diff --git a/core/math/cmplx/cmplx.odin b/core/math/cmplx/cmplx.odin new file mode 100644 index 000000000..c029be30c --- /dev/null +++ b/core/math/cmplx/cmplx.odin @@ -0,0 +1,513 @@ +package math_cmplx + +import "core:builtin" +import "core:math" + +// The original C code, the long comment, and the constants +// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c. +// The go code is a simplified version of the original C. +// +// Cephes Math Library Release 2.8: June, 2000 +// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier +// +// The readme file at http://netlib.sandia.gov/cephes/ says: +// Some software in this archive may be from the book _Methods and +// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster +// International, 1989) or from the Cephes Mathematical Library, a +// commercial product. In either event, it is copyrighted by the author. +// What you see here may be used freely but it comes with no support or +// guarantee. +// +// The two known misprints in the book are repaired here in the +// source listings for the gamma function and the incomplete beta +// integral. +// +// Stephen L. Moshier +// moshier@na-net.ornl.gov + +abs :: builtin.abs +conj :: builtin.conj +real :: builtin.real +imag :: builtin.imag +jmag :: builtin.jmag +kmag :: builtin.kmag + + +sin :: proc{ + sin_complex128, +} +cos :: proc{ + cos_complex128, +} +tan :: proc{ + tan_complex128, +} +cot :: proc{ + cot_complex128, +} + + +sinh :: proc{ + sinh_complex128, +} +cosh :: proc{ + cosh_complex128, +} +tanh :: proc{ + tanh_complex128, +} + + + +// sqrt returns the square root of x. +// The result r is chosen so that real(r) ≥ 0 and imag(r) has the same sign as imag(x). +sqrt :: proc{ + sqrt_complex32, + sqrt_complex64, + sqrt_complex128, +} +ln :: proc{ + ln_complex32, + ln_complex64, + ln_complex128, +} +log10 :: proc{ + log10_complex32, + log10_complex64, + log10_complex128, +} + +exp :: proc{ + exp_complex32, + exp_complex64, + exp_complex128, +} + +pow :: proc{ + pow_complex32, + pow_complex64, + pow_complex128, +} + +phase :: proc{ + phase_complex32, + phase_complex64, + phase_complex128, +} + +polar :: proc{ + polar_complex32, + polar_complex64, + polar_complex128, +} + +is_inf :: proc{ + is_inf_complex32, + is_inf_complex64, + is_inf_complex128, +} + +is_nan :: proc{ + is_nan_complex32, + is_nan_complex64, + is_nan_complex128, +} + + + +// sqrt_complex32 returns the square root of x. +// The result r is chosen so that real(r) ≥ 0 and imag(r) has the same sign as imag(x). +sqrt_complex32 :: proc "contextless" (x: complex32) -> complex32 { + return complex32(sqrt_complex128(complex128(x))) +} + +// sqrt_complex64 returns the square root of x. +// The result r is chosen so that real(r) ≥ 0 and imag(r) has the same sign as imag(x). +sqrt_complex64 :: proc "contextless" (x: complex64) -> complex64 { + return complex64(sqrt_complex128(complex128(x))) +} + + +// sqrt_complex128 returns the square root of x. +// The result r is chosen so that real(r) ≥ 0 and imag(r) has the same sign as imag(x). +sqrt_complex128 :: proc "contextless" (x: complex128) -> complex128 { + // The original C code, the long comment, and the constants + // below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c. + // The go code is a simplified version of the original C. + // + // Cephes Math Library Release 2.8: June, 2000 + // Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier + // + // The readme file at http://netlib.sandia.gov/cephes/ says: + // Some software in this archive may be from the book _Methods and + // Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster + // International, 1989) or from the Cephes Mathematical Library, a + // commercial product. In either event, it is copyrighted by the author. + // What you see here may be used freely but it comes with no support or + // guarantee. + // + // The two known misprints in the book are repaired here in the + // source listings for the gamma function and the incomplete beta + // integral. + // + // Stephen L. Moshier + // moshier@na-net.ornl.gov + + // Complex square root + // + // DESCRIPTION: + // + // If z = x + iy, r = |z|, then + // + // 1/2 + // Re w = [ (r + x)/2 ] , + // + // 1/2 + // Im w = [ (r - x)/2 ] . + // + // Cancellation error in r-x or r+x is avoided by using the + // identity 2 Re w Im w = y. + // + // Note that -w is also a square root of z. The root chosen + // is always in the right half plane and Im w has the same sign as y. + // + // ACCURACY: + // + // Relative error: + // arithmetic domain # trials peak rms + // DEC -10,+10 25000 3.2e-17 9.6e-18 + // IEEE -10,+10 1,000,000 2.9e-16 6.1e-17 + + if imag(x) == 0 { + // Ensure that imag(r) has the same sign as imag(x) for imag(x) == signed zero. + if real(x) == 0 { + return complex(0, imag(x)) + } + if real(x) < 0 { + return complex(0, math.copy_sign(math.sqrt(-real(x)), imag(x))) + } + return complex(math.sqrt(real(x)), imag(x)) + } else if math.is_inf(imag(x), 0) { + return complex(math.inf_f64(1.0), imag(x)) + } + if real(x) == 0 { + if imag(x) < 0 { + r := math.sqrt(-0.5 * imag(x)) + return complex(r, -r) + } + r := math.sqrt(0.5 * imag(x)) + return complex(r, r) + } + a := real(x) + b := imag(x) + scale: f64 + // Rescale to avoid internal overflow or underflow. + if abs(a) > 4 || abs(b) > 4 { + a *= 0.25 + b *= 0.25 + scale = 2 + } else { + a *= 1.8014398509481984e16 // 2**54 + b *= 1.8014398509481984e16 + scale = 7.450580596923828125e-9 // 2**-27 + } + r := math.hypot(a, b) + t: f64 + if a > 0 { + t = math.sqrt(0.5*r + 0.5*a) + r = scale * abs((0.5*b)/t) + t *= scale + } else { + r = math.sqrt(0.5*r - 0.5*a) + t = scale * abs((0.5*b)/r) + r *= scale + } + if b < 0 { + return complex(t, -r) + } + return complex(t, r) +} + +ln_complex32 :: proc "contextless" (x: complex32) -> complex32 { + return complex(math.ln(abs(x)), phase(x)) +} +ln_complex64 :: proc "contextless" (x: complex64) -> complex64 { + return complex(math.ln(abs(x)), phase(x)) +} +ln_complex128 :: proc "contextless" (x: complex128) -> complex128 { + return complex(math.ln(abs(x)), phase(x)) +} + + +exp_complex32 :: proc "contextless" (x: complex32) -> complex32 { + switch re, im := real(x), imag(x); { + case math.is_inf(re, 0): + switch { + case re > 0 && im == 0: + return x + case math.is_inf(im, 0) || math.is_nan(im): + if re < 0 { + return complex(0, math.copy_sign(0, im)) + } else { + return complex(math.inf_f64(1.0), math.nan_f64()) + } + } + case math.is_nan(re): + if im == 0 { + return complex(math.nan_f16(), im) + } + } + r := math.exp(real(x)) + s, c := math.sincos(imag(x)) + return complex(r*c, r*s) +} +exp_complex64 :: proc "contextless" (x: complex64) -> complex64 { + switch re, im := real(x), imag(x); { + case math.is_inf(re, 0): + switch { + case re > 0 && im == 0: + return x + case math.is_inf(im, 0) || math.is_nan(im): + if re < 0 { + return complex(0, math.copy_sign(0, im)) + } else { + return complex(math.inf_f64(1.0), math.nan_f64()) + } + } + case math.is_nan(re): + if im == 0 { + return complex(math.nan_f32(), im) + } + } + r := math.exp(real(x)) + s, c := math.sincos(imag(x)) + return complex(r*c, r*s) +} +exp_complex128 :: proc "contextless" (x: complex128) -> complex128 { + switch re, im := real(x), imag(x); { + case math.is_inf(re, 0): + switch { + case re > 0 && im == 0: + return x + case math.is_inf(im, 0) || math.is_nan(im): + if re < 0 { + return complex(0, math.copy_sign(0, im)) + } else { + return complex(math.inf_f64(1.0), math.nan_f64()) + } + } + case math.is_nan(re): + if im == 0 { + return complex(math.nan_f64(), im) + } + } + r := math.exp(real(x)) + s, c := math.sincos(imag(x)) + return complex(r*c, r*s) +} + + +pow_complex32 :: proc "contextless" (x, y: complex32) -> complex32 { + if x == 0 { // Guaranteed also true for x == -0. + if is_nan(y) { + return nan_complex32() + } + r, i := real(y), imag(y) + switch { + case r == 0: + return 1 + case r < 0: + if i == 0 { + return complex(math.inf_f16(1), 0) + } + return inf_complex32() + case r > 0: + return 0 + } + unreachable() + } + modulus := abs(x) + if modulus == 0 { + return complex(0, 0) + } + r := math.pow(modulus, real(y)) + arg := phase(x) + theta := real(y) * arg + if imag(y) != 0 { + r *= math.exp(-imag(y) * arg) + theta += imag(y) * math.ln(modulus) + } + s, c := math.sincos(theta) + return complex(r*c, r*s) +} +pow_complex64 :: proc "contextless" (x, y: complex64) -> complex64 { + if x == 0 { // Guaranteed also true for x == -0. + if is_nan(y) { + return nan_complex64() + } + r, i := real(y), imag(y) + switch { + case r == 0: + return 1 + case r < 0: + if i == 0 { + return complex(math.inf_f32(1), 0) + } + return inf_complex64() + case r > 0: + return 0 + } + unreachable() + } + modulus := abs(x) + if modulus == 0 { + return complex(0, 0) + } + r := math.pow(modulus, real(y)) + arg := phase(x) + theta := real(y) * arg + if imag(y) != 0 { + r *= math.exp(-imag(y) * arg) + theta += imag(y) * math.ln(modulus) + } + s, c := math.sincos(theta) + return complex(r*c, r*s) +} +pow_complex128 :: proc "contextless" (x, y: complex128) -> complex128 { + if x == 0 { // Guaranteed also true for x == -0. + if is_nan(y) { + return nan_complex128() + } + r, i := real(y), imag(y) + switch { + case r == 0: + return 1 + case r < 0: + if i == 0 { + return complex(math.inf_f64(1), 0) + } + return inf_complex128() + case r > 0: + return 0 + } + unreachable() + } + modulus := abs(x) + if modulus == 0 { + return complex(0, 0) + } + r := math.pow(modulus, real(y)) + arg := phase(x) + theta := real(y) * arg + if imag(y) != 0 { + r *= math.exp(-imag(y) * arg) + theta += imag(y) * math.ln(modulus) + } + s, c := math.sincos(theta) + return complex(r*c, r*s) +} + + + +log10_complex32 :: proc "contextless" (x: complex32) -> complex32 { + return math.LN10*ln(x) +} +log10_complex64 :: proc "contextless" (x: complex64) -> complex64 { + return math.LN10*ln(x) +} +log10_complex128 :: proc "contextless" (x: complex128) -> complex128 { + return math.LN10*ln(x) +} + + +phase_complex32 :: proc "contextless" (x: complex32) -> f16 { + return math.atan2(imag(x), real(x)) +} +phase_complex64 :: proc "contextless" (x: complex64) -> f32 { + return math.atan2(imag(x), real(x)) +} +phase_complex128 :: proc "contextless" (x: complex128) -> f64 { + return math.atan2(imag(x), real(x)) +} + + +rect_complex32 :: proc "contextless" (r, θ: f16) -> complex32 { + s, c := math.sincos(θ) + return complex(r*c, r*s) +} +rect_complex64 :: proc "contextless" (r, θ: f32) -> complex64 { + s, c := math.sincos(θ) + return complex(r*c, r*s) +} +rect_complex128 :: proc "contextless" (r, θ: f64) -> complex128 { + s, c := math.sincos(θ) + return complex(r*c, r*s) +} + +polar_complex32 :: proc "contextless" (x: complex32) -> (r, θ: f16) { + return abs(x), phase(x) +} +polar_complex64 :: proc "contextless" (x: complex64) -> (r, θ: f32) { + return abs(x), phase(x) +} +polar_complex128 :: proc "contextless" (x: complex128) -> (r, θ: f64) { + return abs(x), phase(x) +} + + + + +nan_complex32 :: proc "contextless" () -> complex32 { + return complex(math.nan_f16(), math.nan_f16()) +} +nan_complex64 :: proc "contextless" () -> complex64 { + return complex(math.nan_f32(), math.nan_f32()) +} +nan_complex128 :: proc "contextless" () -> complex128 { + return complex(math.nan_f64(), math.nan_f64()) +} + + +inf_complex32 :: proc "contextless" () -> complex32 { + inf := math.inf_f16(1) + return complex(inf, inf) +} +inf_complex64 :: proc "contextless" () -> complex64 { + inf := math.inf_f32(1) + return complex(inf, inf) +} +inf_complex128 :: proc "contextless" () -> complex128 { + inf := math.inf_f64(1) + return complex(inf, inf) +} + + +is_inf_complex32 :: proc "contextless" (x: complex32) -> bool { + return math.is_inf(real(x), 0) || math.is_inf(imag(x), 0) +} +is_inf_complex64 :: proc "contextless" (x: complex64) -> bool { + return math.is_inf(real(x), 0) || math.is_inf(imag(x), 0) +} +is_inf_complex128 :: proc "contextless" (x: complex128) -> bool { + return math.is_inf(real(x), 0) || math.is_inf(imag(x), 0) +} + + +is_nan_complex32 :: proc "contextless" (x: complex32) -> bool { + if math.is_inf(real(x), 0) || math.is_inf(imag(x), 0) { + return false + } + return math.is_nan(real(x)) || math.is_nan(imag(x)) +} +is_nan_complex64 :: proc "contextless" (x: complex64) -> bool { + if math.is_inf(real(x), 0) || math.is_inf(imag(x), 0) { + return false + } + return math.is_nan(real(x)) || math.is_nan(imag(x)) +} +is_nan_complex128 :: proc "contextless" (x: complex128) -> bool { + if math.is_inf(real(x), 0) || math.is_inf(imag(x), 0) { + return false + } + return math.is_nan(real(x)) || math.is_nan(imag(x)) +} diff --git a/core/math/cmplx/cmplx_invtrig.odin b/core/math/cmplx/cmplx_invtrig.odin new file mode 100644 index 000000000..a746a370f --- /dev/null +++ b/core/math/cmplx/cmplx_invtrig.odin @@ -0,0 +1,273 @@ +package math_cmplx + +import "core:builtin" +import "core:math" + +// The original C code, the long comment, and the constants +// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c. +// The go code is a simplified version of the original C. +// +// Cephes Math Library Release 2.8: June, 2000 +// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier +// +// The readme file at http://netlib.sandia.gov/cephes/ says: +// Some software in this archive may be from the book _Methods and +// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster +// International, 1989) or from the Cephes Mathematical Library, a +// commercial product. In either event, it is copyrighted by the author. +// What you see here may be used freely but it comes with no support or +// guarantee. +// +// The two known misprints in the book are repaired here in the +// source listings for the gamma function and the incomplete beta +// integral. +// +// Stephen L. Moshier +// moshier@na-net.ornl.gov + +acos :: proc{ + acos_complex32, + acos_complex64, + acos_complex128, +} +acosh :: proc{ + acosh_complex32, + acosh_complex64, + acosh_complex128, +} + +asin :: proc{ + asin_complex32, + asin_complex64, + asin_complex128, +} +asinh :: proc{ + asinh_complex32, + asinh_complex64, + asinh_complex128, +} + +atan :: proc{ + atan_complex32, + atan_complex64, + atan_complex128, +} + +atanh :: proc{ + atanh_complex32, + atanh_complex64, + atanh_complex128, +} + + +acos_complex32 :: proc "contextless" (x: complex32) -> complex32 { + w := asin(x) + return complex(math.PI/2 - real(w), -imag(w)) +} +acos_complex64 :: proc "contextless" (x: complex64) -> complex64 { + w := asin(x) + return complex(math.PI/2 - real(w), -imag(w)) +} +acos_complex128 :: proc "contextless" (x: complex128) -> complex128 { + w := asin(x) + return complex(math.PI/2 - real(w), -imag(w)) +} + + +acosh_complex32 :: proc "contextless" (x: complex32) -> complex32 { + if x == 0 { + return complex(0, math.copy_sign(math.PI/2, imag(x))) + } + w := acos(x) + if imag(w) <= 0 { + return complex(-imag(w), real(w)) + } + return complex(imag(w), -real(w)) +} +acosh_complex64 :: proc "contextless" (x: complex64) -> complex64 { + if x == 0 { + return complex(0, math.copy_sign(math.PI/2, imag(x))) + } + w := acos(x) + if imag(w) <= 0 { + return complex(-imag(w), real(w)) + } + return complex(imag(w), -real(w)) +} +acosh_complex128 :: proc "contextless" (x: complex128) -> complex128 { + if x == 0 { + return complex(0, math.copy_sign(math.PI/2, imag(x))) + } + w := acos(x) + if imag(w) <= 0 { + return complex(-imag(w), real(w)) + } + return complex(imag(w), -real(w)) +} + +asin_complex32 :: proc "contextless" (x: complex32) -> complex32 { + return complex32(asin_complex128(complex128(x))) +} +asin_complex64 :: proc "contextless" (x: complex64) -> complex64 { + return complex64(asin_complex128(complex128(x))) +} +asin_complex128 :: proc "contextless" (x: complex128) -> complex128 { + switch re, im := real(x), imag(x); { + case im == 0 && abs(re) <= 1: + return complex(math.asin(re), im) + case re == 0 && abs(im) <= 1: + return complex(re, math.asinh(im)) + case math.is_nan(im): + switch { + case re == 0: + return complex(re, math.nan_f64()) + case math.is_inf(re, 0): + return complex(math.nan_f64(), re) + case: + return nan_complex128() + } + case math.is_inf(im, 0): + switch { + case math.is_nan(re): + return x + case math.is_inf(re, 0): + return complex(math.copy_sign(math.PI/4, re), im) + case: + return complex(math.copy_sign(0, re), im) + } + case math.is_inf(re, 0): + return complex(math.copy_sign(math.PI/2, re), math.copy_sign(re, im)) + } + ct := complex(-imag(x), real(x)) // i * x + xx := x * x + x1 := complex(1-real(xx), -imag(xx)) // 1 - x*x + x2 := sqrt(x1) // x2 = sqrt(1 - x*x) + w := ln(ct + x2) + return complex(imag(w), -real(w)) // -i * w +} + +asinh_complex32 :: proc "contextless" (x: complex32) -> complex32 { + return complex32(asinh_complex128(complex128(x))) +} +asinh_complex64 :: proc "contextless" (x: complex64) -> complex64 { + return complex64(asinh_complex128(complex128(x))) +} +asinh_complex128 :: proc "contextless" (x: complex128) -> complex128 { + switch re, im := real(x), imag(x); { + case im == 0 && abs(re) <= 1: + return complex(math.asinh(re), im) + case re == 0 && abs(im) <= 1: + return complex(re, math.asin(im)) + case math.is_inf(re, 0): + switch { + case math.is_inf(im, 0): + return complex(re, math.copy_sign(math.PI/4, im)) + case math.is_nan(im): + return x + case: + return complex(re, math.copy_sign(0.0, im)) + } + case math.is_nan(re): + switch { + case im == 0: + return x + case math.is_inf(im, 0): + return complex(im, re) + case: + return nan_complex128() + } + case math.is_inf(im, 0): + return complex(math.copy_sign(im, re), math.copy_sign(math.PI/2, im)) + } + xx := x * x + x1 := complex(1+real(xx), imag(xx)) // 1 + x*x + return ln(x + sqrt(x1)) // log(x + sqrt(1 + x*x)) +} + + +atan_complex32 :: proc "contextless" (x: complex32) -> complex32 { + return complex32(atan_complex128(complex128(x))) +} +atan_complex64 :: proc "contextless" (x: complex64) -> complex64 { + return complex64(atan_complex128(complex128(x))) +} +atan_complex128 :: proc "contextless" (x: complex128) -> complex128 { + // Complex circular arc tangent + // + // DESCRIPTION: + // + // If + // z = x + iy, + // + // then + // 1 ( 2x ) + // Re w = - arctan(-----------) + k PI + // 2 ( 2 2) + // (1 - x - y ) + // + // ( 2 2) + // 1 (x + (y+1) ) + // Im w = - log(------------) + // 4 ( 2 2) + // (x + (y-1) ) + // + // Where k is an arbitrary integer. + // + // catan(z) = -i catanh(iz). + // + // ACCURACY: + // + // Relative error: + // arithmetic domain # trials peak rms + // DEC -10,+10 5900 1.3e-16 7.8e-18 + // IEEE -10,+10 30000 2.3e-15 8.5e-17 + // The check catan( ctan(z) ) = z, with |x| and |y| < PI/2, + // had peak relative error 1.5e-16, rms relative error + // 2.9e-17. See also clog(). + + switch re, im := real(x), imag(x); { + case im == 0: + return complex(math.atan(re), im) + case re == 0 && abs(im) <= 1: + return complex(re, math.atanh(im)) + case math.is_inf(im, 0) || math.is_inf(re, 0): + if math.is_nan(re) { + return complex(math.nan_f64(), math.copy_sign(0, im)) + } + return complex(math.copy_sign(math.PI/2, re), math.copy_sign(0, im)) + case math.is_nan(re) || math.is_nan(im): + return nan_complex128() + } + x2 := real(x) * real(x) + a := 1 - x2 - imag(x)*imag(x) + if a == 0 { + return nan_complex128() + } + t := 0.5 * math.atan2(2*real(x), a) + w := _reduce_pi_f64(t) + + t = imag(x) - 1 + b := x2 + t*t + if b == 0 { + return nan_complex128() + } + t = imag(x) + 1 + c := (x2 + t*t) / b + return complex(w, 0.25*math.ln(c)) +} + +atanh_complex32 :: proc "contextless" (x: complex32) -> complex32 { + z := complex(-imag(x), real(x)) // z = i * x + z = atan(z) + return complex(imag(z), -real(z)) // z = -i * z +} +atanh_complex64 :: proc "contextless" (x: complex64) -> complex64 { + z := complex(-imag(x), real(x)) // z = i * x + z = atan(z) + return complex(imag(z), -real(z)) // z = -i * z +} +atanh_complex128 :: proc "contextless" (x: complex128) -> complex128 { + z := complex(-imag(x), real(x)) // z = i * x + z = atan(z) + return complex(imag(z), -real(z)) // z = -i * z +} \ No newline at end of file diff --git a/core/math/cmplx/cmplx_trig.odin b/core/math/cmplx/cmplx_trig.odin new file mode 100644 index 000000000..7ca404fab --- /dev/null +++ b/core/math/cmplx/cmplx_trig.odin @@ -0,0 +1,409 @@ +package math_cmplx + +import "core:math" +import "core:math/bits" + +// The original C code, the long comment, and the constants +// below are from http://netlib.sandia.gov/cephes/c9x-complex/clog.c. +// The go code is a simplified version of the original C. +// +// Cephes Math Library Release 2.8: June, 2000 +// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier +// +// The readme file at http://netlib.sandia.gov/cephes/ says: +// Some software in this archive may be from the book _Methods and +// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster +// International, 1989) or from the Cephes Mathematical Library, a +// commercial product. In either event, it is copyrighted by the author. +// What you see here may be used freely but it comes with no support or +// guarantee. +// +// The two known misprints in the book are repaired here in the +// source listings for the gamma function and the incomplete beta +// integral. +// +// Stephen L. Moshier +// moshier@na-net.ornl.gov + +sin_complex128 :: proc "contextless" (x: complex128) -> complex128 { + // Complex circular sine + // + // DESCRIPTION: + // + // If + // z = x + iy, + // + // then + // + // w = sin x cosh y + i cos x sinh y. + // + // csin(z) = -i csinh(iz). + // + // ACCURACY: + // + // Relative error: + // arithmetic domain # trials peak rms + // DEC -10,+10 8400 5.3e-17 1.3e-17 + // IEEE -10,+10 30000 3.8e-16 1.0e-16 + // Also tested by csin(casin(z)) = z. + + switch re, im := real(x), imag(x); { + case im == 0 && (math.is_inf(re, 0) || math.is_nan(re)): + return complex(math.nan_f64(), im) + case math.is_inf(im, 0): + switch { + case re == 0: + return x + case math.is_inf(re, 0) || math.is_nan(re): + return complex(math.nan_f64(), im) + } + case re == 0 && math.is_nan(im): + return x + } + s, c := math.sincos(real(x)) + sh, ch := _sinhcosh_f64(imag(x)) + return complex(s*ch, c*sh) +} + +cos_complex128 :: proc "contextless" (x: complex128) -> complex128 { + // Complex circular cosine + // + // DESCRIPTION: + // + // If + // z = x + iy, + // + // then + // + // w = cos x cosh y - i sin x sinh y. + // + // ACCURACY: + // + // Relative error: + // arithmetic domain # trials peak rms + // DEC -10,+10 8400 4.5e-17 1.3e-17 + // IEEE -10,+10 30000 3.8e-16 1.0e-16 + + switch re, im := real(x), imag(x); { + case im == 0 && (math.is_inf(re, 0) || math.is_nan(re)): + return complex(math.nan_f64(), -im*math.copy_sign(0, re)) + case math.is_inf(im, 0): + switch { + case re == 0: + return complex(math.inf_f64(1), -re*math.copy_sign(0, im)) + case math.is_inf(re, 0) || math.is_nan(re): + return complex(math.inf_f64(1), math.nan_f64()) + } + case re == 0 && math.is_nan(im): + return complex(math.nan_f64(), 0) + } + s, c := math.sincos(real(x)) + sh, ch := _sinhcosh_f64(imag(x)) + return complex(c*ch, -s*sh) +} + +sinh_complex128 :: proc "contextless" (x: complex128) -> complex128 { + // Complex hyperbolic sine + // + // DESCRIPTION: + // + // csinh z = (cexp(z) - cexp(-z))/2 + // = sinh x * cos y + i cosh x * sin y . + // + // ACCURACY: + // + // Relative error: + // arithmetic domain # trials peak rms + // IEEE -10,+10 30000 3.1e-16 8.2e-17 + + switch re, im := real(x), imag(x); { + case re == 0 && (math.is_inf(im, 0) || math.is_nan(im)): + return complex(re, math.nan_f64()) + case math.is_inf(re, 0): + switch { + case im == 0: + return complex(re, im) + case math.is_inf(im, 0) || math.is_nan(im): + return complex(re, math.nan_f64()) + } + case im == 0 && math.is_nan(re): + return complex(math.nan_f64(), im) + } + s, c := math.sincos(imag(x)) + sh, ch := _sinhcosh_f64(real(x)) + return complex(c*sh, s*ch) +} + +cosh_complex128 :: proc "contextless" (x: complex128) -> complex128 { + // Complex hyperbolic cosine + // + // DESCRIPTION: + // + // ccosh(z) = cosh x cos y + i sinh x sin y . + // + // ACCURACY: + // + // Relative error: + // arithmetic domain # trials peak rms + // IEEE -10,+10 30000 2.9e-16 8.1e-17 + + switch re, im := real(x), imag(x); { + case re == 0 && (math.is_inf(im, 0) || math.is_nan(im)): + return complex(math.nan_f64(), re*math.copy_sign(0, im)) + case math.is_inf(re, 0): + switch { + case im == 0: + return complex(math.inf_f64(1), im*math.copy_sign(0, re)) + case math.is_inf(im, 0) || math.is_nan(im): + return complex(math.inf_f64(1), math.nan_f64()) + } + case im == 0 && math.is_nan(re): + return complex(math.nan_f64(), im) + } + s, c := math.sincos(imag(x)) + sh, ch := _sinhcosh_f64(real(x)) + return complex(c*ch, s*sh) +} + +tan_complex128 :: proc "contextless" (x: complex128) -> complex128 { + // Complex circular tangent + // + // DESCRIPTION: + // + // If + // z = x + iy, + // + // then + // + // sin 2x + i sinh 2y + // w = --------------------. + // cos 2x + cosh 2y + // + // On the real axis the denominator is zero at odd multiples + // of PI/2. The denominator is evaluated by its Taylor + // series near these points. + // + // ctan(z) = -i ctanh(iz). + // + // ACCURACY: + // + // Relative error: + // arithmetic domain # trials peak rms + // DEC -10,+10 5200 7.1e-17 1.6e-17 + // IEEE -10,+10 30000 7.2e-16 1.2e-16 + // Also tested by ctan * ccot = 1 and catan(ctan(z)) = z. + + switch re, im := real(x), imag(x); { + case math.is_inf(im, 0): + switch { + case math.is_inf(re, 0) || math.is_nan(re): + return complex(math.copy_sign(0, re), math.copy_sign(1, im)) + } + return complex(math.copy_sign(0, math.sin(2*re)), math.copy_sign(1, im)) + case re == 0 && math.is_nan(im): + return x + } + d := math.cos(2*real(x)) + math.cosh(2*imag(x)) + if abs(d) < 0.25 { + d = _tan_series_f64(x) + } + if d == 0 { + return inf_complex128() + } + return complex(math.sin(2*real(x))/d, math.sinh(2*imag(x))/d) +} + +tanh_complex128 :: proc "contextless" (x: complex128) -> complex128 { + switch re, im := real(x), imag(x); { + case math.is_inf(re, 0): + switch { + case math.is_inf(im, 0) || math.is_nan(im): + return complex(math.copy_sign(1, re), math.copy_sign(0, im)) + } + return complex(math.copy_sign(1, re), math.copy_sign(0, math.sin(2*im))) + case im == 0 && math.is_nan(re): + return x + } + d := math.cosh(2*real(x)) + math.cos(2*imag(x)) + if d == 0 { + return inf_complex128() + } + return complex(math.sinh(2*real(x))/d, math.sin(2*imag(x))/d) +} + +cot_complex128 :: proc "contextless" (x: complex128) -> complex128 { + d := math.cosh(2*imag(x)) - math.cos(2*real(x)) + if abs(d) < 0.25 { + d = _tan_series_f64(x) + } + if d == 0 { + return inf_complex128() + } + return complex(math.sin(2*real(x))/d, -math.sinh(2*imag(x))/d) +} + + +@(private="file") +_sinhcosh_f64 :: proc "contextless" (x: f64) -> (sh, ch: f64) { + if abs(x) <= 0.5 { + return math.sinh(x), math.cosh(x) + } + e := math.exp(x) + ei := 0.5 / e + e *= 0.5 + return e - ei, e + ei +} + + +// taylor series of cosh(2y) - cos(2x) +@(private) +_tan_series_f64 :: proc "contextless" (z: complex128) -> f64 { + MACH_EPSILON :: 1.0 / (1 << 53) + + x := abs(2 * real(z)) + y := abs(2 * imag(z)) + x = _reduce_pi_f64(x) + x, y = x * x, y * y + x2, y2 := 1.0, 1.0 + f, rn, d := 1.0, 0.0, 0.0 + + for { + rn += 1 + f *= rn + rn += 1 + f *= rn + x2 *= x + y2 *= y + t := y2 + x2 + t /= f + d += t + + rn += 1 + f *= rn + rn += 1 + f *= rn + x2 *= x + y2 *= y + t = y2 - x2 + t /= f + d += t + if !(abs(t/d) > MACH_EPSILON) { // don't use <=, because of floating point nonsense and NaN + break + } + } + return d +} + +// _reduce_pi_f64 reduces the input argument x to the range (-PI/2, PI/2]. +// x must be greater than or equal to 0. For small arguments it +// uses Cody-Waite reduction in 3 f64 parts based on: +// "Elementary Function Evaluation: Algorithms and Implementation" +// Jean-Michel Muller, 1997. +// For very large arguments it uses Payne-Hanek range reduction based on: +// "ARGUMENT REDUCTION FOR HUGE ARGUMENTS: Good to the Last Bit" +@(private) +_reduce_pi_f64 :: proc "contextless" (x: f64) -> f64 #no_bounds_check { + x := x + + // REDUCE_THRESHOLD is the maximum value of x where the reduction using + // Cody-Waite reduction still gives accurate results. This threshold + // is set by t*PIn being representable as a f64 without error + // where t is given by t = floor(x * (1 / PI)) and PIn are the leading partial + // terms of PI. Since the leading terms, PI1 and PI2 below, have 30 and 32 + // trailing zero bits respectively, t should have less than 30 significant bits. + // t < 1<<30 -> floor(x*(1/PI)+0.5) < 1<<30 -> x < (1<<30-1) * PI - 0.5 + // So, conservatively we can take x < 1<<30. + REDUCE_THRESHOLD :: f64(1 << 30) + + if abs(x) < REDUCE_THRESHOLD { + // Use Cody-Waite reduction in three parts. + // PI1, PI2 and PI3 comprise an extended precision value of PI + // such that PI ~= PI1 + PI2 + PI3. The parts are chosen so + // that PI1 and PI2 have an approximately equal number of trailing + // zero bits. This ensures that t*PI1 and t*PI2 are exact for + // large integer values of t. The full precision PI3 ensures the + // approximation of PI is accurate to 102 bits to handle cancellation + // during subtraction. + PI1 :: 0h400921fb40000000 // 3.141592502593994 + PI2 :: 0h3e84442d00000000 // 1.5099578831723193e-07 + PI3 :: 0h3d08469898cc5170 // 1.0780605716316238e-14 + + t := x / math.PI + t += 0.5 + t = f64(i64(t)) // i64(t) = the multiple + return ((x - t*PI1) - t*PI2) - t*PI3 + } + // Must apply Payne-Hanek range reduction + MASK :: 0x7FF + SHIFT :: 64 - 11 - 1 + BIAS :: 1023 + FRAC_MASK :: 1<>SHIFT&MASK) - BIAS - SHIFT + ix &= FRAC_MASK + ix |= 1 << SHIFT + + // bdpi is the binary digits of 1/PI as a u64 array, + // that is, 1/PI = SUM bdpi[i]*2^(-64*i). + // 19 64-bit digits give 1216 bits of precision + // to handle the largest possible f64 exponent. + @static bdpi := [?]u64{ + 0x0000000000000000, + 0x517cc1b727220a94, + 0xfe13abe8fa9a6ee0, + 0x6db14acc9e21c820, + 0xff28b1d5ef5de2b0, + 0xdb92371d2126e970, + 0x0324977504e8c90e, + 0x7f0ef58e5894d39f, + 0x74411afa975da242, + 0x74ce38135a2fbf20, + 0x9cc8eb1cc1a99cfa, + 0x4e422fc5defc941d, + 0x8ffc4bffef02cc07, + 0xf79788c5ad05368f, + 0xb69b3f6793e584db, + 0xa7a31fb34f2ff516, + 0xba93dd63f5f2f8bd, + 0x9e839cfbc5294975, + 0x35fdafd88fc6ae84, + 0x2b0198237e3db5d5, + } + + // Use the exponent to extract the 3 appropriate u64 digits from bdpi, + // B ~ (z0, z1, z2), such that the product leading digit has the exponent -64. + // Note, exp >= 50 since x >= REDUCE_THRESHOLD and exp < 971 for maximum f64. + digit, bitshift := uint(exp+64)/64, uint(exp+64)%64 + z0 := (bdpi[digit] << bitshift) | (bdpi[digit+1] >> (64 - bitshift)) + z1 := (bdpi[digit+1] << bitshift) | (bdpi[digit+2] >> (64 - bitshift)) + z2 := (bdpi[digit+2] << bitshift) | (bdpi[digit+3] >> (64 - bitshift)) + + // Multiply mantissa by the digits and extract the upper two digits (hi, lo). + z2hi, _ := bits.mul(z2, ix) + z1hi, z1lo := bits.mul(z1, ix) + z0lo := z0 * ix + lo, c := bits.add(z1lo, z2hi, 0) + hi, _ := bits.add(z0lo, z1hi, c) + + // Find the magnitude of the fraction. + lz := uint(bits.leading_zeros(hi)) + e := u64(BIAS - (lz + 1)) + + // Clear implicit mantissa bit and shift into place. + hi = (hi << (lz + 1)) | (lo >> (64 - (lz + 1))) + hi >>= 64 - SHIFT + + // Include the exponent and convert to a float. + hi |= e << SHIFT + x = transmute(f64)(hi) + + // map to (-PI/2, PI/2] + if x > 0.5 { + x -= 1 + } + return math.PI * x +} +