1 2 /******************************************************************************* 3 MIT License 4 ----------- 5 6 Copyright (c) 2002-2019 Advanced Micro Devices, Inc. 7 8 Permission is hereby granted, free of charge, to any person obtaining a copy 9 of this Software and associated documentaon files (the "Software"), to deal 10 in the Software without restriction, including without limitation the rights 11 to use, copy, modify, merge, publish, distribute, sublicense, and/or sell 12 copies of the Software, and to permit persons to whom the Software is 13 furnished to do so, subject to the following conditions: 14 15 The above copyright notice and this permission notice shall be included in 16 all copies or substantial portions of the Software. 17 18 THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR 19 IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 20 FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE 21 AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER 22 LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, 23 OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN 24 THE SOFTWARE. 25 *******************************************************************************/ 26 27 #include "libm.h" 28 #include "libm_util.h" 29 30 #define USE_VAL_WITH_FLAGS 31 #define USE_NAN_WITH_FLAGS 32 #define USE_HANDLE_ERROR 33 #include "libm_inlines.h" 34 #undef USE_NAN_WITH_FLAGS 35 #undef USE_VAL_WITH_FLAGS 36 #undef USE_HANDLE_ERROR 37 38 #include "libm_errno.h" 39 40 #ifdef _MSC_VER 41 #pragma function(asin) 42 #endif 43 44 double FN_PROTOTYPE(asin)(double x) 45 { 46 /* Computes arcsin(x). 47 The argument is first reduced by noting that arcsin(x) 48 is invalid for abs(x) > 1 and arcsin(-x) = -arcsin(x). 49 For denormal and small arguments arcsin(x) = x to machine 50 accuracy. Remaining argument ranges are handled as follows. 51 For abs(x) <= 0.5 use 52 arcsin(x) = x + x^3*R(x^2) 53 where R(x^2) is a rational minimax approximation to 54 (arcsin(x) - x)/x^3. 55 For abs(x) > 0.5 exploit the identity: 56 arcsin(x) = pi/2 - 2*arcsin(sqrt(1-x)/2) 57 together with the above rational approximation, and 58 reconstruct the terms carefully. 59 */ 60 61 /* Some constants and split constants. */ 62 63 static const double 64 piby2_tail = 6.1232339957367660e-17, /* 0x3c91a62633145c07 */ 65 hpiby2_head = 7.8539816339744831e-01, /* 0x3fe921fb54442d18 */ 66 piby2 = 1.5707963267948965e+00; /* 0x3ff921fb54442d18 */ 67 double u, v, y, s=0.0, r; 68 int xexp, xnan, transform=0; 69 70 unsigned long long ux, aux, xneg; 71 GET_BITS_DP64(x, ux); 72 aux = ux & ~SIGNBIT_DP64; 73 xneg = (ux & SIGNBIT_DP64); 74 xnan = (aux > PINFBITPATT_DP64); 75 xexp = (int)((ux & EXPBITS_DP64) >> EXPSHIFTBITS_DP64) - EXPBIAS_DP64; 76 77 /* Special cases */ 78 79 if (xnan) 80 { 81 return _handle_error("asin", OP_ASIN, ux|0x0008000000000000, _DOMAIN, 82 0, EDOM, x, 0.0, 1); 83 } 84 else if (xexp < -28) 85 { /* y small enough that arcsin(x) = x */ 86 return val_with_flags(x, AMD_F_INEXACT); 87 } 88 else if (xexp >= 0) 89 { /* abs(x) >= 1.0 */ 90 if (x == 1.0) 91 return val_with_flags(piby2, AMD_F_INEXACT); 92 else if (x == -1.0) 93 return val_with_flags(-piby2, AMD_F_INEXACT); 94 else 95 return _handle_error("asin", OP_ASIN, INDEFBITPATT_DP64, _DOMAIN, 96 AMD_F_INVALID, EDOM, x, 0.0, 1); 97 } 98 99 if (xneg) y = -x; 100 else y = x; 101 102 transform = (xexp >= -1); /* abs(x) >= 0.5 */ 103 104 if (transform) 105 { /* Transform y into the range [0,0.5) */ 106 r = 0.5*(1.0 - y); 107 /* VC++ intrinsic call */ 108 _mm_store_sd(&s, _mm_sqrt_sd(_mm_setzero_pd(), _mm_load_sd(&r))); 109 y = s; 110 } 111 else 112 r = y*y; 113 114 /* Use a rational approximation for [0.0, 0.5] */ 115 116 u = r*(0.227485835556935010735943483075 + 117 (-0.445017216867635649900123110649 + 118 (0.275558175256937652532686256258 + 119 (-0.0549989809235685841612020091328 + 120 (0.00109242697235074662306043804220 + 121 0.0000482901920344786991880522822991*r)*r)*r)*r)*r)/ 122 (1.36491501334161032038194214209 + 123 (-3.28431505720958658909889444194 + 124 (2.76568859157270989520376345954 + 125 (-0.943639137032492685763471240072 + 126 0.105869422087204370341222318533*r)*r)*r)*r); 127 128 if (transform) 129 { /* Reconstruct asin carefully in transformed region */ 130 { 131 double c, s1, p, q; 132 unsigned long long us; 133 GET_BITS_DP64(s, us); 134 PUT_BITS_DP64(0xffffffff00000000 & us, s1); 135 c = (r-s1*s1)/(s+s1); 136 p = 2.0*s*u - (piby2_tail-2.0*c); 137 q = hpiby2_head - 2.0*s1; 138 v = hpiby2_head - (p-q); 139 } 140 } 141 else 142 { 143 /* Use a temporary variable to prevent VC++ rearranging 144 y + y*u 145 into 146 y * (1 + u) 147 and getting an incorrectly rounded result */ 148 double tmp; 149 tmp = y * u; 150 v = y + tmp; 151 } 152 153 if (xneg) return -v; 154 else return v; 155 } 156