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1//===----------------------------------------------------------------------===//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-exception6//7//===----------------------------------------------------------------------===//8//9// Computes arcsin(x).10//11// The incoming argument is first reduced by noting that arcsin(x) is invalid12// for abs(x) > 1 and arcsin(-x) = -arcsin(x).13//14// For denormal and small arguments arcsin(x) = x to machine accuracy.15//16// Remaining argument ranges are handled as follows.17//18// * For abs(x) <= 0.5 use:19// arcsin(x) = x + x^3 * R(x^2)20// where R(x^2) is a rational minimax approximation to (arcsin(x) - x)/x^3.21// * For abs(x) > 0.5 exploit the identity:22// arcsin(x) = pi/2 - 2 * arcsin(sqrt(1 - x)/2)23// together with the above rational approximation, and reconstruct the terms24// carefully.25//26//===----------------------------------------------------------------------===//27 28#if __CLC_FPSIZE == 3229 30_CLC_OVERLOAD _CLC_DEF __CLC_GENTYPE __clc_asinpi(__CLC_GENTYPE x) {31 const __CLC_GENTYPE pi = __CLC_FP_LIT(3.1415926535897933e+00);32 // 0x33a2216833 const __CLC_GENTYPE piby2_tail = __CLC_FP_LIT(7.5497894159e-08);34 // 0x3f490fda35 const __CLC_GENTYPE hpiby2_head = __CLC_FP_LIT(7.8539812565e-01);36 37 __CLC_UINTN ux = __CLC_AS_UINTN(x);38 __CLC_UINTN aux = ux & EXSIGNBIT_SP32;39 __CLC_UINTN xs = ux ^ aux;40 __CLC_GENTYPE shalf =41 __CLC_AS_GENTYPE(xs | __CLC_AS_UINTN(__CLC_FP_LIT(0.5)));42 43 __CLC_INTN xexp = __CLC_AS_INTN(aux >> EXPSHIFTBITS_SP32) - EXPBIAS_SP32;44 45 __CLC_GENTYPE y = __CLC_AS_GENTYPE(aux);46 47 // abs(x) >= 0.548 __CLC_INTN transform = xexp >= -1;49 50 __CLC_GENTYPE y2 = y * y;51 __CLC_GENTYPE rt = 0.5f * (1.0f - y);52 __CLC_GENTYPE r = transform ? rt : y2;53 54 // Use a rational approximation for [0.0, 0.5]55 __CLC_GENTYPE a =56 __clc_mad(r,57 __clc_mad(r,58 __clc_mad(r, -0.00396137437848476485201154797087F,59 -0.0133819288943925804214011424456F),60 -0.0565298683201845211985026327361F),61 0.184161606965100694821398249421F);62 __CLC_GENTYPE b = __clc_mad(r, -0.836411276854206731913362287293F,63 1.10496961524520294485512696706F);64 __CLC_GENTYPE u = r * MATH_DIVIDE(a, b);65 66 __CLC_GENTYPE s = __clc_sqrt(r);67 __CLC_GENTYPE s1 = __CLC_AS_GENTYPE(__CLC_AS_UINTN(s) & 0xffff0000);68 __CLC_GENTYPE c = MATH_DIVIDE(__clc_mad(-s1, s1, r), s + s1);69 __CLC_GENTYPE p = __clc_mad(2.0f * s, u, -__clc_mad(c, -2.0f, piby2_tail));70 __CLC_GENTYPE q = __clc_mad(s1, -2.0f, hpiby2_head);71 __CLC_GENTYPE vt = hpiby2_head - (p - q);72 __CLC_GENTYPE v = __clc_mad(y, u, y);73 v = transform ? vt : v;74 v = MATH_DIVIDE(v, pi);75 __CLC_GENTYPE xbypi = MATH_DIVIDE(x, pi);76 77 __CLC_GENTYPE ret = __CLC_AS_GENTYPE(xs | __CLC_AS_UINTN(v));78 ret = aux > 0x3f800000U ? __CLC_GENTYPE_NAN : ret;79 ret = aux == 0x3f800000U ? shalf : ret;80 ret = xexp < -14 ? xbypi : ret;81 82 return ret;83}84 85#elif __CLC_FPSIZE == 6486 87_CLC_OVERLOAD _CLC_DEF __CLC_GENTYPE __clc_asinpi(__CLC_GENTYPE x) {88 const __CLC_GENTYPE pi = __CLC_FP_LIT(0x1.921fb54442d18p+1);89 // 0x3c91a62633145c0790 const __CLC_GENTYPE piby2_tail = __CLC_FP_LIT(6.1232339957367660e-17);91 // 0x3fe921fb54442d1892 const __CLC_GENTYPE hpiby2_head = __CLC_FP_LIT(7.8539816339744831e-01);93 94 __CLC_GENTYPE y = __clc_fabs(x);95 __CLC_LONGN xneg = x < __CLC_FP_LIT(0.0);96 __CLC_INTN xexp = __CLC_CONVERT_INTN(97 (__CLC_AS_ULONGN(y) >> EXPSHIFTBITS_DP64) - EXPBIAS_DP64);98 99 // abs(x) >= 0.5100 __CLC_LONGN transform = __CLC_CONVERT_LONGN(xexp >= -1);101 102 __CLC_GENTYPE rt = 0.5 * (1.0 - y);103 __CLC_GENTYPE y2 = y * y;104 __CLC_GENTYPE r = transform ? rt : y2;105 106 // Use a rational approximation for [0.0, 0.5]107 __CLC_GENTYPE un = __clc_fma(108 r,109 __clc_fma(110 r,111 __clc_fma(r,112 __clc_fma(r,113 __clc_fma(r, 0.0000482901920344786991880522822991,114 0.00109242697235074662306043804220),115 -0.0549989809235685841612020091328),116 0.275558175256937652532686256258),117 -0.445017216867635649900123110649),118 0.227485835556935010735943483075);119 120 __CLC_GENTYPE ud = __clc_fma(121 r,122 __clc_fma(r,123 __clc_fma(r,124 __clc_fma(r, 0.105869422087204370341222318533,125 -0.943639137032492685763471240072),126 2.76568859157270989520376345954),127 -3.28431505720958658909889444194),128 1.36491501334161032038194214209);129 130 __CLC_GENTYPE u = r * MATH_DIVIDE(un, ud);131 132 // Reconstruct asin carefully in transformed region133 __CLC_GENTYPE s = __clc_sqrt(r);134 __CLC_GENTYPE sh =135 __CLC_AS_GENTYPE(__CLC_AS_ULONGN(s) & 0xffffffff00000000UL);136 __CLC_GENTYPE c = MATH_DIVIDE(__clc_fma(-sh, sh, r), s + sh);137 __CLC_GENTYPE p = __clc_fma(2.0 * s, u, -__clc_fma(-2.0, c, piby2_tail));138 __CLC_GENTYPE q = __clc_fma(-2.0, sh, hpiby2_head);139 __CLC_GENTYPE vt = hpiby2_head - (p - q);140 __CLC_GENTYPE v = __clc_fma(y, u, y);141 v = transform ? vt : v;142 143 v = __CLC_CONVERT_LONGN(xexp < -28) ? y : v;144 v = MATH_DIVIDE(v, pi);145 v = __CLC_CONVERT_LONGN(xexp >= 0) ? __CLC_GENTYPE_NAN : v;146 v = y == 1.0 ? 0.5 : v;147 return xneg ? -v : v;148}149 150#elif __CLC_FPSIZE == 16151 152_CLC_OVERLOAD _CLC_DEF __CLC_GENTYPE __clc_asinpi(__CLC_GENTYPE x) {153 return __CLC_CONVERT_GENTYPE(__clc_asinpi(__CLC_CONVERT_FLOATN(x)));154}155 156#endif157