1/* 2 * Copyright (c) 2014 Advanced Micro Devices, Inc. 3 * 4 * Permission is hereby granted, free of charge, to any person obtaining a copy 5 * of this software and associated documentation files (the "Software"), to deal 6 * in the Software without restriction, including without limitation the rights 7 * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell 8 * copies of the Software, and to permit persons to whom the Software is 9 * furnished to do so, subject to the following conditions: 10 * 11 * The above copyright notice and this permission notice shall be included in 12 * all copies or substantial portions of the Software. 13 * 14 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR 15 * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 16 * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE 17 * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER 18 * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, 19 * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN 20 * THE SOFTWARE. 21 */ 22 23#include <clc/clc.h> 24 25#include "math.h" 26#include "../clcmacro.h" 27 28_CLC_OVERLOAD _CLC_DEF float asin(float x) { 29 // Computes arcsin(x). 30 // The argument is first reduced by noting that arcsin(x) 31 // is invalid for abs(x) > 1 and arcsin(-x) = -arcsin(x). 32 // For denormal and small arguments arcsin(x) = x to machine 33 // accuracy. Remaining argument ranges are handled as follows. 34 // For abs(x) <= 0.5 use 35 // arcsin(x) = x + x^3*R(x^2) 36 // where R(x^2) is a rational minimax approximation to 37 // (arcsin(x) - x)/x^3. 38 // For abs(x) > 0.5 exploit the identity: 39 // arcsin(x) = pi/2 - 2*arcsin(sqrt(1-x)/2) 40 // together with the above rational approximation, and 41 // reconstruct the terms carefully. 42 43 const float piby2_tail = 7.5497894159e-08F; /* 0x33a22168 */ 44 const float hpiby2_head = 7.8539812565e-01F; /* 0x3f490fda */ 45 const float piby2 = 1.5707963705e+00F; /* 0x3fc90fdb */ 46 47 uint ux = as_uint(x); 48 uint aux = ux & EXSIGNBIT_SP32; 49 uint xs = ux ^ aux; 50 float spiby2 = as_float(xs | as_uint(piby2)); 51 int xexp = (int)(aux >> EXPSHIFTBITS_SP32) - EXPBIAS_SP32; 52 float y = as_float(aux); 53 54 // abs(x) >= 0.5 55 int transform = xexp >= -1; 56 57 float y2 = y * y; 58 float rt = 0.5f * (1.0f - y); 59 float r = transform ? rt : y2; 60 61 // Use a rational approximation for [0.0, 0.5] 62 float a = mad(r, 63 mad(r, 64 mad(r, -0.00396137437848476485201154797087F, -0.0133819288943925804214011424456F), 65 -0.0565298683201845211985026327361F), 66 0.184161606965100694821398249421F); 67 68 float b = mad(r, -0.836411276854206731913362287293F, 1.10496961524520294485512696706F); 69 float u = r * MATH_DIVIDE(a, b); 70 71 float s = MATH_SQRT(r); 72 float s1 = as_float(as_uint(s) & 0xffff0000); 73 float c = MATH_DIVIDE(mad(-s1, s1, r), s + s1); 74 float p = mad(2.0f*s, u, -mad(c, -2.0f, piby2_tail)); 75 float q = mad(s1, -2.0f, hpiby2_head); 76 float vt = hpiby2_head - (p - q); 77 float v = mad(y, u, y); 78 v = transform ? vt : v; 79 80 float ret = as_float(xs | as_uint(v)); 81 ret = aux > 0x3f800000U ? as_float(QNANBITPATT_SP32) : ret; 82 ret = aux == 0x3f800000U ? spiby2 : ret; 83 ret = xexp < -14 ? x : ret; 84 85 return ret; 86} 87 88_CLC_UNARY_VECTORIZE(_CLC_OVERLOAD _CLC_DEF, float, asin, float); 89 90#ifdef cl_khr_fp64 91 92#pragma OPENCL EXTENSION cl_khr_fp64 : enable 93 94_CLC_OVERLOAD _CLC_DEF double asin(double x) { 95 // Computes arcsin(x). 96 // The argument is first reduced by noting that arcsin(x) 97 // is invalid for abs(x) > 1 and arcsin(-x) = -arcsin(x). 98 // For denormal and small arguments arcsin(x) = x to machine 99 // accuracy. Remaining argument ranges are handled as follows. 100 // For abs(x) <= 0.5 use 101 // arcsin(x) = x + x^3*R(x^2) 102 // where R(x^2) is a rational minimax approximation to 103 // (arcsin(x) - x)/x^3. 104 // For abs(x) > 0.5 exploit the identity: 105 // arcsin(x) = pi/2 - 2*arcsin(sqrt(1-x)/2) 106 // together with the above rational approximation, and 107 // reconstruct the terms carefully. 108 109 const double piby2_tail = 6.1232339957367660e-17; /* 0x3c91a62633145c07 */ 110 const double hpiby2_head = 7.8539816339744831e-01; /* 0x3fe921fb54442d18 */ 111 const double piby2 = 1.5707963267948965e+00; /* 0x3ff921fb54442d18 */ 112 113 double y = fabs(x); 114 int xneg = as_int2(x).hi < 0; 115 int xexp = (as_int2(y).hi >> 20) - EXPBIAS_DP64; 116 117 // abs(x) >= 0.5 118 int transform = xexp >= -1; 119 120 double rt = 0.5 * (1.0 - y); 121 double y2 = y * y; 122 double r = transform ? rt : y2; 123 124 // Use a rational approximation for [0.0, 0.5] 125 126 double un = fma(r, 127 fma(r, 128 fma(r, 129 fma(r, 130 fma(r, 0.0000482901920344786991880522822991, 131 0.00109242697235074662306043804220), 132 -0.0549989809235685841612020091328), 133 0.275558175256937652532686256258), 134 -0.445017216867635649900123110649), 135 0.227485835556935010735943483075); 136 137 double ud = fma(r, 138 fma(r, 139 fma(r, 140 fma(r, 0.105869422087204370341222318533, 141 -0.943639137032492685763471240072), 142 2.76568859157270989520376345954), 143 -3.28431505720958658909889444194), 144 1.36491501334161032038194214209); 145 146 double u = r * MATH_DIVIDE(un, ud); 147 148 // Reconstruct asin carefully in transformed region 149 double s = sqrt(r); 150 double sh = as_double(as_ulong(s) & 0xffffffff00000000UL); 151 double c = MATH_DIVIDE(fma(-sh, sh, r), s + sh); 152 double p = fma(2.0*s, u, -fma(-2.0, c, piby2_tail)); 153 double q = fma(-2.0, sh, hpiby2_head); 154 double vt = hpiby2_head - (p - q); 155 double v = fma(y, u, y); 156 v = transform ? vt : v; 157 158 v = xexp < -28 ? y : v; 159 v = xexp >= 0 ? as_double(QNANBITPATT_DP64) : v; 160 v = y == 1.0 ? piby2 : v; 161 162 return xneg ? -v : v; 163} 164 165_CLC_UNARY_VECTORIZE(_CLC_OVERLOAD _CLC_DEF, double, asin, double); 166 167#endif // cl_khr_fp64 168