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package math
// The original C code, the long comment, and the constants
// below are from http://netlib.sandia.gov/cephes/cprob/gamma.c.
//
// tgamma.c
//
// Gamma function
//
// SYNOPSIS:
//
// double x, y, tgamma();
// extern int signgam;
//
// y = tgamma( x );
//
// DESCRIPTION:
//
// Returns gamma function of the argument. The result is
// correctly signed, and the sign (+1 or -1) is also
// returned in a global (extern) variable named signgam.
// This variable is also filled in by the logarithmic gamma
// function lgamma().
//
// Arguments |x| <= 34 are reduced by recurrence and the function
// approximated by a rational function of degree 6/7 in the
// interval (2,3). Large arguments are handled by Stirling's
// formula. Large negative arguments are made positive using
// a reflection formula.
//
// ACCURACY:
//
// Relative error:
// arithmetic domain # trials peak rms
// DEC -34, 34 10000 1.3e-16 2.5e-17
// IEEE -170,-33 20000 2.3e-15 3.3e-16
// IEEE -33, 33 20000 9.4e-16 2.2e-16
// IEEE 33, 171.6 20000 2.3e-15 3.2e-16
//
// Error for arguments outside the test range will be larger
// owing to error amplification by the exponential function.
//
// 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
// Gamma function computed by Stirling's formula.
// The pair of results must be multiplied together to get the actual answer.
// The multiplication is left to the caller so that, if careful, the caller can avoid
// infinity for 172 <= x <= 180.
// The polynomial is valid for 33 <= x <= 172; larger values are only used
// in reciprocal and produce denormalized floats. The lower precision there
// masks any imprecision in the polynomial.
@(private="file", require_results)
stirling :: proc "contextless" (x: f64) -> (f64, f64) {
@(static, rodata) gamS := [?]f64{
+7.87311395793093628397e-04,
-2.29549961613378126380e-04,
-2.68132617805781232825e-03,
+3.47222221605458667310e-03,
+8.33333333333482257126e-02,
}
if x > 200 {
return inf_f64(1), 1
}
SQRT_TWO_PI :: 0h40040d931ff62706 // 2.506628274631000502417
MAX_STIRLING :: 143.01608
w := 1 / x
w = 1 + w*((((gamS[0]*w+gamS[1])*w+gamS[2])*w+gamS[3])*w+gamS[4])
y1 := exp(x)
y2 := 1.0
if x > MAX_STIRLING { // avoid pow() overflow
v := pow(x, 0.5*x-0.25)
y1, y2 = v, v/y1
} else {
y1 = pow(x, x-0.5) / y1
}
return y1, SQRT_TWO_PI * w * y2
}
@(require_results)
gamma_f64 :: proc "contextless" (x: f64) -> f64 {
is_neg_int :: proc "contextless" (x: f64) -> bool {
if x < 0 {
_, xf := modf(x)
return xf == 0
}
return false
}
@(static, rodata) gamP := [?]f64{
1.60119522476751861407e-04,
1.19135147006586384913e-03,
1.04213797561761569935e-02,
4.76367800457137231464e-02,
2.07448227648435975150e-01,
4.94214826801497100753e-01,
9.99999999999999996796e-01,
}
@(static, rodata) gamQ := [?]f64{
-2.31581873324120129819e-05,
+5.39605580493303397842e-04,
-4.45641913851797240494e-03,
+1.18139785222060435552e-02,
+3.58236398605498653373e-02,
-2.34591795718243348568e-01,
+7.14304917030273074085e-02,
+1.00000000000000000320e+00,
}
EULER :: 0.57721566490153286060651209008240243104215933593992 // A001620
switch {
case is_neg_int(x) || is_inf(x, -1) || is_nan(x):
return nan_f64()
case is_inf(x, 1):
return inf_f64(1)
case x == 0:
if sign_bit(x) {
return inf_f64(-1)
}
return inf_f64(1)
}
x := x
q := abs(x)
p := floor(q)
if q > 33 {
if x >= 0 {
y1, y2 := stirling(x)
return y1 * y2
}
// Note: x is negative but (checked above) not a negative integer,
// so x must be small enough to be in range for conversion to i64.
// If |x| were >= 2⁶³ it would have to be an integer.
signgam := 1
if ip := i64(p); ip&1 == 0 {
signgam = -1
}
z := q - p
if z > 0.5 {
p = p + 1
z = q - p
}
z = q * sin(PI*z)
if z == 0 {
return inf_f64(signgam)
}
sq1, sq2 := stirling(q)
absz := abs(z)
d := absz * sq1 * sq2
if is_inf(d, 0) {
z = PI / absz / sq1 / sq2
} else {
z = PI / d
}
return f64(signgam) * z
}
// Reduce argument
z := 1.0
for x >= 3 {
x = x - 1
z = z * x
}
for x < 0 {
if x > -1e-09 {
if x == 0 {
return inf_f64(1)
}
return z / ((1 + EULER*x) * x)
}
z = z / x
x = x + 1
}
for x < 2 {
if x < 1e-09 {
if x == 0 {
return inf_f64(1)
}
return z / ((1 + EULER*x) * x)
}
z = z / x
x = x + 1
}
if x == 2 {
return z
}
x = x - 2
p = (((((x*gamP[0]+gamP[1])*x+gamP[2])*x+gamP[3])*x+gamP[4])*x+gamP[5])*x + gamP[6]
q = ((((((x*gamQ[0]+gamQ[1])*x+gamQ[2])*x+gamQ[3])*x+gamQ[4])*x+gamQ[5])*x+gamQ[6])*x + gamQ[7]
return z * p / q
}
@(require_results) gamma_f16 :: proc "contextless" (x: f16) -> f16 { return f16(gamma_f64(f64(x))) }
@(require_results) gamma_f16le :: proc "contextless" (x: f16le) -> f16le { return f16le(gamma_f64(f64(x))) }
@(require_results) gamma_f16be :: proc "contextless" (x: f16be) -> f16be { return f16be(gamma_f64(f64(x))) }
@(require_results) gamma_f32 :: proc "contextless" (x: f32) -> f32 { return f32(gamma_f64(f64(x))) }
@(require_results) gamma_f32le :: proc "contextless" (x: f32le) -> f32le { return f32le(gamma_f64(f64(x))) }
@(require_results) gamma_f32be :: proc "contextless" (x: f32be) -> f32be { return f32be(gamma_f64(f64(x))) }
@(require_results) gamma_f64le :: proc "contextless" (x: f64le) -> f64le { return f64le(gamma_f64(f64(x))) }
@(require_results) gamma_f64be :: proc "contextless" (x: f64be) -> f64be { return f64be(gamma_f64(f64(x))) }
gamma :: proc{
gamma_f16, gamma_f16le, gamma_f16be,
gamma_f32, gamma_f32le, gamma_f32be,
gamma_f64, gamma_f64le, gamma_f64be,
}
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