/*
* Copyright (c) 2014 Advanced Micro Devices, Inc.
*
* Permission is hereby granted, free of charge, to any person obtaining a copy
* of this software and associated documentation files (the "Software"), to deal
* in the Software without restriction, including without limitation the rights
* to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
* copies of the Software, and to permit persons to whom the Software is
* furnished to do so, subject to the following conditions:
*
* The above copyright notice and this permission notice shall be included in
* all copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
* IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
* FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
* AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
* LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
* OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
* THE SOFTWARE.
*/
#include <clc/clc.h>
#include "math.h"
#include "../clcmacro.h"
_CLC_OVERLOAD _CLC_DEF float asin(float x) {
// Computes arcsin(x).
// The argument is first reduced by noting that arcsin(x)
// is invalid for abs(x) > 1 and arcsin(-x) = -arcsin(x).
// For denormal and small arguments arcsin(x) = x to machine
// accuracy. Remaining argument ranges are handled as follows.
// For abs(x) <= 0.5 use
// arcsin(x) = x + x^3*R(x^2)
// where R(x^2) is a rational minimax approximation to
// (arcsin(x) - x)/x^3.
// For abs(x) > 0.5 exploit the identity:
// arcsin(x) = pi/2 - 2*arcsin(sqrt(1-x)/2)
// together with the above rational approximation, and
// reconstruct the terms carefully.
const float piby2_tail = 7.5497894159e-08F; /* 0x33a22168 */
const float hpiby2_head = 7.8539812565e-01F; /* 0x3f490fda */
const float piby2 = 1.5707963705e+00F; /* 0x3fc90fdb */
uint ux = as_uint(x);
uint aux = ux & EXSIGNBIT_SP32;
uint xs = ux ^ aux;
float spiby2 = as_float(xs | as_uint(piby2));
int xexp = (int)(aux >> EXPSHIFTBITS_SP32) - EXPBIAS_SP32;
float y = as_float(aux);
// abs(x) >= 0.5
int transform = xexp >= -1;
float y2 = y * y;
float rt = 0.5f * (1.0f - y);
float r = transform ? rt : y2;
// Use a rational approximation for [0.0, 0.5]
float a = mad(r,
mad(r,
mad(r, -0.00396137437848476485201154797087F, -0.0133819288943925804214011424456F),
-0.0565298683201845211985026327361F),
0.184161606965100694821398249421F);
float b = mad(r, -0.836411276854206731913362287293F, 1.10496961524520294485512696706F);
float u = r * MATH_DIVIDE(a, b);
float s = MATH_SQRT(r);
float s1 = as_float(as_uint(s) & 0xffff0000);
float c = MATH_DIVIDE(mad(-s1, s1, r), s + s1);
float p = mad(2.0f*s, u, -mad(c, -2.0f, piby2_tail));
float q = mad(s1, -2.0f, hpiby2_head);
float vt = hpiby2_head - (p - q);
float v = mad(y, u, y);
v = transform ? vt : v;
float ret = as_float(xs | as_uint(v));
ret = aux > 0x3f800000U ? as_float(QNANBITPATT_SP32) : ret;
ret = aux == 0x3f800000U ? spiby2 : ret;
ret = xexp < -14 ? x : ret;
return ret;
}
_CLC_UNARY_VECTORIZE(_CLC_OVERLOAD _CLC_DEF, float, asin, float);
#ifdef cl_khr_fp64
#pragma OPENCL EXTENSION cl_khr_fp64 : enable
_CLC_OVERLOAD _CLC_DEF double asin(double x) {
// Computes arcsin(x).
// The argument is first reduced by noting that arcsin(x)
// is invalid for abs(x) > 1 and arcsin(-x) = -arcsin(x).
// For denormal and small arguments arcsin(x) = x to machine
// accuracy. Remaining argument ranges are handled as follows.
// For abs(x) <= 0.5 use
// arcsin(x) = x + x^3*R(x^2)
// where R(x^2) is a rational minimax approximation to
// (arcsin(x) - x)/x^3.
// For abs(x) > 0.5 exploit the identity:
// arcsin(x) = pi/2 - 2*arcsin(sqrt(1-x)/2)
// together with the above rational approximation, and
// reconstruct the terms carefully.
const double piby2_tail = 6.1232339957367660e-17; /* 0x3c91a62633145c07 */
const double hpiby2_head = 7.8539816339744831e-01; /* 0x3fe921fb54442d18 */
const double piby2 = 1.5707963267948965e+00; /* 0x3ff921fb54442d18 */
double y = fabs(x);
int xneg = as_int2(x).hi < 0;
int xexp = (as_int2(y).hi >> 20) - EXPBIAS_DP64;
// abs(x) >= 0.5
int transform = xexp >= -1;
double rt = 0.5 * (1.0 - y);
double y2 = y * y;
double r = transform ? rt : y2;
// Use a rational approximation for [0.0, 0.5]
double un = fma(r,
fma(r,
fma(r,
fma(r,
fma(r, 0.0000482901920344786991880522822991,
0.00109242697235074662306043804220),
-0.0549989809235685841612020091328),
0.275558175256937652532686256258),
-0.445017216867635649900123110649),
0.227485835556935010735943483075);
double ud = fma(r,
fma(r,
fma(r,
fma(r, 0.105869422087204370341222318533,
-0.943639137032492685763471240072),
2.76568859157270989520376345954),
-3.28431505720958658909889444194),
1.36491501334161032038194214209);
double u = r * MATH_DIVIDE(un, ud);
// Reconstruct asin carefully in transformed region
double s = sqrt(r);
double sh = as_double(as_ulong(s) & 0xffffffff00000000UL);
double c = MATH_DIVIDE(fma(-sh, sh, r), s + sh);
double p = fma(2.0*s, u, -fma(-2.0, c, piby2_tail));
double q = fma(-2.0, sh, hpiby2_head);
double vt = hpiby2_head - (p - q);
double v = fma(y, u, y);
v = transform ? vt : v;
v = xexp < -28 ? y : v;
v = xexp >= 0 ? as_double(QNANBITPATT_DP64) : v;
v = y == 1.0 ? piby2 : v;
return xneg ? -v : v;
}
_CLC_UNARY_VECTORIZE(_CLC_OVERLOAD _CLC_DEF, double, asin, double);
#endif // cl_khr_fp64
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