/*
* 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 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 erf(float x) {
int hx = as_uint(x);
int ix = hx & 0x7fffffff;
float absx = as_float(ix);
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;
float u, v, tu, tv;
// |x| < 6
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 < 0x1.6db6dcp+1f ? tu : u;
v = absx < 0x1.6db6dcp+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 = 1.0f;
// |x| < 6
float z = as_float(ix & 0xfffff000);
float r = exp(mad(-z, z, -0.5625f)) * exp(mad(z-absx, z+absx, q));
r = 1.0f - MATH_DIVIDE(r, absx);
ret = absx < 6.0f ? r : ret;
r = erx + q;
ret = absx < 1.25f ? r : ret;
ret = as_float((hx & 0x80000000) | as_int(ret));
r = mad(x, q, x);
ret = absx < 0.84375f ? r : ret;
// Prevent underflow
r = 0.125f * mad(8.0f, x, efx8 * x);
ret = absx < 0x1.0p-28f ? r : ret;
ret = isnan(x) ? x : ret;
return ret;
}
_CLC_UNARY_VECTORIZE(_CLC_OVERLOAD _CLC_DEF, float, erf, 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 AV1 3.03380607434824582924e+01
#define AV2 3.25792512996573918826e+02
#define AV3 1.53672958608443695994e+03
#define AV4 3.19985821950859553908e+03
#define AV5 2.55305040643316442583e+03
#define AV6 4.74528541206955367215e+02
#define AV7 -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 BV1 1.96512716674392571292e+01
#define BV2 1.37657754143519042600e+02
#define BV3 4.34565877475229228821e+02
#define BV4 6.45387271733267880336e+02
#define BV5 4.29008140027567833386e+02
#define BV6 1.08635005541779435134e+02
#define BV7 6.57024977031928170135e+00
#define BV8 -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 CV1 1.06420880400844228286e-01
#define CV2 5.40397917702171048937e-01
#define CV3 7.18286544141962662868e-02
#define CV4 1.26171219808761642112e-01
#define CV5 1.36370839120290507362e-02
#define CV6 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 DV1 3.97917223959155352819e-01
#define DV2 6.50222499887672944485e-02
#define DV3 5.08130628187576562776e-03
#define DV4 1.32494738004321644526e-04
#define DV5 -3.96022827877536812320e-06
_CLC_OVERLOAD _CLC_DEF double erf(double y) {
double x = fabs(y);
double x2 = x * x;
double xm1 = x - 1.0;
// Poly variable
double t = 1.0 / x2;
t = x < 1.25 ? xm1 : t;
t = x < 0.84375 ? x2 : t;
double u, ut, v, vt;
// Evaluate rational poly
// XXX We need to see of we can grab 16 coefficents from a table
// faster than evaluating 3 of the poly pairs
// if (x < 6.0)
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, AV7, AV6), AV5), AV4), AV3), AV2), AV1);
ut = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, BU7, BU6), BU5), BU4), BU3), BU2), BU1), BU0);
vt = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, BV8, BV7), BV6), BV5), BV4), BV3), BV2), BV1);
u = x < 0x1.6db6ep+1 ? ut : u;
v = x < 0x1.6db6ep+1 ? vt : v;
ut = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, CU6, CU5), CU4), CU3), CU2), CU1), CU0);
vt = fma(t, fma(t, fma(t, fma(t, fma(t, CV6, CV5), CV4), CV3), CV2), CV1);
u = x < 1.25 ? ut : u;
v = x < 1.25 ? vt : v;
ut = fma(t, fma(t, fma(t, fma(t, DU4, DU3), DU2), DU1), DU0);
vt = fma(t, fma(t, fma(t, fma(t, DV5, DV4), DV3), DV2), DV1);
u = x < 0.84375 ? ut : u;
v = x < 0.84375 ? vt : v;
v = fma(t, v, 1.0);
// Compute rational approximation
double q = u / v;
// Compute results
double z = as_double(as_long(x) & 0xffffffff00000000L);
double r = exp(-z * z - 0.5625) * exp((z - x) * (z + x) + q);
r = 1.0 - r / x;
double ret = x < 6.0 ? r : 1.0;
r = 8.45062911510467529297e-01 + q;
ret = x < 1.25 ? r : ret;
q = x < 0x1.0p-28 ? 1.28379167095512586316e-01 : q;
r = fma(x, q, x);
ret = x < 0.84375 ? r : ret;
ret = isnan(x) ? x : ret;
return y < 0.0 ? -ret : ret;
}
_CLC_UNARY_VECTORIZE(_CLC_OVERLOAD _CLC_DEF, double, erf, double);
#ifdef cl_khr_fp16
#pragma OPENCL EXTENSION cl_khr_fp16 : enable
_CLC_OVERLOAD _CLC_DEF half erf(half h) {
return (half)erf((float)h);
}
_CLC_UNARY_VECTORIZE(_CLC_OVERLOAD _CLC_DEF, half, erf, half);
#endif
#endif
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