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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_acospi(__CLC_GENTYPE x) {30 // Some constants and split constants.31 const __CLC_GENTYPE pi = __CLC_FP_LIT(3.1415926535897933e+00);32 // 0x3ff921fb54442d1833 const __CLC_GENTYPE piby2_head = __CLC_FP_LIT(1.5707963267948965580e+00);34 // 0x3c91a62633145c0735 const __CLC_GENTYPE piby2_tail = __CLC_FP_LIT(6.12323399573676603587e-17);36 37 __CLC_UINTN ux = __CLC_AS_UINTN(x);38 __CLC_UINTN aux = ux & ~SIGNBIT_SP32;39 __CLC_INTN xneg = ux != aux;40 __CLC_INTN xexp = __CLC_AS_INTN(aux >> EXPSHIFTBITS_SP32) - EXPBIAS_SP32;41 42 __CLC_GENTYPE y = __CLC_AS_GENTYPE(aux);43 44 // transform if |x| >= 0.545 __CLC_INTN transform = xexp >= -1;46 47 __CLC_GENTYPE y2 = y * y;48 __CLC_GENTYPE yt = 0.5f * (1.0f - y);49 __CLC_GENTYPE r = transform ? yt : 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 __CLC_GENTYPE b = __clc_mad(r, -0.836411276854206731913362287293F,60 1.10496961524520294485512696706F);61 __CLC_GENTYPE u = r * MATH_DIVIDE(a, b);62 63 __CLC_GENTYPE s = __clc_sqrt(r);64 y = s;65 __CLC_GENTYPE s1 = __CLC_AS_GENTYPE(__CLC_AS_UINTN(s) & 0xffff0000);66 __CLC_GENTYPE c = MATH_DIVIDE(r - s1 * s1, s + s1);67 __CLC_GENTYPE rettn =68 1.0f - MATH_DIVIDE(2.0f * (s + __clc_mad(y, u, -piby2_tail)), pi);69 __CLC_GENTYPE rettp = MATH_DIVIDE(2.0f * (s1 + __clc_mad(y, u, c)), pi);70 __CLC_GENTYPE rett = xneg ? rettn : rettp;71 __CLC_GENTYPE ret =72 MATH_DIVIDE(piby2_head - (x - __clc_mad(x, -u, piby2_tail)), pi);73 74 ret = transform ? rett : ret;75 ret = aux > 0x3f800000U ? __CLC_GENTYPE_NAN : ret;76 ret = ux == 0x3f800000U ? 0.0f : ret;77 ret = ux == 0xbf800000U ? 1.0f : ret;78 ret = xexp < -26 ? 0.5f : ret;79 return ret;80}81 82#elif __CLC_FPSIZE == 6483 84_CLC_OVERLOAD _CLC_DEF __CLC_GENTYPE __clc_acospi(__CLC_GENTYPE x) {85 const __CLC_GENTYPE pi = __CLC_FP_LIT(0x1.921fb54442d18p+1);86 // 0x3c91a62633145c0787 const __CLC_GENTYPE piby2_tail = __CLC_FP_LIT(6.12323399573676603587e-17);88 89 __CLC_GENTYPE y = __clc_fabs(x);90 __CLC_LONGN xneg = x < __CLC_FP_LIT(0.0);91 __CLC_INTN xexp = __CLC_CONVERT_INTN(92 (__CLC_AS_ULONGN(y) >> EXPSHIFTBITS_DP64) - EXPBIAS_DP64);93 94 // abs(x) >= 0.595 __CLC_LONGN transform = __CLC_CONVERT_LONGN(xexp >= -1);96 97 // Transform y into the range [0,0.5)98 __CLC_GENTYPE r1 = 0.5 * (1.0 - y);99 __CLC_GENTYPE s = __clc_sqrt(r1);100 __CLC_GENTYPE r = y * y;101 r = transform ? r1 : r;102 y = transform ? s : y;103 104 // Use a rational approximation for [0.0, 0.5]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 acos carefully in transformed region131 __CLC_GENTYPE res1 =132 __clc_fma(-2.0, MATH_DIVIDE(s + __clc_fma(y, u, -piby2_tail), pi), 1.0);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 res2 =137 MATH_DIVIDE(__clc_fma(2.0, s1, __clc_fma(2.0, c, 2.0 * y * u)), pi);138 res1 = xneg ? res1 : res2;139 res2 = 0.5 - __clc_fma(x, u, x) / pi;140 res1 = transform ? res1 : res2;141 142 res2 = x == 1.0 ? 0.0 : __CLC_GENTYPE_NAN;143 res2 = x == -1.0 ? 1.0 : res2;144 res1 = __CLC_CONVERT_LONGN(xexp >= 0) ? res2 : res1;145 res1 = __CLC_CONVERT_LONGN(xexp < -56) ? 0.5 : res1;146 147 return res1;148}149 150#elif __CLC_FPSIZE == 16151 152_CLC_OVERLOAD _CLC_DEF __CLC_GENTYPE __clc_acospi(__CLC_GENTYPE x) {153 return __CLC_CONVERT_GENTYPE(__clc_acospi(__CLC_CONVERT_FLOATN(x)));154}155 156#endif157