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414 lines
15 KiB
414 lines
15 KiB
/*
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* Copyright (c) 2014 Advanced Micro Devices, Inc.
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*
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* Permission is hereby granted, free of charge, to any person obtaining a copy
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* of this software and associated documentation files (the "Software"), to deal
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* in the Software without restriction, including without limitation the rights
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* to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
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* copies of the Software, and to permit persons to whom the Software is
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* furnished to do so, subject to the following conditions:
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*
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* The above copyright notice and this permission notice shall be included in
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* all copies or substantial portions of the Software.
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*
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* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
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* IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
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* FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
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* AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
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* LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
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* OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
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* THE SOFTWARE.
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*/
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#include <clc/clc.h>
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#include "math.h"
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#include "../clcmacro.h"
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunPro, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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#define erx_f 8.4506291151e-01f /* 0x3f58560b */
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// Coefficients for approximation to erf on [00.84375]
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#define efx 1.2837916613e-01f /* 0x3e0375d4 */
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#define efx8 1.0270333290e+00f /* 0x3f8375d4 */
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#define pp0 1.2837916613e-01f /* 0x3e0375d4 */
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#define pp1 -3.2504209876e-01f /* 0xbea66beb */
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#define pp2 -2.8481749818e-02f /* 0xbce9528f */
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#define pp3 -5.7702702470e-03f /* 0xbbbd1489 */
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#define pp4 -2.3763017452e-05f /* 0xb7c756b1 */
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#define qq1 3.9791721106e-01f /* 0x3ecbbbce */
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#define qq2 6.5022252500e-02f /* 0x3d852a63 */
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#define qq3 5.0813062117e-03f /* 0x3ba68116 */
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#define qq4 1.3249473704e-04f /* 0x390aee49 */
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#define qq5 -3.9602282413e-06f /* 0xb684e21a */
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// Coefficients for approximation to erf in [0.843751.25]
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#define pa0 -2.3621185683e-03f /* 0xbb1acdc6 */
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#define pa1 4.1485610604e-01f /* 0x3ed46805 */
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#define pa2 -3.7220788002e-01f /* 0xbebe9208 */
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#define pa3 3.1834661961e-01f /* 0x3ea2fe54 */
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#define pa4 -1.1089469492e-01f /* 0xbde31cc2 */
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#define pa5 3.5478305072e-02f /* 0x3d1151b3 */
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#define pa6 -2.1663755178e-03f /* 0xbb0df9c0 */
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#define qa1 1.0642088205e-01f /* 0x3dd9f331 */
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#define qa2 5.4039794207e-01f /* 0x3f0a5785 */
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#define qa3 7.1828655899e-02f /* 0x3d931ae7 */
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#define qa4 1.2617121637e-01f /* 0x3e013307 */
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#define qa5 1.3637083583e-02f /* 0x3c5f6e13 */
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#define qa6 1.1984500103e-02f /* 0x3c445aa3 */
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// Coefficients for approximation to erfc in [1.251/0.35]
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#define ra0 -9.8649440333e-03f /* 0xbc21a093 */
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#define ra1 -6.9385856390e-01f /* 0xbf31a0b7 */
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#define ra2 -1.0558626175e+01f /* 0xc128f022 */
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#define ra3 -6.2375331879e+01f /* 0xc2798057 */
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#define ra4 -1.6239666748e+02f /* 0xc322658c */
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#define ra5 -1.8460508728e+02f /* 0xc3389ae7 */
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#define ra6 -8.1287437439e+01f /* 0xc2a2932b */
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#define ra7 -9.8143291473e+00f /* 0xc11d077e */
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#define sa1 1.9651271820e+01f /* 0x419d35ce */
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#define sa2 1.3765776062e+02f /* 0x4309a863 */
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#define sa3 4.3456588745e+02f /* 0x43d9486f */
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#define sa4 6.4538726807e+02f /* 0x442158c9 */
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#define sa5 4.2900814819e+02f /* 0x43d6810b */
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#define sa6 1.0863500214e+02f /* 0x42d9451f */
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#define sa7 6.5702495575e+00f /* 0x40d23f7c */
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#define sa8 -6.0424413532e-02f /* 0xbd777f97 */
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// Coefficients for approximation to erfc in [1/.3528]
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#define rb0 -9.8649431020e-03f /* 0xbc21a092 */
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#define rb1 -7.9928326607e-01f /* 0xbf4c9dd4 */
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#define rb2 -1.7757955551e+01f /* 0xc18e104b */
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#define rb3 -1.6063638306e+02f /* 0xc320a2ea */
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#define rb4 -6.3756646729e+02f /* 0xc41f6441 */
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#define rb5 -1.0250950928e+03f /* 0xc480230b */
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#define rb6 -4.8351919556e+02f /* 0xc3f1c275 */
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#define sb1 3.0338060379e+01f /* 0x41f2b459 */
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#define sb2 3.2579251099e+02f /* 0x43a2e571 */
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#define sb3 1.5367296143e+03f /* 0x44c01759 */
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#define sb4 3.1998581543e+03f /* 0x4547fdbb */
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#define sb5 2.5530502930e+03f /* 0x451f90ce */
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#define sb6 4.7452853394e+02f /* 0x43ed43a7 */
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#define sb7 -2.2440952301e+01f /* 0xc1b38712 */
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_CLC_OVERLOAD _CLC_DEF float erfc(float x) {
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int hx = as_int(x);
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int ix = hx & 0x7fffffff;
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float absx = as_float(ix);
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// Argument for polys
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float x2 = absx * absx;
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float t = 1.0f / x2;
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float tt = absx - 1.0f;
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t = absx < 1.25f ? tt : t;
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t = absx < 0.84375f ? x2 : t;
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// Evaluate polys
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float tu, tv, u, v;
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u = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, rb6, rb5), rb4), rb3), rb2), rb1), rb0);
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v = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, sb7, sb6), sb5), sb4), sb3), sb2), sb1);
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tu = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, ra7, ra6), ra5), ra4), ra3), ra2), ra1), ra0);
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tv = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, sa8, sa7), sa6), sa5), sa4), sa3), sa2), sa1);
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u = absx < 0x1.6db6dap+1f ? tu : u;
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v = absx < 0x1.6db6dap+1f ? tv : v;
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tu = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, pa6, pa5), pa4), pa3), pa2), pa1), pa0);
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tv = mad(t, mad(t, mad(t, mad(t, mad(t, qa6, qa5), qa4), qa3), qa2), qa1);
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u = absx < 1.25f ? tu : u;
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v = absx < 1.25f ? tv : v;
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tu = mad(t, mad(t, mad(t, mad(t, pp4, pp3), pp2), pp1), pp0);
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tv = mad(t, mad(t, mad(t, mad(t, qq5, qq4), qq3), qq2), qq1);
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u = absx < 0.84375f ? tu : u;
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v = absx < 0.84375f ? tv : v;
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v = mad(t, v, 1.0f);
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float q = MATH_DIVIDE(u, v);
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float ret = 0.0f;
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float z = as_float(ix & 0xfffff000);
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float r = exp(mad(-z, z, -0.5625f)) * exp(mad(z - absx, z + absx, q));
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r = MATH_DIVIDE(r, absx);
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t = 2.0f - r;
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r = x < 0.0f ? t : r;
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ret = absx < 28.0f ? r : ret;
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r = 1.0f - erx_f - q;
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t = erx_f + q + 1.0f;
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r = x < 0.0f ? t : r;
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ret = absx < 1.25f ? r : ret;
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r = 0.5f - mad(x, q, x - 0.5f);
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ret = absx < 0.84375f ? r : ret;
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ret = x < -6.0f ? 2.0f : ret;
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ret = isnan(x) ? x : ret;
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return ret;
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}
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_CLC_UNARY_VECTORIZE(_CLC_OVERLOAD _CLC_DEF, float, erfc, float);
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#ifdef cl_khr_fp64
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#pragma OPENCL EXTENSION cl_khr_fp64 : enable
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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*
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* Developed at SunPro, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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/* double erf(double x)
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* double erfc(double x)
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* x
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* 2 |\
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* erf(x) = --------- | exp(-t*t)dt
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* sqrt(pi) \|
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* 0
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*
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* erfc(x) = 1-erf(x)
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* Note that
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* erf(-x) = -erf(x)
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* erfc(-x) = 2 - erfc(x)
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*
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* Method:
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* 1. For |x| in [0, 0.84375]
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* erf(x) = x + x*R(x^2)
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* erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
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* = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
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* where R = P/Q where P is an odd poly of degree 8 and
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* Q is an odd poly of degree 10.
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* -57.90
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* | R - (erf(x)-x)/x | <= 2
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*
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*
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* Remark. The formula is derived by noting
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* erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
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* and that
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* 2/sqrt(pi) = 1.128379167095512573896158903121545171688
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* is close to one. The interval is chosen because the fix
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* point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
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* near 0.6174), and by some experiment, 0.84375 is chosen to
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* guarantee the error is less than one ulp for erf.
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*
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* 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
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* c = 0.84506291151 rounded to single (24 bits)
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* erf(x) = sign(x) * (c + P1(s)/Q1(s))
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* erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
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* 1+(c+P1(s)/Q1(s)) if x < 0
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* |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
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* Remark: here we use the taylor series expansion at x=1.
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* erf(1+s) = erf(1) + s*Poly(s)
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* = 0.845.. + P1(s)/Q1(s)
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* That is, we use rational approximation to approximate
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* erf(1+s) - (c = (single)0.84506291151)
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* Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
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* where
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* P1(s) = degree 6 poly in s
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* Q1(s) = degree 6 poly in s
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*
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* 3. For x in [1.25,1/0.35(~2.857143)],
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* erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
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* erf(x) = 1 - erfc(x)
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* where
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* R1(z) = degree 7 poly in z, (z=1/x^2)
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* S1(z) = degree 8 poly in z
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*
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* 4. For x in [1/0.35,28]
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* erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
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* = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
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* = 2.0 - tiny (if x <= -6)
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* erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
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* erf(x) = sign(x)*(1.0 - tiny)
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* where
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* R2(z) = degree 6 poly in z, (z=1/x^2)
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* S2(z) = degree 7 poly in z
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*
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* Note1:
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* To compute exp(-x*x-0.5625+R/S), let s be a single
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* precision number and s := x; then
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* -x*x = -s*s + (s-x)*(s+x)
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* exp(-x*x-0.5626+R/S) =
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* exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
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* Note2:
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* Here 4 and 5 make use of the asymptotic series
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* exp(-x*x)
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* erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
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* x*sqrt(pi)
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* We use rational approximation to approximate
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* g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
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* Here is the error bound for R1/S1 and R2/S2
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* |R1/S1 - f(x)| < 2**(-62.57)
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* |R2/S2 - f(x)| < 2**(-61.52)
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*
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* 5. For inf > x >= 28
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* erf(x) = sign(x) *(1 - tiny) (raise inexact)
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* erfc(x) = tiny*tiny (raise underflow) if x > 0
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* = 2 - tiny if x<0
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*
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* 7. Special case:
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* erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
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* erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
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* erfc/erf(NaN) is NaN
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*/
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#define AU0 -9.86494292470009928597e-03
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#define AU1 -7.99283237680523006574e-01
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#define AU2 -1.77579549177547519889e+01
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#define AU3 -1.60636384855821916062e+02
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#define AU4 -6.37566443368389627722e+02
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#define AU5 -1.02509513161107724954e+03
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#define AU6 -4.83519191608651397019e+02
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#define AV0 3.03380607434824582924e+01
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#define AV1 3.25792512996573918826e+02
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#define AV2 1.53672958608443695994e+03
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#define AV3 3.19985821950859553908e+03
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#define AV4 2.55305040643316442583e+03
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#define AV5 4.74528541206955367215e+02
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#define AV6 -2.24409524465858183362e+01
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#define BU0 -9.86494403484714822705e-03
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#define BU1 -6.93858572707181764372e-01
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#define BU2 -1.05586262253232909814e+01
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#define BU3 -6.23753324503260060396e+01
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#define BU4 -1.62396669462573470355e+02
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#define BU5 -1.84605092906711035994e+02
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#define BU6 -8.12874355063065934246e+01
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#define BU7 -9.81432934416914548592e+00
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#define BV0 1.96512716674392571292e+01
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#define BV1 1.37657754143519042600e+02
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#define BV2 4.34565877475229228821e+02
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#define BV3 6.45387271733267880336e+02
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#define BV4 4.29008140027567833386e+02
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#define BV5 1.08635005541779435134e+02
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#define BV6 6.57024977031928170135e+00
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#define BV7 -6.04244152148580987438e-02
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#define CU0 -2.36211856075265944077e-03
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#define CU1 4.14856118683748331666e-01
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#define CU2 -3.72207876035701323847e-01
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#define CU3 3.18346619901161753674e-01
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#define CU4 -1.10894694282396677476e-01
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#define CU5 3.54783043256182359371e-02
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#define CU6 -2.16637559486879084300e-03
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#define CV0 1.06420880400844228286e-01
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#define CV1 5.40397917702171048937e-01
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#define CV2 7.18286544141962662868e-02
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#define CV3 1.26171219808761642112e-01
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#define CV4 1.36370839120290507362e-02
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#define CV5 1.19844998467991074170e-02
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#define DU0 1.28379167095512558561e-01
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#define DU1 -3.25042107247001499370e-01
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#define DU2 -2.84817495755985104766e-02
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#define DU3 -5.77027029648944159157e-03
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#define DU4 -2.37630166566501626084e-05
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#define DV0 3.97917223959155352819e-01
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#define DV1 6.50222499887672944485e-02
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#define DV2 5.08130628187576562776e-03
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#define DV3 1.32494738004321644526e-04
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#define DV4 -3.96022827877536812320e-06
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_CLC_OVERLOAD _CLC_DEF double erfc(double x) {
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long lx = as_long(x);
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long ax = lx & 0x7fffffffffffffffL;
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double absx = as_double(ax);
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int xneg = lx != ax;
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// Poly arg
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double x2 = x * x;
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double xm1 = absx - 1.0;
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double t = 1.0 / x2;
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t = absx < 1.25 ? xm1 : t;
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t = absx < 0.84375 ? x2 : t;
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// Evaluate rational poly
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// XXX Need to evaluate if we can grab the 14 coefficients from a
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// table faster than evaluating 3 pairs of polys
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double tu, tv, u, v;
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// |x| < 28
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u = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, AU6, AU5), AU4), AU3), AU2), AU1), AU0);
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v = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, AV6, AV5), AV4), AV3), AV2), AV1), AV0);
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tu = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, BU7, BU6), BU5), BU4), BU3), BU2), BU1), BU0);
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tv = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, BV7, BV6), BV5), BV4), BV3), BV2), BV1), BV0);
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u = absx < 0x1.6db6dp+1 ? tu : u;
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v = absx < 0x1.6db6dp+1 ? tv : v;
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tu = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, CU6, CU5), CU4), CU3), CU2), CU1), CU0);
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tv = fma(t, fma(t, fma(t, fma(t, fma(t, CV5, CV4), CV3), CV2), CV1), CV0);
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u = absx < 1.25 ? tu : u;
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v = absx < 1.25 ? tv : v;
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tu = fma(t, fma(t, fma(t, fma(t, DU4, DU3), DU2), DU1), DU0);
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tv = fma(t, fma(t, fma(t, fma(t, DV4, DV3), DV2), DV1), DV0);
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u = absx < 0.84375 ? tu : u;
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v = absx < 0.84375 ? tv : v;
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|
|
|
v = fma(t, v, 1.0);
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double q = u / v;
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|
|
|
|
|
// Evaluate return value
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|
|
|
// |x| < 28
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|
double z = as_double(ax & 0xffffffff00000000UL);
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double ret = exp(-z * z - 0.5625) * exp((z - absx) * (z + absx) + q) / absx;
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t = 2.0 - ret;
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ret = xneg ? t : ret;
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|
|
|
const double erx = 8.45062911510467529297e-01;
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z = erx + q + 1.0;
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t = 1.0 - erx - q;
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|
t = xneg ? z : t;
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|
ret = absx < 1.25 ? t : ret;
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|
|
|
// z = 1.0 - fma(x, q, x);
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// t = 0.5 - fma(x, q, x - 0.5);
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// t = xneg == 1 | absx < 0.25 ? z : t;
|
|
t = fma(-x, q, 1.0 - x);
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|
ret = absx < 0.84375 ? t : ret;
|
|
|
|
ret = x >= 28.0 ? 0.0 : ret;
|
|
ret = x <= -6.0 ? 2.0 : ret;
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|
ret = ax > 0x7ff0000000000000UL ? x : ret;
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|
|
|
return ret;
|
|
}
|
|
|
|
_CLC_UNARY_VECTORIZE(_CLC_OVERLOAD _CLC_DEF, double, erfc, double);
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#endif
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