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(acos) 42 #endif 43 44 double FN_PROTOTYPE(acos)(double x) 45 { 46 /* Computes arccos(x). 47 The argument is first reduced by noting that arccos(x) 48 is invalid for abs(x) > 1. For denormal and small 49 arguments arccos(x) = pi/2 to machine accuracy. 50 Remaining argument ranges are handled as follows. 51 For abs(x) <= 0.5 use 52 arccos(x) = pi/2 - arcsin(x) 53 = pi/2 - (x + x^3*R(x^2)) 54 where R(x^2) is a rational minimax approximation to 55 (arcsin(x) - x)/x^3. 56 For abs(x) > 0.5 exploit the identity: 57 arccos(x) = pi - 2*arcsin(sqrt(1-x)/2) 58 together with the above rational approximation, and 59 reconstruct the terms carefully. 60 */ 61 62 /* Some constants and split constants. */ 63 64 static const double 65 pi = 3.1415926535897933e+00, /* 0x400921fb54442d18 */ 66 piby2 = 1.5707963267948965580e+00, /* 0x3ff921fb54442d18 */ 67 piby2_head = 1.5707963267948965580e+00, /* 0x3ff921fb54442d18 */ 68 piby2_tail = 6.12323399573676603587e-17; /* 0x3c91a62633145c07 */ 69 70 double u, y, s=0.0, r; 71 int xexp, xnan, transform=0; 72 73 unsigned long long ux, aux, xneg; 74 GET_BITS_DP64(x, ux); 75 aux = ux & ~SIGNBIT_DP64; 76 xneg = (ux & SIGNBIT_DP64); 77 xnan = (aux > PINFBITPATT_DP64); 78 xexp = (int)((ux & EXPBITS_DP64) >> EXPSHIFTBITS_DP64) - EXPBIAS_DP64; 79 80 /* Special cases */ 81 82 if (xnan) 83 { 84 return _handle_error("acos", OP_ACOS, ux|0x0008000000000000, _DOMAIN, 85 0, EDOM, x, 0.0, 1); 86 } 87 else if (xexp < -56) 88 { /* y small enough that arccos(x) = pi/2 */ 89 return val_with_flags(piby2, AMD_F_INEXACT); 90 } 91 else if (xexp >= 0) 92 { /* abs(x) >= 1.0 */ 93 if (x == 1.0) 94 return 0.0; 95 else if (x == -1.0) 96 return val_with_flags(pi, AMD_F_INEXACT); 97 else 98 return _handle_error("acos", OP_ACOS, INDEFBITPATT_DP64, _DOMAIN, 99 AMD_F_INVALID, EDOM, x, 0.0, 1); 100 } 101 102 if (xneg) y = -x; 103 else y = x; 104 105 transform = (xexp >= -1); /* abs(x) >= 0.5 */ 106 107 if (transform) 108 { /* Transform y into the range [0,0.5) */ 109 r = 0.5*(1.0 - y); 110 /* VC++ intrinsic call */ 111 _mm_store_sd(&s, _mm_sqrt_sd(_mm_setzero_pd(), _mm_load_sd(&r))); 112 y = s; 113 } 114 else 115 r = y*y; 116 117 /* Use a rational approximation for [0.0, 0.5] */ 118 119 u = r*(0.227485835556935010735943483075 + 120 (-0.445017216867635649900123110649 + 121 (0.275558175256937652532686256258 + 122 (-0.0549989809235685841612020091328 + 123 (0.00109242697235074662306043804220 + 124 0.0000482901920344786991880522822991*r)*r)*r)*r)*r)/ 125 (1.36491501334161032038194214209 + 126 (-3.28431505720958658909889444194 + 127 (2.76568859157270989520376345954 + 128 (-0.943639137032492685763471240072 + 129 0.105869422087204370341222318533*r)*r)*r)*r); 130 131 if (transform) 132 { /* Reconstruct acos carefully in transformed region */ 133 if (xneg) return pi - 2.0*(s+(y*u - piby2_tail)); 134 else 135 { 136 double c, s1; 137 unsigned long long us; 138 GET_BITS_DP64(s, us); 139 PUT_BITS_DP64(0xffffffff00000000 & us, s1); 140 c = (r-s1*s1)/(s+s1); 141 return 2.0*s1 + (2.0*c+2.0*y*u); 142 } 143 } 144 else 145 return piby2_head - (x - (piby2_tail - x*u)); 146 } 147