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