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 // Written in the D programming language. /** This module contains the \$(LREF Complex) type, which is used to represent complex numbers, along with related mathematical operations and functions. \$(LREF Complex) will eventually \$(DDLINK deprecate, Deprecated Features, replace) the built-in types `cfloat`, `cdouble`, `creal`, `ifloat`, `idouble`, and `ireal`. Macros: TABLE_SV = Special Values \$0 PLUSMN = ± NAN = \$(RED NAN) INFIN = ∞ PI = π Authors: Lars Tandle Kyllingstad, Don Clugston Copyright: Copyright (c) 2010, Lars T. Kyllingstad. License: \$(HTTP boost.org/LICENSE_1_0.txt, Boost License 1.0) Source: \$(PHOBOSSRC std/complex.d) */ module std.complex; import std.traits; /** Helper function that returns a complex number with the specified real and imaginary parts. Params: R = (template parameter) type of real part of complex number I = (template parameter) type of imaginary part of complex number re = real part of complex number to be constructed im = (optional) imaginary part of complex number, 0 if omitted. Returns: `Complex` instance with real and imaginary parts set to the values provided as input. If neither `re` nor `im` are floating-point numbers, the return type will be `Complex!double`. Otherwise, the return type is deduced using \$(D std.traits.CommonType!(R, I)). */ auto complex(R)(const R re) @safe pure nothrow @nogc if (is(R : double)) { static if (isFloatingPoint!R) return Complex!R(re, 0); else return Complex!double(re, 0); } /// ditto auto complex(R, I)(const R re, const I im) @safe pure nothrow @nogc if (is(R : double) && is(I : double)) { static if (isFloatingPoint!R || isFloatingPoint!I) return Complex!(CommonType!(R, I))(re, im); else return Complex!double(re, im); } /// @safe pure nothrow unittest { auto a = complex(1.0); static assert(is(typeof(a) == Complex!double)); assert(a.re == 1.0); assert(a.im == 0.0); auto b = complex(2.0L); static assert(is(typeof(b) == Complex!real)); assert(b.re == 2.0L); assert(b.im == 0.0L); auto c = complex(1.0, 2.0); static assert(is(typeof(c) == Complex!double)); assert(c.re == 1.0); assert(c.im == 2.0); auto d = complex(3.0, 4.0L); static assert(is(typeof(d) == Complex!real)); assert(d.re == 3.0); assert(d.im == 4.0L); auto e = complex(1); static assert(is(typeof(e) == Complex!double)); assert(e.re == 1); assert(e.im == 0); auto f = complex(1L, 2); static assert(is(typeof(f) == Complex!double)); assert(f.re == 1L); assert(f.im == 2); auto g = complex(3, 4.0L); static assert(is(typeof(g) == Complex!real)); assert(g.re == 3); assert(g.im == 4.0L); } /** A complex number parametrised by a type `T`, which must be either `float`, `double` or `real`. */ struct Complex(T) if (isFloatingPoint!T) { import std.format.spec : FormatSpec; import std.range.primitives : isOutputRange; /** The real part of the number. */ T re; /** The imaginary part of the number. */ T im; /** Converts the complex number to a string representation. The second form of this function is usually not called directly; instead, it is used via \$(REF format, std,string), as shown in the examples below. Supported format characters are 'e', 'f', 'g', 'a', and 's'. See the \$(MREF std, format) and \$(REF format, std,string) documentation for more information. */ string toString() const @safe /* TODO: pure nothrow */ { import std.exception : assumeUnique; char[] buf; buf.reserve(100); auto fmt = FormatSpec!char("%s"); toString((const(char)[] s) { buf ~= s; }, fmt); static trustedAssumeUnique(T)(T t) @trusted { return assumeUnique(t); } return trustedAssumeUnique(buf); } static if (is(T == double)) /// @safe unittest { auto c = complex(1.2, 3.4); // Vanilla toString formatting: assert(c.toString() == "1.2+3.4i"); // Formatting with std.string.format specs: the precision and width // specifiers apply to both the real and imaginary parts of the // complex number. import std.format : format; assert(format("%.2f", c) == "1.20+3.40i"); assert(format("%4.1f", c) == " 1.2+ 3.4i"); } /// ditto void toString(Writer, Char)(scope Writer w, scope const ref FormatSpec!Char formatSpec) const if (isOutputRange!(Writer, const(Char)[])) { import std.format.write : formatValue; import std.math.traits : signbit; import std.range.primitives : put; formatValue(w, re, formatSpec); if (signbit(im) == 0) put(w, "+"); formatValue(w, im, formatSpec); put(w, "i"); } @safe pure nothrow @nogc: /** Construct a complex number with the specified real and imaginary parts. In the case where a single argument is passed that is not complex, the imaginary part of the result will be zero. */ this(R : T)(Complex!R z) { re = z.re; im = z.im; } /// ditto this(Rx : T, Ry : T)(const Rx x, const Ry y) { re = x; im = y; } /// ditto this(R : T)(const R r) { re = r; im = 0; } // ASSIGNMENT OPERATORS // this = complex ref Complex opAssign(R : T)(Complex!R z) { re = z.re; im = z.im; return this; } // this = numeric ref Complex opAssign(R : T)(const R r) { re = r; im = 0; return this; } // COMPARISON OPERATORS // this == complex bool opEquals(R : T)(Complex!R z) const { return re == z.re && im == z.im; } // this == numeric bool opEquals(R : T)(const R r) const { return re == r && im == 0; } // UNARY OPERATORS // +complex Complex opUnary(string op)() const if (op == "+") { return this; } // -complex Complex opUnary(string op)() const if (op == "-") { return Complex(-re, -im); } // BINARY OPERATORS // complex op complex Complex!(CommonType!(T,R)) opBinary(string op, R)(Complex!R z) const { alias C = typeof(return); auto w = C(this.re, this.im); return w.opOpAssign!(op)(z); } // complex op numeric Complex!(CommonType!(T,R)) opBinary(string op, R)(const R r) const if (isNumeric!R) { alias C = typeof(return); auto w = C(this.re, this.im); return w.opOpAssign!(op)(r); } // numeric + complex, numeric * complex Complex!(CommonType!(T, R)) opBinaryRight(string op, R)(const R r) const if ((op == "+" || op == "*") && (isNumeric!R)) { return opBinary!(op)(r); } // numeric - complex Complex!(CommonType!(T, R)) opBinaryRight(string op, R)(const R r) const if (op == "-" && isNumeric!R) { return Complex(r - re, -im); } // numeric / complex Complex!(CommonType!(T, R)) opBinaryRight(string op, R)(const R r) const if (op == "/" && isNumeric!R) { version (FastMath) { // Compute norm(this) immutable norm = re * re + im * im; // Compute r * conj(this) immutable prod_re = r * re; immutable prod_im = r * -im; // Divide the product by the norm typeof(return) w = void; w.re = prod_re / norm; w.im = prod_im / norm; return w; } else { import core.math : fabs; typeof(return) w = void; if (fabs(re) < fabs(im)) { immutable ratio = re/im; immutable rdivd = r/(re*ratio + im); w.re = rdivd*ratio; w.im = -rdivd; } else { immutable ratio = im/re; immutable rdivd = r/(re + im*ratio); w.re = rdivd; w.im = -rdivd*ratio; } return w; } } // numeric ^^ complex Complex!(CommonType!(T, R)) opBinaryRight(string op, R)(const R lhs) const if (op == "^^" && isNumeric!R) { import core.math : cos, sin; import std.math.exponential : exp, log; import std.math.constants : PI; Unqual!(CommonType!(T, R)) ab = void, ar = void; if (lhs >= 0) { // r = lhs // theta = 0 ab = lhs ^^ this.re; ar = log(lhs) * this.im; } else { // r = -lhs // theta = PI ab = (-lhs) ^^ this.re * exp(-PI * this.im); ar = PI * this.re + log(-lhs) * this.im; } return typeof(return)(ab * cos(ar), ab * sin(ar)); } // OP-ASSIGN OPERATORS // complex += complex, complex -= complex ref Complex opOpAssign(string op, C)(const C z) if ((op == "+" || op == "-") && is(C R == Complex!R)) { mixin ("re "~op~"= z.re;"); mixin ("im "~op~"= z.im;"); return this; } // complex *= complex ref Complex opOpAssign(string op, C)(const C z) if (op == "*" && is(C R == Complex!R)) { auto temp = re*z.re - im*z.im; im = im*z.re + re*z.im; re = temp; return this; } // complex /= complex ref Complex opOpAssign(string op, C)(const C z) if (op == "/" && is(C R == Complex!R)) { version (FastMath) { // Compute norm(z) immutable norm = z.re * z.re + z.im * z.im; // Compute this * conj(z) immutable prod_re = re * z.re - im * -z.im; immutable prod_im = im * z.re + re * -z.im; // Divide the product by the norm re = prod_re / norm; im = prod_im / norm; return this; } else { import core.math : fabs; if (fabs(z.re) < fabs(z.im)) { immutable ratio = z.re/z.im; immutable denom = z.re*ratio + z.im; immutable temp = (re*ratio + im)/denom; im = (im*ratio - re)/denom; re = temp; } else { immutable ratio = z.im/z.re; immutable denom = z.re + z.im*ratio; immutable temp = (re + im*ratio)/denom; im = (im - re*ratio)/denom; re = temp; } return this; } } // complex ^^= complex ref Complex opOpAssign(string op, C)(const C z) if (op == "^^" && is(C R == Complex!R)) { import core.math : cos, sin; import std.math.exponential : exp, log; immutable r = abs(this); immutable t = arg(this); immutable ab = r^^z.re * exp(-t*z.im); immutable ar = t*z.re + log(r)*z.im; re = ab*cos(ar); im = ab*sin(ar); return this; } // complex += numeric, complex -= numeric ref Complex opOpAssign(string op, U : T)(const U a) if (op == "+" || op == "-") { mixin ("re "~op~"= a;"); return this; } // complex *= numeric, complex /= numeric ref Complex opOpAssign(string op, U : T)(const U a) if (op == "*" || op == "/") { mixin ("re "~op~"= a;"); mixin ("im "~op~"= a;"); return this; } // complex ^^= real ref Complex opOpAssign(string op, R)(const R r) if (op == "^^" && isFloatingPoint!R) { import core.math : cos, sin; immutable ab = abs(this)^^r; immutable ar = arg(this)*r; re = ab*cos(ar); im = ab*sin(ar); return this; } // complex ^^= int ref Complex opOpAssign(string op, U)(const U i) if (op == "^^" && isIntegral!U) { switch (i) { case 0: re = 1.0; im = 0.0; break; case 1: // identity; do nothing break; case 2: this *= this; break; case 3: auto z = this; this *= z; this *= z; break; default: this ^^= cast(real) i; } return this; } } @safe pure nothrow unittest { import std.complex; static import core.math; import std.math; enum EPS = double.epsilon; auto c1 = complex(1.0, 1.0); // Check unary operations. auto c2 = Complex!double(0.5, 2.0); assert(c2 == +c2); assert((-c2).re == -(c2.re)); assert((-c2).im == -(c2.im)); assert(c2 == -(-c2)); // Check complex-complex operations. auto cpc = c1 + c2; assert(cpc.re == c1.re + c2.re); assert(cpc.im == c1.im + c2.im); auto cmc = c1 - c2; assert(cmc.re == c1.re - c2.re); assert(cmc.im == c1.im - c2.im); auto ctc = c1 * c2; assert(isClose(abs(ctc), abs(c1)*abs(c2), EPS)); assert(isClose(arg(ctc), arg(c1)+arg(c2), EPS)); auto cdc = c1 / c2; assert(isClose(abs(cdc), abs(c1)/abs(c2), EPS)); assert(isClose(arg(cdc), arg(c1)-arg(c2), EPS)); auto cec = c1^^c2; assert(isClose(cec.re, 0.1152413197994, 1e-12)); assert(isClose(cec.im, 0.2187079045274, 1e-12)); // Check complex-real operations. double a = 123.456; auto cpr = c1 + a; assert(cpr.re == c1.re + a); assert(cpr.im == c1.im); auto cmr = c1 - a; 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); } /** Params: z = A complex number. Returns: The absolute value (or 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 @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(2.0), 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 issue 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 issue 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 { static import std.math; return log(x) / Complex!T(std.math.log(10.0)); } /// @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 : RealFormat, floatTraits; 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); } }} }