[libclc] r232674 - math: Implement erfc
Aaron Watry
awatry at gmail.com
Wed Mar 18 14:52:08 PDT 2015
Author: awatry
Date: Wed Mar 18 16:52:07 2015
New Revision: 232674
URL: http://llvm.org/viewvc/llvm-project?rev=232674&view=rev
Log:
math: Implement erfc
Signed-off-by: Aaron Watry <awatry at gmail.com>
Reviewed-by: Tom Stellard <thomas.stellard at amd.com>
Added:
libclc/trunk/generic/include/clc/math/erfc.h
libclc/trunk/generic/lib/math/erfc.cl
Modified:
libclc/trunk/generic/include/clc/clc.h
libclc/trunk/generic/lib/SOURCES
Modified: libclc/trunk/generic/include/clc/clc.h
URL: http://llvm.org/viewvc/llvm-project/libclc/trunk/generic/include/clc/clc.h?rev=232674&r1=232673&r2=232674&view=diff
==============================================================================
--- libclc/trunk/generic/include/clc/clc.h (original)
+++ libclc/trunk/generic/include/clc/clc.h Wed Mar 18 16:52:07 2015
@@ -40,6 +40,7 @@
#include <clc/math/cos.h>
#include <clc/math/cospi.h>
#include <clc/math/ceil.h>
+#include <clc/math/erfc.h>
#include <clc/math/exp.h>
#include <clc/math/exp10.h>
#include <clc/math/exp2.h>
Added: libclc/trunk/generic/include/clc/math/erfc.h
URL: http://llvm.org/viewvc/llvm-project/libclc/trunk/generic/include/clc/math/erfc.h?rev=232674&view=auto
==============================================================================
--- libclc/trunk/generic/include/clc/math/erfc.h (added)
+++ libclc/trunk/generic/include/clc/math/erfc.h Wed Mar 18 16:52:07 2015
@@ -0,0 +1,9 @@
+#undef erfc
+
+#define __CLC_BODY <clc/math/unary_decl.inc>
+#define __CLC_FUNCTION erfc
+
+#include <clc/math/gentype.inc>
+
+#undef __CLC_BODY
+#undef __CLC_FUNCTION
Modified: libclc/trunk/generic/lib/SOURCES
URL: http://llvm.org/viewvc/llvm-project/libclc/trunk/generic/lib/SOURCES?rev=232674&r1=232673&r2=232674&view=diff
==============================================================================
--- libclc/trunk/generic/lib/SOURCES (original)
+++ libclc/trunk/generic/lib/SOURCES Wed Mar 18 16:52:07 2015
@@ -63,6 +63,7 @@ math/atan2.cl
math/copysign.cl
math/cos.cl
math/cospi.cl
+math/erfc.cl
math/exp.cl
math/exp10.cl
math/fmax.cl
Added: libclc/trunk/generic/lib/math/erfc.cl
URL: http://llvm.org/viewvc/llvm-project/libclc/trunk/generic/lib/math/erfc.cl?rev=232674&view=auto
==============================================================================
--- libclc/trunk/generic/lib/math/erfc.cl (added)
+++ libclc/trunk/generic/lib/math/erfc.cl Wed Mar 18 16:52:07 2015
@@ -0,0 +1,413 @@
+/*
+ * 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"
+
+/*
+ * ====================================================
+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Developed at SunPro, a Sun Microsystems, Inc. business.
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice
+ * is preserved.
+ * ====================================================
+ */
+
+#define erx_f 8.4506291151e-01f /* 0x3f58560b */
+
+// Coefficients for approximation to erf on [00.84375]
+
+#define efx 1.2837916613e-01f /* 0x3e0375d4 */
+#define efx8 1.0270333290e+00f /* 0x3f8375d4 */
+
+#define pp0 1.2837916613e-01f /* 0x3e0375d4 */
+#define pp1 -3.2504209876e-01f /* 0xbea66beb */
+#define pp2 -2.8481749818e-02f /* 0xbce9528f */
+#define pp3 -5.7702702470e-03f /* 0xbbbd1489 */
+#define pp4 -2.3763017452e-05f /* 0xb7c756b1 */
+#define qq1 3.9791721106e-01f /* 0x3ecbbbce */
+#define qq2 6.5022252500e-02f /* 0x3d852a63 */
+#define qq3 5.0813062117e-03f /* 0x3ba68116 */
+#define qq4 1.3249473704e-04f /* 0x390aee49 */
+#define qq5 -3.9602282413e-06f /* 0xb684e21a */
+
+// Coefficients for approximation to erf in [0.843751.25]
+
+#define pa0 -2.3621185683e-03f /* 0xbb1acdc6 */
+#define pa1 4.1485610604e-01f /* 0x3ed46805 */
+#define pa2 -3.7220788002e-01f /* 0xbebe9208 */
+#define pa3 3.1834661961e-01f /* 0x3ea2fe54 */
+#define pa4 -1.1089469492e-01f /* 0xbde31cc2 */
+#define pa5 3.5478305072e-02f /* 0x3d1151b3 */
+#define pa6 -2.1663755178e-03f /* 0xbb0df9c0 */
+#define qa1 1.0642088205e-01f /* 0x3dd9f331 */
+#define qa2 5.4039794207e-01f /* 0x3f0a5785 */
+#define qa3 7.1828655899e-02f /* 0x3d931ae7 */
+#define qa4 1.2617121637e-01f /* 0x3e013307 */
+#define qa5 1.3637083583e-02f /* 0x3c5f6e13 */
+#define qa6 1.1984500103e-02f /* 0x3c445aa3 */
+
+// Coefficients for approximation to erfc in [1.251/0.35]
+
+#define ra0 -9.8649440333e-03f /* 0xbc21a093 */
+#define ra1 -6.9385856390e-01f /* 0xbf31a0b7 */
+#define ra2 -1.0558626175e+01f /* 0xc128f022 */
+#define ra3 -6.2375331879e+01f /* 0xc2798057 */
+#define ra4 -1.6239666748e+02f /* 0xc322658c */
+#define ra5 -1.8460508728e+02f /* 0xc3389ae7 */
+#define ra6 -8.1287437439e+01f /* 0xc2a2932b */
+#define ra7 -9.8143291473e+00f /* 0xc11d077e */
+#define sa1 1.9651271820e+01f /* 0x419d35ce */
+#define sa2 1.3765776062e+02f /* 0x4309a863 */
+#define sa3 4.3456588745e+02f /* 0x43d9486f */
+#define sa4 6.4538726807e+02f /* 0x442158c9 */
+#define sa5 4.2900814819e+02f /* 0x43d6810b */
+#define sa6 1.0863500214e+02f /* 0x42d9451f */
+#define sa7 6.5702495575e+00f /* 0x40d23f7c */
+#define sa8 -6.0424413532e-02f /* 0xbd777f97 */
+
+// Coefficients for approximation to erfc in [1/.3528]
+
+#define rb0 -9.8649431020e-03f /* 0xbc21a092 */
+#define rb1 -7.9928326607e-01f /* 0xbf4c9dd4 */
+#define rb2 -1.7757955551e+01f /* 0xc18e104b */
+#define rb3 -1.6063638306e+02f /* 0xc320a2ea */
+#define rb4 -6.3756646729e+02f /* 0xc41f6441 */
+#define rb5 -1.0250950928e+03f /* 0xc480230b */
+#define rb6 -4.8351919556e+02f /* 0xc3f1c275 */
+#define sb1 3.0338060379e+01f /* 0x41f2b459 */
+#define sb2 3.2579251099e+02f /* 0x43a2e571 */
+#define sb3 1.5367296143e+03f /* 0x44c01759 */
+#define sb4 3.1998581543e+03f /* 0x4547fdbb */
+#define sb5 2.5530502930e+03f /* 0x451f90ce */
+#define sb6 4.7452853394e+02f /* 0x43ed43a7 */
+#define sb7 -2.2440952301e+01f /* 0xc1b38712 */
+
+_CLC_OVERLOAD _CLC_DEF float erfc(float x) {
+ int hx = as_int(x);
+ int ix = hx & 0x7fffffff;
+ float absx = as_float(ix);
+
+ // Argument for polys
+ float x2 = absx * absx;
+ float t = 1.0f / x2;
+ float tt = absx - 1.0f;
+ t = absx < 1.25f ? tt : t;
+ t = absx < 0.84375f ? x2 : t;
+
+ // Evaluate polys
+ float tu, tv, u, v;
+
+ u = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, rb6, rb5), rb4), rb3), rb2), rb1), rb0);
+ v = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, sb7, sb6), sb5), sb4), sb3), sb2), sb1);
+
+ tu = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, ra7, ra6), ra5), ra4), ra3), ra2), ra1), ra0);
+ tv = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, sa8, sa7), sa6), sa5), sa4), sa3), sa2), sa1);
+ u = absx < 0x1.6db6dap+1f ? tu : u;
+ v = absx < 0x1.6db6dap+1f ? tv : v;
+
+ tu = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, pa6, pa5), pa4), pa3), pa2), pa1), pa0);
+ tv = mad(t, mad(t, mad(t, mad(t, mad(t, qa6, qa5), qa4), qa3), qa2), qa1);
+ u = absx < 1.25f ? tu : u;
+ v = absx < 1.25f ? tv : v;
+
+ tu = mad(t, mad(t, mad(t, mad(t, pp4, pp3), pp2), pp1), pp0);
+ tv = mad(t, mad(t, mad(t, mad(t, qq5, qq4), qq3), qq2), qq1);
+ u = absx < 0.84375f ? tu : u;
+ v = absx < 0.84375f ? tv : v;
+
+ v = mad(t, v, 1.0f);
+
+ float q = MATH_DIVIDE(u, v);
+
+ float ret = 0.0f;
+
+ float z = as_float(ix & 0xfffff000);
+ float r = exp(mad(-z, z, -0.5625f)) * exp(mad(z - absx, z + absx, q));
+ r = MATH_DIVIDE(r, absx);
+ t = 2.0f - r;
+ r = x < 0.0f ? t : r;
+ ret = absx < 28.0f ? r : ret;
+
+ r = 1.0f - erx_f - q;
+ t = erx_f + q + 1.0f;
+ r = x < 0.0f ? t : r;
+ ret = absx < 1.25f ? r : ret;
+
+ r = 0.5f - mad(x, q, x - 0.5f);
+ ret = absx < 0.84375f ? r : ret;
+
+ ret = x < -6.0f ? 2.0f : ret;
+
+ ret = isnan(x) ? x : ret;
+
+ return ret;
+}
+
+_CLC_UNARY_VECTORIZE(_CLC_OVERLOAD _CLC_DEF, float, erfc, float);
+
+#ifdef cl_khr_fp64
+
+#pragma OPENCL EXTENSION cl_khr_fp64 : enable
+
+/*
+ * ====================================================
+ * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
+ *
+ * Developed at SunPro, a Sun Microsystems, Inc. business.
+ * Permission to use, copy, modify, and distribute this
+ * software is freely granted, provided that this notice
+ * is preserved.
+ * ====================================================
+ */
+
+/* double erf(double x)
+ * double erfc(double x)
+ * x
+ * 2 |\
+ * erf(x) = --------- | exp(-t*t)dt
+ * sqrt(pi) \|
+ * 0
+ *
+ * erfc(x) = 1-erf(x)
+ * Note that
+ * erf(-x) = -erf(x)
+ * erfc(-x) = 2 - erfc(x)
+ *
+ * Method:
+ * 1. For |x| in [0, 0.84375]
+ * erf(x) = x + x*R(x^2)
+ * erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
+ * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
+ * where R = P/Q where P is an odd poly of degree 8 and
+ * Q is an odd poly of degree 10.
+ * -57.90
+ * | R - (erf(x)-x)/x | <= 2
+ *
+ *
+ * Remark. The formula is derived by noting
+ * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....)
+ * and that
+ * 2/sqrt(pi) = 1.128379167095512573896158903121545171688
+ * is close to one. The interval is chosen because the fix
+ * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
+ * near 0.6174), and by some experiment, 0.84375 is chosen to
+ * guarantee the error is less than one ulp for erf.
+ *
+ * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and
+ * c = 0.84506291151 rounded to single (24 bits)
+ * erf(x) = sign(x) * (c + P1(s)/Q1(s))
+ * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0
+ * 1+(c+P1(s)/Q1(s)) if x < 0
+ * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
+ * Remark: here we use the taylor series expansion at x=1.
+ * erf(1+s) = erf(1) + s*Poly(s)
+ * = 0.845.. + P1(s)/Q1(s)
+ * That is, we use rational approximation to approximate
+ * erf(1+s) - (c = (single)0.84506291151)
+ * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
+ * where
+ * P1(s) = degree 6 poly in s
+ * Q1(s) = degree 6 poly in s
+ *
+ * 3. For x in [1.25,1/0.35(~2.857143)],
+ * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1)
+ * erf(x) = 1 - erfc(x)
+ * where
+ * R1(z) = degree 7 poly in z, (z=1/x^2)
+ * S1(z) = degree 8 poly in z
+ *
+ * 4. For x in [1/0.35,28]
+ * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0
+ * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0
+ * = 2.0 - tiny (if x <= -6)
+ * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
+ * erf(x) = sign(x)*(1.0 - tiny)
+ * where
+ * R2(z) = degree 6 poly in z, (z=1/x^2)
+ * S2(z) = degree 7 poly in z
+ *
+ * Note1:
+ * To compute exp(-x*x-0.5625+R/S), let s be a single
+ * precision number and s := x; then
+ * -x*x = -s*s + (s-x)*(s+x)
+ * exp(-x*x-0.5626+R/S) =
+ * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S);
+ * Note2:
+ * Here 4 and 5 make use of the asymptotic series
+ * exp(-x*x)
+ * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) )
+ * x*sqrt(pi)
+ * We use rational approximation to approximate
+ * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625
+ * Here is the error bound for R1/S1 and R2/S2
+ * |R1/S1 - f(x)| < 2**(-62.57)
+ * |R2/S2 - f(x)| < 2**(-61.52)
+ *
+ * 5. For inf > x >= 28
+ * erf(x) = sign(x) *(1 - tiny) (raise inexact)
+ * erfc(x) = tiny*tiny (raise underflow) if x > 0
+ * = 2 - tiny if x<0
+ *
+ * 7. Special case:
+ * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
+ * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
+ * erfc/erf(NaN) is NaN
+ */
+
+#define AU0 -9.86494292470009928597e-03
+#define AU1 -7.99283237680523006574e-01
+#define AU2 -1.77579549177547519889e+01
+#define AU3 -1.60636384855821916062e+02
+#define AU4 -6.37566443368389627722e+02
+#define AU5 -1.02509513161107724954e+03
+#define AU6 -4.83519191608651397019e+02
+
+#define AV0 3.03380607434824582924e+01
+#define AV1 3.25792512996573918826e+02
+#define AV2 1.53672958608443695994e+03
+#define AV3 3.19985821950859553908e+03
+#define AV4 2.55305040643316442583e+03
+#define AV5 4.74528541206955367215e+02
+#define AV6 -2.24409524465858183362e+01
+
+#define BU0 -9.86494403484714822705e-03
+#define BU1 -6.93858572707181764372e-01
+#define BU2 -1.05586262253232909814e+01
+#define BU3 -6.23753324503260060396e+01
+#define BU4 -1.62396669462573470355e+02
+#define BU5 -1.84605092906711035994e+02
+#define BU6 -8.12874355063065934246e+01
+#define BU7 -9.81432934416914548592e+00
+
+#define BV0 1.96512716674392571292e+01
+#define BV1 1.37657754143519042600e+02
+#define BV2 4.34565877475229228821e+02
+#define BV3 6.45387271733267880336e+02
+#define BV4 4.29008140027567833386e+02
+#define BV5 1.08635005541779435134e+02
+#define BV6 6.57024977031928170135e+00
+#define BV7 -6.04244152148580987438e-02
+
+#define CU0 -2.36211856075265944077e-03
+#define CU1 4.14856118683748331666e-01
+#define CU2 -3.72207876035701323847e-01
+#define CU3 3.18346619901161753674e-01
+#define CU4 -1.10894694282396677476e-01
+#define CU5 3.54783043256182359371e-02
+#define CU6 -2.16637559486879084300e-03
+
+#define CV0 1.06420880400844228286e-01
+#define CV1 5.40397917702171048937e-01
+#define CV2 7.18286544141962662868e-02
+#define CV3 1.26171219808761642112e-01
+#define CV4 1.36370839120290507362e-02
+#define CV5 1.19844998467991074170e-02
+
+#define DU0 1.28379167095512558561e-01
+#define DU1 -3.25042107247001499370e-01
+#define DU2 -2.84817495755985104766e-02
+#define DU3 -5.77027029648944159157e-03
+#define DU4 -2.37630166566501626084e-05
+
+#define DV0 3.97917223959155352819e-01
+#define DV1 6.50222499887672944485e-02
+#define DV2 5.08130628187576562776e-03
+#define DV3 1.32494738004321644526e-04
+#define DV4 -3.96022827877536812320e-06
+
+_CLC_OVERLOAD _CLC_DEF double erfc(double x) {
+ long lx = as_long(x);
+ long ax = lx & 0x7fffffffffffffffL;
+ double absx = as_double(ax);
+ int xneg = lx != ax;
+
+ // Poly arg
+ double x2 = x * x;
+ double xm1 = absx - 1.0;
+ double t = 1.0 / x2;
+ t = absx < 1.25 ? xm1 : t;
+ t = absx < 0.84375 ? x2 : t;
+
+
+ // Evaluate rational poly
+ // XXX Need to evaluate if we can grab the 14 coefficients from a
+ // table faster than evaluating 3 pairs of polys
+ double tu, tv, u, v;
+
+ // |x| < 28
+ u = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, AU6, AU5), AU4), AU3), AU2), AU1), AU0);
+ v = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, AV6, AV5), AV4), AV3), AV2), AV1), AV0);
+
+ tu = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, BU7, BU6), BU5), BU4), BU3), BU2), BU1), BU0);
+ tv = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, BV7, BV6), BV5), BV4), BV3), BV2), BV1), BV0);
+ u = absx < 0x1.6db6dp+1 ? tu : u;
+ v = absx < 0x1.6db6dp+1 ? tv : v;
+
+ tu = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, CU6, CU5), CU4), CU3), CU2), CU1), CU0);
+ tv = fma(t, fma(t, fma(t, fma(t, fma(t, CV5, CV4), CV3), CV2), CV1), CV0);
+ u = absx < 1.25 ? tu : u;
+ v = absx < 1.25 ? tv : v;
+
+ tu = fma(t, fma(t, fma(t, fma(t, DU4, DU3), DU2), DU1), DU0);
+ tv = fma(t, fma(t, fma(t, fma(t, DV4, DV3), DV2), DV1), DV0);
+ u = absx < 0.84375 ? tu : u;
+ v = absx < 0.84375 ? tv : v;
+
+ v = fma(t, v, 1.0);
+ double q = u / v;
+
+
+ // Evaluate return value
+
+ // |x| < 28
+ double z = as_double(ax & 0xffffffff00000000UL);
+ double ret = exp(-z * z - 0.5625) * exp((z - absx) * (z + absx) + q) / absx;
+ t = 2.0 - ret;
+ ret = xneg ? t : ret;
+
+ const double erx = 8.45062911510467529297e-01;
+ z = erx + q + 1.0;
+ t = 1.0 - erx - q;
+ t = xneg ? z : t;
+ ret = absx < 1.25 ? t : ret;
+
+ // z = 1.0 - fma(x, q, x);
+ // t = 0.5 - fma(x, q, x - 0.5);
+ // t = xneg == 1 | absx < 0.25 ? z : t;
+ t = fma(-x, q, 1.0 - x);
+ ret = absx < 0.84375 ? t : ret;
+
+ ret = x >= 28.0 ? 0.0 : ret;
+ ret = x <= -6.0 ? 2.0 : ret;
+ ret = ax > 0x7ff0000000000000UL ? x : ret;
+
+ return ret;
+}
+
+_CLC_UNARY_VECTORIZE(_CLC_OVERLOAD _CLC_DEF, double, erfc, double);
+
+#endif
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