phobos/std/math/package.d

// Written in the D programming language.

/**
 * Contains the elementary mathematical functions (powers, roots,
 * and trigonometric functions), and low-level floating-point operations.
 * Mathematical special functions are available in `std.mathspecial`.
 *
$(SCRIPT inhibitQuickIndex = 1;)

$(DIVC quickindex,
$(BOOKTABLE ,
$(TR $(TH Category) $(TH Members) )
$(TR $(TDNW $(SUBMODULE Constants, constants)) $(TD
    $(SUBREF constants, E)
    $(SUBREF constants, PI)
    $(SUBREF constants, PI_2)
    $(SUBREF constants, PI_4)
    $(SUBREF constants, M_1_PI)
    $(SUBREF constants, M_2_PI)
    $(SUBREF constants, M_2_SQRTPI)
    $(SUBREF constants, LN10)
    $(SUBREF constants, LN2)
    $(SUBREF constants, LOG2)
    $(SUBREF constants, LOG2E)
    $(SUBREF constants, LOG2T)
    $(SUBREF constants, LOG10E)
    $(SUBREF constants, SQRT2)
    $(SUBREF constants, SQRT1_2)
))
$(TR $(TDNW $(SUBMODULE Algebraic, algebraic)) $(TD
    $(SUBREF algebraic, abs)
    $(SUBREF algebraic, fabs)
    $(SUBREF algebraic, sqrt)
    $(SUBREF algebraic, cbrt)
    $(SUBREF algebraic, hypot)
    $(SUBREF algebraic, poly)
    $(SUBREF algebraic, nextPow2)
    $(SUBREF algebraic, truncPow2)
))
$(TR $(TDNW $(SUBMODULE Trigonometry, trigonometry)) $(TD
    $(SUBREF trigonometry, sin)
    $(SUBREF trigonometry, cos)
    $(SUBREF trigonometry, tan)
    $(SUBREF trigonometry, asin)
    $(SUBREF trigonometry, acos)
    $(SUBREF trigonometry, atan)
    $(SUBREF trigonometry, atan2)
    $(SUBREF trigonometry, sinh)
    $(SUBREF trigonometry, cosh)
    $(SUBREF trigonometry, tanh)
    $(SUBREF trigonometry, asinh)
    $(SUBREF trigonometry, acosh)
    $(SUBREF trigonometry, atanh)
))
$(TR $(TDNW $(SUBMODULE Rounding, rounding)) $(TD
    $(SUBREF rounding, ceil)
    $(SUBREF rounding, floor)
    $(SUBREF rounding, round)
    $(SUBREF rounding, lround)
    $(SUBREF rounding, trunc)
    $(SUBREF rounding, rint)
    $(SUBREF rounding, lrint)
    $(SUBREF rounding, nearbyint)
    $(SUBREF rounding, rndtol)
    $(SUBREF rounding, quantize)
))
$(TR $(TDNW $(SUBMODULE Exponentiation & Logarithms, exponential)) $(TD
    $(SUBREF exponential, pow)
    $(SUBREF exponential, powmod)
    $(SUBREF exponential, exp)
    $(SUBREF exponential, exp2)
    $(SUBREF exponential, expm1)
    $(SUBREF exponential, ldexp)
    $(SUBREF exponential, frexp)
    $(SUBREF exponential, log)
    $(SUBREF exponential, log2)
    $(SUBREF exponential, log10)
    $(SUBREF exponential, logb)
    $(SUBREF exponential, ilogb)
    $(SUBREF exponential, log1p)
    $(SUBREF exponential, scalbn)
))
$(TR $(TDNW $(SUBMODULE Remainder, remainder)) $(TD
    $(SUBREF remainder, fmod)
    $(SUBREF remainder, modf)
    $(SUBREF remainder, remainder)
    $(SUBREF remainder, remquo)
))
$(TR $(TDNW $(SUBMODULE Floating-point operations, operations)) $(TD
    $(SUBREF operations, approxEqual)
    $(SUBREF operations, feqrel)
    $(SUBREF operations, fdim)
    $(SUBREF operations, fmax)
    $(SUBREF operations, fmin)
    $(SUBREF operations, fma)
    $(SUBREF operations, isClose)
    $(SUBREF operations, nextDown)
    $(SUBREF operations, nextUp)
    $(SUBREF operations, nextafter)
    $(SUBREF operations, NaN)
    $(SUBREF operations, getNaNPayload)
    $(SUBREF operations, cmp)
))
$(TR $(TDNW $(SUBMODULE Introspection, traits)) $(TD
    $(SUBREF traits, isFinite)
    $(SUBREF traits, isIdentical)
    $(SUBREF traits, isInfinity)
    $(SUBREF traits, isNaN)
    $(SUBREF traits, isNormal)
    $(SUBREF traits, isSubnormal)
    $(SUBREF traits, signbit)
    $(SUBREF traits, sgn)
    $(SUBREF traits, copysign)
    $(SUBREF traits, isPowerOf2)
))
$(TR $(TDNW $(SUBMODULE Hardware Control, hardware)) $(TD
    $(SUBREF hardware, IeeeFlags)
    $(SUBREF hardware, ieeeFlags)
    $(SUBREF hardware, resetIeeeFlags)
    $(SUBREF hardware, FloatingPointControl)
))
)
)

 * The functionality closely follows the IEEE754-2008 standard for
 * floating-point arithmetic, including the use of camelCase names rather
 * than C99-style lower case names. All of these functions behave correctly
 * when presented with an infinity or NaN.
 *
 * The following IEEE 'real' formats are currently supported:
 * $(UL
 * $(LI 64 bit Big-endian  'double' (eg PowerPC))
 * $(LI 128 bit Big-endian 'quadruple' (eg SPARC))
 * $(LI 64 bit Little-endian 'double' (eg x86-SSE2))
 * $(LI 80 bit Little-endian, with implied bit 'real80' (eg x87, Itanium))
 * $(LI 128 bit Little-endian 'quadruple' (not implemented on any known processor!))
 * $(LI Non-IEEE 128 bit Big-endian 'doubledouble' (eg PowerPC) has partial support)
 * )
 * Unlike C, there is no global 'errno' variable. Consequently, almost all of
 * these functions are pure nothrow.
 *
 * Macros:
 *      SUBMODULE = $(MREF_ALTTEXT $1, std, math, $2)
 *      SUBREF = $(REF_ALTTEXT $(TT $2), $2, std, math, $1)$(NBSP)
 *
 * Copyright: Copyright The D Language Foundation 2000 - 2011.
 *            D implementations of tan, atan, atan2, exp, expm1, exp2, log, log10, log1p,
 *            log2, floor, ceil and lrint functions are based on the CEPHES math library,
 *            which is Copyright (C) 2001 Stephen L. Moshier $(LT)steve@moshier.net$(GT)
 *            and are incorporated herein by permission of the author.  The author
 *            reserves the right to distribute this material elsewhere under different
 *            copying permissions.  These modifications are distributed here under
 *            the following terms:
 * License:   $(HTTP www.boost.org/LICENSE_1_0.txt, Boost License 1.0).
 * Authors:   $(HTTP digitalmars.com, Walter Bright), Don Clugston,
 *            Conversion of CEPHES math library to D by Iain Buclaw and David Nadlinger
 * Source: $(PHOBOSSRC std/math/package.d)
 */
module std.math;

public import std.math.algebraic;
public import std.math.constants;
public import std.math.exponential;
public import std.math.operations;
public import std.math.hardware;
public import std.math.remainder;
public import std.math.rounding;
public import std.math.traits;
public import std.math.trigonometry;

// @@@DEPRECATED_2.102@@@
// Note: Exposed accidentally, should be deprecated / removed
deprecated("std.meta.AliasSeq was unintentionally available from std.math "
           ~ "and will be removed after 2.102. Please import std.meta instead")
public import std.meta : AliasSeq;

package(std): // 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);
}
do
{
    // 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 ||
               F.realFormat == RealFormat.ieeeExtended53)
    {
        // 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));
}


// The following IEEE 'real' formats are currently supported.
version (LittleEndian)
{
    static assert(real.mant_dig == 53 || real.mant_dig == 64
               || real.mant_dig == 113,
      "Only 64-bit, 80-bit, and 128-bit reals"~
      " are supported for LittleEndian CPUs");
}
else
{
    static assert(real.mant_dig == 53 || real.mant_dig == 113,
    "Only 64-bit and 128-bit reals are supported for BigEndian CPUs.");
}

// Underlying format exposed through floatTraits
enum RealFormat
{
    ieeeHalf,
    ieeeSingle,
    ieeeDouble,
    ieeeExtended,   // x87 80-bit real
    ieeeExtended53, // x87 real rounded to precision of double.
    ibmExtended,    // IBM 128-bit extended
    ieeeQuadruple,
}

// Constants used for extracting the components of the representation.
// They supplement the built-in floating point properties.
template floatTraits(T)
{
    import std.traits : Unqual;

    // EXPMASK is a ushort mask to select the exponent portion (without sign)
    // EXPSHIFT is the number of bits the exponent is left-shifted by in its ushort
    // EXPBIAS is the exponent bias - 1 (exp == EXPBIAS yields ×2^-1).
    // EXPPOS_SHORT is the index of the exponent when represented as a ushort array.
    // SIGNPOS_BYTE is the index of the sign when represented as a ubyte array.
    // RECIP_EPSILON is the value such that (smallest_subnormal) * RECIP_EPSILON == T.min_normal
    enum Unqual!T RECIP_EPSILON = (1/T.epsilon);
    static if (T.mant_dig == 24)
    {
        // Single precision float
        enum ushort EXPMASK = 0x7F80;
        enum ushort EXPSHIFT = 7;
        enum ushort EXPBIAS = 0x3F00;
        enum uint EXPMASK_INT = 0x7F80_0000;
        enum uint MANTISSAMASK_INT = 0x007F_FFFF;
        enum realFormat = RealFormat.ieeeSingle;
        version (LittleEndian)
        {
            enum EXPPOS_SHORT = 1;
            enum SIGNPOS_BYTE = 3;
        }
        else
        {
            enum EXPPOS_SHORT = 0;
            enum SIGNPOS_BYTE = 0;
        }
    }
    else static if (T.mant_dig == 53)
    {
        static if (T.sizeof == 8)
        {
            // Double precision float, or real == double
            enum ushort EXPMASK = 0x7FF0;
            enum ushort EXPSHIFT = 4;
            enum ushort EXPBIAS = 0x3FE0;
            enum uint EXPMASK_INT = 0x7FF0_0000;
            enum uint MANTISSAMASK_INT = 0x000F_FFFF; // for the MSB only
            enum realFormat = RealFormat.ieeeDouble;
            version (LittleEndian)
            {
                enum EXPPOS_SHORT = 3;
                enum SIGNPOS_BYTE = 7;
            }
            else
            {
                enum EXPPOS_SHORT = 0;
                enum SIGNPOS_BYTE = 0;
            }
        }
        else static if (T.sizeof == 12)
        {
            // Intel extended real80 rounded to double
            enum ushort EXPMASK = 0x7FFF;
            enum ushort EXPSHIFT = 0;
            enum ushort EXPBIAS = 0x3FFE;
            enum realFormat = RealFormat.ieeeExtended53;
            version (LittleEndian)
            {
                enum EXPPOS_SHORT = 4;
                enum SIGNPOS_BYTE = 9;
            }
            else
            {
                enum EXPPOS_SHORT = 0;
                enum SIGNPOS_BYTE = 0;
            }
        }
        else
            static assert(false, "No traits support for " ~ T.stringof);
    }
    else static if (T.mant_dig == 64)
    {
        // Intel extended real80
        enum ushort EXPMASK = 0x7FFF;
        enum ushort EXPSHIFT = 0;
        enum ushort EXPBIAS = 0x3FFE;
        enum realFormat = RealFormat.ieeeExtended;
        version (LittleEndian)
        {
            enum EXPPOS_SHORT = 4;
            enum SIGNPOS_BYTE = 9;
        }
        else
        {
            enum EXPPOS_SHORT = 0;
            enum SIGNPOS_BYTE = 0;
        }
    }
    else static if (T.mant_dig == 113)
    {
        // Quadruple precision float
        enum ushort EXPMASK = 0x7FFF;
        enum ushort EXPSHIFT = 0;
        enum ushort EXPBIAS = 0x3FFE;
        enum realFormat = RealFormat.ieeeQuadruple;
        version (LittleEndian)
        {
            enum EXPPOS_SHORT = 7;
            enum SIGNPOS_BYTE = 15;
        }
        else
        {
            enum EXPPOS_SHORT = 0;
            enum SIGNPOS_BYTE = 0;
        }
    }
    else static if (T.mant_dig == 106)
    {
        // IBM Extended doubledouble
        enum ushort EXPMASK = 0x7FF0;
        enum ushort EXPSHIFT = 4;
        enum realFormat = RealFormat.ibmExtended;

        // For IBM doubledouble the larger magnitude double comes first.
        // It's really a double[2] and arrays don't index differently
        // between little and big-endian targets.
        enum DOUBLEPAIR_MSB = 0;
        enum DOUBLEPAIR_LSB = 1;

        // The exponent/sign byte is for most significant part.
        version (LittleEndian)
        {
            enum EXPPOS_SHORT = 3;
            enum SIGNPOS_BYTE = 7;
        }
        else
        {
            enum EXPPOS_SHORT = 0;
            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;
}