generic/lib/math/acospi.cl - llvm-project/libclc - Git at Google

 /*
  * 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
	/*
	* 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