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// Ported from musl, which is licensed under the MIT license:
// https://git.musl-libc.org/cgit/musl/tree/COPYRIGHT
//
// https://git.musl-libc.org/cgit/musl/tree/src/math/tgamma.c
const builtin = @import("builtin");
const std = @import("../std.zig");
/// Returns the gamma function of x,
/// gamma(x) = factorial(x - 1) for integer x.
///
/// Special Cases:
/// - gamma(+-nan) = nan
/// - gamma(-inf) = nan
/// - gamma(n) = nan for negative integers
/// - gamma(-0.0) = -inf
/// - gamma(+0.0) = +inf
/// - gamma(+inf) = +inf
pub fn gamma(comptime T: type, x: T) T {
if (T != f32 and T != f64) {
@compileError("gamma not implemented for " ++ @typeName(T));
}
// common integer case first
if (x == @trunc(x)) {
// gamma(-inf) = nan
// gamma(n) = nan for negative integers
if (x < 0) {
return std.math.nan(T);
}
// gamma(-0.0) = -inf
// gamma(+0.0) = +inf
if (x == 0) {
return 1 / x;
}
if (x < integer_result_table.len) {
const i = @as(u8, @intFromFloat(x));
return @floatCast(integer_result_table[i]);
}
}
// below this, result underflows, but has a sign
// negative for (-1, 0)
// positive for (-2, -1)
// negative for (-3, -2)
// ...
const lower_bound = if (T == f64) -184 else -42;
if (x < lower_bound) {
return if (@mod(x, 2) > 1) -0.0 else 0.0;
}
// above this, result overflows
// gamma(+inf) = +inf
const upper_bound = if (T == f64) 172 else 36;
if (x > upper_bound) {
return std.math.inf(T);
}
const abs = @abs(x);
// perfect precision here
if (abs < 0x1p-54) {
return 1 / x;
}
const base = abs + lanczos_minus_half;
const exponent = abs - 0.5;
// error of y for correction, see
// https://github.com/python/cpython/blob/5dc79e3d7f26a6a871a89ce3efc9f1bcee7bb447/Modules/mathmodule.c#L286-L324
const e = if (abs > lanczos_minus_half)
base - abs - lanczos_minus_half
else
base - lanczos_minus_half - abs;
const correction = lanczos * e / base;
const initial = series(T, abs) * @exp(-base);
// use reflection formula for negatives
if (x < 0) {
const reflected = -std.math.pi / (abs * sinpi(T, abs) * initial);
const corrected = reflected - reflected * correction;
const half_pow = std.math.pow(T, base, -0.5 * exponent);
return corrected * half_pow * half_pow;
} else {
const corrected = initial + initial * correction;
const half_pow = std.math.pow(T, base, 0.5 * exponent);
return corrected * half_pow * half_pow;
}
}
/// Returns the natural logarithm of the absolute value of the gamma function.
///
/// Special Cases:
/// - lgamma(+-nan) = nan
/// - lgamma(+-inf) = +inf
/// - lgamma(n) = +inf for negative integers
/// - lgamma(+-0.0) = +inf
/// - lgamma(1) = +0.0
/// - lgamma(2) = +0.0
pub fn lgamma(comptime T: type, x: T) T {
if (T != f32 and T != f64) {
@compileError("gamma not implemented for " ++ @typeName(T));
}
// common integer case first
if (x == @trunc(x)) {
// lgamma(-inf) = +inf
// lgamma(n) = +inf for negative integers
// lgamma(+-0.0) = +inf
if (x <= 0) {
return std.math.inf(T);
}
// lgamma(1) = +0.0
// lgamma(2) = +0.0
if (x < integer_result_table.len) {
const i = @as(u8, @intFromFloat(x));
return @log(@as(T, @floatCast(integer_result_table[i])));
}
// lgamma(+inf) = +inf
if (std.math.isPositiveInf(x)) {
return x;
}
}
const abs = @abs(x);
// perfect precision here
if (abs < 0x1p-54) {
return -@log(abs);
}
// obvious approach when overflow is not a problem
const upper_bound = if (T == f64) 128 else 26;
if (abs < upper_bound) {
return @log(@abs(gamma(T, x)));
}
const log_base = @log(abs + lanczos_minus_half) - 1;
const exponent = abs - 0.5;
const log_series = @log(series(T, abs));
const initial = exponent * log_base + log_series - lanczos;
// use reflection formula for negatives
if (x < 0) {
const reflected = std.math.pi / (abs * sinpi(T, abs));
return @log(@abs(reflected)) - initial;
}
return initial;
}
// table of factorials for integer early return
// stops at 22 because 23 isn't representable with full precision on f64
const integer_result_table = [_]f64{
std.math.inf(f64), // gamma(+0.0)
1, // gamma(1)
1, // ...
2,
6,
24,
120,
720,
5040,
40320,
362880,
3628800,
39916800,
479001600,
6227020800,
87178291200,
1307674368000,
20922789888000,
355687428096000,
6402373705728000,
121645100408832000,
2432902008176640000,
51090942171709440000, // gamma(22)
};
// "g" constant, arbitrary
const lanczos = 6.024680040776729583740234375;
const lanczos_minus_half = lanczos - 0.5;
fn series(comptime T: type, abs: T) T {
const numerator = [_]T{
23531376880.410759688572007674451636754734846804940,
42919803642.649098768957899047001988850926355848959,
35711959237.355668049440185451547166705960488635843,
17921034426.037209699919755754458931112671403265390,
6039542586.3520280050642916443072979210699388420708,
1439720407.3117216736632230727949123939715485786772,
248874557.86205415651146038641322942321632125127801,
31426415.585400194380614231628318205362874684987640,
2876370.6289353724412254090516208496135991145378768,
186056.26539522349504029498971604569928220784236328,
8071.6720023658162106380029022722506138218516325024,
210.82427775157934587250973392071336271166969580291,
2.5066282746310002701649081771338373386264310793408,
};
const denominator = [_]T{
0,
39916800,
120543840,
150917976,
105258076,
45995730,
13339535,
2637558,
357423,
32670,
1925,
66,
1,
};
var num: T = 0;
var den: T = 0;
// split to avoid overflow
if (abs < 8) {
// big abs would overflow here
for (0..numerator.len) |i| {
num = num * abs + numerator[numerator.len - 1 - i];
den = den * abs + denominator[numerator.len - 1 - i];
}
} else {
// small abs would overflow here
for (0..numerator.len) |i| {
num = num / abs + numerator[i];
den = den / abs + denominator[i];
}
}
return num / den;
}
// precise sin(pi * x)
// but not for integer x or |x| < 2^-54, we handle those already
fn sinpi(comptime T: type, x: T) T {
const xmod2 = @mod(x, 2); // [0, 2]
const n = (@as(u8, @intFromFloat(4 * xmod2)) + 1) / 2; // {0, 1, 2, 3, 4}
const y = xmod2 - 0.5 * @as(T, @floatFromInt(n)); // [-0.25, 0.25]
return switch (n) {
0, 4 => @sin(std.math.pi * y),
1 => @cos(std.math.pi * y),
2 => -@sin(std.math.pi * y),
3 => -@cos(std.math.pi * y),
else => unreachable,
};
}
const expect = std.testing.expect;
const expectEqual = std.testing.expectEqual;
const expectApproxEqRel = std.testing.expectApproxEqRel;
test gamma {
inline for (&.{ f32, f64 }) |T| {
const eps = @sqrt(std.math.floatEps(T));
try expectApproxEqRel(@as(T, 120), gamma(T, 6), eps);
try expectApproxEqRel(@as(T, 362880), gamma(T, 10), eps);
try expectApproxEqRel(@as(T, 6402373705728000), gamma(T, 19), eps);
try expectApproxEqRel(@as(T, 332.7590766955334570), gamma(T, 0.003), eps);
try expectApproxEqRel(@as(T, 1.377260301981044573), gamma(T, 0.654), eps);
try expectApproxEqRel(@as(T, 1.025393882573518478), gamma(T, 0.959), eps);
try expectApproxEqRel(@as(T, 7.361898021467681690), gamma(T, 4.16), eps);
try expectApproxEqRel(@as(T, 198337.2940287730753), gamma(T, 9.73), eps);
try expectApproxEqRel(@as(T, 113718145797241.1666), gamma(T, 17.6), eps);
try expectApproxEqRel(@as(T, -1.13860211111081424930673), gamma(T, -2.80), eps);
try expectApproxEqRel(@as(T, 0.00018573407931875070158), gamma(T, -7.74), eps);
try expectApproxEqRel(@as(T, -0.00000001647990903942825), gamma(T, -12.1), eps);
}
}
test "gamma.special" {
if (builtin.cpu.arch.isArmOrThumb() and builtin.target.floatAbi() == .soft) return error.SkipZigTest; // https://github.com/ziglang/zig/issues/21234
inline for (&.{ f32, f64 }) |T| {
try expect(std.math.isNan(gamma(T, -std.math.nan(T))));
try expect(std.math.isNan(gamma(T, std.math.nan(T))));
try expect(std.math.isNan(gamma(T, -std.math.inf(T))));
try expect(std.math.isNan(gamma(T, -4)));
try expect(std.math.isNan(gamma(T, -11)));
try expect(std.math.isNan(gamma(T, -78)));
try expectEqual(-std.math.inf(T), gamma(T, -0.0));
try expectEqual(std.math.inf(T), gamma(T, 0.0));
try expect(std.math.isNegativeZero(gamma(T, -200.5)));
try expect(std.math.isPositiveZero(gamma(T, -201.5)));
try expect(std.math.isNegativeZero(gamma(T, -202.5)));
try expectEqual(std.math.inf(T), gamma(T, 200));
try expectEqual(std.math.inf(T), gamma(T, 201));
try expectEqual(std.math.inf(T), gamma(T, 202));
try expectEqual(std.math.inf(T), gamma(T, std.math.inf(T)));
}
}
test lgamma {
inline for (&.{ f32, f64 }) |T| {
const eps = @sqrt(std.math.floatEps(T));
try expectApproxEqRel(@as(T, @log(24.0)), lgamma(T, 5), eps);
try expectApproxEqRel(@as(T, @log(20922789888000.0)), lgamma(T, 17), eps);
try expectApproxEqRel(@as(T, @log(2432902008176640000.0)), lgamma(T, 21), eps);
try expectApproxEqRel(@as(T, 2.201821590438859327), lgamma(T, 0.105), eps);
try expectApproxEqRel(@as(T, 1.275416975248413231), lgamma(T, 0.253), eps);
try expectApproxEqRel(@as(T, 0.130463884049976732), lgamma(T, 0.823), eps);
try expectApproxEqRel(@as(T, 43.24395772148497989), lgamma(T, 21.3), eps);
try expectApproxEqRel(@as(T, 110.6908958012102623), lgamma(T, 41.1), eps);
try expectApproxEqRel(@as(T, 215.2123266224689711), lgamma(T, 67.4), eps);
try expectApproxEqRel(@as(T, -122.605958469563489), lgamma(T, -43.6), eps);
try expectApproxEqRel(@as(T, -278.633885462703133), lgamma(T, -81.4), eps);
try expectApproxEqRel(@as(T, -333.247676253238363), lgamma(T, -93.6), eps);
}
}
test "lgamma.special" {
inline for (&.{ f32, f64 }) |T| {
try expect(std.math.isNan(lgamma(T, -std.math.nan(T))));
try expect(std.math.isNan(lgamma(T, std.math.nan(T))));
try expectEqual(std.math.inf(T), lgamma(T, -std.math.inf(T)));
try expectEqual(std.math.inf(T), lgamma(T, std.math.inf(T)));
try expectEqual(std.math.inf(T), lgamma(T, -5));
try expectEqual(std.math.inf(T), lgamma(T, -8));
try expectEqual(std.math.inf(T), lgamma(T, -15));
try expectEqual(std.math.inf(T), lgamma(T, -0.0));
try expectEqual(std.math.inf(T), lgamma(T, 0.0));
try expect(std.math.isPositiveZero(lgamma(T, 1)));
try expect(std.math.isPositiveZero(lgamma(T, 2)));
}
}
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