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https://github.com/dlang/phobos.git
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91aa5feead
This ports the fixes from abs(T)(ComplexT) regarding NaN and skipping potentially inaccuracy inducing mathematical no-ops when one arg is 0 to hypot. Both functions do the same thing and should be deduplicated. hypot also has some logic regarding under and overflows, and while I don't fully understand it, it should probably not be removed for complex numbers.
1972 lines
53 KiB
D
1972 lines
53 KiB
D
// Written in the D programming language.
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/** This module contains the $(LREF Complex) type, which is used to represent
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complex numbers, along with related mathematical operations and functions.
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$(LREF Complex) will eventually
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$(DDLINK deprecate, Deprecated Features, replace)
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the built-in types `cfloat`, `cdouble`, `creal`, `ifloat`,
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`idouble`, and `ireal`.
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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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PLUSMN = ±
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NAN = $(RED NAN)
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INFIN = ∞
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PI = π
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Authors: Lars Tandle Kyllingstad, Don Clugston
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Copyright: Copyright (c) 2010, Lars T. Kyllingstad.
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License: $(HTTP boost.org/LICENSE_1_0.txt, Boost License 1.0)
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Source: $(PHOBOSSRC std/complex.d)
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*/
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module std.complex;
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import std.traits;
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/** Helper function that returns a complex number with the specified
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real and imaginary parts.
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Params:
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R = (template parameter) type of real part of complex number
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I = (template parameter) type of imaginary part of complex number
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re = real part of complex number to be constructed
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im = (optional) imaginary part of complex number, 0 if omitted.
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Returns:
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`Complex` instance with real and imaginary parts set
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to the values provided as input. If neither `re` nor
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`im` are floating-point numbers, the return type will
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be `Complex!double`. Otherwise, the return type is
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deduced using $(D std.traits.CommonType!(R, I)).
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*/
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auto complex(R)(const R re) @safe pure nothrow @nogc
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if (is(R : double))
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{
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static if (isFloatingPoint!R)
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return Complex!R(re, 0);
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else
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return Complex!double(re, 0);
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}
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/// ditto
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auto complex(R, I)(const R re, const I im) @safe pure nothrow @nogc
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if (is(R : double) && is(I : double))
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{
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static if (isFloatingPoint!R || isFloatingPoint!I)
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return Complex!(CommonType!(R, I))(re, im);
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else
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return Complex!double(re, im);
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}
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///
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@safe pure nothrow unittest
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{
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auto a = complex(1.0);
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static assert(is(typeof(a) == Complex!double));
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assert(a.re == 1.0);
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assert(a.im == 0.0);
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auto b = complex(2.0L);
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static assert(is(typeof(b) == Complex!real));
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assert(b.re == 2.0L);
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assert(b.im == 0.0L);
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auto c = complex(1.0, 2.0);
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static assert(is(typeof(c) == Complex!double));
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assert(c.re == 1.0);
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assert(c.im == 2.0);
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auto d = complex(3.0, 4.0L);
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static assert(is(typeof(d) == Complex!real));
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assert(d.re == 3.0);
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assert(d.im == 4.0L);
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auto e = complex(1);
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static assert(is(typeof(e) == Complex!double));
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assert(e.re == 1);
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assert(e.im == 0);
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auto f = complex(1L, 2);
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static assert(is(typeof(f) == Complex!double));
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assert(f.re == 1L);
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assert(f.im == 2);
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auto g = complex(3, 4.0L);
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static assert(is(typeof(g) == Complex!real));
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assert(g.re == 3);
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assert(g.im == 4.0L);
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}
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/** A complex number parametrised by a type `T`, which must be either
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`float`, `double` or `real`.
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*/
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struct Complex(T)
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if (isFloatingPoint!T)
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{
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import std.format.spec : FormatSpec;
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import std.range.primitives : isOutputRange;
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/** The real part of the number. */
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T re;
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/** The imaginary part of the number. */
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T im;
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/** Converts the complex number to a string representation.
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The second form of this function is usually not called directly;
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instead, it is used via $(REF format, std,string), as shown in the examples
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below. Supported format characters are 'e', 'f', 'g', 'a', and 's'.
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See the $(MREF std, format) and $(REF format, std,string)
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documentation for more information.
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*/
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string toString() const @safe /* TODO: pure nothrow */
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{
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import std.exception : assumeUnique;
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char[] buf;
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buf.reserve(100);
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auto fmt = FormatSpec!char("%s");
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toString((const(char)[] s) { buf ~= s; }, fmt);
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static trustedAssumeUnique(T)(T t) @trusted { return assumeUnique(t); }
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return trustedAssumeUnique(buf);
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}
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static if (is(T == double))
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///
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@safe unittest
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{
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auto c = complex(1.2, 3.4);
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// Vanilla toString formatting:
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assert(c.toString() == "1.2+3.4i");
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// Formatting with std.string.format specs: the precision and width
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// specifiers apply to both the real and imaginary parts of the
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// complex number.
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import std.format : format;
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assert(format("%.2f", c) == "1.20+3.40i");
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assert(format("%4.1f", c) == " 1.2+ 3.4i");
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}
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/// ditto
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void toString(Writer, Char)(scope Writer w, scope const ref FormatSpec!Char formatSpec) const
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if (isOutputRange!(Writer, const(Char)[]))
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{
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import std.format.write : formatValue;
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import std.math.traits : signbit;
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import std.range.primitives : put;
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formatValue(w, re, formatSpec);
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if (signbit(im) == 0)
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put(w, "+");
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formatValue(w, im, formatSpec);
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put(w, "i");
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}
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@safe pure nothrow @nogc:
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/** Construct a complex number with the specified real and
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imaginary parts. In the case where a single argument is passed
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that is not complex, the imaginary part of the result will be
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zero.
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*/
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this(R : T)(Complex!R z)
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{
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re = z.re;
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im = z.im;
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}
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/// ditto
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this(Rx : T, Ry : T)(const Rx x, const Ry y)
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{
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re = x;
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im = y;
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}
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/// ditto
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this(R : T)(const R r)
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{
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re = r;
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im = 0;
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}
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// ASSIGNMENT OPERATORS
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// this = complex
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ref Complex opAssign(R : T)(Complex!R z)
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{
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re = z.re;
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im = z.im;
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return this;
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}
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// this = numeric
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ref Complex opAssign(R : T)(const R r)
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{
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re = r;
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im = 0;
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return this;
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}
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// COMPARISON OPERATORS
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// this == complex
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bool opEquals(R : T)(Complex!R z) const
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{
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return re == z.re && im == z.im;
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}
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// this == numeric
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bool opEquals(R : T)(const R r) const
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{
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return re == r && im == 0;
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}
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// UNARY OPERATORS
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// +complex
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Complex opUnary(string op)() const
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if (op == "+")
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{
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return this;
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}
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// -complex
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Complex opUnary(string op)() const
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if (op == "-")
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{
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return Complex(-re, -im);
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}
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// BINARY OPERATORS
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// complex op complex
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Complex!(CommonType!(T,R)) opBinary(string op, R)(Complex!R z) const
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{
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alias C = typeof(return);
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auto w = C(this.re, this.im);
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return w.opOpAssign!(op)(z);
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}
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// complex op numeric
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Complex!(CommonType!(T,R)) opBinary(string op, R)(const R r) const
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if (isNumeric!R)
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{
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alias C = typeof(return);
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auto w = C(this.re, this.im);
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return w.opOpAssign!(op)(r);
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}
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// numeric + complex, numeric * complex
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Complex!(CommonType!(T, R)) opBinaryRight(string op, R)(const R r) const
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if ((op == "+" || op == "*") && (isNumeric!R))
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{
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return opBinary!(op)(r);
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}
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// numeric - complex
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Complex!(CommonType!(T, R)) opBinaryRight(string op, R)(const R r) const
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if (op == "-" && isNumeric!R)
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{
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return Complex(r - re, -im);
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}
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// numeric / complex
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Complex!(CommonType!(T, R)) opBinaryRight(string op, R)(const R r) const
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if (op == "/" && isNumeric!R)
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{
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version (FastMath)
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{
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// Compute norm(this)
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immutable norm = re * re + im * im;
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// Compute r * conj(this)
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immutable prod_re = r * re;
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immutable prod_im = r * -im;
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// Divide the product by the norm
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typeof(return) w = void;
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w.re = prod_re / norm;
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w.im = prod_im / norm;
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return w;
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}
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else
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{
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import core.math : fabs;
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typeof(return) w = void;
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if (fabs(re) < fabs(im))
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{
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immutable ratio = re/im;
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immutable rdivd = r/(re*ratio + im);
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w.re = rdivd*ratio;
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w.im = -rdivd;
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}
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else
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{
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immutable ratio = im/re;
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immutable rdivd = r/(re + im*ratio);
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w.re = rdivd;
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w.im = -rdivd*ratio;
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}
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return w;
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}
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}
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// numeric ^^ complex
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Complex!(CommonType!(T, R)) opBinaryRight(string op, R)(const R lhs) const
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if (op == "^^" && isNumeric!R)
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{
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import core.math : cos, sin;
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import std.math.exponential : exp, log;
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import std.math.constants : PI;
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Unqual!(CommonType!(T, R)) ab = void, ar = void;
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if (lhs >= 0)
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{
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// r = lhs
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// theta = 0
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ab = lhs ^^ this.re;
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ar = log(lhs) * this.im;
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}
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else
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{
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// r = -lhs
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// theta = PI
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ab = (-lhs) ^^ this.re * exp(-PI * this.im);
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ar = PI * this.re + log(-lhs) * this.im;
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}
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return typeof(return)(ab * cos(ar), ab * sin(ar));
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}
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// OP-ASSIGN OPERATORS
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// complex += complex, complex -= complex
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ref Complex opOpAssign(string op, C)(const C z)
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if ((op == "+" || op == "-") && is(C R == Complex!R))
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{
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mixin ("re "~op~"= z.re;");
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mixin ("im "~op~"= z.im;");
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return this;
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}
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// complex *= complex
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ref Complex opOpAssign(string op, C)(const C z)
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if (op == "*" && is(C R == Complex!R))
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{
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auto temp = re*z.re - im*z.im;
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im = im*z.re + re*z.im;
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re = temp;
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return this;
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}
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// complex /= complex
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ref Complex opOpAssign(string op, C)(const C z)
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if (op == "/" && is(C R == Complex!R))
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{
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version (FastMath)
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{
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// Compute norm(z)
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immutable norm = z.re * z.re + z.im * z.im;
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// Compute this * conj(z)
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immutable prod_re = re * z.re - im * -z.im;
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immutable prod_im = im * z.re + re * -z.im;
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// Divide the product by the norm
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re = prod_re / norm;
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im = prod_im / norm;
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return this;
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}
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else
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{
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import core.math : fabs;
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if (fabs(z.re) < fabs(z.im))
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{
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immutable ratio = z.re/z.im;
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immutable denom = z.re*ratio + z.im;
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immutable temp = (re*ratio + im)/denom;
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im = (im*ratio - re)/denom;
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re = temp;
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}
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else
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{
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immutable ratio = z.im/z.re;
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immutable denom = z.re + z.im*ratio;
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immutable temp = (re + im*ratio)/denom;
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im = (im - re*ratio)/denom;
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re = temp;
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}
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return this;
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}
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}
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// complex ^^= complex
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ref Complex opOpAssign(string op, C)(const C z)
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if (op == "^^" && is(C R == Complex!R))
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{
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import core.math : cos, sin;
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import std.math.exponential : exp, log;
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immutable r = abs(this);
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immutable t = arg(this);
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immutable ab = r^^z.re * exp(-t*z.im);
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immutable ar = t*z.re + log(r)*z.im;
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re = ab*cos(ar);
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im = ab*sin(ar);
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return this;
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}
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// complex += numeric, complex -= numeric
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ref Complex opOpAssign(string op, U : T)(const U a)
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if (op == "+" || op == "-")
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{
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mixin ("re "~op~"= a;");
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return this;
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}
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// complex *= numeric, complex /= numeric
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ref Complex opOpAssign(string op, U : T)(const U a)
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if (op == "*" || op == "/")
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{
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mixin ("re "~op~"= a;");
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mixin ("im "~op~"= a;");
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return this;
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}
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// complex ^^= real
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ref Complex opOpAssign(string op, R)(const R r)
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if (op == "^^" && isFloatingPoint!R)
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{
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import core.math : cos, sin;
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immutable ab = abs(this)^^r;
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immutable ar = arg(this)*r;
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re = ab*cos(ar);
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im = ab*sin(ar);
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return this;
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}
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// complex ^^= int
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ref Complex opOpAssign(string op, U)(const U i)
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if (op == "^^" && isIntegral!U)
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{
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switch (i)
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{
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case 0:
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re = 1.0;
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im = 0.0;
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break;
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case 1:
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// identity; do nothing
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break;
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case 2:
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this *= this;
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break;
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case 3:
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auto z = this;
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this *= z;
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this *= z;
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break;
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default:
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this ^^= cast(real) i;
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}
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return this;
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}
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/** Returns a complex number instance that correponds in size and in ABI
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to the associated C compiler's `_Complex` type.
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*/
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auto toNative()
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{
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import core.stdc.config : c_complex_float, c_complex_double, c_complex_real;
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static if (is(T == float))
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return c_complex_float(re, im);
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else static if (is(T == double))
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return c_complex_double(re, im);
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else
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return c_complex_real(re, im);
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}
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}
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@safe pure nothrow unittest
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{
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import std.complex;
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static import core.math;
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import std.math;
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enum EPS = double.epsilon;
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auto c1 = complex(1.0, 1.0);
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// Check unary operations.
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auto c2 = Complex!double(0.5, 2.0);
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assert(c2 == +c2);
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assert((-c2).re == -(c2.re));
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assert((-c2).im == -(c2.im));
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assert(c2 == -(-c2));
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// Check complex-complex operations.
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auto cpc = c1 + c2;
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assert(cpc.re == c1.re + c2.re);
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assert(cpc.im == c1.im + c2.im);
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auto cmc = c1 - c2;
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assert(cmc.re == c1.re - c2.re);
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assert(cmc.im == c1.im - c2.im);
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auto ctc = c1 * c2;
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assert(isClose(abs(ctc), abs(c1)*abs(c2), EPS));
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assert(isClose(arg(ctc), arg(c1)+arg(c2), EPS));
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auto cdc = c1 / c2;
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assert(isClose(abs(cdc), abs(c1)/abs(c2), EPS));
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assert(isClose(arg(cdc), arg(c1)-arg(c2), EPS));
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auto cec = c1^^c2;
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assert(isClose(cec.re, 0.1152413197994, 1e-12));
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assert(isClose(cec.im, 0.2187079045274, 1e-12));
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// Check complex-real operations.
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double a = 123.456;
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auto cpr = c1 + a;
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assert(cpr.re == c1.re + a);
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assert(cpr.im == c1.im);
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auto cmr = c1 - a;
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assert(cmr.re == c1.re - a);
|
|
assert(cmr.im == c1.im);
|
|
|
|
auto ctr = c1 * a;
|
|
assert(ctr.re == c1.re*a);
|
|
assert(ctr.im == c1.im*a);
|
|
|
|
auto cdr = c1 / a;
|
|
assert(isClose(abs(cdr), abs(c1)/a, EPS));
|
|
assert(isClose(arg(cdr), arg(c1), EPS));
|
|
|
|
auto cer = c1^^3.0;
|
|
assert(isClose(abs(cer), abs(c1)^^3, EPS));
|
|
assert(isClose(arg(cer), arg(c1)*3, EPS));
|
|
|
|
auto rpc = a + c1;
|
|
assert(rpc == cpr);
|
|
|
|
auto rmc = a - c1;
|
|
assert(rmc.re == a-c1.re);
|
|
assert(rmc.im == -c1.im);
|
|
|
|
auto rtc = a * c1;
|
|
assert(rtc == ctr);
|
|
|
|
auto rdc = a / c1;
|
|
assert(isClose(abs(rdc), a/abs(c1), EPS));
|
|
assert(isClose(arg(rdc), -arg(c1), EPS));
|
|
|
|
rdc = a / c2;
|
|
assert(isClose(abs(rdc), a/abs(c2), EPS));
|
|
assert(isClose(arg(rdc), -arg(c2), EPS));
|
|
|
|
auto rec1a = 1.0 ^^ c1;
|
|
assert(rec1a.re == 1.0);
|
|
assert(rec1a.im == 0.0);
|
|
|
|
auto rec2a = 1.0 ^^ c2;
|
|
assert(rec2a.re == 1.0);
|
|
assert(rec2a.im == 0.0);
|
|
|
|
auto rec1b = (-1.0) ^^ c1;
|
|
assert(isClose(abs(rec1b), std.math.exp(-PI * c1.im), EPS));
|
|
auto arg1b = arg(rec1b);
|
|
/* The argument _should_ be PI, but floating-point rounding error
|
|
* means that in fact the imaginary part is very slightly negative.
|
|
*/
|
|
assert(isClose(arg1b, PI, EPS) || isClose(arg1b, -PI, EPS));
|
|
|
|
auto rec2b = (-1.0) ^^ c2;
|
|
assert(isClose(abs(rec2b), std.math.exp(-2 * PI), EPS));
|
|
assert(isClose(arg(rec2b), PI_2, EPS));
|
|
|
|
auto rec3a = 0.79 ^^ complex(6.8, 5.7);
|
|
auto rec3b = complex(0.79, 0.0) ^^ complex(6.8, 5.7);
|
|
assert(isClose(rec3a.re, rec3b.re, 1e-14));
|
|
assert(isClose(rec3a.im, rec3b.im, 1e-14));
|
|
|
|
auto rec4a = (-0.79) ^^ complex(6.8, 5.7);
|
|
auto rec4b = complex(-0.79, 0.0) ^^ complex(6.8, 5.7);
|
|
assert(isClose(rec4a.re, rec4b.re, 1e-14));
|
|
assert(isClose(rec4a.im, rec4b.im, 1e-14));
|
|
|
|
auto rer = a ^^ complex(2.0, 0.0);
|
|
auto rcheck = a ^^ 2.0;
|
|
static assert(is(typeof(rcheck) == double));
|
|
assert(feqrel(rer.re, rcheck) == double.mant_dig);
|
|
assert(isIdentical(rer.re, rcheck));
|
|
assert(rer.im == 0.0);
|
|
|
|
auto rer2 = (-a) ^^ complex(2.0, 0.0);
|
|
rcheck = (-a) ^^ 2.0;
|
|
assert(feqrel(rer2.re, rcheck) == double.mant_dig);
|
|
assert(isIdentical(rer2.re, rcheck));
|
|
assert(isClose(rer2.im, 0.0, 0.0, 1e-10));
|
|
|
|
auto rer3 = (-a) ^^ complex(-2.0, 0.0);
|
|
rcheck = (-a) ^^ (-2.0);
|
|
assert(feqrel(rer3.re, rcheck) == double.mant_dig);
|
|
assert(isIdentical(rer3.re, rcheck));
|
|
assert(isClose(rer3.im, 0.0, 0.0, EPS));
|
|
|
|
auto rer4 = a ^^ complex(-2.0, 0.0);
|
|
rcheck = a ^^ (-2.0);
|
|
assert(feqrel(rer4.re, rcheck) == double.mant_dig);
|
|
assert(isIdentical(rer4.re, rcheck));
|
|
assert(rer4.im == 0.0);
|
|
|
|
// Check Complex-int operations.
|
|
foreach (i; 0 .. 6)
|
|
{
|
|
auto cei = c1^^i;
|
|
assert(isClose(abs(cei), abs(c1)^^i, 1e-14));
|
|
// Use cos() here to deal with arguments that go outside
|
|
// the (-pi,pi] interval (only an issue for i>3).
|
|
assert(isClose(core.math.cos(arg(cei)), core.math.cos(arg(c1)*i), 1e-14));
|
|
}
|
|
|
|
// Check operations between different complex types.
|
|
auto cf = Complex!float(1.0, 1.0);
|
|
auto cr = Complex!real(1.0, 1.0);
|
|
auto c1pcf = c1 + cf;
|
|
auto c1pcr = c1 + cr;
|
|
static assert(is(typeof(c1pcf) == Complex!double));
|
|
static assert(is(typeof(c1pcr) == Complex!real));
|
|
assert(c1pcf.re == c1pcr.re);
|
|
assert(c1pcf.im == c1pcr.im);
|
|
|
|
auto c1c = c1;
|
|
auto c2c = c2;
|
|
|
|
c1c /= c1;
|
|
assert(isClose(c1c.re, 1.0, EPS));
|
|
assert(isClose(c1c.im, 0.0, 0.0, EPS));
|
|
|
|
c1c = c1;
|
|
c1c /= c2;
|
|
assert(isClose(c1c.re, 0.5882352941177, 1e-12));
|
|
assert(isClose(c1c.im, -0.3529411764706, 1e-12));
|
|
|
|
c2c /= c1;
|
|
assert(isClose(c2c.re, 1.25, EPS));
|
|
assert(isClose(c2c.im, 0.75, EPS));
|
|
|
|
c2c = c2;
|
|
c2c /= c2;
|
|
assert(isClose(c2c.re, 1.0, EPS));
|
|
assert(isClose(c2c.im, 0.0, 0.0, EPS));
|
|
}
|
|
|
|
@safe pure nothrow unittest
|
|
{
|
|
// Initialization
|
|
Complex!double a = 1;
|
|
assert(a.re == 1 && a.im == 0);
|
|
Complex!double b = 1.0;
|
|
assert(b.re == 1.0 && b.im == 0);
|
|
Complex!double c = Complex!real(1.0, 2);
|
|
assert(c.re == 1.0 && c.im == 2);
|
|
}
|
|
|
|
@safe pure nothrow unittest
|
|
{
|
|
// Assignments and comparisons
|
|
Complex!double z;
|
|
|
|
z = 1;
|
|
assert(z == 1);
|
|
assert(z.re == 1.0 && z.im == 0.0);
|
|
|
|
z = 2.0;
|
|
assert(z == 2.0);
|
|
assert(z.re == 2.0 && z.im == 0.0);
|
|
|
|
z = 1.0L;
|
|
assert(z == 1.0L);
|
|
assert(z.re == 1.0 && z.im == 0.0);
|
|
|
|
auto w = Complex!real(1.0, 1.0);
|
|
z = w;
|
|
assert(z == w);
|
|
assert(z.re == 1.0 && z.im == 1.0);
|
|
|
|
auto c = Complex!float(2.0, 2.0);
|
|
z = c;
|
|
assert(z == c);
|
|
assert(z.re == 2.0 && z.im == 2.0);
|
|
}
|
|
|
|
|
|
/* Makes Complex!(Complex!T) fold to Complex!T.
|
|
|
|
The rationale for this is that just like the real line is a
|
|
subspace of the complex plane, the complex plane is a subspace
|
|
of itself. Example of usage:
|
|
---
|
|
Complex!T addI(T)(T x)
|
|
{
|
|
return x + Complex!T(0.0, 1.0);
|
|
}
|
|
---
|
|
The above will work if T is both real and complex.
|
|
*/
|
|
template Complex(T)
|
|
if (is(T R == Complex!R))
|
|
{
|
|
alias Complex = T;
|
|
}
|
|
|
|
@safe pure nothrow unittest
|
|
{
|
|
static assert(is(Complex!(Complex!real) == Complex!real));
|
|
|
|
Complex!T addI(T)(T x)
|
|
{
|
|
return x + Complex!T(0.0, 1.0);
|
|
}
|
|
|
|
auto z1 = addI(1.0);
|
|
assert(z1.re == 1.0 && z1.im == 1.0);
|
|
|
|
enum one = Complex!double(1.0, 0.0);
|
|
auto z2 = addI(one);
|
|
assert(z1 == z2);
|
|
}
|
|
|
|
|
|
/**
|
|
* Calculates the absolute value (or modulus) of a complex number.
|
|
*
|
|
* $(TABLE_SV
|
|
* $(TR $(TH $(I z)) $(TH abs(z)) $(TH Notes))
|
|
* $(TR $(TD (0, 0)) $(TD 0) $(TD ))
|
|
* $(TR $(TD (NaN, any) or (any, NaN)) $(TD NaN) $(TD ))
|
|
* $(TR $(TD (Inf, any) or (any, Inf)) $(TD Inf) $(TD ))
|
|
* $(TR $(TD (a, b)) normal case $(TD hypot(a, b)) $(TD Uses algorithm to prevent overflow/underflow ))
|
|
* )
|
|
*
|
|
* Params:
|
|
* z = A complex number of type Complex!T
|
|
*
|
|
* Returns:
|
|
* The absolute value (modulus) of `z`
|
|
*/
|
|
T abs(T)(Complex!T z) @safe pure nothrow @nogc
|
|
{
|
|
import std.math.algebraic : hypot;
|
|
return hypot(z.re, z.im);
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow unittest
|
|
{
|
|
static import core.math;
|
|
assert(abs(complex(1.0)) == 1.0);
|
|
assert(abs(complex(0.0, 1.0)) == 1.0);
|
|
assert(abs(complex(1.0L, -2.0L)) == core.math.sqrt(5.0L));
|
|
}
|
|
|
|
@safe pure nothrow unittest
|
|
{
|
|
{
|
|
auto x = Complex!float(-5.016556e-20, 0);
|
|
assert(x.abs == 5.016556e-20f);
|
|
auto x1 = Complex!float(-5.016556e-20f, 0);
|
|
assert(x1.abs == 5.016556e-20f);
|
|
auto x2 = Complex!float(5.016556e-20f, 0);
|
|
assert(x2.abs == 5.016556e-20f);
|
|
}
|
|
{
|
|
import std.math.traits : isNaN, isInfinity;
|
|
assert(Complex!double(double.nan, 0).abs.isNaN);
|
|
assert(Complex!double(double.nan, double.nan).abs.isNaN);
|
|
assert(Complex!double(double.infinity, 0).abs.isInfinity);
|
|
assert(Complex!double(0, double.infinity).abs.isInfinity);
|
|
assert(Complex!double(0, 0).abs == 0);
|
|
}
|
|
}
|
|
|
|
@safe pure nothrow @nogc unittest
|
|
{
|
|
static import core.math;
|
|
assert(abs(complex(0.0L, -3.2L)) == 3.2L);
|
|
assert(abs(complex(0.0L, 71.6L)) == 71.6L);
|
|
assert(abs(complex(-1.0L, 1.0L)) == core.math.sqrt(2.0L));
|
|
}
|
|
|
|
@safe pure nothrow @nogc unittest
|
|
{
|
|
import std.meta : AliasSeq;
|
|
static foreach (T; AliasSeq!(float, double, real))
|
|
{{
|
|
static import std.math;
|
|
Complex!T a = complex(T(-12), T(3));
|
|
T b = std.math.hypot(a.re, a.im);
|
|
assert(std.math.isClose(abs(a), b));
|
|
assert(std.math.isClose(abs(-a), b));
|
|
}}
|
|
}
|
|
|
|
|
|
/++
|
|
Params:
|
|
z = A complex number.
|
|
x = A real number.
|
|
Returns: The squared modulus of `z`.
|
|
For genericity, if called on a real number, returns its square.
|
|
+/
|
|
T sqAbs(T)(Complex!T z) @safe pure nothrow @nogc
|
|
{
|
|
return z.re*z.re + z.im*z.im;
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow unittest
|
|
{
|
|
import std.math.operations : isClose;
|
|
assert(sqAbs(complex(0.0)) == 0.0);
|
|
assert(sqAbs(complex(1.0)) == 1.0);
|
|
assert(sqAbs(complex(0.0, 1.0)) == 1.0);
|
|
assert(isClose(sqAbs(complex(1.0L, -2.0L)), 5.0L));
|
|
assert(isClose(sqAbs(complex(-3.0L, 1.0L)), 10.0L));
|
|
assert(isClose(sqAbs(complex(1.0f,-1.0f)), 2.0f));
|
|
}
|
|
|
|
|
|
/// ditto
|
|
T sqAbs(T)(const T x) @safe pure nothrow @nogc
|
|
if (isFloatingPoint!T)
|
|
{
|
|
return x*x;
|
|
}
|
|
|
|
@safe pure nothrow unittest
|
|
{
|
|
import std.math.operations : isClose;
|
|
assert(sqAbs(0.0) == 0.0);
|
|
assert(sqAbs(-1.0) == 1.0);
|
|
assert(isClose(sqAbs(-3.0L), 9.0L));
|
|
assert(isClose(sqAbs(-5.0f), 25.0f));
|
|
}
|
|
|
|
|
|
/**
|
|
Params: z = A complex number.
|
|
Returns: The argument (or phase) of `z`.
|
|
*/
|
|
T arg(T)(Complex!T z) @safe pure nothrow @nogc
|
|
{
|
|
import std.math.trigonometry : atan2;
|
|
return atan2(z.im, z.re);
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow unittest
|
|
{
|
|
import std.math.constants : PI_2, PI_4;
|
|
assert(arg(complex(1.0)) == 0.0);
|
|
assert(arg(complex(0.0L, 1.0L)) == PI_2);
|
|
assert(arg(complex(1.0L, 1.0L)) == PI_4);
|
|
}
|
|
|
|
|
|
/**
|
|
* Extracts the norm of a complex number.
|
|
* Params:
|
|
* z = A complex number
|
|
* Returns:
|
|
* The squared magnitude of `z`.
|
|
*/
|
|
T norm(T)(Complex!T z) @safe pure nothrow @nogc
|
|
{
|
|
return z.re * z.re + z.im * z.im;
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow @nogc unittest
|
|
{
|
|
import std.math.operations : isClose;
|
|
import std.math.constants : PI;
|
|
assert(norm(complex(3.0, 4.0)) == 25.0);
|
|
assert(norm(fromPolar(5.0, 0.0)) == 25.0);
|
|
assert(isClose(norm(fromPolar(5.0L, PI / 6)), 25.0L));
|
|
assert(isClose(norm(fromPolar(5.0L, 13 * PI / 6)), 25.0L));
|
|
}
|
|
|
|
|
|
/**
|
|
Params: z = A complex number.
|
|
Returns: The complex conjugate of `z`.
|
|
*/
|
|
Complex!T conj(T)(Complex!T z) @safe pure nothrow @nogc
|
|
{
|
|
return Complex!T(z.re, -z.im);
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow unittest
|
|
{
|
|
assert(conj(complex(1.0)) == complex(1.0));
|
|
assert(conj(complex(1.0, 2.0)) == complex(1.0, -2.0));
|
|
}
|
|
|
|
@safe pure nothrow @nogc unittest
|
|
{
|
|
import std.meta : AliasSeq;
|
|
static foreach (T; AliasSeq!(float, double, real))
|
|
{{
|
|
auto c = Complex!T(7, 3L);
|
|
assert(conj(c) == Complex!T(7, -3L));
|
|
auto z = Complex!T(0, -3.2L);
|
|
assert(conj(z) == -z);
|
|
}}
|
|
}
|
|
|
|
/**
|
|
* Returns the projection of `z` onto the Riemann sphere.
|
|
* Params:
|
|
* z = A complex number
|
|
* Returns:
|
|
* The projection of `z` onto the Riemann sphere.
|
|
*/
|
|
Complex!T proj(T)(Complex!T z)
|
|
{
|
|
static import std.math;
|
|
|
|
if (std.math.isInfinity(z.re) || std.math.isInfinity(z.im))
|
|
return Complex!T(T.infinity, std.math.copysign(0.0, z.im));
|
|
|
|
return z;
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow unittest
|
|
{
|
|
assert(proj(complex(1.0)) == complex(1.0));
|
|
assert(proj(complex(double.infinity, 5.0)) == complex(double.infinity, 0.0));
|
|
assert(proj(complex(5.0, -double.infinity)) == complex(double.infinity, -0.0));
|
|
}
|
|
|
|
|
|
/**
|
|
Constructs a complex number given its absolute value and argument.
|
|
Params:
|
|
modulus = The modulus
|
|
argument = The argument
|
|
Returns: The complex number with the given modulus and argument.
|
|
*/
|
|
Complex!(CommonType!(T, U)) fromPolar(T, U)(const T modulus, const U argument)
|
|
@safe pure nothrow @nogc
|
|
{
|
|
import core.math : sin, cos;
|
|
return Complex!(CommonType!(T,U))
|
|
(modulus*cos(argument), modulus*sin(argument));
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow unittest
|
|
{
|
|
import core.math;
|
|
import std.math.operations : isClose;
|
|
import std.math.algebraic : sqrt;
|
|
import std.math.constants : PI_4;
|
|
auto z = fromPolar(core.math.sqrt(2.0L), PI_4);
|
|
assert(isClose(z.re, 1.0L));
|
|
assert(isClose(z.im, 1.0L));
|
|
}
|
|
|
|
version (StdUnittest)
|
|
{
|
|
// Helper function for comparing two Complex numbers.
|
|
int ceqrel(T)(const Complex!T x, const Complex!T y) @safe pure nothrow @nogc
|
|
{
|
|
import std.math.operations : feqrel;
|
|
const r = feqrel(x.re, y.re);
|
|
const i = feqrel(x.im, y.im);
|
|
return r < i ? r : i;
|
|
}
|
|
}
|
|
|
|
/**
|
|
Trigonometric functions on complex numbers.
|
|
|
|
Params: z = A complex number.
|
|
Returns: The sine, cosine and tangent of `z`, respectively.
|
|
*/
|
|
Complex!T sin(T)(Complex!T z) @safe pure nothrow @nogc
|
|
{
|
|
auto cs = expi(z.re);
|
|
auto csh = coshisinh(z.im);
|
|
return typeof(return)(cs.im * csh.re, cs.re * csh.im);
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow unittest
|
|
{
|
|
static import core.math;
|
|
assert(sin(complex(0.0)) == 0.0);
|
|
assert(sin(complex(2.0, 0)) == core.math.sin(2.0));
|
|
}
|
|
|
|
@safe pure nothrow unittest
|
|
{
|
|
static import core.math;
|
|
assert(ceqrel(sin(complex(2.0L, 0)), complex(core.math.sin(2.0L))) >= real.mant_dig - 1);
|
|
}
|
|
|
|
/// ditto
|
|
Complex!T cos(T)(Complex!T z) @safe pure nothrow @nogc
|
|
{
|
|
auto cs = expi(z.re);
|
|
auto csh = coshisinh(z.im);
|
|
return typeof(return)(cs.re * csh.re, - cs.im * csh.im);
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow unittest
|
|
{
|
|
static import core.math;
|
|
static import std.math;
|
|
assert(cos(complex(0.0)) == 1.0);
|
|
assert(cos(complex(1.3, 0.0)) == core.math.cos(1.3));
|
|
assert(cos(complex(0.0, 5.2)) == std.math.cosh(5.2));
|
|
}
|
|
|
|
@safe pure nothrow unittest
|
|
{
|
|
static import core.math;
|
|
static import std.math;
|
|
assert(ceqrel(cos(complex(0, 5.2L)), complex(std.math.cosh(5.2L), 0.0L)) >= real.mant_dig - 1);
|
|
assert(ceqrel(cos(complex(1.3L)), complex(core.math.cos(1.3L))) >= real.mant_dig - 1);
|
|
}
|
|
|
|
/// ditto
|
|
Complex!T tan(T)(Complex!T z) @safe pure nothrow @nogc
|
|
{
|
|
return sin(z) / cos(z);
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow @nogc unittest
|
|
{
|
|
static import std.math;
|
|
|
|
int ceqrel(T)(const Complex!T x, const Complex!T y) @safe pure nothrow @nogc
|
|
{
|
|
import std.math.operations : feqrel;
|
|
const r = feqrel(x.re, y.re);
|
|
const i = feqrel(x.im, y.im);
|
|
return r < i ? r : i;
|
|
}
|
|
assert(ceqrel(tan(complex(1.0, 0.0)), complex(std.math.tan(1.0), 0.0)) >= double.mant_dig - 2);
|
|
assert(ceqrel(tan(complex(0.0, 1.0)), complex(0.0, std.math.tanh(1.0))) >= double.mant_dig - 2);
|
|
}
|
|
|
|
/**
|
|
Inverse trigonometric functions on complex numbers.
|
|
|
|
Params: z = A complex number.
|
|
Returns: The arcsine, arccosine and arctangent of `z`, respectively.
|
|
*/
|
|
Complex!T asin(T)(Complex!T z) @safe pure nothrow @nogc
|
|
{
|
|
auto ash = asinh(Complex!T(-z.im, z.re));
|
|
return Complex!T(ash.im, -ash.re);
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow unittest
|
|
{
|
|
import std.math.operations : isClose;
|
|
import std.math.constants : PI;
|
|
assert(asin(complex(0.0)) == 0.0);
|
|
assert(isClose(asin(complex(0.5L)), PI / 6));
|
|
}
|
|
|
|
@safe pure nothrow unittest
|
|
{
|
|
import std.math.operations : isClose;
|
|
import std.math.constants : PI;
|
|
version (DigitalMars) {} else // Disabled because of https://issues.dlang.org/show_bug.cgi?id=21376
|
|
assert(isClose(asin(complex(0.5f)), float(PI) / 6));
|
|
}
|
|
|
|
/// ditto
|
|
Complex!T acos(T)(Complex!T z) @safe pure nothrow @nogc
|
|
{
|
|
static import std.math;
|
|
auto as = asin(z);
|
|
return Complex!T(T(std.math.PI_2) - as.re, as.im);
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow unittest
|
|
{
|
|
import std.math.operations : isClose;
|
|
import std.math.constants : PI;
|
|
import std.math.trigonometry : std_math_acos = acos;
|
|
assert(acos(complex(0.0)) == std_math_acos(0.0));
|
|
assert(isClose(acos(complex(0.5L)), PI / 3));
|
|
}
|
|
|
|
@safe pure nothrow unittest
|
|
{
|
|
import std.math.operations : isClose;
|
|
import std.math.constants : PI;
|
|
version (DigitalMars) {} else // Disabled because of https://issues.dlang.org/show_bug.cgi?id=21376
|
|
assert(isClose(acos(complex(0.5f)), float(PI) / 3));
|
|
}
|
|
|
|
/// ditto
|
|
Complex!T atan(T)(Complex!T z) @safe pure nothrow @nogc
|
|
{
|
|
static import std.math;
|
|
const T re2 = z.re * z.re;
|
|
const T x = 1 - re2 - z.im * z.im;
|
|
|
|
T num = z.im + 1;
|
|
T den = z.im - 1;
|
|
|
|
num = re2 + num * num;
|
|
den = re2 + den * den;
|
|
|
|
return Complex!T(T(0.5) * std.math.atan2(2 * z.re, x),
|
|
T(0.25) * std.math.log(num / den));
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow @nogc unittest
|
|
{
|
|
import std.math.operations : isClose;
|
|
import std.math.constants : PI;
|
|
assert(atan(complex(0.0)) == 0.0);
|
|
assert(isClose(atan(sqrt(complex(3.0L))), PI / 3));
|
|
assert(isClose(atan(sqrt(complex(3.0f))), float(PI) / 3));
|
|
}
|
|
|
|
/**
|
|
Hyperbolic trigonometric functions on complex numbers.
|
|
|
|
Params: z = A complex number.
|
|
Returns: The hyperbolic sine, cosine and tangent of `z`, respectively.
|
|
*/
|
|
Complex!T sinh(T)(Complex!T z) @safe pure nothrow @nogc
|
|
{
|
|
static import core.math, std.math;
|
|
return Complex!T(std.math.sinh(z.re) * core.math.cos(z.im),
|
|
std.math.cosh(z.re) * core.math.sin(z.im));
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow unittest
|
|
{
|
|
static import std.math;
|
|
assert(sinh(complex(0.0)) == 0.0);
|
|
assert(sinh(complex(1.0L)) == std.math.sinh(1.0L));
|
|
assert(sinh(complex(1.0f)) == std.math.sinh(1.0f));
|
|
}
|
|
|
|
/// ditto
|
|
Complex!T cosh(T)(Complex!T z) @safe pure nothrow @nogc
|
|
{
|
|
static import core.math, std.math;
|
|
return Complex!T(std.math.cosh(z.re) * core.math.cos(z.im),
|
|
std.math.sinh(z.re) * core.math.sin(z.im));
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow unittest
|
|
{
|
|
static import std.math;
|
|
assert(cosh(complex(0.0)) == 1.0);
|
|
assert(cosh(complex(1.0L)) == std.math.cosh(1.0L));
|
|
assert(cosh(complex(1.0f)) == std.math.cosh(1.0f));
|
|
}
|
|
|
|
/// ditto
|
|
Complex!T tanh(T)(Complex!T z) @safe pure nothrow @nogc
|
|
{
|
|
return sinh(z) / cosh(z);
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow @nogc unittest
|
|
{
|
|
import std.math.operations : isClose;
|
|
import std.math.trigonometry : std_math_tanh = tanh;
|
|
assert(tanh(complex(0.0)) == 0.0);
|
|
assert(isClose(tanh(complex(1.0L)), std_math_tanh(1.0L)));
|
|
assert(isClose(tanh(complex(1.0f)), std_math_tanh(1.0f)));
|
|
}
|
|
|
|
/**
|
|
Inverse hyperbolic trigonometric functions on complex numbers.
|
|
|
|
Params: z = A complex number.
|
|
Returns: The hyperbolic arcsine, arccosine and arctangent of `z`, respectively.
|
|
*/
|
|
Complex!T asinh(T)(Complex!T z) @safe pure nothrow @nogc
|
|
{
|
|
auto t = Complex!T((z.re - z.im) * (z.re + z.im) + 1, 2 * z.re * z.im);
|
|
return log(sqrt(t) + z);
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow unittest
|
|
{
|
|
import std.math.operations : isClose;
|
|
import std.math.trigonometry : std_math_asinh = asinh;
|
|
assert(asinh(complex(0.0)) == 0.0);
|
|
assert(isClose(asinh(complex(1.0L)), std_math_asinh(1.0L)));
|
|
assert(isClose(asinh(complex(1.0f)), std_math_asinh(1.0f)));
|
|
}
|
|
|
|
/// ditto
|
|
Complex!T acosh(T)(Complex!T z) @safe pure nothrow @nogc
|
|
{
|
|
return 2 * log(sqrt(T(0.5) * (z + 1)) + sqrt(T(0.5) * (z - 1)));
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow unittest
|
|
{
|
|
import std.math.operations : isClose;
|
|
import std.math.trigonometry : std_math_acosh = acosh;
|
|
assert(acosh(complex(1.0)) == 0.0);
|
|
assert(isClose(acosh(complex(3.0L)), std_math_acosh(3.0L)));
|
|
assert(isClose(acosh(complex(3.0f)), std_math_acosh(3.0f)));
|
|
}
|
|
|
|
/// ditto
|
|
Complex!T atanh(T)(Complex!T z) @safe pure nothrow @nogc
|
|
{
|
|
static import std.math;
|
|
const T im2 = z.im * z.im;
|
|
const T x = 1 - im2 - z.re * z.re;
|
|
|
|
T num = 1 + z.re;
|
|
T den = 1 - z.re;
|
|
|
|
num = im2 + num * num;
|
|
den = im2 + den * den;
|
|
|
|
return Complex!T(T(0.25) * (std.math.log(num) - std.math.log(den)),
|
|
T(0.5) * std.math.atan2(2 * z.im, x));
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow @nogc unittest
|
|
{
|
|
import std.math.operations : isClose;
|
|
import std.math.trigonometry : std_math_atanh = atanh;
|
|
assert(atanh(complex(0.0)) == 0.0);
|
|
assert(isClose(atanh(complex(0.5L)), std_math_atanh(0.5L)));
|
|
assert(isClose(atanh(complex(0.5f)), std_math_atanh(0.5f)));
|
|
}
|
|
|
|
/**
|
|
Params: y = A real number.
|
|
Returns: The value of cos(y) + i sin(y).
|
|
|
|
Note:
|
|
`expi` is included here for convenience and for easy migration of code.
|
|
*/
|
|
Complex!real expi(real y) @trusted pure nothrow @nogc
|
|
{
|
|
import core.math : cos, sin;
|
|
return Complex!real(cos(y), sin(y));
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow unittest
|
|
{
|
|
import core.math : cos, sin;
|
|
assert(expi(0.0L) == 1.0L);
|
|
assert(expi(1.3e5L) == complex(cos(1.3e5L), sin(1.3e5L)));
|
|
}
|
|
|
|
/**
|
|
Params: y = A real number.
|
|
Returns: The value of cosh(y) + i sinh(y)
|
|
|
|
Note:
|
|
`coshisinh` is included here for convenience and for easy migration of code.
|
|
*/
|
|
Complex!real coshisinh(real y) @safe pure nothrow @nogc
|
|
{
|
|
static import core.math;
|
|
static import std.math;
|
|
if (core.math.fabs(y) <= 0.5)
|
|
return Complex!real(std.math.cosh(y), std.math.sinh(y));
|
|
else
|
|
{
|
|
auto z = std.math.exp(y);
|
|
auto zi = 0.5 / z;
|
|
z = 0.5 * z;
|
|
return Complex!real(z + zi, z - zi);
|
|
}
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow @nogc unittest
|
|
{
|
|
import std.math.trigonometry : cosh, sinh;
|
|
assert(coshisinh(3.0L) == complex(cosh(3.0L), sinh(3.0L)));
|
|
}
|
|
|
|
/**
|
|
Params: z = A complex number.
|
|
Returns: The square root of `z`.
|
|
*/
|
|
Complex!T sqrt(T)(Complex!T z) @safe pure nothrow @nogc
|
|
{
|
|
static import core.math;
|
|
typeof(return) c;
|
|
real x,y,w,r;
|
|
|
|
if (z == 0)
|
|
{
|
|
c = typeof(return)(0, 0);
|
|
}
|
|
else
|
|
{
|
|
real z_re = z.re;
|
|
real z_im = z.im;
|
|
|
|
x = core.math.fabs(z_re);
|
|
y = core.math.fabs(z_im);
|
|
if (x >= y)
|
|
{
|
|
r = y / x;
|
|
w = core.math.sqrt(x)
|
|
* core.math.sqrt(0.5 * (1 + core.math.sqrt(1 + r * r)));
|
|
}
|
|
else
|
|
{
|
|
r = x / y;
|
|
w = core.math.sqrt(y)
|
|
* core.math.sqrt(0.5 * (r + core.math.sqrt(1 + r * r)));
|
|
}
|
|
|
|
if (z_re >= 0)
|
|
{
|
|
c = typeof(return)(w, z_im / (w + w));
|
|
}
|
|
else
|
|
{
|
|
if (z_im < 0)
|
|
w = -w;
|
|
c = typeof(return)(z_im / (w + w), w);
|
|
}
|
|
}
|
|
return c;
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow unittest
|
|
{
|
|
static import core.math;
|
|
assert(sqrt(complex(0.0)) == 0.0);
|
|
assert(sqrt(complex(1.0L, 0)) == core.math.sqrt(1.0L));
|
|
assert(sqrt(complex(-1.0L, 0)) == complex(0, 1.0L));
|
|
assert(sqrt(complex(-8.0, -6.0)) == complex(1.0, -3.0));
|
|
}
|
|
|
|
@safe pure nothrow unittest
|
|
{
|
|
import std.math.operations : isClose;
|
|
|
|
auto c1 = complex(1.0, 1.0);
|
|
auto c2 = Complex!double(0.5, 2.0);
|
|
|
|
auto c1s = sqrt(c1);
|
|
assert(isClose(c1s.re, 1.09868411347));
|
|
assert(isClose(c1s.im, 0.455089860562));
|
|
|
|
auto c2s = sqrt(c2);
|
|
assert(isClose(c2s.re, 1.13171392428));
|
|
assert(isClose(c2s.im, 0.883615530876));
|
|
}
|
|
|
|
// support %f formatting of complex numbers
|
|
// https://issues.dlang.org/show_bug.cgi?id=10881
|
|
@safe unittest
|
|
{
|
|
import std.format : format;
|
|
|
|
auto x = complex(1.2, 3.4);
|
|
assert(format("%.2f", x) == "1.20+3.40i");
|
|
|
|
auto y = complex(1.2, -3.4);
|
|
assert(format("%.2f", y) == "1.20-3.40i");
|
|
}
|
|
|
|
@safe unittest
|
|
{
|
|
// Test wide string formatting
|
|
import std.format.write : formattedWrite;
|
|
wstring wformat(T)(string format, Complex!T c)
|
|
{
|
|
import std.array : appender;
|
|
auto w = appender!wstring();
|
|
auto n = formattedWrite(w, format, c);
|
|
return w.data;
|
|
}
|
|
|
|
auto x = complex(1.2, 3.4);
|
|
assert(wformat("%.2f", x) == "1.20+3.40i"w);
|
|
}
|
|
|
|
@safe unittest
|
|
{
|
|
// Test ease of use (vanilla toString() should be supported)
|
|
assert(complex(1.2, 3.4).toString() == "1.2+3.4i");
|
|
}
|
|
|
|
@safe pure nothrow @nogc unittest
|
|
{
|
|
auto c = complex(3.0L, 4.0L);
|
|
c = sqrt(c);
|
|
assert(c.re == 2.0L);
|
|
assert(c.im == 1.0L);
|
|
}
|
|
|
|
/**
|
|
* Calculates e$(SUPERSCRIPT x).
|
|
* Params:
|
|
* x = A complex number
|
|
* Returns:
|
|
* The complex base e exponential of `x`
|
|
*
|
|
* $(TABLE_SV
|
|
* $(TR $(TH x) $(TH exp(x)))
|
|
* $(TR $(TD ($(PLUSMN)0, +0)) $(TD (1, +0)))
|
|
* $(TR $(TD (any, +$(INFIN))) $(TD ($(NAN), $(NAN))))
|
|
* $(TR $(TD (any, $(NAN)) $(TD ($(NAN), $(NAN)))))
|
|
* $(TR $(TD (+$(INFIN), +0)) $(TD (+$(INFIN), +0)))
|
|
* $(TR $(TD (-$(INFIN), any)) $(TD ($(PLUSMN)0, cis(x.im))))
|
|
* $(TR $(TD (+$(INFIN), any)) $(TD ($(PLUSMN)$(INFIN), cis(x.im))))
|
|
* $(TR $(TD (-$(INFIN), +$(INFIN))) $(TD ($(PLUSMN)0, $(PLUSMN)0)))
|
|
* $(TR $(TD (+$(INFIN), +$(INFIN))) $(TD ($(PLUSMN)$(INFIN), $(NAN))))
|
|
* $(TR $(TD (-$(INFIN), $(NAN))) $(TD ($(PLUSMN)0, $(PLUSMN)0)))
|
|
* $(TR $(TD (+$(INFIN), $(NAN))) $(TD ($(PLUSMN)$(INFIN), $(NAN))))
|
|
* $(TR $(TD ($(NAN), +0)) $(TD ($(NAN), +0)))
|
|
* $(TR $(TD ($(NAN), any)) $(TD ($(NAN), $(NAN))))
|
|
* $(TR $(TD ($(NAN), $(NAN))) $(TD ($(NAN), $(NAN))))
|
|
* )
|
|
*/
|
|
Complex!T exp(T)(Complex!T x) @trusted pure nothrow @nogc // TODO: @safe
|
|
{
|
|
static import std.math;
|
|
|
|
// Handle special cases explicitly here, as fromPolar will otherwise
|
|
// cause them to return Complex!T(NaN, NaN), or with the wrong sign.
|
|
if (std.math.isInfinity(x.re))
|
|
{
|
|
if (std.math.isNaN(x.im))
|
|
{
|
|
if (std.math.signbit(x.re))
|
|
return Complex!T(0, std.math.copysign(0, x.im));
|
|
else
|
|
return x;
|
|
}
|
|
if (std.math.isInfinity(x.im))
|
|
{
|
|
if (std.math.signbit(x.re))
|
|
return Complex!T(0, std.math.copysign(0, x.im));
|
|
else
|
|
return Complex!T(T.infinity, -T.nan);
|
|
}
|
|
if (x.im == 0.0)
|
|
{
|
|
if (std.math.signbit(x.re))
|
|
return Complex!T(0.0);
|
|
else
|
|
return Complex!T(T.infinity);
|
|
}
|
|
}
|
|
if (std.math.isNaN(x.re))
|
|
{
|
|
if (std.math.isNaN(x.im) || std.math.isInfinity(x.im))
|
|
return Complex!T(T.nan, T.nan);
|
|
if (x.im == 0.0)
|
|
return x;
|
|
}
|
|
if (x.re == 0.0)
|
|
{
|
|
if (std.math.isNaN(x.im) || std.math.isInfinity(x.im))
|
|
return Complex!T(T.nan, T.nan);
|
|
if (x.im == 0.0)
|
|
return Complex!T(1.0, 0.0);
|
|
}
|
|
|
|
return fromPolar!(T, T)(std.math.exp(x.re), x.im);
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow @nogc unittest
|
|
{
|
|
import std.math.operations : isClose;
|
|
import std.math.constants : PI;
|
|
|
|
assert(exp(complex(0.0, 0.0)) == complex(1.0, 0.0));
|
|
|
|
auto a = complex(2.0, 1.0);
|
|
assert(exp(conj(a)) == conj(exp(a)));
|
|
|
|
auto b = exp(complex(0.0L, 1.0L) * PI);
|
|
assert(isClose(b, -1.0L, 0.0, 1e-15));
|
|
}
|
|
|
|
@safe pure nothrow @nogc unittest
|
|
{
|
|
import std.math.traits : isNaN, isInfinity;
|
|
|
|
auto a = exp(complex(0.0, double.infinity));
|
|
assert(a.re.isNaN && a.im.isNaN);
|
|
auto b = exp(complex(0.0, double.infinity));
|
|
assert(b.re.isNaN && b.im.isNaN);
|
|
auto c = exp(complex(0.0, double.nan));
|
|
assert(c.re.isNaN && c.im.isNaN);
|
|
|
|
auto d = exp(complex(+double.infinity, 0.0));
|
|
assert(d == complex(double.infinity, 0.0));
|
|
auto e = exp(complex(-double.infinity, 0.0));
|
|
assert(e == complex(0.0));
|
|
auto f = exp(complex(-double.infinity, 1.0));
|
|
assert(f == complex(0.0));
|
|
auto g = exp(complex(+double.infinity, 1.0));
|
|
assert(g == complex(double.infinity, double.infinity));
|
|
auto h = exp(complex(-double.infinity, +double.infinity));
|
|
assert(h == complex(0.0));
|
|
auto i = exp(complex(+double.infinity, +double.infinity));
|
|
assert(i.re.isInfinity && i.im.isNaN);
|
|
auto j = exp(complex(-double.infinity, double.nan));
|
|
assert(j == complex(0.0));
|
|
auto k = exp(complex(+double.infinity, double.nan));
|
|
assert(k.re.isInfinity && k.im.isNaN);
|
|
|
|
auto l = exp(complex(double.nan, 0));
|
|
assert(l.re.isNaN && l.im == 0.0);
|
|
auto m = exp(complex(double.nan, 1));
|
|
assert(m.re.isNaN && m.im.isNaN);
|
|
auto n = exp(complex(double.nan, double.nan));
|
|
assert(n.re.isNaN && n.im.isNaN);
|
|
}
|
|
|
|
@safe pure nothrow @nogc unittest
|
|
{
|
|
import std.math.constants : PI;
|
|
import std.math.operations : isClose;
|
|
|
|
auto a = exp(complex(0.0, -PI));
|
|
assert(isClose(a, -1.0, 0.0, 1e-15));
|
|
|
|
auto b = exp(complex(0.0, -2.0 * PI / 3.0));
|
|
assert(isClose(b, complex(-0.5L, -0.866025403784438646763L)));
|
|
|
|
auto c = exp(complex(0.0, PI / 3.0));
|
|
assert(isClose(c, complex(0.5L, 0.866025403784438646763L)));
|
|
|
|
auto d = exp(complex(0.0, 2.0 * PI / 3.0));
|
|
assert(isClose(d, complex(-0.5L, 0.866025403784438646763L)));
|
|
|
|
auto e = exp(complex(0.0, PI));
|
|
assert(isClose(e, -1.0, 0.0, 1e-15));
|
|
}
|
|
|
|
/**
|
|
* Calculate the natural logarithm of x.
|
|
* The branch cut is along the negative axis.
|
|
* Params:
|
|
* x = A complex number
|
|
* Returns:
|
|
* The complex natural logarithm of `x`
|
|
*
|
|
* $(TABLE_SV
|
|
* $(TR $(TH x) $(TH log(x)))
|
|
* $(TR $(TD (-0, +0)) $(TD (-$(INFIN), $(PI))))
|
|
* $(TR $(TD (+0, +0)) $(TD (-$(INFIN), +0)))
|
|
* $(TR $(TD (any, +$(INFIN))) $(TD (+$(INFIN), $(PI)/2)))
|
|
* $(TR $(TD (any, $(NAN))) $(TD ($(NAN), $(NAN))))
|
|
* $(TR $(TD (-$(INFIN), any)) $(TD (+$(INFIN), $(PI))))
|
|
* $(TR $(TD (+$(INFIN), any)) $(TD (+$(INFIN), +0)))
|
|
* $(TR $(TD (-$(INFIN), +$(INFIN))) $(TD (+$(INFIN), 3$(PI)/4)))
|
|
* $(TR $(TD (+$(INFIN), +$(INFIN))) $(TD (+$(INFIN), $(PI)/4)))
|
|
* $(TR $(TD ($(PLUSMN)$(INFIN), $(NAN))) $(TD (+$(INFIN), $(NAN))))
|
|
* $(TR $(TD ($(NAN), any)) $(TD ($(NAN), $(NAN))))
|
|
* $(TR $(TD ($(NAN), +$(INFIN))) $(TD (+$(INFIN), $(NAN))))
|
|
* $(TR $(TD ($(NAN), $(NAN))) $(TD ($(NAN), $(NAN))))
|
|
* )
|
|
*/
|
|
Complex!T log(T)(Complex!T x) @safe pure nothrow @nogc
|
|
{
|
|
static import std.math;
|
|
|
|
// Handle special cases explicitly here for better accuracy.
|
|
// The order here is important, so that the correct path is chosen.
|
|
if (std.math.isNaN(x.re))
|
|
{
|
|
if (std.math.isInfinity(x.im))
|
|
return Complex!T(T.infinity, T.nan);
|
|
else
|
|
return Complex!T(T.nan, T.nan);
|
|
}
|
|
if (std.math.isInfinity(x.re))
|
|
{
|
|
if (std.math.isNaN(x.im))
|
|
return Complex!T(T.infinity, T.nan);
|
|
else if (std.math.isInfinity(x.im))
|
|
{
|
|
if (std.math.signbit(x.re))
|
|
return Complex!T(T.infinity, std.math.copysign(3.0 * std.math.PI_4, x.im));
|
|
else
|
|
return Complex!T(T.infinity, std.math.copysign(std.math.PI_4, x.im));
|
|
}
|
|
else
|
|
{
|
|
if (std.math.signbit(x.re))
|
|
return Complex!T(T.infinity, std.math.copysign(std.math.PI, x.im));
|
|
else
|
|
return Complex!T(T.infinity, std.math.copysign(0.0, x.im));
|
|
}
|
|
}
|
|
if (std.math.isNaN(x.im))
|
|
return Complex!T(T.nan, T.nan);
|
|
if (std.math.isInfinity(x.im))
|
|
return Complex!T(T.infinity, std.math.copysign(std.math.PI_2, x.im));
|
|
if (x.re == 0.0 && x.im == 0.0)
|
|
{
|
|
if (std.math.signbit(x.re))
|
|
return Complex!T(-T.infinity, std.math.copysign(std.math.PI, x.im));
|
|
else
|
|
return Complex!T(-T.infinity, std.math.copysign(0.0, x.im));
|
|
}
|
|
|
|
return Complex!T(std.math.log(abs(x)), arg(x));
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow @nogc unittest
|
|
{
|
|
import core.math : sqrt;
|
|
import std.math.constants : PI;
|
|
import std.math.operations : isClose;
|
|
|
|
auto a = complex(2.0, 1.0);
|
|
assert(log(conj(a)) == conj(log(a)));
|
|
|
|
auto b = 2.0 * log10(complex(0.0, 1.0));
|
|
auto c = 4.0 * log10(complex(sqrt(2.0) / 2, sqrt(2.0) / 2));
|
|
assert(isClose(b, c, 0.0, 1e-15));
|
|
|
|
assert(log(complex(-1.0L, 0.0L)) == complex(0.0L, PI));
|
|
assert(log(complex(-1.0L, -0.0L)) == complex(0.0L, -PI));
|
|
}
|
|
|
|
@safe pure nothrow @nogc unittest
|
|
{
|
|
import std.math.traits : isNaN, isInfinity;
|
|
import std.math.constants : PI, PI_2, PI_4;
|
|
|
|
auto a = log(complex(-0.0L, 0.0L));
|
|
assert(a == complex(-real.infinity, PI));
|
|
auto b = log(complex(0.0L, 0.0L));
|
|
assert(b == complex(-real.infinity, +0.0L));
|
|
auto c = log(complex(1.0L, real.infinity));
|
|
assert(c == complex(real.infinity, PI_2));
|
|
auto d = log(complex(1.0L, real.nan));
|
|
assert(d.re.isNaN && d.im.isNaN);
|
|
|
|
auto e = log(complex(-real.infinity, 1.0L));
|
|
assert(e == complex(real.infinity, PI));
|
|
auto f = log(complex(real.infinity, 1.0L));
|
|
assert(f == complex(real.infinity, 0.0L));
|
|
auto g = log(complex(-real.infinity, real.infinity));
|
|
assert(g == complex(real.infinity, 3.0 * PI_4));
|
|
auto h = log(complex(real.infinity, real.infinity));
|
|
assert(h == complex(real.infinity, PI_4));
|
|
auto i = log(complex(real.infinity, real.nan));
|
|
assert(i.re.isInfinity && i.im.isNaN);
|
|
|
|
auto j = log(complex(real.nan, 1.0L));
|
|
assert(j.re.isNaN && j.im.isNaN);
|
|
auto k = log(complex(real.nan, real.infinity));
|
|
assert(k.re.isInfinity && k.im.isNaN);
|
|
auto l = log(complex(real.nan, real.nan));
|
|
assert(l.re.isNaN && l.im.isNaN);
|
|
}
|
|
|
|
@safe pure nothrow @nogc unittest
|
|
{
|
|
import std.math.constants : PI;
|
|
import std.math.operations : isClose;
|
|
|
|
auto a = log(fromPolar(1.0, PI / 6.0));
|
|
assert(isClose(a, complex(0.0L, 0.523598775598298873077L), 0.0, 1e-15));
|
|
|
|
auto b = log(fromPolar(1.0, PI / 3.0));
|
|
assert(isClose(b, complex(0.0L, 1.04719755119659774615L), 0.0, 1e-15));
|
|
|
|
auto c = log(fromPolar(1.0, PI / 2.0));
|
|
assert(isClose(c, complex(0.0L, 1.57079632679489661923L), 0.0, 1e-15));
|
|
|
|
auto d = log(fromPolar(1.0, 2.0 * PI / 3.0));
|
|
assert(isClose(d, complex(0.0L, 2.09439510239319549230L), 0.0, 1e-15));
|
|
|
|
auto e = log(fromPolar(1.0, 5.0 * PI / 6.0));
|
|
assert(isClose(e, complex(0.0L, 2.61799387799149436538L), 0.0, 1e-15));
|
|
|
|
auto f = log(complex(-1.0L, 0.0L));
|
|
assert(isClose(f, complex(0.0L, PI), 0.0, 1e-15));
|
|
}
|
|
|
|
/**
|
|
* Calculate the base-10 logarithm of x.
|
|
* Params:
|
|
* x = A complex number
|
|
* Returns:
|
|
* The complex base 10 logarithm of `x`
|
|
*/
|
|
Complex!T log10(T)(Complex!T x) @safe pure nothrow @nogc
|
|
{
|
|
import std.math.constants : LN10;
|
|
|
|
return log(x) / Complex!T(LN10);
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow @nogc unittest
|
|
{
|
|
import core.math : sqrt;
|
|
import std.math.constants : LN10, PI;
|
|
import std.math.operations : isClose;
|
|
|
|
auto a = complex(2.0, 1.0);
|
|
assert(log10(a) == log(a) / log(complex(10.0)));
|
|
|
|
auto b = log10(complex(0.0, 1.0)) * 2.0;
|
|
auto c = log10(complex(sqrt(2.0) / 2, sqrt(2.0) / 2)) * 4.0;
|
|
assert(isClose(b, c, 0.0, 1e-15));
|
|
}
|
|
|
|
@safe pure nothrow @nogc unittest
|
|
{
|
|
import std.math.constants : LN10, PI;
|
|
import std.math.operations : isClose;
|
|
|
|
auto a = log10(fromPolar(1.0, PI / 6.0));
|
|
assert(isClose(a, complex(0.0L, 0.227396058973640224580L), 0.0, 1e-15));
|
|
|
|
auto b = log10(fromPolar(1.0, PI / 3.0));
|
|
assert(isClose(b, complex(0.0L, 0.454792117947280449161L), 0.0, 1e-15));
|
|
|
|
auto c = log10(fromPolar(1.0, PI / 2.0));
|
|
assert(isClose(c, complex(0.0L, 0.682188176920920673742L), 0.0, 1e-15));
|
|
|
|
auto d = log10(fromPolar(1.0, 2.0 * PI / 3.0));
|
|
assert(isClose(d, complex(0.0L, 0.909584235894560898323L), 0.0, 1e-15));
|
|
|
|
auto e = log10(fromPolar(1.0, 5.0 * PI / 6.0));
|
|
assert(isClose(e, complex(0.0L, 1.13698029486820112290L), 0.0, 1e-15));
|
|
|
|
auto f = log10(complex(-1.0L, 0.0L));
|
|
assert(isClose(f, complex(0.0L, 1.36437635384184134748L), 0.0, 1e-15));
|
|
|
|
assert(ceqrel(log10(complex(-100.0L, 0.0L)), complex(2.0L, PI / LN10)) >= real.mant_dig - 1);
|
|
assert(ceqrel(log10(complex(-100.0L, -0.0L)), complex(2.0L, -PI / LN10)) >= real.mant_dig - 1);
|
|
}
|
|
|
|
/**
|
|
* Calculates x$(SUPERSCRIPT n).
|
|
* The branch cut is on the negative axis.
|
|
* Params:
|
|
* x = base
|
|
* n = exponent
|
|
* Returns:
|
|
* `x` raised to the power of `n`
|
|
*/
|
|
Complex!T pow(T, Int)(Complex!T x, const Int n) @safe pure nothrow @nogc
|
|
if (isIntegral!Int)
|
|
{
|
|
alias UInt = Unsigned!(Unqual!Int);
|
|
|
|
UInt m = (n < 0) ? -cast(UInt) n : n;
|
|
Complex!T y = (m % 2) ? x : Complex!T(1);
|
|
|
|
while (m >>= 1)
|
|
{
|
|
x *= x;
|
|
if (m % 2)
|
|
y *= x;
|
|
}
|
|
|
|
return (n < 0) ? Complex!T(1) / y : y;
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow @nogc unittest
|
|
{
|
|
import std.math.operations : isClose;
|
|
|
|
auto a = complex(1.0, 2.0);
|
|
assert(pow(a, 2) == a * a);
|
|
assert(pow(a, 3) == a * a * a);
|
|
assert(pow(a, -2) == 1.0 / (a * a));
|
|
assert(isClose(pow(a, -3), 1.0 / (a * a * a)));
|
|
}
|
|
|
|
/// ditto
|
|
Complex!T pow(T)(Complex!T x, const T n) @trusted pure nothrow @nogc
|
|
{
|
|
static import std.math;
|
|
|
|
if (x == 0.0)
|
|
return Complex!T(0.0);
|
|
|
|
if (x.im == 0 && x.re > 0.0)
|
|
return Complex!T(std.math.pow(x.re, n));
|
|
|
|
Complex!T t = log(x);
|
|
return fromPolar!(T, T)(std.math.exp(n * t.re), n * t.im);
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow @nogc unittest
|
|
{
|
|
import std.math.operations : isClose;
|
|
assert(pow(complex(0.0), 2.0) == complex(0.0));
|
|
assert(pow(complex(5.0), 2.0) == complex(25.0));
|
|
|
|
auto a = pow(complex(-1.0, 0.0), 0.5);
|
|
assert(isClose(a, complex(0.0, +1.0), 0.0, 1e-16));
|
|
|
|
auto b = pow(complex(-1.0, -0.0), 0.5);
|
|
assert(isClose(b, complex(0.0, -1.0), 0.0, 1e-16));
|
|
}
|
|
|
|
/// ditto
|
|
Complex!T pow(T)(Complex!T x, Complex!T y) @trusted pure nothrow @nogc
|
|
{
|
|
return (x == 0) ? Complex!T(0) : exp(y * log(x));
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow @nogc unittest
|
|
{
|
|
import std.math.operations : isClose;
|
|
import std.math.exponential : exp;
|
|
import std.math.constants : PI;
|
|
auto a = complex(0.0);
|
|
auto b = complex(2.0);
|
|
assert(pow(a, b) == complex(0.0));
|
|
|
|
auto c = complex(0.0L, 1.0L);
|
|
assert(isClose(pow(c, c), exp((-PI) / 2)));
|
|
}
|
|
|
|
/// ditto
|
|
Complex!T pow(T)(const T x, Complex!T n) @trusted pure nothrow @nogc
|
|
{
|
|
static import std.math;
|
|
|
|
return (x > 0.0)
|
|
? fromPolar!(T, T)(std.math.pow(x, n.re), n.im * std.math.log(x))
|
|
: pow(Complex!T(x), n);
|
|
}
|
|
|
|
///
|
|
@safe pure nothrow @nogc unittest
|
|
{
|
|
import std.math.operations : isClose;
|
|
assert(pow(2.0, complex(0.0)) == complex(1.0));
|
|
assert(pow(2.0, complex(5.0)) == complex(32.0));
|
|
|
|
auto a = pow(-2.0, complex(-1.0));
|
|
assert(isClose(a, complex(-0.5), 0.0, 1e-16));
|
|
|
|
auto b = pow(-0.5, complex(-1.0));
|
|
assert(isClose(b, complex(-2.0), 0.0, 1e-15));
|
|
}
|
|
|
|
@safe pure nothrow @nogc unittest
|
|
{
|
|
import std.math.constants : PI;
|
|
import std.math.operations : isClose;
|
|
|
|
auto a = pow(complex(3.0, 4.0), 2);
|
|
assert(isClose(a, complex(-7.0, 24.0)));
|
|
|
|
auto b = pow(complex(3.0, 4.0), PI);
|
|
assert(ceqrel(b, complex(-152.91512205297134, 35.547499631917738)) >= double.mant_dig - 3);
|
|
|
|
auto c = pow(complex(3.0, 4.0), complex(-2.0, 1.0));
|
|
assert(ceqrel(c, complex(0.015351734187477306, -0.0038407695456661503)) >= double.mant_dig - 3);
|
|
|
|
auto d = pow(PI, complex(2.0, -1.0));
|
|
assert(ceqrel(d, complex(4.0790296880118296, -8.9872469554541869)) >= double.mant_dig - 1);
|
|
|
|
auto e = complex(2.0);
|
|
assert(ceqrel(pow(e, 3), exp(3 * log(e))) >= double.mant_dig - 1);
|
|
}
|
|
|
|
@safe pure nothrow @nogc unittest
|
|
{
|
|
import std.meta : AliasSeq;
|
|
import std.math.traits : floatTraits, RealFormat;
|
|
static foreach (T; AliasSeq!(float, double, real))
|
|
{{
|
|
static if (floatTraits!T.realFormat == RealFormat.ibmExtended)
|
|
{
|
|
/* For IBM real, epsilon is too small (since 1.0 plus any double is
|
|
representable) to be able to expect results within epsilon * 100. */
|
|
}
|
|
else
|
|
{
|
|
T eps = T.epsilon * 100;
|
|
|
|
T a = -1.0;
|
|
T b = 0.5;
|
|
Complex!T ref1 = pow(complex(a), complex(b));
|
|
Complex!T res1 = pow(a, complex(b));
|
|
Complex!T res2 = pow(complex(a), b);
|
|
assert(abs(ref1 - res1) < eps);
|
|
assert(abs(ref1 - res2) < eps);
|
|
assert(abs(res1 - res2) < eps);
|
|
|
|
T c = -3.2;
|
|
T d = 1.4;
|
|
Complex!T ref2 = pow(complex(a), complex(b));
|
|
Complex!T res3 = pow(a, complex(b));
|
|
Complex!T res4 = pow(complex(a), b);
|
|
assert(abs(ref2 - res3) < eps);
|
|
assert(abs(ref2 - res4) < eps);
|
|
assert(abs(res3 - res4) < eps);
|
|
}
|
|
}}
|
|
}
|
|
|
|
@safe pure nothrow @nogc unittest
|
|
{
|
|
import std.meta : AliasSeq;
|
|
static foreach (T; AliasSeq!(float, double, real))
|
|
{{
|
|
auto c = Complex!T(123, 456);
|
|
auto n = c.toNative();
|
|
assert(c.re == n.re && c.im == n.im);
|
|
}}
|
|
}
|