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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 arccos(x).10//11// The incoming argument is first reduced by noting that arccos(x) is invalid12// for abs(x) > 1.13//14// For denormal and small arguments arccos(x) = pi/2 to machine accuracy.15//16// Remaining argument ranges are handled as follows:17// * For abs(x) <= 0.5 use:18// arccos(x) = pi/2 - arcsin(x) = pi/2 - (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// arccos(x) = pi - 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_acos(__CLC_GENTYPE x) {30 // Some constants and split constants.31 const __CLC_GENTYPE piby2 = __CLC_FP_LIT(1.5707963705e+00);32 const __CLC_GENTYPE pi = __CLC_FP_LIT(3.1415926535897933e+00);33 const __CLC_GENTYPE piby2_head = __CLC_FP_LIT(1.5707963267948965580e+00);34 const __CLC_GENTYPE piby2_tail = __CLC_FP_LIT(6.12323399573676603587e-17);35 36 __CLC_UINTN ux = __CLC_AS_UINTN(x);37 __CLC_UINTN aux = ux & ~SIGNBIT_SP32;38 __CLC_INTN xneg = ux != aux;39 __CLC_INTN xexp = __CLC_AS_INTN(aux >> EXPSHIFTBITS_SP32) - EXPBIAS_SP32;40 __CLC_GENTYPE y = __CLC_AS_GENTYPE(aux);41 42 // transform if |x| >= 0.543 __CLC_INTN transform = xexp >= -1;44 45 __CLC_GENTYPE y2 = y * y;46 __CLC_GENTYPE yt = 0.5f * (1.0f - y);47 __CLC_GENTYPE r = transform ? yt : y2;48 49 // Use a rational approximation for [0.0, 0.5]50 __CLC_GENTYPE a =51 __clc_mad(r,52 __clc_mad(r,53 __clc_mad(r, -0.00396137437848476485201154797087F,54 -0.0133819288943925804214011424456F),55 -0.0565298683201845211985026327361F),56 0.184161606965100694821398249421F);57 58 __CLC_GENTYPE b = __clc_mad(r, -0.836411276854206731913362287293F,59 1.10496961524520294485512696706F);60 __CLC_GENTYPE u = r * MATH_DIVIDE(a, b);61 62 __CLC_GENTYPE s = __clc_sqrt(r);63 y = s;64 __CLC_GENTYPE s1 = __CLC_AS_GENTYPE(__CLC_AS_UINTN(s) & 0xffff0000);65 __CLC_GENTYPE c = MATH_DIVIDE(__clc_mad(s1, -s1, r), s + s1);66 __CLC_GENTYPE rettn = __clc_mad(s + __clc_mad(y, u, -piby2_tail), -2.0f, pi);67 __CLC_GENTYPE rettp = 2.0F * (s1 + __clc_mad(y, u, c));68 __CLC_GENTYPE rett = xneg ? rettn : rettp;69 __CLC_GENTYPE ret = piby2_head - (x - __clc_mad(x, -u, piby2_tail));70 71 ret = transform ? rett : ret;72 ret = aux > 0x3f800000U ? __CLC_GENTYPE_NAN : ret;73 ret = ux == 0x3f800000U ? 0.0f : ret;74 ret = ux == 0xbf800000U ? pi : ret;75 ret = xexp < -26 ? piby2 : ret;76 return ret;77}78 79#elif __CLC_FPSIZE == 6480 81_CLC_OVERLOAD _CLC_DEF __CLC_GENTYPE __clc_acos(__CLC_GENTYPE x) {82 // 0x400921fb54442d1883 const __CLC_GENTYPE pi = __CLC_FP_LIT(3.1415926535897933e+00);84 // 0x3ff921fb54442d1885 const __CLC_GENTYPE piby2 = __CLC_FP_LIT(1.5707963267948965580e+00);86 // 0x3ff921fb54442d1887 const __CLC_GENTYPE piby2_head = __CLC_FP_LIT(1.5707963267948965580e+00);88 // 0x3c91a62633145c0789 const __CLC_GENTYPE piby2_tail = __CLC_FP_LIT(6.12323399573676603587e-17);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 __CLC_GENTYPE un = __clc_fma(105 r,106 __clc_fma(107 r,108 __clc_fma(r,109 __clc_fma(r,110 __clc_fma(r, 0.0000482901920344786991880522822991,111 0.00109242697235074662306043804220),112 -0.0549989809235685841612020091328),113 0.275558175256937652532686256258),114 -0.445017216867635649900123110649),115 0.227485835556935010735943483075);116 117 __CLC_GENTYPE ud = __clc_fma(118 r,119 __clc_fma(r,120 __clc_fma(r,121 __clc_fma(r, 0.105869422087204370341222318533,122 -0.943639137032492685763471240072),123 2.76568859157270989520376345954),124 -3.28431505720958658909889444194),125 1.36491501334161032038194214209);126 127 __CLC_GENTYPE u = r * MATH_DIVIDE(un, ud);128 129 // Reconstruct acos carefully in transformed region130 __CLC_GENTYPE s = __clc_sqrt(r);131 __CLC_GENTYPE ztn = __clc_fma(-2.0, (s + __clc_fma(s, u, -piby2_tail)), pi);132 133 __CLC_GENTYPE s1 =134 __CLC_AS_GENTYPE(__CLC_AS_ULONGN(s) & 0xffffffff00000000UL);135 __CLC_GENTYPE c = MATH_DIVIDE(__clc_fma(-s1, s1, r), s + s1);136 __CLC_GENTYPE ztp = 2.0 * (s1 + __clc_fma(s, u, c));137 __CLC_GENTYPE zt = xneg ? ztn : ztp;138 __CLC_GENTYPE z = piby2_head - (x - __clc_fma(-x, u, piby2_tail));139 140 z = transform ? zt : z;141 142 z = __CLC_CONVERT_LONGN(xexp < -56) ? piby2 : z;143 z = __clc_isnan(x) ? __CLC_AS_GENTYPE((__CLC_AS_ULONGN(x) |144 (__CLC_ULONGN)QNANBITPATT_DP64))145 : z;146 z = x == 1.0 ? 0.0 : z;147 z = x == -1.0 ? pi : z;148 149 return z;150}151 152#elif __CLC_FPSIZE == 16153 154_CLC_OVERLOAD _CLC_DEF __CLC_GENTYPE __clc_acos(__CLC_GENTYPE x) {155 return __CLC_CONVERT_GENTYPE(__clc_acos(__CLC_CONVERT_FLOATN(x)));156}157 158#endif159