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#include <clc/clc.h>
23
24#include "math.h"
25#include "../clcmacro.h"
26
27_CLC_OVERLOAD _CLC_DEF float acos(float x) {
28    // Computes arccos(x).
29    // The argument is first reduced by noting that arccos(x)
30    // is invalid for abs(x) > 1. For denormal and small
31    // arguments arccos(x) = pi/2 to machine accuracy.
32    // Remaining argument ranges are handled as follows.
33    // For abs(x) <= 0.5 use
34    // arccos(x) = pi/2 - arcsin(x)
35    // = pi/2 - (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    // arccos(x) = pi - 2*arcsin(sqrt(1-x)/2)
40    // together with the above rational approximation, and
41    // reconstruct the terms carefully.
42
43
44    // Some constants and split constants.
45    const float piby2 = 1.5707963705e+00F;
46    const float pi = 3.1415926535897933e+00F;
47    const float piby2_head = 1.5707963267948965580e+00F;
48    const float piby2_tail = 6.12323399573676603587e-17F;
49
50    uint ux = as_uint(x);
51    uint aux = ux & ~SIGNBIT_SP32;
52    int xneg = ux != aux;
53    int xexp = (int)(aux >> EXPSHIFTBITS_SP32) - EXPBIAS_SP32;
54    float y = as_float(aux);
55
56    // transform if |x| >= 0.5
57    int transform = xexp >= -1;
58
59    float y2 = y * y;
60    float yt = 0.5f * (1.0f - y);
61    float r = transform ? yt : y2;
62
63    // Use a rational approximation for [0.0, 0.5]
64    float a = mad(r,
65                  mad(r,
66                      mad(r, -0.00396137437848476485201154797087F, -0.0133819288943925804214011424456F),
67                      -0.0565298683201845211985026327361F),
68                  0.184161606965100694821398249421F);
69
70    float b = mad(r, -0.836411276854206731913362287293F, 1.10496961524520294485512696706F);
71    float u = r * MATH_DIVIDE(a, b);
72
73    float s = MATH_SQRT(r);
74    y = s;
75    float s1 = as_float(as_uint(s) & 0xffff0000);
76    float c = MATH_DIVIDE(mad(s1, -s1, r), s + s1);
77    float rettn = mad(s + mad(y, u, -piby2_tail), -2.0f, pi);
78    float rettp = 2.0F * (s1 + mad(y, u, c));
79    float rett = xneg ? rettn : rettp;
80    float ret = piby2_head - (x - mad(x, -u, piby2_tail));
81
82    ret = transform ? rett : ret;
83    ret = aux > 0x3f800000U ? as_float(QNANBITPATT_SP32) : ret;
84    ret = ux == 0x3f800000U ? 0.0f : ret;
85    ret = ux == 0xbf800000U ? pi : ret;
86    ret = xexp < -26 ? piby2 : ret;
87    return ret;
88}
89
90_CLC_UNARY_VECTORIZE(_CLC_OVERLOAD _CLC_DEF, float, acos, float);
91
92#ifdef cl_khr_fp64
93
94#pragma OPENCL EXTENSION cl_khr_fp64 : enable
95
96_CLC_OVERLOAD _CLC_DEF double acos(double x) {
97    // Computes arccos(x).
98    // The argument is first reduced by noting that arccos(x)
99    // is invalid for abs(x) > 1. For denormal and small
100    // arguments arccos(x) = pi/2 to machine accuracy.
101    // Remaining argument ranges are handled as follows.
102    // For abs(x) <= 0.5 use
103    // arccos(x) = pi/2 - arcsin(x)
104    // = pi/2 - (x + x^3*R(x^2))
105    // where R(x^2) is a rational minimax approximation to
106    // (arcsin(x) - x)/x^3.
107    // For abs(x) > 0.5 exploit the identity:
108    // arccos(x) = pi - 2*arcsin(sqrt(1-x)/2)
109    // together with the above rational approximation, and
110    // reconstruct the terms carefully.
111
112    const double pi = 3.1415926535897933e+00;             /* 0x400921fb54442d18 */
113    const double piby2 = 1.5707963267948965580e+00;       /* 0x3ff921fb54442d18 */
114    const double piby2_head = 1.5707963267948965580e+00;  /* 0x3ff921fb54442d18 */
115    const double piby2_tail = 6.12323399573676603587e-17; /* 0x3c91a62633145c07 */
116
117    double y = fabs(x);
118    int xneg = as_int2(x).hi < 0;
119    int xexp = (as_int2(y).hi >> 20) - EXPBIAS_DP64;
120
121    // abs(x) >= 0.5
122    int transform = xexp >= -1;
123
124    double rt = 0.5 * (1.0 - y);
125    double y2 = y * y;
126    double r = transform ? rt : y2;
127
128    // Use a rational approximation for [0.0, 0.5]
129    double un = fma(r,
130                    fma(r,
131                        fma(r,
132                            fma(r,
133                                fma(r, 0.0000482901920344786991880522822991,
134                                       0.00109242697235074662306043804220),
135                                -0.0549989809235685841612020091328),
136                            0.275558175256937652532686256258),
137                        -0.445017216867635649900123110649),
138                    0.227485835556935010735943483075);
139
140    double ud = fma(r,
141                    fma(r,
142                        fma(r,
143                            fma(r, 0.105869422087204370341222318533,
144                                   -0.943639137032492685763471240072),
145                            2.76568859157270989520376345954),
146                        -3.28431505720958658909889444194),
147                    1.36491501334161032038194214209);
148
149    double u = r * MATH_DIVIDE(un, ud);
150
151    // Reconstruct acos carefully in transformed region
152    double s = sqrt(r);
153    double ztn =  fma(-2.0, (s + fma(s, u, -piby2_tail)), pi);
154
155    double s1 = as_double(as_ulong(s) & 0xffffffff00000000UL);
156    double c = MATH_DIVIDE(fma(-s1, s1, r), s + s1);
157    double ztp = 2.0 * (s1 + fma(s, u, c));
158    double zt =  xneg ? ztn : ztp;
159    double z = piby2_head - (x - fma(-x, u, piby2_tail));
160
161    z =  transform ? zt : z;
162
163    z = xexp < -56 ? piby2 : z;
164    z = isnan(x) ? as_double((as_ulong(x) | QNANBITPATT_DP64)) : z;
165    z = x == 1.0 ? 0.0 : z;
166    z = x == -1.0 ? pi : z;
167
168    return z;
169}
170
171_CLC_UNARY_VECTORIZE(_CLC_OVERLOAD _CLC_DEF, double, acos, double);
172
173#endif // cl_khr_fp64
174