1//===-- Collection of utils for asin/acos -----------------------*- 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_MATH_GENERIC_ASIN_UTILS_H
10#define LLVM_LIBC_SRC_MATH_GENERIC_ASIN_UTILS_H
11
12#include "src/__support/FPUtil/PolyEval.h"
13#include "src/__support/FPUtil/double_double.h"
14#include "src/__support/FPUtil/dyadic_float.h"
15#include "src/__support/FPUtil/multiply_add.h"
16#include "src/__support/FPUtil/nearest_integer.h"
17#include "src/__support/integer_literals.h"
18#include "src/__support/macros/config.h"
19#include "src/__support/macros/optimization.h"
20
21namespace LIBC_NAMESPACE_DECL {
22
23namespace {
24
25using DoubleDouble = fputil::DoubleDouble;
26using Float128 = fputil::DyadicFloat<128>;
27
28constexpr DoubleDouble PI = {0x1.1a62633145c07p-53, 0x1.921fb54442d18p1};
29
30constexpr DoubleDouble PI_OVER_TWO = {0x1.1a62633145c07p-54,
31 0x1.921fb54442d18p0};
32
33#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
34
35// When correct rounding is not needed, we use a degree-22 minimax polynomial to
36// approximate asin(x)/x on [0, 0.5] using Sollya with:
37// > P = fpminimax(asin(x)/x, [|0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22|],
38// [|1, D...|], [0, 0.5]);
39// > dirtyinfnorm(asin(x)/x - P, [0, 0.5]);
40// 0x1.1a71ef0a0f26a9fb7ed7e41dee788b13d1770db3dp-52
41
42constexpr double ASIN_COEFFS[12] = {
43 0x1.0000000000000p0, 0x1.5555555556dcfp-3, 0x1.3333333082e11p-4,
44 0x1.6db6dd14099edp-5, 0x1.f1c69b35bf81fp-6, 0x1.6e97194225a67p-6,
45 0x1.1babddb82ce12p-6, 0x1.d55bd078600d6p-7, 0x1.33328959e63d6p-7,
46 0x1.2b5993bda1d9bp-6, -0x1.806aff270bf25p-7, 0x1.02614e5ed3936p-5,
47};
48
49LIBC_INLINE double asin_eval(double u) {
50 double u2 = u * u;
51 double c0 = fputil::multiply_add(u, ASIN_COEFFS[1], ASIN_COEFFS[0]);
52 double c1 = fputil::multiply_add(u, ASIN_COEFFS[3], ASIN_COEFFS[2]);
53 double c2 = fputil::multiply_add(u, ASIN_COEFFS[5], ASIN_COEFFS[4]);
54 double c3 = fputil::multiply_add(u, ASIN_COEFFS[7], ASIN_COEFFS[6]);
55 double c4 = fputil::multiply_add(u, ASIN_COEFFS[9], ASIN_COEFFS[8]);
56 double c5 = fputil::multiply_add(u, ASIN_COEFFS[11], ASIN_COEFFS[10]);
57
58 double u4 = u2 * u2;
59 double d0 = fputil::multiply_add(u2, c1, c0);
60 double d1 = fputil::multiply_add(u2, c3, c2);
61 double d2 = fputil::multiply_add(u2, c5, c4);
62
63 return fputil::polyeval(u4, d0, d1, d2);
64}
65
66#else
67
68// The Taylor expansion of asin(x) around 0 is:
69// asin(x) = x + x^3/6 + 3x^5/40 + ...
70// ~ x * P(x^2).
71// Let u = x^2, then P(x^2) = P(u), and |x| = sqrt(u). Note that when
72// |x| <= 0.5, we have |u| <= 0.25.
73// We approximate P(u) by breaking it down by performing range reduction mod
74// 2^-5 = 1/32.
75// So for:
76// k = round(u * 32),
77// y = u - k/32,
78// we have that:
79// x = sqrt(u) = sqrt(k/32 + y),
80// |y| <= 2^-5 = 1/32,
81// and:
82// P(u) = P(k/32 + y) = Q_k(y).
83// Hence :
84// asin(x) = sqrt(k/32 + y) * Q_k(y),
85// Or equivalently:
86// Q_k(y) = asin(sqrt(k/32 + y)) / sqrt(k/32 + y).
87// We generate the coefficients of Q_k by Sollya as following:
88// > procedure ASIN_APPROX(N, Deg) {
89// abs_error = 0;
90// rel_error = 0;
91// deg = [||];
92// for i from 2 to Deg do deg = deg :. i;
93// for i from 1 to N/4 do {
94// F = asin(sqrt(i/N + x))/sqrt(i/N + x);
95// T = taylor(F, 1, 0);
96// T_DD = roundcoefficients(T, [|DD...|]);
97// I = [-1/(2*N), 1/(2*N)];
98// Q = fpminimax(F, deg, [|D...|], I, T_DD);
99// abs_err = dirtyinfnorm(F - Q, I);
100// rel_err = dirtyinfnorm((F - Q)/x^2, I);
101// if (abs_err > abs_error) then abs_error = abs_err;
102// if (rel_err > rel_error) then rel_error = rel_err;
103// d0 = D(coeff(Q, 0));
104// d1 = coeff(Q, 0) - d0;
105// write("{", d0, ", ", d1);
106// d0 = D(coeff(Q, 1)); d1 = coeff(Q, 1) - d0; write(", ", d0, ", ", d1);
107// for j from 2 to Deg do {
108// write(", ", coeff(Q, j));
109// };
110// print("},");
111// };
112// print("Absolute Errors:", D(abs_error));
113// print("Relative Errors:", D(rel_error));
114// };
115// > ASIN_APPROX(32, 9);
116// Absolute Errors: 0x1.69837b5183654p-72
117// Relative Errors: 0x1.4d7f82835bf64p-55
118
119// For k = 0, we use the degree-18 Taylor polynomial of asin(x)/x:
120//
121// > P = 1 + x^2 * DD(1/6) + x^4 * D(3/40) + x^6 * D(5/112) + x^8 * D(35/1152) +
122// x^10 * D(63/2816) + x^12 * D(231/13312) + x^14 * D(143/10240) +
123// x^16 * D(6435/557056) + x^18 * D(12155/1245184);
124// > dirtyinfnorm(asin(x)/x - P, [-1/64, 1/64]);
125// 0x1.999075402cafp-83
126
127constexpr double ASIN_COEFFS[9][12] = {
128 {1.0, 0.0, 0x1.5555555555555p-3, 0x1.5555555555555p-57,
129 0x1.3333333333333p-4, 0x1.6db6db6db6db7p-5, 0x1.f1c71c71c71c7p-6,
130 0x1.6e8ba2e8ba2e9p-6, 0x1.1c4ec4ec4ec4fp-6, 0x1.c99999999999ap-7,
131 0x1.7a87878787878p-7, 0x1.3fde50d79435ep-7},
132 {0x1.015a397cf0f1cp0, -0x1.eebd6ccfe3ee3p-55, 0x1.5f3581be7b08bp-3,
133 -0x1.5df80d0e7237dp-57, 0x1.4519ddf1ae53p-4, 0x1.8eb4b6eeb1696p-5,
134 0x1.17bc85420fec8p-5, 0x1.a8e39b5dcad81p-6, 0x1.53f8df127539bp-6,
135 0x1.1a485a0b0130ap-6, 0x1.e20e6e493002p-7, 0x1.a466a7030f4c9p-7},
136 {0x1.02be9ce0b87cdp0, 0x1.e5d09da2e0f04p-56, 0x1.69ab5325bc359p-3,
137 -0x1.92f480cfede2dp-57, 0x1.58a4c3097aab1p-4, 0x1.b3db36068dd8p-5,
138 0x1.3b9482184625p-5, 0x1.eedc823765d21p-6, 0x1.98e35d756be6bp-6,
139 0x1.5ea4f1b32731ap-6, 0x1.355115764148ep-6, 0x1.16a5853847c91p-6},
140 {0x1.042dc6a65ffbfp0, -0x1.c7ea28dce95d1p-55, 0x1.74c4bd7412f9dp-3,
141 0x1.447024c0a3c87p-58, 0x1.6e09c6d2b72b9p-4, 0x1.ddd9dcdae5315p-5,
142 0x1.656f1f64058b8p-5, 0x1.21a42e4437101p-5, 0x1.eed0350b7edb2p-6,
143 0x1.b6bc877e58c52p-6, 0x1.903a0872eb2a4p-6, 0x1.74da839ddd6d8p-6},
144 {0x1.05a8621feb16bp0, -0x1.e5b33b1407c5fp-56, 0x1.809186c2e57ddp-3,
145 -0x1.3dcb4d6069407p-60, 0x1.8587d99442dc5p-4, 0x1.06c23d1e75be3p-4,
146 0x1.969024051c67dp-5, 0x1.54e4f934aacfdp-5, 0x1.2d60a732dbc9cp-5,
147 0x1.149f0c046eac7p-5, 0x1.053a56dba1fbap-5, 0x1.f7face3343992p-6},
148 {0x1.072f2b6f1e601p0, -0x1.2dcbb0541997p-54, 0x1.8d2397127aebap-3,
149 0x1.ead0c497955fbp-57, 0x1.9f68df88da518p-4, 0x1.21ee26a5900d7p-4,
150 0x1.d08e7081b53a9p-5, 0x1.938dd661713f7p-5, 0x1.71b9f299b72e6p-5,
151 0x1.5fbc7d2450527p-5, 0x1.58573247ec325p-5, 0x1.585a174a6a4cep-5},
152 {0x1.08c2f1d638e4cp0, 0x1.b47c159534a3dp-56, 0x1.9a8f592078624p-3,
153 -0x1.ea339145b65cdp-57, 0x1.bc04165b57aabp-4, 0x1.410df5f58441dp-4,
154 0x1.0ab6bdf5f8f7p-4, 0x1.e0b92eea1fce1p-5, 0x1.c9094e443a971p-5,
155 0x1.c34651d64bc74p-5, 0x1.caa008d1af08p-5, 0x1.dc165bc0c4fc5p-5},
156 {0x1.0a649a73e61f2p0, 0x1.74ac0d817e9c7p-55, 0x1.a8ec30dc9389p-3,
157 -0x1.8ab1c0eef300cp-59, 0x1.dbc11ea95061bp-4, 0x1.64e371d661328p-4,
158 0x1.33e0023b3d895p-4, 0x1.2042269c243cep-4, 0x1.1cce74bda223p-4,
159 0x1.244d425572ce9p-4, 0x1.34d475c7f1e3ep-4, 0x1.4d4e653082ad3p-4},
160 {0x1.0c152382d7366p0, -0x1.ee6913347c2a6p-54, 0x1.b8550d62bfb6dp-3,
161 -0x1.d10aec3f116d5p-57, 0x1.ff1bde0fa3cap-4, 0x1.8e5f3ab69f6a4p-4,
162 0x1.656be8b6527cep-4, 0x1.5c39755dc041ap-4, 0x1.661e6ebd40599p-4,
163 0x1.7ea3dddee2a4fp-4, 0x1.a4f439abb4869p-4, 0x1.d9181c0fda658p-4},
164};
165
166// We calculate the lower part of the approximation P(u).
167LIBC_INLINE DoubleDouble asin_eval(const DoubleDouble &u, unsigned &idx,
168 double &err) {
169 using fputil::multiply_add;
170 // k = round(u * 32).
171 double k = fputil::nearest_integer(u.hi * 0x1.0p5);
172 idx = static_cast<unsigned>(k);
173 // y = u - k/32.
174 double y_hi = multiply_add(k, -0x1.0p-5, u.hi); // Exact
175 DoubleDouble y = fputil::exact_add(y_hi, u.lo);
176 double y2 = y.hi * y.hi;
177 // Add double-double errors in addition to the relative errors from y2.
178 err = fputil::multiply_add(err, y2, 0x1.0p-102);
179 DoubleDouble c0 = fputil::quick_mult(
180 y, DoubleDouble{ASIN_COEFFS[idx][3], ASIN_COEFFS[idx][2]});
181 double c1 = multiply_add(y.hi, ASIN_COEFFS[idx][5], ASIN_COEFFS[idx][4]);
182 double c2 = multiply_add(y.hi, ASIN_COEFFS[idx][7], ASIN_COEFFS[idx][6]);
183 double c3 = multiply_add(y.hi, ASIN_COEFFS[idx][9], ASIN_COEFFS[idx][8]);
184 double c4 = multiply_add(y.hi, ASIN_COEFFS[idx][11], ASIN_COEFFS[idx][10]);
185
186 double y4 = y2 * y2;
187 double d0 = multiply_add(y2, c2, c1);
188 double d1 = multiply_add(y2, c4, c3);
189
190 DoubleDouble r = fputil::exact_add(ASIN_COEFFS[idx][0], c0.hi);
191
192 double e1 = multiply_add(y4, d1, d0);
193
194 r.lo = multiply_add(y2, e1, ASIN_COEFFS[idx][1] + c0.lo + r.lo);
195
196 return r;
197}
198
199// Follow the discussion above, we generate the coefficients of Q_k by Sollya as
200// following:
201// > procedure PRINTF128(a) {
202// write("{");
203// if (a < 0)
204// then write("Sign::NEG, ") else write("Sign::POS, ");
205// a_exp = floor(log2(a)) + 1;
206// write((a + 2 ^ a_exp) * 2 ^ -128);
207// print("},");
208// };
209// > verbosity = 0;
210// > procedure ASIN_APPROX(N, Deg) {
211// abs_error = 0;
212// rel_error = 0;
213// for i from 1 to N / 4 do {
214// Q = fpminimax(asin(sqrt(i / N + x)) / sqrt(i / N + x), Deg,
215// [| 128... | ], [ -1 / (2 * N), 1 / (2 * N) ]);
216// abs_err = dirtyinfnorm(asin(sqrt(i / N + x)) - sqrt(i / N + x) * Q,
217// [ -1 / (2 * N), 1 / (2 * N) ]);
218// rel_err = dirtyinfnorm(asin(sqrt(i / N + x)) / sqrt(i / N + x) - Q,
219// [ -1 / (2 * N), 1 / (2 * N) ]);
220// if (abs_err > abs_error) then abs_error = abs_err;
221// if (rel_err > rel_error) then rel_error = rel_err;
222// write("{");
223// for j from 0 to Deg do PRINTF128(coeff(Q, j));
224// print("},");
225// };
226// print("Absolute Errors:", abs_error);
227// print("Relative Errors:", rel_error);
228// };
229// > ASIN_APPROX(64, 15);
230// ...
231// Absolute Errors: 0x1.0b3...p-129
232// Relative Errors: 0x1.1db...p-128
233//
234// For k = 0, we use Taylor polynomial of asin(x)/x around x = 0.
235// asin(x)/x ~ 1 + x^2/6 + (3 x^4)/40 + (5 x^6)/112 + (35 x^8)/1152 +
236// + (63 x^10)/2816 + (231 x^12)/13312 + (143 x^14)/10240 +
237// + (6435 x^16)/557056 + (12155 x^18)/1245184 +
238// + (46189 x^20)/5505024 + (88179 x^22)/12058624 +
239// + (676039 x^24)/104857600 + (1300075 x^26)/226492416 +
240// + (5014575 x^28)/973078528 + (9694845 x^30)/2080374784.
241
242constexpr Float128 ASIN_COEFFS_F128[17][16] = {
243 {
244 {Sign::POS, -127, 0x80000000'00000000'00000000'00000000_u128},
245 {Sign::POS, -130, 0xaaaaaaaa'aaaaaaaa'aaaaaaaa'aaaaaaab_u128},
246 {Sign::POS, -131, 0x99999999'99999999'99999999'9999999a_u128},
247 {Sign::POS, -132, 0xb6db6db6'db6db6db'6db6db6d'b6db6db7_u128},
248 {Sign::POS, -133, 0xf8e38e38'e38e38e3'8e38e38e'38e38e39_u128},
249 {Sign::POS, -133, 0xb745d174'5d1745d1'745d1745'd1745d17_u128},
250 {Sign::POS, -133, 0x8e276276'27627627'62762762'76276276_u128},
251 {Sign::POS, -134, 0xe4cccccc'cccccccc'cccccccc'cccccccd_u128},
252 {Sign::POS, -134, 0xbd43c3c3'c3c3c3c3'c3c3c3c3'c3c3c3c4_u128},
253 {Sign::POS, -134, 0x9fef286b'ca1af286'bca1af28'6bca1af3_u128},
254 {Sign::POS, -134, 0x89779e79'e79e79e7'9e79e79e'79e79e7a_u128},
255 {Sign::POS, -135, 0xef9de9bd'37a6f4de'9bd37a6f'4de9bd38_u128},
256 {Sign::POS, -135, 0xd3431eb8'51eb851e'b851eb85'1eb851ec_u128},
257 {Sign::POS, -135, 0xbc16ed09'7b425ed0'97b425ed'097b425f_u128},
258 {Sign::POS, -135, 0xa8dd1846'9ee58469'ee58469e'e58469ee_u128},
259 {Sign::POS, -135, 0x98b41def'7bdef7bd'ef7bdef7'bdef7bdf_u128},
260 },
261 {
262 {Sign::POS, -127, 0x8055f060'94f0f05f'3ac3b927'50a701d9_u128},
263 {Sign::POS, -130, 0xad19c2ea'e3dd2429'8d04f71d'b965ee1b_u128},
264 {Sign::POS, -131, 0x9dfa882b'7b31af17'f9f19d33'0c45d24b_u128},
265 {Sign::POS, -132, 0xbedd3b58'c9e605ef'1404e1f0'4ba57940_u128},
266 {Sign::POS, -132, 0x83df2581'cb4fea82'b406f201'2fde6d5c_u128},
267 {Sign::POS, -133, 0xc534fe61'9b82dd16'ed5d8a43'f7710526_u128},
268 {Sign::POS, -133, 0x9b56fa62'88295ddf'ce8425fe'a04d733e_u128},
269 {Sign::POS, -134, 0xfdeddb19'4a030da7'27158080'd24caf46_u128},
270 {Sign::POS, -134, 0xd55827db'ff416ea8'042c4d8c'07cddeeb_u128},
271 {Sign::POS, -134, 0xb71d73a9'f2ba0688'5eaeeae9'413a0f5f_u128},
272 {Sign::POS, -134, 0x9fde87e2'ace91274'38f82666'd619c1ba_u128},
273 {Sign::POS, -134, 0x8d876557'5e4626a1'1b621336'93587847_u128},
274 {Sign::POS, -135, 0xfd801840'c8710595'6880fe13'a9657f8f_u128},
275 {Sign::POS, -135, 0xe54245a9'4c8c2ebb'30488494'64b0e34d_u128},
276 {Sign::POS, -135, 0xd11eb46f'4095a661'8890d123'15c96482_u128},
277 {Sign::POS, -135, 0xc01a4201'467fbc0b'960618d5'ec2adaa8_u128},
278 },
279 {
280 {Sign::POS, -127, 0x80ad1cbe'7878de11'4293301c'11ce9d49_u128},
281 {Sign::POS, -130, 0xaf9ac0df'3d845544'0fe5e31b'9051d03e_u128},
282 {Sign::POS, -131, 0xa28ceef8'd7297e05'f94773ad'f4a695c6_u128},
283 {Sign::POS, -132, 0xc75a5b77'58b4b11d'396c68ad'6733022b_u128},
284 {Sign::POS, -132, 0x8bde42a1'084a6674'50c5bceb'005d4b62_u128},
285 {Sign::POS, -133, 0xd471cdae'e2f35a96'bd4bc513'e0ccdf2c_u128},
286 {Sign::POS, -133, 0xa9fc6fd5'd204a4e3'e609940c'6b991b67_u128},
287 {Sign::POS, -133, 0x8d242d97'ba12b492'e25c7e7c'0c3fcf60_u128},
288 {Sign::POS, -134, 0xf0f1ba74'b149afc3'2f0bbab5'a20c6199_u128},
289 {Sign::POS, -134, 0xd21b42fb'd8e9098d'19612692'9a043332_u128},
290 {Sign::POS, -134, 0xba5e5492'7896a3e7'193a74d5'78631587_u128},
291 {Sign::POS, -134, 0xa7a17ae7'fc707f45'910e7a5d'c95251f4_u128},
292 {Sign::POS, -134, 0x98889a6a'b0370464'50c950d3'61d79ed7_u128},
293 {Sign::POS, -134, 0x8c29330e'4318fd29'25c5b528'84e39e7c_u128},
294 {Sign::POS, -134, 0x81e7bf48'b25bc7c0'b9204a4f'd4f5fa8b_u128},
295 {Sign::POS, -135, 0xf2801b09'11bf0768'773996dd'5224d852_u128},
296 },
297 {
298 {Sign::POS, -127, 0x81058e3e'f82ba622'ab81cd63'e1a91d57_u128},
299 {Sign::POS, -130, 0xb22e7055'c80dd354'8a2f2e8e'860d3f33_u128},
300 {Sign::POS, -131, 0xa753ce1a'7e3d1f57'247b37e6'03f93624_u128},
301 {Sign::POS, -132, 0xd05c5604'8eca8d18'dcdd76b7'f4b1f185_u128},
302 {Sign::POS, -132, 0x947cdd5e'f1d64df0'84f78df1'e2ecb854_u128},
303 {Sign::POS, -133, 0xe5218370'2ebbf6e8'3727a755'57843b93_u128},
304 {Sign::POS, -133, 0xba482553'383b92eb'186f78f1'8c35d6af_u128},
305 {Sign::POS, -133, 0x9d2b034a'7266c6a1'54b78a98'1a547429_u128},
306 {Sign::POS, -133, 0x8852f723'feea6046'e125f5a9'64e168e6_u128},
307 {Sign::POS, -134, 0xf19c9891'6c896c99'732052fe'5c54e992_u128},
308 {Sign::POS, -134, 0xd9cc81a5'c5ddf0f0'd651011e'a8ecd936_u128},
309 {Sign::POS, -134, 0xc7173169'dcb6095f'a6160847'b595aaff_u128},
310 {Sign::POS, -134, 0xb81cd3f6'4a422ebe'07aeb734'e4dcf3a1_u128},
311 {Sign::POS, -134, 0xabf01b1c'd15932aa'698d4382'512318a9_u128},
312 {Sign::POS, -134, 0xa1f1cf1b'd889a1ac'7120ca2f'bbbc1745_u128},
313 {Sign::POS, -134, 0x99a1b838'e38fbf11'429a4350'76b7d191_u128},
314 },
315 {
316 {Sign::POS, -127, 0x815f4e70'5c3e68f2'e84ed170'78211dfd_u128},
317 {Sign::POS, -130, 0xb4d5a992'de1ac4da'16fe6024'3a6cc371_u128},
318 {Sign::POS, -131, 0xac526184'bd558c65'66642dce'edc4b04a_u128},
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333 {
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387 {
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531 {
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543 {Sign::POS, -130, 0x9cbb5447'af8574f1'21fa4cda'93d82b7e_u128},
544 {Sign::POS, -130, 0xb7f5a6c0'430a347f'11b22cde'91de0885_u128},
545 {Sign::POS, -130, 0xda153cc4'14abdb96'840df7c2'3299fec0_u128},
546 {Sign::POS, -129, 0x826c129b'3e4a2612'b2cd11f1'4d2ba60c_u128},
547 {Sign::POS, -129, 0x9d19c289'fc0e8aa4'f351418b'b760ce90_u128},
548 },
549};
550
551constexpr Float128 PI_OVER_TWO_F128 = {
552 Sign::POS, -127, 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128};
553
554constexpr Float128 PI_F128 = {Sign::POS, -126,
555 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128};
556
557LIBC_INLINE Float128 asin_eval(const Float128 &u, unsigned idx) {
558 return fputil::polyeval(u, ASIN_COEFFS_F128[idx][0], ASIN_COEFFS_F128[idx][1],
559 ASIN_COEFFS_F128[idx][2], ASIN_COEFFS_F128[idx][3],
560 ASIN_COEFFS_F128[idx][4], ASIN_COEFFS_F128[idx][5],
561 ASIN_COEFFS_F128[idx][6], ASIN_COEFFS_F128[idx][7],
562 ASIN_COEFFS_F128[idx][8], ASIN_COEFFS_F128[idx][9],
563 ASIN_COEFFS_F128[idx][10], ASIN_COEFFS_F128[idx][11],
564 ASIN_COEFFS_F128[idx][12], ASIN_COEFFS_F128[idx][13],
565 ASIN_COEFFS_F128[idx][14], ASIN_COEFFS_F128[idx][15]);
566}
567
568#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
569
570} // anonymous namespace
571
572} // namespace LIBC_NAMESPACE_DECL
573
574#endif // LLVM_LIBC_SRC_MATH_GENERIC_ASIN_UTILS_H
575

source code of libc/src/math/generic/asin_utils.h