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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// * For abs(x) <= 0.5 use:18// arcsin(x) = x + x^3 * R(x^2)19// where R(x^2) is a rational minimax approximation to (arcsin(x) - x)/x^3.20// * For abs(x) > 0.5 exploit the identity:21// arcsin(x) = pi/2 - 2 * arcsin(sqrt(1 - x)/2)22// together with the above rational approximation, and reconstruct the terms23// carefully.24//25//===----------------------------------------------------------------------===//26 27#if __CLC_FPSIZE == 3228 29_CLC_OVERLOAD _CLC_DEF __CLC_GENTYPE __clc_asin(__CLC_GENTYPE x) {30 // 0x33a2216831 const __CLC_GENTYPE piby2_tail = __CLC_FP_LIT(7.5497894159e-08);32 // 0x3f490fda33 const __CLC_GENTYPE hpiby2_head = __CLC_FP_LIT(7.8539812565e-01);34 // 0x3fc90fdb35 const __CLC_GENTYPE piby2 = __CLC_FP_LIT(1.5707963705e+00);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 spiby2 = __CLC_AS_GENTYPE(xs | __CLC_AS_UINTN(piby2));41 __CLC_INTN xexp = __CLC_AS_INTN(aux >> EXPSHIFTBITS_SP32) - EXPBIAS_SP32;42 __CLC_GENTYPE y = __CLC_AS_GENTYPE(aux);43 44 // abs(x) >= 0.545 __CLC_INTN transform = xexp >= -1;46 47 __CLC_GENTYPE y2 = y * y;48 __CLC_GENTYPE rt = __CLC_FP_LIT(0.5) * (__CLC_FP_LIT(1.0) - y);49 __CLC_GENTYPE r = transform ? rt : y2;50 51 // Use a rational approximation for [0.0, 0.5]52 __CLC_GENTYPE a =53 __clc_mad(r,54 __clc_mad(r,55 __clc_mad(r, -0.00396137437848476485201154797087F,56 -0.0133819288943925804214011424456F),57 -0.0565298683201845211985026327361F),58 0.184161606965100694821398249421F);59 60 __CLC_GENTYPE b = __clc_mad(r, -0.836411276854206731913362287293F,61 1.10496961524520294485512696706F);62 __CLC_GENTYPE u = r * MATH_DIVIDE(a, b);63 64 __CLC_GENTYPE s = __clc_sqrt(r);65 __CLC_GENTYPE s1 = __CLC_AS_GENTYPE(__CLC_AS_UINTN(s) & 0xffff0000);66 __CLC_GENTYPE c = MATH_DIVIDE(__clc_mad(-s1, s1, r), s + s1);67 __CLC_GENTYPE p = __clc_mad(2.0f * s, u, -__clc_mad(c, -2.0f, piby2_tail));68 __CLC_GENTYPE q = __clc_mad(s1, -2.0f, hpiby2_head);69 __CLC_GENTYPE vt = hpiby2_head - (p - q);70 __CLC_GENTYPE v = __clc_mad(y, u, y);71 v = transform ? vt : v;72 73 __CLC_GENTYPE ret = __CLC_AS_GENTYPE(xs | __CLC_AS_UINTN(v));74 ret = aux > 0x3f800000U ? __CLC_GENTYPE_NAN : ret;75 ret = aux == 0x3f800000U ? spiby2 : ret;76 ret = xexp < -14 ? x : ret;77 78 return ret;79}80 81#elif __CLC_FPSIZE == 6482 83_CLC_OVERLOAD _CLC_DEF __CLC_GENTYPE __clc_asin(__CLC_GENTYPE x) {84 // 0x3c91a62633145c0785 const __CLC_GENTYPE piby2_tail = __CLC_FP_LIT(6.1232339957367660e-17);86 // 0x3fe921fb54442d1887 const __CLC_GENTYPE hpiby2_head = 7.8539816339744831e-01;88 // 0x3ff921fb54442d1889 const __CLC_GENTYPE piby2 = 1.5707963267948965e+00;90 91 __CLC_GENTYPE y = __clc_fabs(x);92 __CLC_LONGN xneg = x < __CLC_FP_LIT(0.0);93 __CLC_INTN xexp = __CLC_CONVERT_INTN(94 (__CLC_AS_ULONGN(y) >> EXPSHIFTBITS_DP64) - EXPBIAS_DP64);95 96 // abs(x) >= 0.597 __CLC_LONGN transform = __CLC_CONVERT_LONGN(xexp >= -1);98 99 __CLC_GENTYPE rt = __CLC_FP_LIT(0.5) * (__CLC_FP_LIT(1.0) - y);100 __CLC_GENTYPE y2 = y * y;101 __CLC_GENTYPE r = transform ? rt : y2;102 103 // Use a rational approximation for [0.0, 0.5]104 105 __CLC_GENTYPE un = __clc_fma(106 r,107 __clc_fma(108 r,109 __clc_fma(r,110 __clc_fma(r,111 __clc_fma(r, 0.0000482901920344786991880522822991,112 0.00109242697235074662306043804220),113 -0.0549989809235685841612020091328),114 0.275558175256937652532686256258),115 -0.445017216867635649900123110649),116 0.227485835556935010735943483075);117 118 __CLC_GENTYPE ud = __clc_fma(119 r,120 __clc_fma(r,121 __clc_fma(r,122 __clc_fma(r, 0.105869422087204370341222318533,123 -0.943639137032492685763471240072),124 2.76568859157270989520376345954),125 -3.28431505720958658909889444194),126 1.36491501334161032038194214209);127 128 __CLC_GENTYPE u = r * MATH_DIVIDE(un, ud);129 130 // Reconstruct asin carefully in transformed region131 __CLC_GENTYPE s = __clc_sqrt(r);132 __CLC_GENTYPE sh =133 __CLC_AS_GENTYPE(__CLC_AS_ULONGN(s) & 0xffffffff00000000UL);134 __CLC_GENTYPE c = MATH_DIVIDE(__clc_fma(-sh, sh, r), s + sh);135 __CLC_GENTYPE p = __clc_fma(2.0 * s, u, -__clc_fma(-2.0, c, piby2_tail));136 __CLC_GENTYPE q = __clc_fma(-2.0, sh, hpiby2_head);137 __CLC_GENTYPE vt = hpiby2_head - (p - q);138 __CLC_GENTYPE v = __clc_fma(y, u, y);139 v = transform ? vt : v;140 141 v = __CLC_CONVERT_LONGN(xexp < -28) ? y : v;142 v = __CLC_CONVERT_LONGN(xexp >= 0) ? __CLC_GENTYPE_NAN : v;143 v = y == 1.0 ? piby2 : v;144 145 return xneg ? -v : v;146}147 148#elif __CLC_FPSIZE == 16149 150_CLC_OVERLOAD _CLC_DEF __CLC_GENTYPE __clc_asin(__CLC_GENTYPE x) {151 return __CLC_CONVERT_GENTYPE(__clc_asin(__CLC_CONVERT_FLOATN(x)));152}153 154#endif155