
||
assert(isInfinity(d));
|
||
assert(isInfinity(-d));
|
||
d = (-1.0 / 0.0);
|
||
assert(isInfinity(d));
|
||
|
||
shared real e;
|
||
e = real.init;
|
||
assert(!isInfinity(e));
|
||
assert(!isInfinity(-e));
|
||
e = real.nan;
|
||
assert(!isInfinity(e));
|
||
assert(!isInfinity(-e));
|
||
e = real.infinity;
|
||
assert(isInfinity(e));
|
||
assert(isInfinity(-e));
|
||
e = (-1.0L / 0.0L);
|
||
assert(isInfinity(e));
|
||
}
|
||
|
||
/*********************************
|
||
* Is the binary representation of x identical to y?
|
||
*
|
||
* Same as ==, except that positive and negative zero are not identical,
|
||
* and two $(NAN)s are identical if they have the same 'payload'.
|
||
*/
|
||
bool isIdentical(real x, real y) @trusted pure nothrow @nogc
|
||
{
|
||
// We're doing a bitwise comparison so the endianness is irrelevant.
|
||
long* pxs = cast(long *)&x;
|
||
long* pys = cast(long *)&y;
|
||
alias F = floatTraits!(real);
|
||
static if (F.realFormat == RealFormat.ieeeDouble)
|
||
{
|
||
return pxs[0] == pys[0];
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeQuadruple
|
||
|| F.realFormat == RealFormat.ibmExtended)
|
||
{
|
||
return pxs[0] == pys[0] && pxs[1] == pys[1];
|
||
}
|
||
else
|
||
{
|
||
ushort* pxe = cast(ushort *)&x;
|
||
ushort* pye = cast(ushort *)&y;
|
||
return pxe[4] == pye[4] && pxs[0] == pys[0];
|
||
}
|
||
}
|
||
|
||
/*********************************
|
||
* Return 1 if sign bit of e is set, 0 if not.
|
||
*/
|
||
int signbit(X)(X x) @nogc @trusted pure nothrow
|
||
{
|
||
alias F = floatTraits!(X);
|
||
return ((cast(ubyte *)&x)[F.SIGNPOS_BYTE] & 0x80) != 0;
|
||
}
|
||
|
||
///
|
||
@nogc @safe pure nothrow unittest
|
||
{
|
||
assert(!signbit(float.nan));
|
||
assert(signbit(-float.nan));
|
||
assert(!signbit(168.1234f));
|
||
assert(signbit(-168.1234f));
|
||
assert(!signbit(0.0f));
|
||
assert(signbit(-0.0f));
|
||
assert(signbit(-float.max));
|
||
assert(!signbit(float.max));
|
||
|
||
assert(!signbit(double.nan));
|
||
assert(signbit(-double.nan));
|
||
assert(!signbit(168.1234));
|
||
assert(signbit(-168.1234));
|
||
assert(!signbit(0.0));
|
||
assert(signbit(-0.0));
|
||
assert(signbit(-double.max));
|
||
assert(!signbit(double.max));
|
||
|
||
assert(!signbit(real.nan));
|
||
assert(signbit(-real.nan));
|
||
assert(!signbit(168.1234L));
|
||
assert(signbit(-168.1234L));
|
||
assert(!signbit(0.0L));
|
||
assert(signbit(-0.0L));
|
||
assert(signbit(-real.max));
|
||
assert(!signbit(real.max));
|
||
}
|
||
|
||
|
||
/*********************************
|
||
* Return a value composed of to with from's sign bit.
|
||
*/
|
||
R copysign(R, X)(R to, X from) @trusted pure nothrow @nogc
|
||
if (isFloatingPoint!(R) && isFloatingPoint!(X))
|
||
{
|
||
ubyte* pto = cast(ubyte *)&to;
|
||
const ubyte* pfrom = cast(ubyte *)&from;
|
||
|
||
alias T = floatTraits!(R);
|
||
alias F = floatTraits!(X);
|
||
pto[T.SIGNPOS_BYTE] &= 0x7F;
|
||
pto[T.SIGNPOS_BYTE] |= pfrom[F.SIGNPOS_BYTE] & 0x80;
|
||
return to;
|
||
}
|
||
|
||
// ditto
|
||
R copysign(R, X)(X to, R from) @trusted pure nothrow @nogc
|
||
if (isIntegral!(X) && isFloatingPoint!(R))
|
||
{
|
||
return copysign(cast(R) to, from);
|
||
}
|
||
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
import std.meta : AliasSeq;
|
||
|
||
foreach (X; AliasSeq!(float, double, real, int, long))
|
||
{
|
||
foreach (Y; AliasSeq!(float, double, real))
|
||
(){ // avoid slow optimizations for large functions @@@BUG@@@ 2396
|
||
X x = 21;
|
||
Y y = 23.8;
|
||
Y e = void;
|
||
|
||
e = copysign(x, y);
|
||
assert(e == 21.0);
|
||
|
||
e = copysign(-x, y);
|
||
assert(e == 21.0);
|
||
|
||
e = copysign(x, -y);
|
||
assert(e == -21.0);
|
||
|
||
e = copysign(-x, -y);
|
||
assert(e == -21.0);
|
||
|
||
static if (isFloatingPoint!X)
|
||
{
|
||
e = copysign(X.nan, y);
|
||
assert(isNaN(e) && !signbit(e));
|
||
|
||
e = copysign(X.nan, -y);
|
||
assert(isNaN(e) && signbit(e));
|
||
}
|
||
}();
|
||
}
|
||
}
|
||
|
||
/*********************************
|
||
Returns $(D -1) if $(D x < 0), $(D x) if $(D x == 0), $(D 1) if
|
||
$(D x > 0), and $(NAN) if x==$(NAN).
|
||
*/
|
||
F sgn(F)(F x) @safe pure nothrow @nogc
|
||
{
|
||
// @@@TODO@@@: make this faster
|
||
return x > 0 ? 1 : x < 0 ? -1 : x;
|
||
}
|
||
|
||
///
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
assert(sgn(168.1234) == 1);
|
||
assert(sgn(-168.1234) == -1);
|
||
assert(sgn(0.0) == 0);
|
||
assert(sgn(-0.0) == 0);
|
||
}
|
||
|
||
// Functions for NaN payloads
|
||
/*
|
||
* A 'payload' can be stored in the significand of a $(NAN). One bit is required
|
||
* to distinguish between a quiet and a signalling $(NAN). This leaves 22 bits
|
||
* of payload for a float; 51 bits for a double; 62 bits for an 80-bit real;
|
||
* and 111 bits for a 128-bit quad.
|
||
*/
|
||
/**
|
||
* Create a quiet $(NAN), storing an integer inside the payload.
|
||
*
|
||
* For floats, the largest possible payload is 0x3F_FFFF.
|
||
* For doubles, it is 0x3_FFFF_FFFF_FFFF.
|
||
* For 80-bit or 128-bit reals, it is 0x3FFF_FFFF_FFFF_FFFF.
|
||
*/
|
||
real NaN(ulong payload) @trusted pure nothrow @nogc
|
||
{
|
||
alias F = floatTraits!(real);
|
||
static if (F.realFormat == RealFormat.ieeeExtended)
|
||
{
|
||
// real80 (in x86 real format, the implied bit is actually
|
||
// not implied but a real bit which is stored in the real)
|
||
ulong v = 3; // implied bit = 1, quiet bit = 1
|
||
}
|
||
else
|
||
{
|
||
ulong v = 1; // no implied bit. quiet bit = 1
|
||
}
|
||
|
||
ulong a = payload;
|
||
|
||
// 22 Float bits
|
||
ulong w = a & 0x3F_FFFF;
|
||
a -= w;
|
||
|
||
v <<=22;
|
||
v |= w;
|
||
a >>=22;
|
||
|
||
// 29 Double bits
|
||
v <<=29;
|
||
w = a & 0xFFF_FFFF;
|
||
v |= w;
|
||
a -= w;
|
||
a >>=29;
|
||
|
||
static if (F.realFormat == RealFormat.ieeeDouble)
|
||
{
|
||
v |= 0x7FF0_0000_0000_0000;
|
||
real x;
|
||
* cast(ulong *)(&x) = v;
|
||
return x;
|
||
}
|
||
else
|
||
{
|
||
v <<=11;
|
||
a &= 0x7FF;
|
||
v |= a;
|
||
real x = real.nan;
|
||
|
||
// Extended real bits
|
||
static if (F.realFormat == RealFormat.ieeeQuadruple)
|
||
{
|
||
v <<= 1; // there's no implicit bit
|
||
|
||
version(LittleEndian)
|
||
{
|
||
*cast(ulong*)(6+cast(ubyte*)(&x)) = v;
|
||
}
|
||
else
|
||
{
|
||
*cast(ulong*)(2+cast(ubyte*)(&x)) = v;
|
||
}
|
||
}
|
||
else
|
||
{
|
||
*cast(ulong *)(&x) = v;
|
||
}
|
||
return x;
|
||
}
|
||
}
|
||
|
||
@system pure nothrow @nogc unittest // not @safe because taking address of local.
|
||
{
|
||
static if (floatTraits!(real).realFormat == RealFormat.ieeeDouble)
|
||
{
|
||
auto x = NaN(1);
|
||
auto xl = *cast(ulong*)&x;
|
||
assert(xl & 0x8_0000_0000_0000UL); //non-signaling bit, bit 52
|
||
assert((xl & 0x7FF0_0000_0000_0000UL) == 0x7FF0_0000_0000_0000UL); //all exp bits set
|
||
}
|
||
}
|
||
|
||
/**
|
||
* Extract an integral payload from a $(NAN).
|
||
*
|
||
* Returns:
|
||
* the integer payload as a ulong.
|
||
*
|
||
* For floats, the largest possible payload is 0x3F_FFFF.
|
||
* For doubles, it is 0x3_FFFF_FFFF_FFFF.
|
||
* For 80-bit or 128-bit reals, it is 0x3FFF_FFFF_FFFF_FFFF.
|
||
*/
|
||
ulong getNaNPayload(real x) @trusted pure nothrow @nogc
|
||
{
|
||
// assert(isNaN(x));
|
||
alias F = floatTraits!(real);
|
||
static if (F.realFormat == RealFormat.ieeeDouble)
|
||
{
|
||
ulong m = *cast(ulong *)(&x);
|
||
// Make it look like an 80-bit significand.
|
||
// Skip exponent, and quiet bit
|
||
m &= 0x0007_FFFF_FFFF_FFFF;
|
||
m <<= 11;
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeQuadruple)
|
||
{
|
||
version(LittleEndian)
|
||
{
|
||
ulong m = *cast(ulong*)(6+cast(ubyte*)(&x));
|
||
}
|
||
else
|
||
{
|
||
ulong m = *cast(ulong*)(2+cast(ubyte*)(&x));
|
||
}
|
||
|
||
m >>= 1; // there's no implicit bit
|
||
}
|
||
else
|
||
{
|
||
ulong m = *cast(ulong *)(&x);
|
||
}
|
||
|
||
// ignore implicit bit and quiet bit
|
||
|
||
const ulong f = m & 0x3FFF_FF00_0000_0000L;
|
||
|
||
ulong w = f >>> 40;
|
||
w |= (m & 0x00FF_FFFF_F800L) << (22 - 11);
|
||
w |= (m & 0x7FF) << 51;
|
||
return w;
|
||
}
|
||
|
||
debug(UnitTest)
|
||
{
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
real nan4 = NaN(0x789_ABCD_EF12_3456);
|
||
static if (floatTraits!(real).realFormat == RealFormat.ieeeExtended
|
||
|| floatTraits!(real).realFormat == RealFormat.ieeeQuadruple)
|
||
{
|
||
assert(getNaNPayload(nan4) == 0x789_ABCD_EF12_3456);
|
||
}
|
||
else
|
||
{
|
||
assert(getNaNPayload(nan4) == 0x1_ABCD_EF12_3456);
|
||
}
|
||
double nan5 = nan4;
|
||
assert(getNaNPayload(nan5) == 0x1_ABCD_EF12_3456);
|
||
float nan6 = nan4;
|
||
assert(getNaNPayload(nan6) == 0x12_3456);
|
||
nan4 = NaN(0xFABCD);
|
||
assert(getNaNPayload(nan4) == 0xFABCD);
|
||
nan6 = nan4;
|
||
assert(getNaNPayload(nan6) == 0xFABCD);
|
||
nan5 = NaN(0x100_0000_0000_3456);
|
||
assert(getNaNPayload(nan5) == 0x0000_0000_3456);
|
||
}
|
||
}
|
||
|
||
/**
|
||
* Calculate the next largest floating point value after x.
|
||
*
|
||
* Return the least number greater than x that is representable as a real;
|
||
* thus, it gives the next point on the IEEE number line.
|
||
*
|
||
* $(TABLE_SV
|
||
* $(SVH x, nextUp(x) )
|
||
* $(SV -$(INFIN), -real.max )
|
||
* $(SV $(PLUSMN)0.0, real.min_normal*real.epsilon )
|
||
* $(SV real.max, $(INFIN) )
|
||
* $(SV $(INFIN), $(INFIN) )
|
||
* $(SV $(NAN), $(NAN) )
|
||
* )
|
||
*/
|
||
real nextUp(real x) @trusted pure nothrow @nogc
|
||
{
|
||
alias F = floatTraits!(real);
|
||
static if (F.realFormat == RealFormat.ieeeDouble)
|
||
{
|
||
return nextUp(cast(double) x);
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeQuadruple)
|
||
{
|
||
ushort e = F.EXPMASK & (cast(ushort *)&x)[F.EXPPOS_SHORT];
|
||
if (e == F.EXPMASK)
|
||
{
|
||
// NaN or Infinity
|
||
if (x == -real.infinity) return -real.max;
|
||
return x; // +Inf and NaN are unchanged.
|
||
}
|
||
|
||
auto ps = cast(ulong *)&x;
|
||
if (ps[MANTISSA_MSB] & 0x8000_0000_0000_0000)
|
||
{
|
||
// Negative number
|
||
if (ps[MANTISSA_LSB] == 0 && ps[MANTISSA_MSB] == 0x8000_0000_0000_0000)
|
||
{
|
||
// it was negative zero, change to smallest subnormal
|
||
ps[MANTISSA_LSB] = 1;
|
||
ps[MANTISSA_MSB] = 0;
|
||
return x;
|
||
}
|
||
if (ps[MANTISSA_LSB] == 0) --ps[MANTISSA_MSB];
|
||
--ps[MANTISSA_LSB];
|
||
}
|
||
else
|
||
{
|
||
// Positive number
|
||
++ps[MANTISSA_LSB];
|
||
if (ps[MANTISSA_LSB] == 0) ++ps[MANTISSA_MSB];
|
||
}
|
||
return x;
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeExtended)
|
||
{
|
||
// For 80-bit reals, the "implied bit" is a nuisance...
|
||
ushort *pe = cast(ushort *)&x;
|
||
ulong *ps = cast(ulong *)&x;
|
||
|
||
if ((pe[F.EXPPOS_SHORT] & F.EXPMASK) == F.EXPMASK)
|
||
{
|
||
// First, deal with NANs and infinity
|
||
if (x == -real.infinity) return -real.max;
|
||
return x; // +Inf and NaN are unchanged.
|
||
}
|
||
if (pe[F.EXPPOS_SHORT] & 0x8000)
|
||
{
|
||
// Negative number -- need to decrease the significand
|
||
--*ps;
|
||
// Need to mask with 0x7FFF... so subnormals are treated correctly.
|
||
if ((*ps & 0x7FFF_FFFF_FFFF_FFFF) == 0x7FFF_FFFF_FFFF_FFFF)
|
||
{
|
||
if (pe[F.EXPPOS_SHORT] == 0x8000) // it was negative zero
|
||
{
|
||
*ps = 1;
|
||
pe[F.EXPPOS_SHORT] = 0; // smallest subnormal.
|
||
return x;
|
||
}
|
||
|
||
--pe[F.EXPPOS_SHORT];
|
||
|
||
if (pe[F.EXPPOS_SHORT] == 0x8000)
|
||
return x; // it's become a subnormal, implied bit stays low.
|
||
|
||
*ps = 0xFFFF_FFFF_FFFF_FFFF; // set the implied bit
|
||
return x;
|
||
}
|
||
return x;
|
||
}
|
||
else
|
||
{
|
||
// Positive number -- need to increase the significand.
|
||
// Works automatically for positive zero.
|
||
++*ps;
|
||
if ((*ps & 0x7FFF_FFFF_FFFF_FFFF) == 0)
|
||
{
|
||
// change in exponent
|
||
++pe[F.EXPPOS_SHORT];
|
||
*ps = 0x8000_0000_0000_0000; // set the high bit
|
||
}
|
||
}
|
||
return x;
|
||
}
|
||
else // static if (F.realFormat == RealFormat.ibmExtended)
|
||
{
|
||
assert(0, "nextUp not implemented");
|
||
}
|
||
}
|
||
|
||
/** ditto */
|
||
double nextUp(double x) @trusted pure nothrow @nogc
|
||
{
|
||
ulong *ps = cast(ulong *)&x;
|
||
|
||
if ((*ps & 0x7FF0_0000_0000_0000) == 0x7FF0_0000_0000_0000)
|
||
{
|
||
// First, deal with NANs and infinity
|
||
if (x == -x.infinity) return -x.max;
|
||
return x; // +INF and NAN are unchanged.
|
||
}
|
||
if (*ps & 0x8000_0000_0000_0000) // Negative number
|
||
{
|
||
if (*ps == 0x8000_0000_0000_0000) // it was negative zero
|
||
{
|
||
*ps = 0x0000_0000_0000_0001; // change to smallest subnormal
|
||
return x;
|
||
}
|
||
--*ps;
|
||
}
|
||
else
|
||
{ // Positive number
|
||
++*ps;
|
||
}
|
||
return x;
|
||
}
|
||
|
||
/** ditto */
|
||
float nextUp(float x) @trusted pure nothrow @nogc
|
||
{
|
||
uint *ps = cast(uint *)&x;
|
||
|
||
if ((*ps & 0x7F80_0000) == 0x7F80_0000)
|
||
{
|
||
// First, deal with NANs and infinity
|
||
if (x == -x.infinity) return -x.max;
|
||
|
||
return x; // +INF and NAN are unchanged.
|
||
}
|
||
if (*ps & 0x8000_0000) // Negative number
|
||
{
|
||
if (*ps == 0x8000_0000) // it was negative zero
|
||
{
|
||
*ps = 0x0000_0001; // change to smallest subnormal
|
||
return x;
|
||
}
|
||
|
||
--*ps;
|
||
}
|
||
else
|
||
{
|
||
// Positive number
|
||
++*ps;
|
||
}
|
||
return x;
|
||
}
|
||
|
||
/**
|
||
* Calculate the next smallest floating point value before x.
|
||
*
|
||
* Return the greatest number less than x that is representable as a real;
|
||
* thus, it gives the previous point on the IEEE number line.
|
||
*
|
||
* $(TABLE_SV
|
||
* $(SVH x, nextDown(x) )
|
||
* $(SV $(INFIN), real.max )
|
||
* $(SV $(PLUSMN)0.0, -real.min_normal*real.epsilon )
|
||
* $(SV -real.max, -$(INFIN) )
|
||
* $(SV -$(INFIN), -$(INFIN) )
|
||
* $(SV $(NAN), $(NAN) )
|
||
* )
|
||
*/
|
||
real nextDown(real x) @safe pure nothrow @nogc
|
||
{
|
||
return -nextUp(-x);
|
||
}
|
||
|
||
/** ditto */
|
||
double nextDown(double x) @safe pure nothrow @nogc
|
||
{
|
||
return -nextUp(-x);
|
||
}
|
||
|
||
/** ditto */
|
||
float nextDown(float x) @safe pure nothrow @nogc
|
||
{
|
||
return -nextUp(-x);
|
||
}
|
||
|
||
///
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
assert( nextDown(1.0 + real.epsilon) == 1.0);
|
||
}
|
||
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
static if (floatTraits!(real).realFormat == RealFormat.ieeeExtended)
|
||
{
|
||
|
||
// Tests for 80-bit reals
|
||
assert(isIdentical(nextUp(NaN(0xABC)), NaN(0xABC)));
|
||
// negative numbers
|
||
assert( nextUp(-real.infinity) == -real.max );
|
||
assert( nextUp(-1.0L-real.epsilon) == -1.0 );
|
||
assert( nextUp(-2.0L) == -2.0 + real.epsilon);
|
||
// subnormals and zero
|
||
assert( nextUp(-real.min_normal) == -real.min_normal*(1-real.epsilon) );
|
||
assert( nextUp(-real.min_normal*(1-real.epsilon)) == -real.min_normal*(1-2*real.epsilon) );
|
||
assert( isIdentical(-0.0L, nextUp(-real.min_normal*real.epsilon)) );
|
||
assert( nextUp(-0.0L) == real.min_normal*real.epsilon );
|
||
assert( nextUp(0.0L) == real.min_normal*real.epsilon );
|
||
assert( nextUp(real.min_normal*(1-real.epsilon)) == real.min_normal );
|
||
assert( nextUp(real.min_normal) == real.min_normal*(1+real.epsilon) );
|
||
// positive numbers
|
||
assert( nextUp(1.0L) == 1.0 + real.epsilon );
|
||
assert( nextUp(2.0L-real.epsilon) == 2.0 );
|
||
assert( nextUp(real.max) == real.infinity );
|
||
assert( nextUp(real.infinity)==real.infinity );
|
||
}
|
||
|
||
double n = NaN(0xABC);
|
||
assert(isIdentical(nextUp(n), n));
|
||
// negative numbers
|
||
assert( nextUp(-double.infinity) == -double.max );
|
||
assert( nextUp(-1-double.epsilon) == -1.0 );
|
||
assert( nextUp(-2.0) == -2.0 + double.epsilon);
|
||
// subnormals and zero
|
||
|
||
assert( nextUp(-double.min_normal) == -double.min_normal*(1-double.epsilon) );
|
||
assert( nextUp(-double.min_normal*(1-double.epsilon)) == -double.min_normal*(1-2*double.epsilon) );
|
||
assert( isIdentical(-0.0, nextUp(-double.min_normal*double.epsilon)) );
|
||
assert( nextUp(0.0) == double.min_normal*double.epsilon );
|
||
assert( nextUp(-0.0) == double.min_normal*double.epsilon );
|
||
assert( nextUp(double.min_normal*(1-double.epsilon)) == double.min_normal );
|
||
assert( nextUp(double.min_normal) == double.min_normal*(1+double.epsilon) );
|
||
// positive numbers
|
||
assert( nextUp(1.0) == 1.0 + double.epsilon );
|
||
assert( nextUp(2.0-double.epsilon) == 2.0 );
|
||
assert( nextUp(double.max) == double.infinity );
|
||
|
||
float fn = NaN(0xABC);
|
||
assert(isIdentical(nextUp(fn), fn));
|
||
float f = -float.min_normal*(1-float.epsilon);
|
||
float f1 = -float.min_normal;
|
||
assert( nextUp(f1) == f);
|
||
f = 1.0f+float.epsilon;
|
||
f1 = 1.0f;
|
||
assert( nextUp(f1) == f );
|
||
f1 = -0.0f;
|
||
assert( nextUp(f1) == float.min_normal*float.epsilon);
|
||
assert( nextUp(float.infinity)==float.infinity );
|
||
|
||
assert(nextDown(1.0L+real.epsilon)==1.0);
|
||
assert(nextDown(1.0+double.epsilon)==1.0);
|
||
f = 1.0f+float.epsilon;
|
||
assert(nextDown(f)==1.0);
|
||
assert(nextafter(1.0+real.epsilon, -real.infinity)==1.0);
|
||
}
|
||
|
||
|
||
|
||
/******************************************
|
||
* Calculates the next representable value after x in the direction of y.
|
||
*
|
||
* If y > x, the result will be the next largest floating-point value;
|
||
* if y < x, the result will be the next smallest value.
|
||
* If x == y, the result is y.
|
||
*
|
||
* Remarks:
|
||
* This function is not generally very useful; it's almost always better to use
|
||
* the faster functions nextUp() or nextDown() instead.
|
||
*
|
||
* The FE_INEXACT and FE_OVERFLOW exceptions will be raised if x is finite and
|
||
* the function result is infinite. The FE_INEXACT and FE_UNDERFLOW
|
||
* exceptions will be raised if the function value is subnormal, and x is
|
||
* not equal to y.
|
||
*/
|
||
T nextafter(T)(const T x, const T y) @safe pure nothrow @nogc
|
||
{
|
||
if (x == y) return y;
|
||
return ((y>x) ? nextUp(x) : nextDown(x));
|
||
}
|
||
|
||
///
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
float a = 1;
|
||
assert(is(typeof(nextafter(a, a)) == float));
|
||
assert(nextafter(a, a.infinity) > a);
|
||
|
||
double b = 2;
|
||
assert(is(typeof(nextafter(b, b)) == double));
|
||
assert(nextafter(b, b.infinity) > b);
|
||
|
||
real c = 3;
|
||
assert(is(typeof(nextafter(c, c)) == real));
|
||
assert(nextafter(c, c.infinity) > c);
|
||
}
|
||
|
||
//real nexttoward(real x, real y) { return core.stdc.math.nexttowardl(x, y); }
|
||
|
||
/*******************************************
|
||
* Returns the positive difference between x and y.
|
||
* Returns:
|
||
* $(TABLE_SV
|
||
* $(TR $(TH x, y) $(TH fdim(x, y)))
|
||
* $(TR $(TD x $(GT) y) $(TD x - y))
|
||
* $(TR $(TD x $(LT)= y) $(TD +0.0))
|
||
* )
|
||
*/
|
||
real fdim(real x, real y) @safe pure nothrow @nogc { return (x > y) ? x - y : +0.0; }
|
||
|
||
/****************************************
|
||
* Returns the larger of x and y.
|
||
*/
|
||
real fmax(real x, real y) @safe pure nothrow @nogc { return x > y ? x : y; }
|
||
|
||
/****************************************
|
||
* Returns the smaller of x and y.
|
||
*/
|
||
real fmin(real x, real y) @safe pure nothrow @nogc { return x < y ? x : y; }
|
||
|
||
/**************************************
|
||
* Returns (x * y) + z, rounding only once according to the
|
||
* current rounding mode.
|
||
*
|
||
* BUGS: Not currently implemented - rounds twice.
|
||
*/
|
||
real fma(real x, real y, real z) @safe pure nothrow @nogc { return (x * y) + z; }
|
||
|
||
/*******************************************************************
|
||
* Compute the value of x $(SUPERSCRIPT n), where n is an integer
|
||
*/
|
||
Unqual!F pow(F, G)(F x, G n) @nogc @trusted pure nothrow
|
||
if (isFloatingPoint!(F) && isIntegral!(G))
|
||
{
|
||
import std.traits : Unsigned;
|
||
real p = 1.0, v = void;
|
||
Unsigned!(Unqual!G) m = n;
|
||
if (n < 0)
|
||
{
|
||
switch (n)
|
||
{
|
||
case -1:
|
||
return 1 / x;
|
||
case -2:
|
||
return 1 / (x * x);
|
||
default:
|
||
}
|
||
|
||
m = -n;
|
||
v = p / x;
|
||
}
|
||
else
|
||
{
|
||
switch (n)
|
||
{
|
||
case 0:
|
||
return 1.0;
|
||
case 1:
|
||
return x;
|
||
case 2:
|
||
return x * x;
|
||
default:
|
||
}
|
||
|
||
v = x;
|
||
}
|
||
|
||
while (1)
|
||
{
|
||
if (m & 1)
|
||
p *= v;
|
||
m >>= 1;
|
||
if (!m)
|
||
break;
|
||
v *= v;
|
||
}
|
||
return p;
|
||
}
|
||
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
// Make sure it instantiates and works properly on immutable values and
|
||
// with various integer and float types.
|
||
immutable real x = 46;
|
||
immutable float xf = x;
|
||
immutable double xd = x;
|
||
immutable uint one = 1;
|
||
immutable ushort two = 2;
|
||
immutable ubyte three = 3;
|
||
immutable ulong eight = 8;
|
||
|
||
immutable int neg1 = -1;
|
||
immutable short neg2 = -2;
|
||
immutable byte neg3 = -3;
|
||
immutable long neg8 = -8;
|
||
|
||
|
||
assert(pow(x,0) == 1.0);
|
||
assert(pow(xd,one) == x);
|
||
assert(pow(xf,two) == x * x);
|
||
assert(pow(x,three) == x * x * x);
|
||
assert(pow(x,eight) == (x * x) * (x * x) * (x * x) * (x * x));
|
||
|
||
assert(pow(x, neg1) == 1 / x);
|
||
|
||
version(X86_64)
|
||
{
|
||
pragma(msg, "test disabled on x86_64, see bug 5628");
|
||
}
|
||
else version(ARM)
|
||
{
|
||
pragma(msg, "test disabled on ARM, see bug 5628");
|
||
}
|
||
else
|
||
{
|
||
assert(pow(xd, neg2) == 1 / (x * x));
|
||
assert(pow(xf, neg8) == 1 / ((x * x) * (x * x) * (x * x) * (x * x)));
|
||
}
|
||
|
||
assert(feqrel(pow(x, neg3), 1 / (x * x * x)) >= real.mant_dig - 1);
|
||
}
|
||
|
||
@system unittest
|
||
{
|
||
assert(equalsDigit(pow(2.0L, 10.0L), 1024, 19));
|
||
}
|
||
|
||
/** Compute the value of an integer x, raised to the power of a positive
|
||
* integer n.
|
||
*
|
||
* If both x and n are 0, the result is 1.
|
||
* If n is negative, an integer divide error will occur at runtime,
|
||
* regardless of the value of x.
|
||
*/
|
||
typeof(Unqual!(F).init * Unqual!(G).init) pow(F, G)(F x, G n) @nogc @trusted pure nothrow
|
||
if (isIntegral!(F) && isIntegral!(G))
|
||
{
|
||
if (n<0) return x/0; // Only support positive powers
|
||
typeof(return) p, v = void;
|
||
Unqual!G m = n;
|
||
|
||
switch (m)
|
||
{
|
||
case 0:
|
||
p = 1;
|
||
break;
|
||
|
||
case 1:
|
||
p = x;
|
||
break;
|
||
|
||
case 2:
|
||
p = x * x;
|
||
break;
|
||
|
||
default:
|
||
v = x;
|
||
p = 1;
|
||
while (1)
|
||
{
|
||
if (m & 1)
|
||
p *= v;
|
||
m >>= 1;
|
||
if (!m)
|
||
break;
|
||
v *= v;
|
||
}
|
||
break;
|
||
}
|
||
return p;
|
||
}
|
||
|
||
///
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
immutable int one = 1;
|
||
immutable byte two = 2;
|
||
immutable ubyte three = 3;
|
||
immutable short four = 4;
|
||
immutable long ten = 10;
|
||
|
||
assert(pow(two, three) == 8);
|
||
assert(pow(two, ten) == 1024);
|
||
assert(pow(one, ten) == 1);
|
||
assert(pow(ten, four) == 10_000);
|
||
assert(pow(four, 10) == 1_048_576);
|
||
assert(pow(three, four) == 81);
|
||
|
||
}
|
||
|
||
/**Computes integer to floating point powers.*/
|
||
real pow(I, F)(I x, F y) @nogc @trusted pure nothrow
|
||
if (isIntegral!I && isFloatingPoint!F)
|
||
{
|
||
return pow(cast(real) x, cast(Unqual!F) y);
|
||
}
|
||
|
||
/*********************************************
|
||
* Calculates x$(SUPERSCRIPT y).
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH x) $(TH y) $(TH pow(x, y))
|
||
* $(TH div 0) $(TH invalid?))
|
||
* $(TR $(TD anything) $(TD $(PLUSMN)0.0) $(TD 1.0)
|
||
* $(TD no) $(TD no) )
|
||
* $(TR $(TD |x| $(GT) 1) $(TD +$(INFIN)) $(TD +$(INFIN))
|
||
* $(TD no) $(TD no) )
|
||
* $(TR $(TD |x| $(LT) 1) $(TD +$(INFIN)) $(TD +0.0)
|
||
* $(TD no) $(TD no) )
|
||
* $(TR $(TD |x| $(GT) 1) $(TD -$(INFIN)) $(TD +0.0)
|
||
* $(TD no) $(TD no) )
|
||
* $(TR $(TD |x| $(LT) 1) $(TD -$(INFIN)) $(TD +$(INFIN))
|
||
* $(TD no) $(TD no) )
|
||
* $(TR $(TD +$(INFIN)) $(TD $(GT) 0.0) $(TD +$(INFIN))
|
||
* $(TD no) $(TD no) )
|
||
* $(TR $(TD +$(INFIN)) $(TD $(LT) 0.0) $(TD +0.0)
|
||
* $(TD no) $(TD no) )
|
||
* $(TR $(TD -$(INFIN)) $(TD odd integer $(GT) 0.0) $(TD -$(INFIN))
|
||
* $(TD no) $(TD no) )
|
||
* $(TR $(TD -$(INFIN)) $(TD $(GT) 0.0, not odd integer) $(TD +$(INFIN))
|
||
* $(TD no) $(TD no))
|
||
* $(TR $(TD -$(INFIN)) $(TD odd integer $(LT) 0.0) $(TD -0.0)
|
||
* $(TD no) $(TD no) )
|
||
* $(TR $(TD -$(INFIN)) $(TD $(LT) 0.0, not odd integer) $(TD +0.0)
|
||
* $(TD no) $(TD no) )
|
||
* $(TR $(TD $(PLUSMN)1.0) $(TD $(PLUSMN)$(INFIN)) $(TD $(NAN))
|
||
* $(TD no) $(TD yes) )
|
||
* $(TR $(TD $(LT) 0.0) $(TD finite, nonintegral) $(TD $(NAN))
|
||
* $(TD no) $(TD yes))
|
||
* $(TR $(TD $(PLUSMN)0.0) $(TD odd integer $(LT) 0.0) $(TD $(PLUSMNINF))
|
||
* $(TD yes) $(TD no) )
|
||
* $(TR $(TD $(PLUSMN)0.0) $(TD $(LT) 0.0, not odd integer) $(TD +$(INFIN))
|
||
* $(TD yes) $(TD no))
|
||
* $(TR $(TD $(PLUSMN)0.0) $(TD odd integer $(GT) 0.0) $(TD $(PLUSMN)0.0)
|
||
* $(TD no) $(TD no) )
|
||
* $(TR $(TD $(PLUSMN)0.0) $(TD $(GT) 0.0, not odd integer) $(TD +0.0)
|
||
* $(TD no) $(TD no) )
|
||
* )
|
||
*/
|
||
Unqual!(Largest!(F, G)) pow(F, G)(F x, G y) @nogc @trusted pure nothrow
|
||
if (isFloatingPoint!(F) && isFloatingPoint!(G))
|
||
{
|
||
alias Float = typeof(return);
|
||
|
||
static real impl(real x, real y) @nogc pure nothrow
|
||
{
|
||
// Special cases.
|
||
if (isNaN(y))
|
||
return y;
|
||
if (isNaN(x) && y != 0.0)
|
||
return x;
|
||
|
||
// Even if x is NaN.
|
||
if (y == 0.0)
|
||
return 1.0;
|
||
if (y == 1.0)
|
||
return x;
|
||
|
||
if (isInfinity(y))
|
||
{
|
||
if (fabs(x) > 1)
|
||
{
|
||
if (signbit(y))
|
||
return +0.0;
|
||
else
|
||
return F.infinity;
|
||
}
|
||
else if (fabs(x) == 1)
|
||
{
|
||
return y * 0; // generate NaN.
|
||
}
|
||
else // < 1
|
||
{
|
||
if (signbit(y))
|
||
return F.infinity;
|
||
else
|
||
return +0.0;
|
||
}
|
||
}
|
||
if (isInfinity(x))
|
||
{
|
||
if (signbit(x))
|
||
{
|
||
long i = cast(long) y;
|
||
if (y > 0.0)
|
||
{
|
||
if (i == y && i & 1)
|
||
return -F.infinity;
|
||
else
|
||
return F.infinity;
|
||
}
|
||
else if (y < 0.0)
|
||
{
|
||
if (i == y && i & 1)
|
||
return -0.0;
|
||
else
|
||
return +0.0;
|
||
}
|
||
}
|
||
else
|
||
{
|
||
if (y > 0.0)
|
||
return F.infinity;
|
||
else if (y < 0.0)
|
||
return +0.0;
|
||
}
|
||
}
|
||
|
||
if (x == 0.0)
|
||
{
|
||
if (signbit(x))
|
||
{
|
||
long i = cast(long) y;
|
||
if (y > 0.0)
|
||
{
|
||
if (i == y && i & 1)
|
||
return -0.0;
|
||
else
|
||
return +0.0;
|
||
}
|
||
else if (y < 0.0)
|
||
{
|
||
if (i == y && i & 1)
|
||
return -F.infinity;
|
||
else
|
||
return F.infinity;
|
||
}
|
||
}
|
||
else
|
||
{
|
||
if (y > 0.0)
|
||
return +0.0;
|
||
else if (y < 0.0)
|
||
return F.infinity;
|
||
}
|
||
}
|
||
if (x == 1.0)
|
||
return 1.0;
|
||
|
||
if (y >= F.max)
|
||
{
|
||
if ((x > 0.0 && x < 1.0) || (x > -1.0 && x < 0.0))
|
||
return 0.0;
|
||
if (x > 1.0 || x < -1.0)
|
||
return F.infinity;
|
||
}
|
||
if (y <= -F.max)
|
||
{
|
||
if ((x > 0.0 && x < 1.0) || (x > -1.0 && x < 0.0))
|
||
return F.infinity;
|
||
if (x > 1.0 || x < -1.0)
|
||
return 0.0;
|
||
}
|
||
|
||
if (x >= F.max)
|
||
{
|
||
if (y > 0.0)
|
||
return F.infinity;
|
||
else
|
||
return 0.0;
|
||
}
|
||
if (x <= -F.max)
|
||
{
|
||
long i = cast(long) y;
|
||
if (y > 0.0)
|
||
{
|
||
if (i == y && i & 1)
|
||
return -F.infinity;
|
||
else
|
||
return F.infinity;
|
||
}
|
||
else if (y < 0.0)
|
||
{
|
||
if (i == y && i & 1)
|
||
return -0.0;
|
||
else
|
||
return +0.0;
|
||
}
|
||
}
|
||
|
||
// Integer power of x.
|
||
long iy = cast(long) y;
|
||
if (iy == y && fabs(y) < 32_768.0)
|
||
return pow(x, iy);
|
||
|
||
real sign = 1.0;
|
||
if (x < 0)
|
||
{
|
||
// Result is real only if y is an integer
|
||
// Check for a non-zero fractional part
|
||
enum maxOdd = pow(2.0L, real.mant_dig) - 1.0L;
|
||
static if (maxOdd > ulong.max)
|
||
{
|
||
// Generic method, for any FP type
|
||
if (floor(y) != y)
|
||
return sqrt(x); // Complex result -- create a NaN
|
||
|
||
const hy = ldexp(y, -1);
|
||
if (floor(hy) != hy)
|
||
sign = -1.0;
|
||
}
|
||
else
|
||
{
|
||
// Much faster, if ulong has enough precision
|
||
const absY = fabs(y);
|
||
if (absY <= maxOdd)
|
||
{
|
||
const uy = cast(ulong) absY;
|
||
if (uy != absY)
|
||
return sqrt(x); // Complex result -- create a NaN
|
||
|
||
if (uy & 1)
|
||
sign = -1.0;
|
||
}
|
||
}
|
||
x = -x;
|
||
}
|
||
version(INLINE_YL2X)
|
||
{
|
||
// If x > 0, x ^^ y == 2 ^^ ( y * log2(x) )
|
||
// TODO: This is not accurate in practice. A fast and accurate
|
||
// (though complicated) method is described in:
|
||
// "An efficient rounding boundary test for pow(x, y)
|
||
// in double precision", C.Q. Lauter and V. Lefèvre, INRIA (2007).
|
||
return sign * exp2( core.math.yl2x(x, y) );
|
||
}
|
||
else
|
||
{
|
||
// If x > 0, x ^^ y == 2 ^^ ( y * log2(x) )
|
||
// TODO: This is not accurate in practice. A fast and accurate
|
||
// (though complicated) method is described in:
|
||
// "An efficient rounding boundary test for pow(x, y)
|
||
// in double precision", C.Q. Lauter and V. Lefèvre, INRIA (2007).
|
||
Float w = exp2(y * log2(x));
|
||
return sign * w;
|
||
}
|
||
}
|
||
return impl(x, y);
|
||
}
|
||
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
// Test all the special values. These unittests can be run on Windows
|
||
// by temporarily changing the version(linux) to version(all).
|
||
immutable float zero = 0;
|
||
immutable real one = 1;
|
||
immutable double two = 2;
|
||
immutable float three = 3;
|
||
immutable float fnan = float.nan;
|
||
immutable double dnan = double.nan;
|
||
immutable real rnan = real.nan;
|
||
immutable dinf = double.infinity;
|
||
immutable rninf = -real.infinity;
|
||
|
||
assert(pow(fnan, zero) == 1);
|
||
assert(pow(dnan, zero) == 1);
|
||
assert(pow(rnan, zero) == 1);
|
||
|
||
assert(pow(two, dinf) == double.infinity);
|
||
assert(isIdentical(pow(0.2f, dinf), +0.0));
|
||
assert(pow(0.99999999L, rninf) == real.infinity);
|
||
assert(isIdentical(pow(1.000000001, rninf), +0.0));
|
||
assert(pow(dinf, 0.001) == dinf);
|
||
assert(isIdentical(pow(dinf, -0.001), +0.0));
|
||
assert(pow(rninf, 3.0L) == rninf);
|
||
assert(pow(rninf, 2.0L) == real.infinity);
|
||
assert(isIdentical(pow(rninf, -3.0), -0.0));
|
||
assert(isIdentical(pow(rninf, -2.0), +0.0));
|
||
|
||
// @@@BUG@@@ somewhere
|
||
version(OSX) {} else assert(isNaN(pow(one, dinf)));
|
||
version(OSX) {} else assert(isNaN(pow(-one, dinf)));
|
||
assert(isNaN(pow(-0.2, PI)));
|
||
// boundary cases. Note that epsilon == 2^^-n for some n,
|
||
// so 1/epsilon == 2^^n is always even.
|
||
assert(pow(-1.0L, 1/real.epsilon - 1.0L) == -1.0L);
|
||
assert(pow(-1.0L, 1/real.epsilon) == 1.0L);
|
||
assert(isNaN(pow(-1.0L, 1/real.epsilon-0.5L)));
|
||
assert(isNaN(pow(-1.0L, -1/real.epsilon+0.5L)));
|
||
|
||
assert(pow(0.0, -3.0) == double.infinity);
|
||
assert(pow(-0.0, -3.0) == -double.infinity);
|
||
assert(pow(0.0, -PI) == double.infinity);
|
||
assert(pow(-0.0, -PI) == double.infinity);
|
||
assert(isIdentical(pow(0.0, 5.0), 0.0));
|
||
assert(isIdentical(pow(-0.0, 5.0), -0.0));
|
||
assert(isIdentical(pow(0.0, 6.0), 0.0));
|
||
assert(isIdentical(pow(-0.0, 6.0), 0.0));
|
||
|
||
// Issue #14786 fixed
|
||
immutable real maxOdd = pow(2.0L, real.mant_dig) - 1.0L;
|
||
assert(pow(-1.0L, maxOdd) == -1.0L);
|
||
assert(pow(-1.0L, -maxOdd) == -1.0L);
|
||
assert(pow(-1.0L, maxOdd + 1.0L) == 1.0L);
|
||
assert(pow(-1.0L, -maxOdd + 1.0L) == 1.0L);
|
||
assert(pow(-1.0L, maxOdd - 1.0L) == 1.0L);
|
||
assert(pow(-1.0L, -maxOdd - 1.0L) == 1.0L);
|
||
|
||
// Now, actual numbers.
|
||
assert(approxEqual(pow(two, three), 8.0));
|
||
assert(approxEqual(pow(two, -2.5), 0.1767767));
|
||
|
||
// Test integer to float power.
|
||
immutable uint twoI = 2;
|
||
assert(approxEqual(pow(twoI, three), 8.0));
|
||
}
|
||
|
||
/**************************************
|
||
* To what precision is x equal to y?
|
||
*
|
||
* Returns: the number of mantissa bits which are equal in x and y.
|
||
* eg, 0x1.F8p+60 and 0x1.F1p+60 are equal to 5 bits of precision.
|
||
*
|
||
* $(TABLE_SV
|
||
* $(TR $(TH x) $(TH y) $(TH feqrel(x, y)))
|
||
* $(TR $(TD x) $(TD x) $(TD real.mant_dig))
|
||
* $(TR $(TD x) $(TD $(GT)= 2*x) $(TD 0))
|
||
* $(TR $(TD x) $(TD $(LT)= x/2) $(TD 0))
|
||
* $(TR $(TD $(NAN)) $(TD any) $(TD 0))
|
||
* $(TR $(TD any) $(TD $(NAN)) $(TD 0))
|
||
* )
|
||
*/
|
||
int feqrel(X)(const X x, const X y) @trusted pure nothrow @nogc
|
||
if (isFloatingPoint!(X))
|
||
{
|
||
/* Public Domain. Author: Don Clugston, 18 Aug 2005.
|
||
*/
|
||
alias F = floatTraits!(X);
|
||
static if (F.realFormat == RealFormat.ibmExtended)
|
||
{
|
||
if (cast(double*)(&x)[MANTISSA_MSB] == cast(double*)(&y)[MANTISSA_MSB])
|
||
{
|
||
return double.mant_dig
|
||
+ feqrel(cast(double*)(&x)[MANTISSA_LSB],
|
||
cast(double*)(&y)[MANTISSA_LSB]);
|
||
}
|
||
else
|
||
{
|
||
return feqrel(cast(double*)(&x)[MANTISSA_MSB],
|
||
cast(double*)(&y)[MANTISSA_MSB]);
|
||
}
|
||
}
|
||
else
|
||
{
|
||
static assert(F.realFormat == RealFormat.ieeeSingle
|
||
|| F.realFormat == RealFormat.ieeeDouble
|
||
|| F.realFormat == RealFormat.ieeeExtended
|
||
|| F.realFormat == RealFormat.ieeeQuadruple);
|
||
|
||
if (x == y)
|
||
return X.mant_dig; // ensure diff != 0, cope with INF.
|
||
|
||
Unqual!X diff = fabs(x - y);
|
||
|
||
ushort *pa = cast(ushort *)(&x);
|
||
ushort *pb = cast(ushort *)(&y);
|
||
ushort *pd = cast(ushort *)(&diff);
|
||
|
||
|
||
// The difference in abs(exponent) between x or y and abs(x-y)
|
||
// is equal to the number of significand bits of x which are
|
||
// equal to y. If negative, x and y have different exponents.
|
||
// If positive, x and y are equal to 'bitsdiff' bits.
|
||
// AND with 0x7FFF to form the absolute value.
|
||
// To avoid out-by-1 errors, we subtract 1 so it rounds down
|
||
// if the exponents were different. This means 'bitsdiff' is
|
||
// always 1 lower than we want, except that if bitsdiff == 0,
|
||
// they could have 0 or 1 bits in common.
|
||
|
||
int bitsdiff = ((( (pa[F.EXPPOS_SHORT] & F.EXPMASK)
|
||
+ (pb[F.EXPPOS_SHORT] & F.EXPMASK)
|
||
- (1 << F.EXPSHIFT)) >> 1)
|
||
- (pd[F.EXPPOS_SHORT] & F.EXPMASK)) >> F.EXPSHIFT;
|
||
if ( (pd[F.EXPPOS_SHORT] & F.EXPMASK) == 0)
|
||
{ // Difference is subnormal
|
||
// For subnormals, we need to add the number of zeros that
|
||
// lie at the start of diff's significand.
|
||
// We do this by multiplying by 2^^real.mant_dig
|
||
diff *= F.RECIP_EPSILON;
|
||
return bitsdiff + X.mant_dig - ((pd[F.EXPPOS_SHORT] & F.EXPMASK) >> F.EXPSHIFT);
|
||
}
|
||
|
||
if (bitsdiff > 0)
|
||
return bitsdiff + 1; // add the 1 we subtracted before
|
||
|
||
// Avoid out-by-1 errors when factor is almost 2.
|
||
if (bitsdiff == 0
|
||
&& ((pa[F.EXPPOS_SHORT] ^ pb[F.EXPPOS_SHORT]) & F.EXPMASK) == 0)
|
||
{
|
||
return 1;
|
||
} else return 0;
|
||
}
|
||
}
|
||
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
void testFeqrel(F)()
|
||
{
|
||
// Exact equality
|
||
assert(feqrel(F.max, F.max) == F.mant_dig);
|
||
assert(feqrel!(F)(0.0, 0.0) == F.mant_dig);
|
||
assert(feqrel(F.infinity, F.infinity) == F.mant_dig);
|
||
|
||
// a few bits away from exact equality
|
||
F w=1;
|
||
for (int i = 1; i < F.mant_dig - 1; ++i)
|
||
{
|
||
assert(feqrel!(F)(1.0 + w * F.epsilon, 1.0) == F.mant_dig-i);
|
||
assert(feqrel!(F)(1.0 - w * F.epsilon, 1.0) == F.mant_dig-i);
|
||
assert(feqrel!(F)(1.0, 1 + (w-1) * F.epsilon) == F.mant_dig - i + 1);
|
||
w*=2;
|
||
}
|
||
|
||
assert(feqrel!(F)(1.5+F.epsilon, 1.5) == F.mant_dig-1);
|
||
assert(feqrel!(F)(1.5-F.epsilon, 1.5) == F.mant_dig-1);
|
||
assert(feqrel!(F)(1.5-F.epsilon, 1.5+F.epsilon) == F.mant_dig-2);
|
||
|
||
|
||
// Numbers that are close
|
||
assert(feqrel!(F)(0x1.Bp+84, 0x1.B8p+84) == 5);
|
||
assert(feqrel!(F)(0x1.8p+10, 0x1.Cp+10) == 2);
|
||
assert(feqrel!(F)(1.5 * (1 - F.epsilon), 1.0L) == 2);
|
||
assert(feqrel!(F)(1.5, 1.0) == 1);
|
||
assert(feqrel!(F)(2 * (1 - F.epsilon), 1.0L) == 1);
|
||
|
||
// Factors of 2
|
||
assert(feqrel(F.max, F.infinity) == 0);
|
||
assert(feqrel!(F)(2 * (1 - F.epsilon), 1.0L) == 1);
|
||
assert(feqrel!(F)(1.0, 2.0) == 0);
|
||
assert(feqrel!(F)(4.0, 1.0) == 0);
|
||
|
||
// Extreme inequality
|
||
assert(feqrel(F.nan, F.nan) == 0);
|
||
assert(feqrel!(F)(0.0L, -F.nan) == 0);
|
||
assert(feqrel(F.nan, F.infinity) == 0);
|
||
assert(feqrel(F.infinity, -F.infinity) == 0);
|
||
assert(feqrel(F.max, -F.max) == 0);
|
||
|
||
assert(feqrel(F.min_normal / 8, F.min_normal / 17) == 3);
|
||
|
||
const F Const = 2;
|
||
immutable F Immutable = 2;
|
||
auto Compiles = feqrel(Const, Immutable);
|
||
}
|
||
|
||
assert(feqrel(7.1824L, 7.1824L) == real.mant_dig);
|
||
|
||
testFeqrel!(real)();
|
||
testFeqrel!(double)();
|
||
testFeqrel!(float)();
|
||
}
|
||
|
||
package: // Not public yet
|
||
/* Return the value that lies halfway between x and y on the IEEE number line.
|
||
*
|
||
* Formally, the result is the arithmetic mean of the binary significands of x
|
||
* and y, multiplied by the geometric mean of the binary exponents of x and y.
|
||
* x and y must have the same sign, and must not be NaN.
|
||
* Note: this function is useful for ensuring O(log n) behaviour in algorithms
|
||
* involving a 'binary chop'.
|
||
*
|
||
* Special cases:
|
||
* If x and y are within a factor of 2, (ie, feqrel(x, y) > 0), the return value
|
||
* is the arithmetic mean (x + y) / 2.
|
||
* If x and y are even powers of 2, the return value is the geometric mean,
|
||
* ieeeMean(x, y) = sqrt(x * y).
|
||
*
|
||
*/
|
||
T ieeeMean(T)(const T x, const T y) @trusted pure nothrow @nogc
|
||
in
|
||
{
|
||
// both x and y must have the same sign, and must not be NaN.
|
||
assert(signbit(x) == signbit(y));
|
||
assert(x == x && y == y);
|
||
}
|
||
body
|
||
{
|
||
// Runtime behaviour for contract violation:
|
||
// If signs are opposite, or one is a NaN, return 0.
|
||
if (!((x >= 0 && y >= 0) || (x <= 0 && y <= 0))) return 0.0;
|
||
|
||
// The implementation is simple: cast x and y to integers,
|
||
// average them (avoiding overflow), and cast the result back to a floating-point number.
|
||
|
||
alias F = floatTraits!(T);
|
||
T u;
|
||
static if (F.realFormat == RealFormat.ieeeExtended)
|
||
{
|
||
// There's slight additional complexity because they are actually
|
||
// 79-bit reals...
|
||
ushort *ue = cast(ushort *)&u;
|
||
ulong *ul = cast(ulong *)&u;
|
||
ushort *xe = cast(ushort *)&x;
|
||
ulong *xl = cast(ulong *)&x;
|
||
ushort *ye = cast(ushort *)&y;
|
||
ulong *yl = cast(ulong *)&y;
|
||
|
||
// Ignore the useless implicit bit. (Bonus: this prevents overflows)
|
||
ulong m = ((*xl) & 0x7FFF_FFFF_FFFF_FFFFL) + ((*yl) & 0x7FFF_FFFF_FFFF_FFFFL);
|
||
|
||
// @@@ BUG? @@@
|
||
// Cast shouldn't be here
|
||
ushort e = cast(ushort) ((xe[F.EXPPOS_SHORT] & F.EXPMASK)
|
||
+ (ye[F.EXPPOS_SHORT] & F.EXPMASK));
|
||
if (m & 0x8000_0000_0000_0000L)
|
||
{
|
||
++e;
|
||
m &= 0x7FFF_FFFF_FFFF_FFFFL;
|
||
}
|
||
// Now do a multi-byte right shift
|
||
const uint c = e & 1; // carry
|
||
e >>= 1;
|
||
m >>>= 1;
|
||
if (c)
|
||
m |= 0x4000_0000_0000_0000L; // shift carry into significand
|
||
if (e)
|
||
*ul = m | 0x8000_0000_0000_0000L; // set implicit bit...
|
||
else
|
||
*ul = m; // ... unless exponent is 0 (subnormal or zero).
|
||
|
||
ue[4]= e | (xe[F.EXPPOS_SHORT]& 0x8000); // restore sign bit
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeQuadruple)
|
||
{
|
||
// This would be trivial if 'ucent' were implemented...
|
||
ulong *ul = cast(ulong *)&u;
|
||
ulong *xl = cast(ulong *)&x;
|
||
ulong *yl = cast(ulong *)&y;
|
||
|
||
// Multi-byte add, then multi-byte right shift.
|
||
import core.checkedint : addu;
|
||
bool carry;
|
||
ulong ml = addu(xl[MANTISSA_LSB], yl[MANTISSA_LSB], carry);
|
||
|
||
ulong mh = carry + (xl[MANTISSA_MSB] & 0x7FFF_FFFF_FFFF_FFFFL) +
|
||
(yl[MANTISSA_MSB] & 0x7FFF_FFFF_FFFF_FFFFL);
|
||
|
||
ul[MANTISSA_MSB] = (mh >>> 1) | (xl[MANTISSA_MSB] & 0x8000_0000_0000_0000);
|
||
ul[MANTISSA_LSB] = (ml >>> 1) | (mh & 1) << 63;
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeDouble)
|
||
{
|
||
ulong *ul = cast(ulong *)&u;
|
||
ulong *xl = cast(ulong *)&x;
|
||
ulong *yl = cast(ulong *)&y;
|
||
ulong m = (((*xl) & 0x7FFF_FFFF_FFFF_FFFFL)
|
||
+ ((*yl) & 0x7FFF_FFFF_FFFF_FFFFL)) >>> 1;
|
||
m |= ((*xl) & 0x8000_0000_0000_0000L);
|
||
*ul = m;
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeSingle)
|
||
{
|
||
uint *ul = cast(uint *)&u;
|
||
uint *xl = cast(uint *)&x;
|
||
uint *yl = cast(uint *)&y;
|
||
uint m = (((*xl) & 0x7FFF_FFFF) + ((*yl) & 0x7FFF_FFFF)) >>> 1;
|
||
m |= ((*xl) & 0x8000_0000);
|
||
*ul = m;
|
||
}
|
||
else
|
||
{
|
||
assert(0, "Not implemented");
|
||
}
|
||
return u;
|
||
}
|
||
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
assert(ieeeMean(-0.0,-1e-20)<0);
|
||
assert(ieeeMean(0.0,1e-20)>0);
|
||
|
||
assert(ieeeMean(1.0L,4.0L)==2L);
|
||
assert(ieeeMean(2.0*1.013,8.0*1.013)==4*1.013);
|
||
assert(ieeeMean(-1.0L,-4.0L)==-2L);
|
||
assert(ieeeMean(-1.0,-4.0)==-2);
|
||
assert(ieeeMean(-1.0f,-4.0f)==-2f);
|
||
assert(ieeeMean(-1.0,-2.0)==-1.5);
|
||
assert(ieeeMean(-1*(1+8*real.epsilon),-2*(1+8*real.epsilon))
|
||
==-1.5*(1+5*real.epsilon));
|
||
assert(ieeeMean(0x1p60,0x1p-10)==0x1p25);
|
||
|
||
static if (floatTraits!(real).realFormat == RealFormat.ieeeExtended)
|
||
{
|
||
assert(ieeeMean(1.0L,real.infinity)==0x1p8192L);
|
||
assert(ieeeMean(0.0L,real.infinity)==1.5);
|
||
}
|
||
assert(ieeeMean(0.5*real.min_normal*(1-4*real.epsilon),0.5*real.min_normal)
|
||
== 0.5*real.min_normal*(1-2*real.epsilon));
|
||
}
|
||
|
||
public:
|
||
|
||
|
||
/***********************************
|
||
* Evaluate polynomial A(x) = $(SUB a, 0) + $(SUB a, 1)x + $(SUB a, 2)$(POWER x,2)
|
||
* + $(SUB a,3)$(POWER x,3); ...
|
||
*
|
||
* Uses Horner's rule A(x) = $(SUB a, 0) + x($(SUB a, 1) + x($(SUB a, 2)
|
||
* + x($(SUB a, 3) + ...)))
|
||
* Params:
|
||
* x = the value to evaluate.
|
||
* A = array of coefficients $(SUB a, 0), $(SUB a, 1), etc.
|
||
*/
|
||
Unqual!(CommonType!(T1, T2)) poly(T1, T2)(T1 x, in T2[] A) @trusted pure nothrow @nogc
|
||
if (isFloatingPoint!T1 && isFloatingPoint!T2)
|
||
in
|
||
{
|
||
assert(A.length > 0);
|
||
}
|
||
body
|
||
{
|
||
static if (is(Unqual!T2 == real))
|
||
{
|
||
return polyImpl(x, A);
|
||
}
|
||
else
|
||
{
|
||
return polyImplBase(x, A);
|
||
}
|
||
}
|
||
|
||
///
|
||
@safe nothrow @nogc unittest
|
||
{
|
||
real x = 3.1;
|
||
static real[] pp = [56.1, 32.7, 6];
|
||
|
||
assert(poly(x, pp) == (56.1L + (32.7L + 6.0L * x) * x));
|
||
}
|
||
|
||
@safe nothrow @nogc unittest
|
||
{
|
||
double x = 3.1;
|
||
static double[] pp = [56.1, 32.7, 6];
|
||
double y = x;
|
||
y *= 6.0;
|
||
y += 32.7;
|
||
y *= x;
|
||
y += 56.1;
|
||
assert(poly(x, pp) == y);
|
||
}
|
||
|
||
@safe unittest
|
||
{
|
||
static assert(poly(3.0, [1.0, 2.0, 3.0]) == 34);
|
||
}
|
||
|
||
private Unqual!(CommonType!(T1, T2)) polyImplBase(T1, T2)(T1 x, in T2[] A) @trusted pure nothrow @nogc
|
||
if (isFloatingPoint!T1 && isFloatingPoint!T2)
|
||
{
|
||
ptrdiff_t i = A.length - 1;
|
||
typeof(return) r = A[i];
|
||
while (--i >= 0)
|
||
{
|
||
r *= x;
|
||
r += A[i];
|
||
}
|
||
return r;
|
||
}
|
||
|
||
private real polyImpl(real x, in real[] A) @trusted pure nothrow @nogc
|
||
{
|
||
version (D_InlineAsm_X86)
|
||
{
|
||
if (__ctfe)
|
||
{
|
||
return polyImplBase(x, A);
|
||
}
|
||
version (Windows)
|
||
{
|
||
// BUG: This code assumes a frame pointer in EBP.
|
||
asm pure nothrow @nogc // assembler by W. Bright
|
||
{
|
||
// EDX = (A.length - 1) * real.sizeof
|
||
mov ECX,A[EBP] ; // ECX = A.length
|
||
dec ECX ;
|
||
lea EDX,[ECX][ECX*8] ;
|
||
add EDX,ECX ;
|
||
add EDX,A+4[EBP] ;
|
||
fld real ptr [EDX] ; // ST0 = coeff[ECX]
|
||
jecxz return_ST ;
|
||
fld x[EBP] ; // ST0 = x
|
||
fxch ST(1) ; // ST1 = x, ST0 = r
|
||
align 4 ;
|
||
L2: fmul ST,ST(1) ; // r *= x
|
||
fld real ptr -10[EDX] ;
|
||
sub EDX,10 ; // deg--
|
||
faddp ST(1),ST ;
|
||
dec ECX ;
|
||
jne L2 ;
|
||
fxch ST(1) ; // ST1 = r, ST0 = x
|
||
fstp ST(0) ; // dump x
|
||
align 4 ;
|
||
return_ST: ;
|
||
;
|
||
}
|
||
}
|
||
else version (linux)
|
||
{
|
||
asm pure nothrow @nogc // assembler by W. Bright
|
||
{
|
||
// EDX = (A.length - 1) * real.sizeof
|
||
mov ECX,A[EBP] ; // ECX = A.length
|
||
dec ECX ;
|
||
lea EDX,[ECX*8] ;
|
||
lea EDX,[EDX][ECX*4] ;
|
||
add EDX,A+4[EBP] ;
|
||
fld real ptr [EDX] ; // ST0 = coeff[ECX]
|
||
jecxz return_ST ;
|
||
fld x[EBP] ; // ST0 = x
|
||
fxch ST(1) ; // ST1 = x, ST0 = r
|
||
align 4 ;
|
||
L2: fmul ST,ST(1) ; // r *= x
|
||
fld real ptr -12[EDX] ;
|
||
sub EDX,12 ; // deg--
|
||
faddp ST(1),ST ;
|
||
dec ECX ;
|
||
jne L2 ;
|
||
fxch ST(1) ; // ST1 = r, ST0 = x
|
||
fstp ST(0) ; // dump x
|
||
align 4 ;
|
||
return_ST: ;
|
||
;
|
||
}
|
||
}
|
||
else version (OSX)
|
||
{
|
||
asm pure nothrow @nogc // assembler by W. Bright
|
||
{
|
||
// EDX = (A.length - 1) * real.sizeof
|
||
mov ECX,A[EBP] ; // ECX = A.length
|
||
dec ECX ;
|
||
lea EDX,[ECX*8] ;
|
||
add EDX,EDX ;
|
||
add EDX,A+4[EBP] ;
|
||
fld real ptr [EDX] ; // ST0 = coeff[ECX]
|
||
jecxz return_ST ;
|
||
fld x[EBP] ; // ST0 = x
|
||
fxch ST(1) ; // ST1 = x, ST0 = r
|
||
align 4 ;
|
||
L2: fmul ST,ST(1) ; // r *= x
|
||
fld real ptr -16[EDX] ;
|
||
sub EDX,16 ; // deg--
|
||
faddp ST(1),ST ;
|
||
dec ECX ;
|
||
jne L2 ;
|
||
fxch ST(1) ; // ST1 = r, ST0 = x
|
||
fstp ST(0) ; // dump x
|
||
align 4 ;
|
||
return_ST: ;
|
||
;
|
||
}
|
||
}
|
||
else version (FreeBSD)
|
||
{
|
||
asm pure nothrow @nogc // assembler by W. Bright
|
||
{
|
||
// EDX = (A.length - 1) * real.sizeof
|
||
mov ECX,A[EBP] ; // ECX = A.length
|
||
dec ECX ;
|
||
lea EDX,[ECX*8] ;
|
||
lea EDX,[EDX][ECX*4] ;
|
||
add EDX,A+4[EBP] ;
|
||
fld real ptr [EDX] ; // ST0 = coeff[ECX]
|
||
jecxz return_ST ;
|
||
fld x[EBP] ; // ST0 = x
|
||
fxch ST(1) ; // ST1 = x, ST0 = r
|
||
align 4 ;
|
||
L2: fmul ST,ST(1) ; // r *= x
|
||
fld real ptr -12[EDX] ;
|
||
sub EDX,12 ; // deg--
|
||
faddp ST(1),ST ;
|
||
dec ECX ;
|
||
jne L2 ;
|
||
fxch ST(1) ; // ST1 = r, ST0 = x
|
||
fstp ST(0) ; // dump x
|
||
align 4 ;
|
||
return_ST: ;
|
||
;
|
||
}
|
||
}
|
||
else version (Solaris)
|
||
{
|
||
asm pure nothrow @nogc // assembler by W. Bright
|
||
{
|
||
// EDX = (A.length - 1) * real.sizeof
|
||
mov ECX,A[EBP] ; // ECX = A.length
|
||
dec ECX ;
|
||
lea EDX,[ECX*8] ;
|
||
lea EDX,[EDX][ECX*4] ;
|
||
add EDX,A+4[EBP] ;
|
||
fld real ptr [EDX] ; // ST0 = coeff[ECX]
|
||
jecxz return_ST ;
|
||
fld x[EBP] ; // ST0 = x
|
||
fxch ST(1) ; // ST1 = x, ST0 = r
|
||
align 4 ;
|
||
L2: fmul ST,ST(1) ; // r *= x
|
||
fld real ptr -12[EDX] ;
|
||
sub EDX,12 ; // deg--
|
||
faddp ST(1),ST ;
|
||
dec ECX ;
|
||
jne L2 ;
|
||
fxch ST(1) ; // ST1 = r, ST0 = x
|
||
fstp ST(0) ; // dump x
|
||
align 4 ;
|
||
return_ST: ;
|
||
;
|
||
}
|
||
}
|
||
else
|
||
{
|
||
static assert(0);
|
||
}
|
||
}
|
||
else
|
||
{
|
||
return polyImplBase(x, A);
|
||
}
|
||
}
|
||
|
||
|
||
/**
|
||
Computes whether two values are approximately equal, admitting a maximum
|
||
relative difference, and a maximum absolute difference.
|
||
|
||
Params:
|
||
lhs = First item to compare.
|
||
rhs = Second item to compare.
|
||
maxRelDiff = Maximum allowable difference relative to `rhs`.
|
||
maxAbsDiff = Maximum absolute difference.
|
||
|
||
Returns:
|
||
`true` if the two items are approximately equal under either criterium.
|
||
If one item is a range, and the other is a single value, then the result
|
||
is the logical and-ing of calling `approxEqual` on each element of the
|
||
ranged item against the single item. If both items are ranges, then
|
||
`approxEqual` returns `true` if and only if the ranges have the same
|
||
number of elements and if `approxEqual` evaluates to `true` for each
|
||
pair of elements.
|
||
*/
|
||
bool approxEqual(T, U, V)(T lhs, U rhs, V maxRelDiff, V maxAbsDiff = 1e-5)
|
||
{
|
||
import std.range.primitives : empty, front, isInputRange, popFront;
|
||
static if (isInputRange!T)
|
||
{
|
||
static if (isInputRange!U)
|
||
{
|
||
// Two ranges
|
||
for (;; lhs.popFront(), rhs.popFront())
|
||
{
|
||
if (lhs.empty) return rhs.empty;
|
||
if (rhs.empty) return lhs.empty;
|
||
if (!approxEqual(lhs.front, rhs.front, maxRelDiff, maxAbsDiff))
|
||
return false;
|
||
}
|
||
}
|
||
else static if (isIntegral!U)
|
||
{
|
||
// convert rhs to real
|
||
return approxEqual(lhs, real(rhs), maxRelDiff, maxAbsDiff);
|
||
}
|
||
else
|
||
{
|
||
// lhs is range, rhs is number
|
||
for (; !lhs.empty; lhs.popFront())
|
||
{
|
||
if (!approxEqual(lhs.front, rhs, maxRelDiff, maxAbsDiff))
|
||
return false;
|
||
}
|
||
return true;
|
||
}
|
||
}
|
||
else
|
||
{
|
||
static if (isInputRange!U)
|
||
{
|
||
// lhs is number, rhs is range
|
||
for (; !rhs.empty; rhs.popFront())
|
||
{
|
||
if (!approxEqual(lhs, rhs.front, maxRelDiff, maxAbsDiff))
|
||
return false;
|
||
}
|
||
return true;
|
||
}
|
||
else static if (isIntegral!T || isIntegral!U)
|
||
{
|
||
// convert both lhs and rhs to real
|
||
return approxEqual(real(lhs), real(rhs), maxRelDiff, maxAbsDiff);
|
||
}
|
||
else
|
||
{
|
||
// two numbers
|
||
//static assert(is(T : real) && is(U : real));
|
||
if (rhs == 0)
|
||
{
|
||
return fabs(lhs) <= maxAbsDiff;
|
||
}
|
||
static if (is(typeof(lhs.infinity)) && is(typeof(rhs.infinity)))
|
||
{
|
||
if (lhs == lhs.infinity && rhs == rhs.infinity ||
|
||
lhs == -lhs.infinity && rhs == -rhs.infinity) return true;
|
||
}
|
||
return fabs((lhs - rhs) / rhs) <= maxRelDiff
|
||
|| maxAbsDiff != 0 && fabs(lhs - rhs) <= maxAbsDiff;
|
||
}
|
||
}
|
||
}
|
||
|
||
/**
|
||
Returns $(D approxEqual(lhs, rhs, 1e-2, 1e-5)).
|
||
*/
|
||
bool approxEqual(T, U)(T lhs, U rhs)
|
||
{
|
||
return approxEqual(lhs, rhs, 1e-2, 1e-5);
|
||
}
|
||
|
||
///
|
||
@safe pure nothrow unittest
|
||
{
|
||
assert(approxEqual(1.0, 1.0099));
|
||
assert(!approxEqual(1.0, 1.011));
|
||
float[] arr1 = [ 1.0, 2.0, 3.0 ];
|
||
double[] arr2 = [ 1.001, 1.999, 3 ];
|
||
assert(approxEqual(arr1, arr2));
|
||
|
||
real num = real.infinity;
|
||
assert(num == real.infinity); // Passes.
|
||
assert(approxEqual(num, real.infinity)); // Fails.
|
||
num = -real.infinity;
|
||
assert(num == -real.infinity); // Passes.
|
||
assert(approxEqual(num, -real.infinity)); // Fails.
|
||
|
||
assert(!approxEqual(3, 0));
|
||
assert(approxEqual(3, 3));
|
||
assert(approxEqual(3.0, 3));
|
||
assert(approxEqual([3, 3, 3], 3.0));
|
||
assert(approxEqual([3.0, 3.0, 3.0], 3));
|
||
int a = 10;
|
||
assert(approxEqual(10, a));
|
||
}
|
||
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
real num = real.infinity;
|
||
assert(num == real.infinity); // Passes.
|
||
assert(approxEqual(num, real.infinity)); // Fails.
|
||
}
|
||
|
||
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
float f = sqrt(2.0f);
|
||
assert(fabs(f * f - 2.0f) < .00001);
|
||
|
||
double d = sqrt(2.0);
|
||
assert(fabs(d * d - 2.0) < .00001);
|
||
|
||
real r = sqrt(2.0L);
|
||
assert(fabs(r * r - 2.0) < .00001);
|
||
}
|
||
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
float f = fabs(-2.0f);
|
||
assert(f == 2);
|
||
|
||
double d = fabs(-2.0);
|
||
assert(d == 2);
|
||
|
||
real r = fabs(-2.0L);
|
||
assert(r == 2);
|
||
}
|
||
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
float f = sin(-2.0f);
|
||
assert(fabs(f - -0.909297f) < .00001);
|
||
|
||
double d = sin(-2.0);
|
||
assert(fabs(d - -0.909297f) < .00001);
|
||
|
||
real r = sin(-2.0L);
|
||
assert(fabs(r - -0.909297f) < .00001);
|
||
}
|
||
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
float f = cos(-2.0f);
|
||
assert(fabs(f - -0.416147f) < .00001);
|
||
|
||
double d = cos(-2.0);
|
||
assert(fabs(d - -0.416147f) < .00001);
|
||
|
||
real r = cos(-2.0L);
|
||
assert(fabs(r - -0.416147f) < .00001);
|
||
}
|
||
|
||
@safe pure nothrow @nogc unittest
|
||
{
|
||
float f = tan(-2.0f);
|
||
assert(fabs(f - 2.18504f) < .00001);
|
||
|
||
double d = tan(-2.0);
|
||
assert(fabs(d - 2.18504f) < .00001);
|
||
|
||
real r = tan(-2.0L);
|
||
assert(fabs(r - 2.18504f) < .00001);
|
||
|
||
// Verify correct behavior for large inputs
|
||
assert(!isNaN(tan(0x1p63)));
|
||
assert(!isNaN(tan(0x1p300L)));
|
||
assert(!isNaN(tan(-0x1p63)));
|
||
assert(!isNaN(tan(-0x1p300L)));
|
||
}
|
||
|
||
@safe pure nothrow unittest
|
||
{
|
||
// issue 6381: floor/ceil should be usable in pure function.
|
||
auto x = floor(1.2);
|
||
auto y = ceil(1.2);
|
||
}
|
||
|
||
@safe pure nothrow unittest
|
||
{
|
||
// relative comparison depends on rhs, make sure proper side is used when
|
||
// comparing range to single value. Based on bugzilla issue 15763
|
||
auto a = [2e-3 - 1e-5];
|
||
auto b = 2e-3 + 1e-5;
|
||
assert(a[0].approxEqual(b));
|
||
assert(!b.approxEqual(a[0]));
|
||
assert(a.approxEqual(b));
|
||
assert(!b.approxEqual(a));
|
||
}
|
||
|
||
/***********************************
|
||
* Defines a total order on all floating-point numbers.
|
||
*
|
||
* The order is defined as follows:
|
||
* $(UL
|
||
* $(LI All numbers in [-$(INFIN), +$(INFIN)] are ordered
|
||
* the same way as by built-in comparison, with the exception of
|
||
* -0.0, which is less than +0.0;)
|
||
* $(LI If the sign bit is set (that is, it's 'negative'), $(NAN) is less
|
||
* than any number; if the sign bit is not set (it is 'positive'),
|
||
* $(NAN) is greater than any number;)
|
||
* $(LI $(NAN)s of the same sign are ordered by the payload ('negative'
|
||
* ones - in reverse order).)
|
||
* )
|
||
*
|
||
* Returns:
|
||
* negative value if $(D x) precedes $(D y) in the order specified above;
|
||
* 0 if $(D x) and $(D y) are identical, and positive value otherwise.
|
||
*
|
||
* See_Also:
|
||
* $(MYREF isIdentical)
|
||
* Standards: Conforms to IEEE 754-2008
|
||
*/
|
||
int cmp(T)(const(T) x, const(T) y) @nogc @trusted pure nothrow
|
||
if (isFloatingPoint!T)
|
||
{
|
||
alias F = floatTraits!T;
|
||
|
||
static if (F.realFormat == RealFormat.ieeeSingle
|
||
|| F.realFormat == RealFormat.ieeeDouble)
|
||
{
|
||
static if (T.sizeof == 4)
|
||
alias UInt = uint;
|
||
else
|
||
alias UInt = ulong;
|
||
|
||
union Repainter
|
||
{
|
||
T number;
|
||
UInt bits;
|
||
}
|
||
|
||
enum msb = ~(UInt.max >>> 1);
|
||
|
||
import std.typecons : Tuple;
|
||
Tuple!(Repainter, Repainter) vars = void;
|
||
vars[0].number = x;
|
||
vars[1].number = y;
|
||
|
||
foreach (ref var; vars)
|
||
if (var.bits & msb)
|
||
var.bits = ~var.bits;
|
||
else
|
||
var.bits |= msb;
|
||
|
||
if (vars[0].bits < vars[1].bits)
|
||
return -1;
|
||
else if (vars[0].bits > vars[1].bits)
|
||
return 1;
|
||
else
|
||
return 0;
|
||
}
|
||
else static if (F.realFormat == RealFormat.ieeeExtended53
|
||
|| F.realFormat == RealFormat.ieeeExtended
|
||
|| F.realFormat == RealFormat.ieeeQuadruple)
|
||
{
|
||
static if (F.realFormat == RealFormat.ieeeQuadruple)
|
||
alias RemT = ulong;
|
||
else
|
||
alias RemT = ushort;
|
||
|
||
struct Bits
|
||
{
|
||
ulong bulk;
|
||
RemT rem;
|
||
}
|
||
|
||
union Repainter
|
||
{
|
||
T number;
|
||
Bits bits;
|
||
ubyte[T.sizeof] bytes;
|
||
}
|
||
|
||
import std.typecons : Tuple;
|
||
Tuple!(Repainter, Repainter) vars = void;
|
||
vars[0].number = x;
|
||
vars[1].number = y;
|
||
|
||
foreach (ref var; vars)
|
||
if (var.bytes[F.SIGNPOS_BYTE] & 0x80)
|
||
{
|
||
var.bits.bulk = ~var.bits.bulk;
|
||
var.bits.rem = ~var.bits.rem;
|
||
}
|
||
else
|
||
{
|
||
var.bytes[F.SIGNPOS_BYTE] |= 0x80;
|
||
}
|
||
|
||
version(LittleEndian)
|
||
{
|
||
if (vars[0].bits.rem < vars[1].bits.rem)
|
||
return -1;
|
||
else if (vars[0].bits.rem > vars[1].bits.rem)
|
||
return 1;
|
||
else if (vars[0].bits.bulk < vars[1].bits.bulk)
|
||
return -1;
|
||
else if (vars[0].bits.bulk > vars[1].bits.bulk)
|
||
return 1;
|
||
else
|
||
return 0;
|
||
}
|
||
else
|
||
{
|
||
if (vars[0].bits.bulk < vars[1].bits.bulk)
|
||
return -1;
|
||
else if (vars[0].bits.bulk > vars[1].bits.bulk)
|
||
return 1;
|
||
else if (vars[0].bits.rem < vars[1].bits.rem)
|
||
return -1;
|
||
else if (vars[0].bits.rem > vars[1].bits.rem)
|
||
return 1;
|
||
else
|
||
return 0;
|
||
}
|
||
}
|
||
else
|
||
{
|
||
// IBM Extended doubledouble does not follow the general
|
||
// sign-exponent-significand layout, so has to be handled generically
|
||
|
||
const int xSign = signbit(x),
|
||
ySign = signbit(y);
|
||
|
||
if (xSign == 1 && ySign == 1)
|
||
return cmp(-y, -x);
|
||
else if (xSign == 1)
|
||
return -1;
|
||
else if (ySign == 1)
|
||
return 1;
|
||
else if (x < y)
|
||
return -1;
|
||
else if (x == y)
|
||
return 0;
|
||
else if (x > y)
|
||
return 1;
|
||
else if (isNaN(x) && !isNaN(y))
|
||
return 1;
|
||
else if (isNaN(y) && !isNaN(x))
|
||
return -1;
|
||
else if (getNaNPayload(x) < getNaNPayload(y))
|
||
return -1;
|
||
else if (getNaNPayload(x) > getNaNPayload(y))
|
||
return 1;
|
||
else
|
||
return 0;
|
||
}
|
||
}
|
||
|
||
/// Most numbers are ordered naturally.
|
||
@safe unittest
|
||
{
|
||
assert(cmp(-double.infinity, -double.max) < 0);
|
||
assert(cmp(-double.max, -100.0) < 0);
|
||
assert(cmp(-100.0, -0.5) < 0);
|
||
assert(cmp(-0.5, 0.0) < 0);
|
||
assert(cmp(0.0, 0.5) < 0);
|
||
assert(cmp(0.5, 100.0) < 0);
|
||
assert(cmp(100.0, double.max) < 0);
|
||
assert(cmp(double.max, double.infinity) < 0);
|
||
|
||
assert(cmp(1.0, 1.0) == 0);
|
||
}
|
||
|
||
/// Positive and negative zeroes are distinct.
|
||
@safe unittest
|
||
{
|
||
assert(cmp(-0.0, +0.0) < 0);
|
||
assert(cmp(+0.0, -0.0) > 0);
|
||
}
|
||
|
||
/// Depending on the sign, $(NAN)s go to either end of the spectrum.
|
||
@safe unittest
|
||
{
|
||
assert(cmp(-double.nan, -double.infinity) < 0);
|
||
assert(cmp(double.infinity, double.nan) < 0);
|
||
assert(cmp(-double.nan, double.nan) < 0);
|
||
}
|
||
|
||
/// $(NAN)s of the same sign are ordered by the payload.
|
||
@safe unittest
|
||
{
|
||
assert(cmp(NaN(10), NaN(20)) < 0);
|
||
assert(cmp(-NaN(20), -NaN(10)) < 0);
|
||
}
|
||
|
||
@safe unittest
|
||
{
|
||
import std.meta : AliasSeq;
|
||
foreach (T; AliasSeq!(float, double, real))
|
||
{
|
||
T[] values = [-cast(T) NaN(20), -cast(T) NaN(10), -T.nan, -T.infinity,
|
||
-T.max, -T.max / 2, T(-16.0), T(-1.0).nextDown,
|
||
T(-1.0), T(-1.0).nextUp,
|
||
T(-0.5), -T.min_normal, (-T.min_normal).nextUp,
|
||
-2 * T.min_normal * T.epsilon,
|
||
-T.min_normal * T.epsilon,
|
||
T(-0.0), T(0.0),
|
||
T.min_normal * T.epsilon,
|
||
2 * T.min_normal * T.epsilon,
|
||
T.min_normal.nextDown, T.min_normal, T(0.5),
|
||
T(1.0).nextDown, T(1.0),
|
||
T(1.0).nextUp, T(16.0), T.max / 2, T.max,
|
||
T.infinity, T.nan, cast(T) NaN(10), cast(T) NaN(20)];
|
||
|
||
foreach (i, x; values)
|
||
{
|
||
foreach (y; values[i + 1 .. $])
|
||
{
|
||
assert(cmp(x, y) < 0);
|
||
assert(cmp(y, x) > 0);
|
||
}
|
||
assert(cmp(x, x) == 0);
|
||
}
|
||
}
|
||
}
|
||
|
||
private enum PowType
|
||
{
|
||
floor,
|
||
ceil
|
||
}
|
||
|
||
pragma(inline, true)
|
||
private T powIntegralImpl(PowType type, T)(T val)
|
||
{
|
||
import core.bitop : bsr;
|
||
|
||
if (val == 0 || (type == PowType.ceil && (val > T.max / 2 || val == T.min)))
|
||
return 0;
|
||
else
|
||
{
|
||
static if (isSigned!T)
|
||
return cast(Unqual!T) (val < 0 ? -(T(1) << bsr(-val) + type) : T(1) << bsr(val) + type);
|
||
else
|
||
return cast(Unqual!T) (T(1) << bsr(val) + type);
|
||
}
|
||
}
|
||
|
||
private T powFloatingPointImpl(PowType type, T)(T x)
|
||
{
|
||
if (!x.isFinite)
|
||
return x;
|
||
|
||
if (!x)
|
||
return x;
|
||
|
||
int exp;
|
||
auto y = frexp(x, exp);
|
||
|
||
static if (type == PowType.ceil)
|
||
y = ldexp(cast(T) 0.5, exp + 1);
|
||
else
|
||
y = ldexp(cast(T) 0.5, exp);
|
||
|
||
if (!y.isFinite)
|
||
return cast(T) 0.0;
|
||
|
||
y = copysign(y, x);
|
||
|
||
return y;
|
||
}
|
||
|
||
/**
|
||
* Gives the next power of two after $(D val). `T` can be any built-in
|
||
* numerical type.
|
||
*
|
||
* If the operation would lead to an over/underflow, this function will
|
||
* return `0`.
|
||
*
|
||
* Params:
|
||
* val = any number
|
||
*
|
||
* Returns:
|
||
* the next power of two after $(D val)
|
||
*/
|
||
T nextPow2(T)(const T val)
|
||
if (isIntegral!T)
|
||
{
|
||
return powIntegralImpl!(PowType.ceil)(val);
|
||
}
|
||
|
||
/// ditto
|
||
T nextPow2(T)(const T val)
|
||
if (isFloatingPoint!T)
|
||
{
|
||
return powFloatingPointImpl!(PowType.ceil)(val);
|
||
}
|
||
|
||
///
|
||
@safe @nogc pure nothrow unittest
|
||
{
|
||
assert(nextPow2(2) == 4);
|
||
assert(nextPow2(10) == 16);
|
||
assert(nextPow2(4000) == 4096);
|
||
|
||
assert(nextPow2(-2) == -4);
|
||
assert(nextPow2(-10) == -16);
|
||
|
||
assert(nextPow2(uint.max) == 0);
|
||
assert(nextPow2(uint.min) == 0);
|
||
assert(nextPow2(size_t.max) == 0);
|
||
assert(nextPow2(size_t.min) == 0);
|
||
|
||
assert(nextPow2(int.max) == 0);
|
||
assert(nextPow2(int.min) == 0);
|
||
assert(nextPow2(long.max) == 0);
|
||
assert(nextPow2(long.min) == 0);
|
||
}
|
||
|
||
///
|
||
@safe @nogc pure nothrow unittest
|
||
{
|
||
assert(nextPow2(2.1) == 4.0);
|
||
assert(nextPow2(-2.0) == -4.0);
|
||
assert(nextPow2(0.25) == 0.5);
|
||
assert(nextPow2(-4.0) == -8.0);
|
||
|
||
assert(nextPow2(double.max) == 0.0);
|
||
assert(nextPow2(double.infinity) == double.infinity);
|
||
}
|
||
|
||
@safe @nogc pure nothrow unittest
|
||
{
|
||
assert(nextPow2(ubyte(2)) == 4);
|
||
assert(nextPow2(ubyte(10)) == 16);
|
||
|
||
assert(nextPow2(byte(2)) == 4);
|
||
assert(nextPow2(byte(10)) == 16);
|
||
|
||
assert(nextPow2(short(2)) == 4);
|
||
assert(nextPow2(short(10)) == 16);
|
||
assert(nextPow2(short(4000)) == 4096);
|
||
|
||
assert(nextPow2(ushort(2)) == 4);
|
||
assert(nextPow2(ushort(10)) == 16);
|
||
assert(nextPow2(ushort(4000)) == 4096);
|
||
}
|
||
|
||
@safe @nogc pure nothrow unittest
|
||
{
|
||
foreach (ulong i; 1 .. 62)
|
||
{
|
||
assert(nextPow2(1UL << i) == 2UL << i);
|
||
assert(nextPow2((1UL << i) - 1) == 1UL << i);
|
||
assert(nextPow2((1UL << i) + 1) == 2UL << i);
|
||
assert(nextPow2((1UL << i) + (1UL<<(i-1))) == 2UL << i);
|
||
}
|
||
}
|
||
|
||
@safe @nogc pure nothrow unittest
|
||
{
|
||
import std.meta : AliasSeq;
|
||
|
||
foreach (T; AliasSeq!(float, double, real))
|
||
{
|
||
enum T subNormal = T.min_normal / 2;
|
||
|
||
static if (subNormal) assert(nextPow2(subNormal) == T.min_normal);
|
||
|
||
assert(nextPow2(T(0.0)) == 0.0);
|
||
|
||
assert(nextPow2(T(2.0)) == 4.0);
|
||
assert(nextPow2(T(2.1)) == 4.0);
|
||
assert(nextPow2(T(3.1)) == 4.0);
|
||
assert(nextPow2(T(4.0)) == 8.0);
|
||
assert(nextPow2(T(0.25)) == 0.5);
|
||
|
||
assert(nextPow2(T(-2.0)) == -4.0);
|
||
assert(nextPow2(T(-2.1)) == -4.0);
|
||
assert(nextPow2(T(-3.1)) == -4.0);
|
||
assert(nextPow2(T(-4.0)) == -8.0);
|
||
assert(nextPow2(T(-0.25)) == -0.5);
|
||
|
||
assert(nextPow2(T.max) == 0);
|
||
assert(nextPow2(-T.max) == 0);
|
||
|
||
assert(nextPow2(T.infinity) == T.infinity);
|
||
assert(nextPow2(T.init).isNaN);
|
||
}
|
||
}
|
||
|
||
@safe @nogc pure nothrow unittest // Issue 15973
|
||
{
|
||
assert(nextPow2(uint.max / 2) == uint.max / 2 + 1);
|
||
assert(nextPow2(uint.max / 2 + 2) == 0);
|
||
assert(nextPow2(int.max / 2) == int.max / 2 + 1);
|
||
assert(nextPow2(int.max / 2 + 2) == 0);
|
||
assert(nextPow2(int.min + 1) == int.min);
|
||
}
|
||
|
||
/**
|
||
* Gives the last power of two before $(D val). $(T) can be any built-in
|
||
* numerical type.
|
||
*
|
||
* Params:
|
||
* val = any number
|
||
*
|
||
* Returns:
|
||
* the last power of two before $(D val)
|
||
*/
|
||
T truncPow2(T)(const T val)
|
||
if (isIntegral!T)
|
||
{
|
||
return powIntegralImpl!(PowType.floor)(val);
|
||
}
|
||
|
||
/// ditto
|
||
T truncPow2(T)(const T val)
|
||
if (isFloatingPoint!T)
|
||
{
|
||
return powFloatingPointImpl!(PowType.floor)(val);
|
||
}
|
||
|
||
///
|
||
@safe @nogc pure nothrow unittest
|
||
{
|
||
assert(truncPow2(3) == 2);
|
||
assert(truncPow2(4) == 4);
|
||
assert(truncPow2(10) == 8);
|
||
assert(truncPow2(4000) == 2048);
|
||
|
||
assert(truncPow2(-5) == -4);
|
||
assert(truncPow2(-20) == -16);
|
||
|
||
assert(truncPow2(uint.max) == int.max + 1);
|
||
assert(truncPow2(uint.min) == 0);
|
||
assert(truncPow2(ulong.max) == long.max + 1);
|
||
assert(truncPow2(ulong.min) == 0);
|
||
|
||
assert(truncPow2(int.max) == (int.max / 2) + 1);
|
||
assert(truncPow2(int.min) == int.min);
|
||
assert(truncPow2(long.max) == (long.max / 2) + 1);
|
||
assert(truncPow2(long.min) == long.min);
|
||
}
|
||
|
||
///
|
||
@safe @nogc pure nothrow unittest
|
||
{
|
||
assert(truncPow2(2.1) == 2.0);
|
||
assert(truncPow2(7.0) == 4.0);
|
||
assert(truncPow2(-1.9) == -1.0);
|
||
assert(truncPow2(0.24) == 0.125);
|
||
assert(truncPow2(-7.0) == -4.0);
|
||
|
||
assert(truncPow2(double.infinity) == double.infinity);
|
||
}
|
||
|
||
@safe @nogc pure nothrow unittest
|
||
{
|
||
assert(truncPow2(ubyte(3)) == 2);
|
||
assert(truncPow2(ubyte(4)) == 4);
|
||
assert(truncPow2(ubyte(10)) == 8);
|
||
|
||
assert(truncPow2(byte(3)) == 2);
|
||
assert(truncPow2(byte(4)) == 4);
|
||
assert(truncPow2(byte(10)) == 8);
|
||
|
||
assert(truncPow2(ushort(3)) == 2);
|
||
assert(truncPow2(ushort(4)) == 4);
|
||
assert(truncPow2(ushort(10)) == 8);
|
||
assert(truncPow2(ushort(4000)) == 2048);
|
||
|
||
assert(truncPow2(short(3)) == 2);
|
||
assert(truncPow2(short(4)) == 4);
|
||
assert(truncPow2(short(10)) == 8);
|
||
assert(truncPow2(short(4000)) == 2048);
|
||
}
|
||
|
||
@safe @nogc pure nothrow unittest
|
||
{
|
||
foreach (ulong i; 1 .. 62)
|
||
{
|
||
assert(truncPow2(2UL << i) == 2UL << i);
|
||
assert(truncPow2((2UL << i) + 1) == 2UL << i);
|
||
assert(truncPow2((2UL << i) - 1) == 1UL << i);
|
||
assert(truncPow2((2UL << i) - (2UL<<(i-1))) == 1UL << i);
|
||
}
|
||
}
|
||
|
||
@safe @nogc pure nothrow unittest
|
||
{
|
||
import std.meta : AliasSeq;
|
||
|
||
foreach (T; AliasSeq!(float, double, real))
|
||
{
|
||
assert(truncPow2(T(0.0)) == 0.0);
|
||
|
||
assert(truncPow2(T(4.0)) == 4.0);
|
||
assert(truncPow2(T(2.1)) == 2.0);
|
||
assert(truncPow2(T(3.5)) == 2.0);
|
||
assert(truncPow2(T(7.0)) == 4.0);
|
||
assert(truncPow2(T(0.24)) == 0.125);
|
||
|
||
assert(truncPow2(T(-2.0)) == -2.0);
|
||
assert(truncPow2(T(-2.1)) == -2.0);
|
||
assert(truncPow2(T(-3.1)) == -2.0);
|
||
assert(truncPow2(T(-7.0)) == -4.0);
|
||
assert(truncPow2(T(-0.24)) == -0.125);
|
||
|
||
assert(truncPow2(T.infinity) == T.infinity);
|
||
assert(truncPow2(T.init).isNaN);
|
||
}
|
||
}
|
||
|
||
/**
|
||
Check whether a number is an integer power of two.
|
||
|
||
Note that only positive numbers can be integer powers of two. This
|
||
function always return `false` if `x` is negative or zero.
|
||
|
||
Params:
|
||
x = the number to test
|
||
|
||
Returns:
|
||
`true` if `x` is an integer power of two.
|
||
*/
|
||
bool isPowerOf2(X)(const X x) pure @safe nothrow @nogc
|
||
if (isNumeric!X)
|
||
{
|
||
static if (isFloatingPoint!X)
|
||
{
|
||
int exp;
|
||
const X sig = frexp(x, exp);
|
||
|
||
return (exp != int.min) && (sig is cast(X) 0.5L);
|
||
}
|
||
else
|
||
{
|
||
static if (isSigned!X)
|
||
{
|
||
auto y = cast(typeof(x + 0))x;
|
||
return y > 0 && !(y & (y - 1));
|
||
}
|
||
else
|
||
{
|
||
auto y = cast(typeof(x + 0u))x;
|
||
return (y & -y) > (y - 1);
|
||
}
|
||
}
|
||
}
|
||
///
|
||
@safe unittest
|
||
{
|
||
assert( isPowerOf2(1.0L));
|
||
assert( isPowerOf2(2.0L));
|
||
assert( isPowerOf2(0.5L));
|
||
assert( isPowerOf2(pow(2.0L, 96)));
|
||
assert( isPowerOf2(pow(2.0L, -77)));
|
||
|
||
assert(!isPowerOf2(-2.0L));
|
||
assert(!isPowerOf2(-0.5L));
|
||
assert(!isPowerOf2(0.0L));
|
||
assert(!isPowerOf2(4.315));
|
||
assert(!isPowerOf2(1.0L / 3.0L));
|
||
|
||
assert(!isPowerOf2(real.nan));
|
||
assert(!isPowerOf2(real.infinity));
|
||
}
|
||
///
|
||
@safe unittest
|
||
{
|
||
assert( isPowerOf2(1));
|
||
assert( isPowerOf2(2));
|
||
assert( isPowerOf2(1uL << 63));
|
||
|
||
assert(!isPowerOf2(-4));
|
||
assert(!isPowerOf2(0));
|
||
assert(!isPowerOf2(1337u));
|
||
}
|
||
|
||
@safe unittest
|
||
{
|
||
import std.meta : AliasSeq;
|
||
|
||
immutable smallP2 = pow(2.0L, -62);
|
||
immutable bigP2 = pow(2.0L, 50);
|
||
immutable smallP7 = pow(7.0L, -35);
|
||
immutable bigP7 = pow(7.0L, 30);
|
||
|
||
foreach (X; AliasSeq!(float, double, real))
|
||
{
|
||
immutable min_sub = X.min_normal * X.epsilon;
|
||
|
||
foreach (x; AliasSeq!(smallP2, min_sub, X.min_normal, .25L, 0.5L, 1.0L,
|
||
2.0L, 8.0L, pow(2.0L, X.max_exp - 1), bigP2))
|
||
{
|
||
assert( isPowerOf2(cast(X) x));
|
||
assert(!isPowerOf2(cast(X)-x));
|
||
}
|
||
|
||
foreach (x; AliasSeq!(0.0L, 3 * min_sub, smallP7, 0.1L, 1337.0L, bigP7, X.max, real.nan, real.infinity))
|
||
{
|
||
assert(!isPowerOf2(cast(X) x));
|
||
assert(!isPowerOf2(cast(X)-x));
|
||
}
|
||
}
|
||
|
||
foreach (X; AliasSeq!(byte, ubyte, short, ushort, int, uint, long, ulong))
|
||
{
|
||
foreach (x; [1, 2, 4, 8, (X.max >>> 1) + 1])
|
||
{
|
||
assert( isPowerOf2(cast(X) x));
|
||
static if (isSigned!X)
|
||
assert(!isPowerOf2(cast(X)-x));
|
||
}
|
||
|
||
foreach (x; [0, 3, 5, 13, 77, X.min, X.max])
|
||
assert(!isPowerOf2(cast(X) x));
|
||
}
|
||
}
|