1//===-- Implementation header for lgammaf16 ---------------------*- C++ -*-===//
2//
3// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
4// See https://llvm.org/LICENSE.txt for license information.
5// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
6//
7//===----------------------------------------------------------------------===//
8
9#ifndef LLVM_LIBC_SRC___SUPPORT_MATH_LGAMMAF16_H
10#define LLVM_LIBC_SRC___SUPPORT_MATH_LGAMMAF16_H
11
12#include "include/llvm-libc-macros/float16-macros.h"
13
14#ifdef LIBC_TYPES_HAS_FLOAT16
15
16#include "src/__support/CPP/bit.h"
17#include "src/__support/FPUtil/FEnvImpl.h"
18#include "src/__support/FPUtil/FPBits.h"
19#include "src/__support/FPUtil/NearestIntegerOperations.h"
20#include "src/__support/FPUtil/PolyEval.h"
21#include "src/__support/FPUtil/cast.h"
22#include "src/__support/FPUtil/multiply_add.h"
23#include "src/__support/macros/config.h"
24#include "src/__support/macros/optimization.h"
25#include "src/__support/macros/properties/types.h"
26
27namespace LIBC_NAMESPACE_DECL {
28
29namespace math {
30
31namespace lgammaf16_internal {
32
33LIBC_INLINE constexpr bool is_integer(float16 x) {
34 using FPBits = fputil::FPBits<float16>;
35 FPBits xbits(x);
36 uint16_t x_u = xbits.uintval();
37 unsigned x_e = static_cast<unsigned>(xbits.get_biased_exponent());
38 unsigned lsb = static_cast<unsigned>(
39 cpp::countr_zero(value: static_cast<uint16_t>(x_u | FPBits::EXP_MASK)));
40 constexpr unsigned UNIT_EXPONENT =
41 static_cast<unsigned>(FPBits::EXP_BIAS + FPBits::FRACTION_LEN);
42 return x_e + lsb >= UNIT_EXPONENT;
43}
44
45// sin(pi*x) for x in (0, 1), written as (0.25 - u^2) * P(u^2) where u = x-0.5.
46// Coefficients for P are a degree-7 polynomial in u^2
47// approximating sin(pi*(u+0.5)) / (pi*(0.25-u^2))
48// with max error ~2^{-55}, generated by Sollya with:
49// > P = fpminimax(sin(pi*(x+0.5))/(pi*(0.25-x^2)),
50// [|0,2,4,6,8,10,12,14|], [|D...|], [-0.5, 0.5]);
51LIBC_INLINE constexpr double lg_sinpi(double x) {
52 constexpr double COEFFS[8] = {0x000000000000001p+2, -0x1.de9e64df22ea4p+1,
53 0x1.472be122401f8p+0, -0x1.d4fcd82df91bp-3,
54 0x1.9f05c97e0aab2p-6, -0x1.f3091c427b611p-10,
55 0x1.b22c9bfdca547p-14, -0x1.15484325ef569p-18};
56 double u = x - 0.5;
57 double u2 = u * u, u4 = u2 * u2, u8 = u4 * u4;
58 double p01 = fputil::multiply_add(x: u2, y: COEFFS[1], z: COEFFS[0]);
59 double p23 = fputil::multiply_add(x: u2, y: COEFFS[3], z: COEFFS[2]);
60 double p45 = fputil::multiply_add(x: u2, y: COEFFS[5], z: COEFFS[4]);
61 double p67 = fputil::multiply_add(x: u2, y: COEFFS[7], z: COEFFS[6]);
62 double p03 = fputil::multiply_add(x: u4, y: p23, z: p01);
63 double p47 = fputil::multiply_add(x: u4, y: p67, z: p45);
64 return (0.25 - u2) * fputil::multiply_add(x: u8, y: p47, z: p03);
65}
66
67// Natural logarithm of x (x > 0), using 16-entry table + degree-7 polynomial.
68// Range reduction: x = 2^e * m where m in [1, 2), decomposed as
69// m = (1+i/16)*(1+z) so log(x) = e*log(2) + log(1+i/16) + log(1+z)
70// = e*log(2) + IL[i] + z*P(z).
71// P approximates log(1+z)/z on z in [-1/16, 1/16] with max
72// error ~2^{-54}, generated by Sollya with:
73// > P = fpminimax(log(1+x)/x, [|0,1,2,3,4,5,6,7|], [|D...|], [-1/16, 1/16]);
74LIBC_INLINE double lg_ln(double x) {
75 using FPBits = fputil::FPBits<double>;
76 uint64_t u = FPBits(x).uintval();
77 int e = static_cast<int>(FPBits(x).get_biased_exponent()) - 0x3ff;
78
79 // Coefficients for log(1 + z)/z on z in [-1/16, 1/16]
80 constexpr double COEFFS[8] = {0x1.fffffffffff24p-1, -0x1.ffffffffd1d67p-2,
81 0x1.55555537802dep-2, -0x1.ffffeca81b866p-3,
82 0x1.999611761d772p-3, -0x1.54f3e581b61bfp-3,
83 0x1.1e642b4cb5143p-3, -0x1.9115a5af1e1edp-4};
84 // IL[i] = log(1 + i/16) for i = 0..15
85 constexpr double IL[16] = {
86 0x1.59caeec280116p-57, 0x1.f0a30c01162aap-5, 0x1.e27076e2af2ebp-4,
87 0x1.5ff3070a793d6p-3, 0x1.c8ff7c79a9a2p-3, 0x1.1675cababa60fp-2,
88 0x1.4618bc21c5ec2p-2, 0x1.739d7f6bbd007p-2, 0x1.9f323ecbf984dp-2,
89 0x1.c8ff7c79a9a21p-2, 0x1.f128f5faf06ecp-2, 0x1.0be72e4252a83p-1,
90 0x1.1e85f5e7040d1p-1, 0x1.307d7334f10bep-1, 0x1.41d8fe84672afp-1,
91 0x1.52a2d265bc5abp-1};
92 // IX[i] = 1 / (1 + i/16) for i = 0..15
93 constexpr double IX[16] = {
94 0x000000000000001p+0, 0x1.e1e1e1e1e1e1ep-1, 0x1.c71c71c71c71cp-1,
95 0x1.af286bca1af28p-1, 0x1.999999999999ap-1, 0x1.8618618618618p-1,
96 0x1.745d1745d1746p-1, 0x1.642c8590b2164p-1, 0x1.5555555555555p-1,
97 0x1.47ae147ae147bp-1, 0x1.3b13b13b13b14p-1, 0x1.2f684bda12f68p-1,
98 0x1.2492492492492p-1, 0x1.1a7b9611a7b96p-1, 0x1.1111111111111p-1,
99 0x1.0842108421084p-1};
100
101 int i = static_cast<int>((u >> 48) & 0xf);
102 // Reduce to mantissa in [1, 2)
103 uint64_t mant_u = (u & (~uint64_t(0) >> 12)) | (uint64_t(0x3ff) << 52);
104 double mant = FPBits(mant_u).get_val();
105 double z = IX[i] * mant - 1.0, z2 = z * z, z4 = z2 * z2;
106 double q01 = fputil::multiply_add(x: z, y: COEFFS[1], z: COEFFS[0]);
107 double q23 = fputil::multiply_add(x: z, y: COEFFS[3], z: COEFFS[2]);
108 double q45 = fputil::multiply_add(x: z, y: COEFFS[5], z: COEFFS[4]);
109 double q67 = fputil::multiply_add(x: z, y: COEFFS[7], z: COEFFS[6]);
110 double q03 = fputil::multiply_add(x: z2, y: q23, z: q01);
111 double q47 = fputil::multiply_add(x: z2, y: q67, z: q45);
112 return e * 0x1.62e42fefa39efp-1 + IL[i] +
113 z * fputil::multiply_add(x: z4, y: q47, z: q03);
114}
115
116} // namespace lgammaf16_internal
117
118LIBC_INLINE float16 lgammaf16(float16 x) {
119 using namespace lgammaf16_internal;
120 using FPBits = fputil::FPBits<float16>;
121
122 FPBits xbits(x);
123 uint16_t x_abs = xbits.abs().uintval();
124
125 // Handle NaN and Inf
126 if (LIBC_UNLIKELY(x_abs >= 0x7c00u)) {
127 if (x_abs == 0x7c00u)
128 return FPBits::inf().get_val();
129 if (xbits.is_signaling_nan()) {
130 fputil::raise_except_if_required(FE_INVALID);
131 return FPBits::quiet_nan().get_val();
132 }
133 return x;
134 }
135
136 // +-0 pole error
137 if (LIBC_UNLIKELY(x_abs == 0)) {
138 fputil::raise_except_if_required(FE_DIVBYZERO);
139 fputil::set_errno_if_required(ERANGE);
140 return FPBits::inf().get_val();
141 }
142
143 if (LIBC_UNLIKELY(is_integer(x))) {
144 if (xbits.is_neg()) {
145 // pole -> +Inf
146 fputil::raise_except_if_required(FE_DIVBYZERO);
147 fputil::set_errno_if_required(ERANGE);
148 return FPBits::inf().get_val();
149 }
150 // lgamma(1) = lgamma(2) = 0
151 if (x_abs == 0x3c00u || x_abs == 0x4000u)
152 return FPBits::zero().get_val();
153 }
154
155 double xd = fputil::cast<double>(x);
156 double abs_xd = xd < 0.0 ? -xd : xd;
157 double lgamma_val;
158
159 // |x| = 0.66015625
160 if (LIBC_UNLIKELY(x_abs < 0x3948u)) {
161 // Small path: |x| < 0.66015625
162 // For t = |x|: lgamma(t) = t*P(u) - log(t), where P(u) approximates
163 // h(t) = (lgamma(t) + log(t)) / t, u = (t - MID_S) / HW_S in [-1, 1]
164 // For x < 0: lgamma(-t) = log(pi) - log(sin(pi*t)) - lgamma(t) - log(t)
165 // = log(pi) - lg_ln(lg_sinpi(t)) - t*P(u)
166 // Degree-12 Chebyshev approximation for h in u; max lgamma error ~1.5e-13
167 // Coefficients generated by DCT at 13 Chebyshev nodes on
168 // [1e-5, 0.66015625].
169 // Evaluated via Clenshaw: sum_{k=0}^{12} CHEB_H[k] * T_k(u)
170 constexpr double MID_S = 0x1.52014f8b588e3p-2;
171 constexpr double HW_S = 0x1.51feb074a771dp-2;
172 constexpr double CHEB_H[13] = {
173 -0x1.6ab0351d4e02fp-2, 0x1.ac55f136e49cep-3, -0x1.9fc5280cd41bfp-7,
174 0x1.1825d37c81c1ep-10, -0x1.ae2f107c31c9ep-14, 0x1.6174bdf95e9abp-17,
175 -0x1.2e6abd22bc4ecp-20, 0x1.09b66d4f14b06p-23, -0x1.dbb938f189d8ap-27,
176 0x1.afcb748e20e4ep-30, -0x1.8c221a9d89d8ap-33, 0x1.6e708e49b3aeep-36,
177 -0x1.50accec4ec4ecp-39};
178 // Clenshaw's algorithm: p = sum_k CHEB_H[k] * T_k(u_s)
179 double u_s = (abs_xd - MID_S) / HW_S, u_s2 = 2.0 * u_s;
180 double bh1 = CHEB_H[12], bh2 = 0.0;
181 for (int k = 11; k >= 1; k--) {
182 double bh0 = fputil::multiply_add(x: u_s2, y: bh1, z: CHEB_H[k] - bh2);
183 bh2 = bh1;
184 bh1 = bh0;
185 }
186 double poly_h = fputil::multiply_add(x: u_s, y: bh1, z: CHEB_H[0] - bh2);
187 // abs_xd * poly_h = lgamma(abs_xd) + log(abs_xd)
188 double poly_val = abs_xd * poly_h;
189 if (xbits.is_neg())
190 lgamma_val = 0x1.250d048e7a1bdp+0 - lg_ln(x: lg_sinpi(x: abs_xd)) - poly_val;
191 else
192 lgamma_val = poly_val - lg_ln(x: abs_xd);
193 } else if (x_abs > 0x42BEu) { // 3.3710938
194 double log_abs_xd = lg_ln(x: abs_xd);
195 // lgamma(x) = (x-0.5)*log(x) - x + log(2*pi)/2 + Bernoulli corrections
196 // Stirling base: (x-0.5)*(log(x)-1) + (log(2*pi)-1)/2
197 lgamma_val = (abs_xd - 0.5) * (log_abs_xd - 1.0) + 0x1.acfe390c97d69p-2;
198
199 double inv_x = 1.0 / abs_xd, inv_x2 = inv_x * inv_x;
200
201 if (x_abs > 0x64AEu) {
202 // |x| > 1198 1-term Stirling-Bernoulli: 1/(12*x)
203 lgamma_val += inv_x * 0x1.5555555555555p-4;
204 } else if (x_abs > 0x549Eu) {
205 // |x| in (73.875, 1198] 2-term correction
206 // Minimax polynomial in 1/x^2 for R(x) = lgamma(x) - stirling_base(x),
207 // where stirling_base(x) = (x-0.5)*log(x) - x + log(2*pi)/2
208 // Exact Bernoulli: B_2/(1*2) = 1/12, B_4/(3*4) = -1/360
209 // Generated by Sollya with:
210 // > stir = proc(x) { return lgamma(x)-((x-0.5)*log(x)-x+log(2*pi)/2);
211 // }; > P = fpminimax(stir(x), [|1,3|], [|D...|], [73.875, 1198]);
212 constexpr double BERN2[2] = {0x1.555555547fbadp-4, -0x1.6c0fd270c465p-9};
213 lgamma_val += inv_x * fputil::polyeval(x: inv_x2, a0: BERN2[0], a: BERN2[1]);
214 } else if (x_abs > 0x4955u) {
215 // |x| in (10.664, 73.875] 4-term correction
216 // Minimax polynomial in 1/x^2 for R(x); exact Bernoulli terms are
217 // 1/12, -1/360, 1/1260, -1/1680
218 // Generated by Sollya with:
219 // > display = hexadecimal;
220 // > stir = proc(x) { return lgamma(x)-((x-0.5)*log(x)-x+log(2*pi)/2);
221 // }; > P = fpminimax(stir(x), [|1,3,5,7|], [|D...|], [10.664, 73.875]);
222 constexpr double BERN4[4] = {0x1.555555554de0bp-4, -0x1.6c16bdc45944fp-9,
223 0x1.a0077f300ecb3p-11,
224 -0x1.2e9cfff3b29c2p-11};
225 double inv_x4 = inv_x2 * inv_x2;
226 lgamma_val +=
227 inv_x * fputil::multiply_add(
228 x: inv_x4, y: fputil::multiply_add(x: inv_x2, y: BERN4[3], z: BERN4[2]),
229 z: fputil::multiply_add(x: inv_x2, y: BERN4[1], z: BERN4[0]));
230 } else {
231 // |x| in (3.373, 10.664]: 8-term correction.
232 // Degree-7 minimax polynomial in 1/x^2 for R(x), generated by Sollya
233 // with:
234 // > stir = proc(x) { return lgamma(x)-((x-0.5)*log(x)-x+log(2*pi)/2);
235 // }; > P = fpminimax(stir(x), [|1,3,5,7,9,11,13,15|], [|D...|],
236 // [3.373, 10.664]);
237 constexpr double BERN8[8] = {
238 0x1.5555555551286p-4, -0x1.6c16c0e7c4cf4p-9, 0x1.a0193267fe6f2p-11,
239 -0x1.37e87ec19cb45p-11, 0x1.b40011dfff081p-11, -0x1.c16c8946b19b6p-10,
240 0x1.e9f47ace150d8p-9, -0x1.4f5843a71a338p-8};
241 double inv_x4 = inv_x2 * inv_x2, inv_x8 = inv_x4 * inv_x4;
242 double p01 = fputil::multiply_add(x: inv_x2, y: BERN8[1], z: BERN8[0]);
243 double p23 = fputil::multiply_add(x: inv_x2, y: BERN8[3], z: BERN8[2]);
244 double p45 = fputil::multiply_add(x: inv_x2, y: BERN8[5], z: BERN8[4]);
245 double p67 = fputil::multiply_add(x: inv_x2, y: BERN8[7], z: BERN8[6]);
246 double p03 = fputil::multiply_add(x: inv_x4, y: p23, z: p01);
247 double p47 = fputil::multiply_add(x: inv_x4, y: p67, z: p45);
248 lgamma_val += inv_x * fputil::multiply_add(x: inv_x8, y: p47, z: p03);
249 }
250
251 if (xbits.is_neg()) {
252 // Reflection formula
253 // lgamma(x) = log(pi) - log|x| - log|sin(pi*frac_x)| - lgamma|x|
254 double frac_x = xd - fputil::floor(x: xd);
255 lgamma_val = 0x1.250d048e7a1bdp+0 - lgamma_val - log_abs_xd;
256 lgamma_val -= lg_ln(x: lg_sinpi(x: frac_x));
257 }
258 } else {
259 // Medium path: |x| in [0.66015625, 3.373046875]
260 // lgamma(|x|) = (|x|-1)*(|x|-2) * P(u), where P(u) approximates
261 // g(t) = lgamma(t) / ((t-1)*(t-2)), u = (t - MID_M) / HW_M in [-1, 1]
262 // Degree-20 Chebyshev approximation for g in u; max lgamma error ~1.4e-10
263 // Coefficients generated by DCT at 21 Chebyshev nodes on
264 // [0.66015625, 3.373046875]. Evaluated via Clenshaw: sum_{k=0}^{20}
265 // CHEB_G[k] * T_k(u)
266 constexpr double MID_M = 0x1.0220000000000p+1;
267 constexpr double HW_M = 0x1.5b40000000000p+0;
268 constexpr double CHEB_G[21] = {
269 0x1.d690d4de8b355p-2, -0x1.53439726f7d8bp-3, 0x1.568fc6653c026p-5,
270 -0x1.8d074f73f36efp-7, 0x1.eefebf4ca8e86p-9, -0x1.4275913d85ff5p-10,
271 0x1.b075f0d96e492p-12, -0x1.27ed35f357339p-13, 0x1.9b12cbe71a800p-15,
272 -0x1.20c949c2829b5p-16, 0x1.996eb95241555p-18, -0x1.2460d6c60f800p-19,
273 0x1.a42297aae1862p-21, -0x1.2f6c7d28230a5p-22, 0x1.b8394aac30c31p-24,
274 -0x1.409726ce1110ep-25, 0x1.d484ac42db6dbp-27, -0x1.574dae55861abp-28,
275 0x1.f782605555555p-30, -0x1.6cefb8f65aa2ep-31, 0x1.d9f268f3cf3cfp-33};
276 // Clenshaw's algorithm: poly_g = sum_k CHEB_G[k] * T_k(u_m)
277 double u_m = (abs_xd - MID_M) / HW_M, u_m2 = 2.0 * u_m;
278 double bg1 = CHEB_G[20], bg2 = 0.0;
279 for (int k = 19; k >= 1; k--) {
280 double bg0 = fputil::multiply_add(x: u_m2, y: bg1, z: CHEB_G[k] - bg2);
281 bg2 = bg1;
282 bg1 = bg0;
283 }
284 double poly_g = fputil::multiply_add(x: u_m, y: bg1, z: CHEB_G[0] - bg2);
285 lgamma_val = (abs_xd - 1.0) * (abs_xd - 2.0) * poly_g;
286
287 if (xbits.is_neg()) {
288 // Reflection
289 // (x) = log(pi) - lgamma|x| - log(|x|*|sin(pi*frac_x)|)
290 double frac_x = xd - fputil::floor(x: xd);
291 lgamma_val = 0x1.250d048e7a1bdp+0 - lgamma_val;
292 lgamma_val -= lg_ln(x: lg_sinpi(x: frac_x) * abs_xd);
293 }
294 }
295
296 float16 result = fputil::cast<float16>(x: lgamma_val);
297 if (LIBC_UNLIKELY(fputil::FPBits<float16>(result).is_inf()))
298 fputil::set_errno_if_required(ERANGE);
299 return result;
300}
301
302} // namespace math
303
304} // namespace LIBC_NAMESPACE_DECL
305
306#endif // LIBC_TYPES_HAS_FLOAT16
307
308#endif // LLVM_LIBC_SRC___SUPPORT_MATH_LGAMMAF16_H
309