/* * Copyright (c) 2014,2015 Advanced Micro Devices, Inc. * * Permission is hereby granted, free of charge, to any person obtaining a copy * of this software and associated documentation files (the "Software"), to deal * in the Software without restriction, including without limitation the rights * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell * copies of the Software, and to permit persons to whom the Software is * furnished to do so, subject to the following conditions: * * The above copyright notice and this permission notice shall be included in * all copies or substantial portions of the Software. * * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN * THE SOFTWARE. */ #include #include "math.h" #include "../clcmacro.h" _CLC_OVERLOAD _CLC_DEF float acospi(float x) { // Computes arccos(x). // The argument is first reduced by noting that arccos(x) // is invalid for abs(x) > 1. For denormal and small // arguments arccos(x) = pi/2 to machine accuracy. // Remaining argument ranges are handled as follows. // For abs(x) <= 0.5 use // arccos(x) = pi/2 - arcsin(x) // = pi/2 - (x + x^3*R(x^2)) // where R(x^2) is a rational minimax approximation to // (arcsin(x) - x)/x^3. // For abs(x) > 0.5 exploit the identity: // arccos(x) = pi - 2*arcsin(sqrt(1-x)/2) // together with the above rational approximation, and // reconstruct the terms carefully. // Some constants and split constants. const float pi = 3.1415926535897933e+00f; const float piby2_head = 1.5707963267948965580e+00f; /* 0x3ff921fb54442d18 */ const float piby2_tail = 6.12323399573676603587e-17f; /* 0x3c91a62633145c07 */ uint ux = as_uint(x); uint aux = ux & ~SIGNBIT_SP32; int xneg = ux != aux; int xexp = (int)(aux >> EXPSHIFTBITS_SP32) - EXPBIAS_SP32; float y = as_float(aux); // transform if |x| >= 0.5 int transform = xexp >= -1; float y2 = y * y; float yt = 0.5f * (1.0f - y); float r = transform ? yt : y2; // Use a rational approximation for [0.0, 0.5] float a = mad(r, mad(r, mad(r, -0.00396137437848476485201154797087F, -0.0133819288943925804214011424456F), -0.0565298683201845211985026327361F), 0.184161606965100694821398249421F); float b = mad(r, -0.836411276854206731913362287293F, 1.10496961524520294485512696706F); float u = r * MATH_DIVIDE(a, b); float s = MATH_SQRT(r); y = s; float s1 = as_float(as_uint(s) & 0xffff0000); float c = MATH_DIVIDE(r - s1 * s1, s + s1); // float rettn = 1.0f - MATH_DIVIDE(2.0f * (s + (y * u - piby2_tail)), pi); float rettn = 1.0f - MATH_DIVIDE(2.0f * (s + mad(y, u, -piby2_tail)), pi); // float rettp = MATH_DIVIDE(2.0F * s1 + (2.0F * c + 2.0F * y * u), pi); float rettp = MATH_DIVIDE(2.0f*(s1 + mad(y, u, c)), pi); float rett = xneg ? rettn : rettp; // float ret = MATH_DIVIDE(piby2_head - (x - (piby2_tail - x * u)), pi); float ret = MATH_DIVIDE(piby2_head - (x - mad(x, -u, piby2_tail)), pi); ret = transform ? rett : ret; ret = aux > 0x3f800000U ? as_float(QNANBITPATT_SP32) : ret; ret = ux == 0x3f800000U ? 0.0f : ret; ret = ux == 0xbf800000U ? 1.0f : ret; ret = xexp < -26 ? 0.5f : ret; return ret; } _CLC_UNARY_VECTORIZE(_CLC_OVERLOAD _CLC_DEF, float, acospi, float) #ifdef cl_khr_fp64 #pragma OPENCL EXTENSION cl_khr_fp64 : enable _CLC_OVERLOAD _CLC_DEF double acospi(double x) { // Computes arccos(x). // The argument is first reduced by noting that arccos(x) // is invalid for abs(x) > 1. For denormal and small // arguments arccos(x) = pi/2 to machine accuracy. // Remaining argument ranges are handled as follows. // For abs(x) <= 0.5 use // arccos(x) = pi/2 - arcsin(x) // = pi/2 - (x + x^3*R(x^2)) // where R(x^2) is a rational minimax approximation to // (arcsin(x) - x)/x^3. // For abs(x) > 0.5 exploit the identity: // arccos(x) = pi - 2*arcsin(sqrt(1-x)/2) // together with the above rational approximation, and // reconstruct the terms carefully. const double pi = 0x1.921fb54442d18p+1; const double piby2_tail = 6.12323399573676603587e-17; /* 0x3c91a62633145c07 */ double y = fabs(x); int xneg = as_int2(x).hi < 0; int xexp = (as_int2(y).hi >> 20) - EXPBIAS_DP64; // abs(x) >= 0.5 int transform = xexp >= -1; // Transform y into the range [0,0.5) double r1 = 0.5 * (1.0 - y); double s = sqrt(r1); double r = y * y; r = transform ? r1 : r; y = transform ? s : y; // Use a rational approximation for [0.0, 0.5] double un = fma(r, fma(r, fma(r, fma(r, fma(r, 0.0000482901920344786991880522822991, 0.00109242697235074662306043804220), -0.0549989809235685841612020091328), 0.275558175256937652532686256258), -0.445017216867635649900123110649), 0.227485835556935010735943483075); double ud = fma(r, fma(r, fma(r, fma(r, 0.105869422087204370341222318533, -0.943639137032492685763471240072), 2.76568859157270989520376345954), -3.28431505720958658909889444194), 1.36491501334161032038194214209); double u = r * MATH_DIVIDE(un, ud); // Reconstruct acos carefully in transformed region double res1 = fma(-2.0, MATH_DIVIDE(s + fma(y, u, -piby2_tail), pi), 1.0); double s1 = as_double(as_ulong(s) & 0xffffffff00000000UL); double c = MATH_DIVIDE(fma(-s1, s1, r), s + s1); double res2 = MATH_DIVIDE(fma(2.0, s1, fma(2.0, c, 2.0 * y * u)), pi); res1 = xneg ? res1 : res2; res2 = 0.5 - fma(x, u, x) / pi; res1 = transform ? res1 : res2; const double qnan = as_double(QNANBITPATT_DP64); res2 = x == 1.0 ? 0.0 : qnan; res2 = x == -1.0 ? 1.0 : res2; res1 = xexp >= 0 ? res2 : res1; res1 = xexp < -56 ? 0.5 : res1; return res1; } _CLC_UNARY_VECTORIZE(_CLC_OVERLOAD _CLC_DEF, double, acospi, double) #endif