/*
* Copyright (c) 2014,2015 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 acospi(float x) {
// Computes arccos(x).
// The argument is first reduced by noting that arccos(x)
// is invalid for abs(x) > 1. For denormal and small
// arguments arccos(x) = pi/2 to machine accuracy.
// Remaining argument ranges are handled as follows.
// For abs(x) <= 0.5 use
// arccos(x) = pi/2 - arcsin(x)
// = pi/2 - (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:
// arccos(x) = pi - 2*arcsin(sqrt(1-x)/2)
// together with the above rational approximation, and
// reconstruct the terms carefully.
// Some constants and split constants.
const float pi = 3.1415926535897933e+00f;
const float piby2_head = 1.5707963267948965580e+00f; /* 0x3ff921fb54442d18 */
const float piby2_tail = 6.12323399573676603587e-17f; /* 0x3c91a62633145c07 */
uint ux = as_uint(x);
uint aux = ux & ~SIGNBIT_SP32;
int xneg = ux != aux;
int xexp = (int)(aux >> EXPSHIFTBITS_SP32) - EXPBIAS_SP32;
float y = as_float(aux);
// transform if |x| >= 0.5
int transform = xexp >= -1;
float y2 = y * y;
float yt = 0.5f * (1.0f - y);
float r = transform ? yt : 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);
y = s;
float s1 = as_float(as_uint(s) & 0xffff0000);
float c = MATH_DIVIDE(r - s1 * s1, s + s1);
// float rettn = 1.0f - MATH_DIVIDE(2.0f * (s + (y * u - piby2_tail)), pi);
float rettn = 1.0f - MATH_DIVIDE(2.0f * (s + mad(y, u, -piby2_tail)), pi);
// float rettp = MATH_DIVIDE(2.0F * s1 + (2.0F * c + 2.0F * y * u), pi);
float rettp = MATH_DIVIDE(2.0f*(s1 + mad(y, u, c)), pi);
float rett = xneg ? rettn : rettp;
// float ret = MATH_DIVIDE(piby2_head - (x - (piby2_tail - x * u)), pi);
float ret = MATH_DIVIDE(piby2_head - (x - mad(x, -u, piby2_tail)), pi);
ret = transform ? rett : ret;
ret = aux > 0x3f800000U ? as_float(QNANBITPATT_SP32) : ret;
ret = ux == 0x3f800000U ? 0.0f : ret;
ret = ux == 0xbf800000U ? 1.0f : ret;
ret = xexp < -26 ? 0.5f : ret;
return ret;
}
_CLC_UNARY_VECTORIZE(_CLC_OVERLOAD _CLC_DEF, float, acospi, float)
#ifdef cl_khr_fp64
#pragma OPENCL EXTENSION cl_khr_fp64 : enable
_CLC_OVERLOAD _CLC_DEF double acospi(double x) {
// Computes arccos(x).
// The argument is first reduced by noting that arccos(x)
// is invalid for abs(x) > 1. For denormal and small
// arguments arccos(x) = pi/2 to machine accuracy.
// Remaining argument ranges are handled as follows.
// For abs(x) <= 0.5 use
// arccos(x) = pi/2 - arcsin(x)
// = pi/2 - (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:
// arccos(x) = pi - 2*arcsin(sqrt(1-x)/2)
// together with the above rational approximation, and
// reconstruct the terms carefully.
const double pi = 0x1.921fb54442d18p+1;
const double piby2_tail = 6.12323399573676603587e-17; /* 0x3c91a62633145c07 */
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;
// Transform y into the range [0,0.5)
double r1 = 0.5 * (1.0 - y);
double s = sqrt(r1);
double r = y * y;
r = transform ? r1 : r;
y = transform ? s : y;
// 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 acos carefully in transformed region
double res1 = fma(-2.0, MATH_DIVIDE(s + fma(y, u, -piby2_tail), pi), 1.0);
double s1 = as_double(as_ulong(s) & 0xffffffff00000000UL);
double c = MATH_DIVIDE(fma(-s1, s1, r), s + s1);
double res2 = MATH_DIVIDE(fma(2.0, s1, fma(2.0, c, 2.0 * y * u)), pi);
res1 = xneg ? res1 : res2;
res2 = 0.5 - fma(x, u, x) / pi;
res1 = transform ? res1 : res2;
const double qnan = as_double(QNANBITPATT_DP64);
res2 = x == 1.0 ? 0.0 : qnan;
res2 = x == -1.0 ? 1.0 : res2;
res1 = xexp >= 0 ? res2 : res1;
res1 = xexp < -56 ? 0.5 : res1;
return res1;
}
_CLC_UNARY_VECTORIZE(_CLC_OVERLOAD _CLC_DEF, double, acospi, double)
#endif
_CLC_DEFINE_UNARY_BUILTIN_FP16(acospi)
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