Cycles: Add fast math function module
It is based on fmath.h from OIIO and could be used to give some speedup in areas where absolute accuracy is not so critical.
This commit is contained in:
parent
3dbfebc42c
commit
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@ -1010,6 +1010,7 @@ if env['OURPLATFORM']!='darwin':
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source.append('intern/cycles/util/util_color.h')
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source.append('intern/cycles/util/util_half.h')
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source.append('intern/cycles/util/util_math.h')
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source.append('intern/cycles/util/util_math_fast.h')
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source.append('intern/cycles/util/util_transform.h')
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source.append('intern/cycles/util/util_types.h')
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scriptinstall.append(env.Install(dir=dir,source=source))
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@ -137,6 +137,7 @@ set(SRC_UTIL_HEADERS
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../util/util_color.h
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../util/util_half.h
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../util/util_math.h
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../util/util_math_fast.h
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../util/util_transform.h
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../util/util_types.h
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)
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@ -19,6 +19,7 @@
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#include "util_color.h"
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#include "util_math.h"
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#include "util_math_fast.h"
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#include "util_transform.h"
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#endif /* __KERNEL_MATH_H__ */
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@ -47,6 +47,7 @@ set(SRC_HEADERS
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util_logging.h
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util_map.h
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util_math.h
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util_math_fast.h
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util_md5.h
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util_opengl.h
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util_optimization.h
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@ -0,0 +1,611 @@
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/*
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* Adapted from OpenImageIO library with this license:
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*
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* Copyright 2008-2014 Larry Gritz and the other authors and contributors.
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* All Rights Reserved.
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* Redistribution and use in source and binary forms, with or without
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* modification, are permitted provided that the following conditions are
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* met:
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* * Redistributions of source code must retain the above copyright
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* notice, this list of conditions and the following disclaimer.
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* * Redistributions in binary form must reproduce the above copyright
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* notice, this list of conditions and the following disclaimer in the
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* documentation and/or other materials provided with the distribution.
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* * Neither the name of the software's owners nor the names of its
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* contributors may be used to endorse or promote products derived from
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* this software without specific prior written permission.
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* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
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* "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
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* LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
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* A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
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* OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
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* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
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* LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
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* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
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* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
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* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
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* OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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*
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* (This is the Modified BSD License)
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*
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* A few bits here are based upon code from NVIDIA that was also released
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* under the same modified BSD license, and marked as:
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* Copyright 2004 NVIDIA Corporation. All Rights Reserved.
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*
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* Some parts of this file were first open-sourced in Open Shading Language,
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* then later moved here. The original copyright notice was:
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* Copyright (c) 2009-2014 Sony Pictures Imageworks Inc., et al.
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*
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* Many of the math functions were copied from or inspired by other
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* public domain sources or open source packages with compatible licenses.
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* The individual functions give references were applicable.
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*/
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#ifndef __UTIL_FAST_MATH__
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#define __UTIL_FAST_MATH__
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CCL_NAMESPACE_BEGIN
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/* TODO(sergey): Make sure it does not conflict with SSE intrinsics. */
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ccl_device_inline float madd(const float a, const float b, const float c)
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{
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/* NOTE: In the future we may want to explicitly ask for a fused
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* multiply-add in a specialized version for float.
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*
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* NOTE: GCC/ICC will turn this (for float) into a FMA unless
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* explicitly asked not to, clang seems to leave the code alone.
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*/
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return a * b + c;
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}
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/*
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* FAST & APPROXIMATE MATH
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*
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* The functions named "fast_*" provide a set of replacements to libm that
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* are much faster at the expense of some accuracy and robust handling of
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* extreme values. One design goal for these approximation was to avoid
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* branches as much as possible and operate on single precision values only
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* so that SIMD versions should be straightforward ports We also try to
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* implement "safe" semantics (ie: clamp to valid range where possible)
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* natively since wrapping these inline calls in another layer would be
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* wasteful.
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*
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* Some functions are fast_safe_*, which is both a faster approximation as
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* well as clamped input domain to ensure no NaN, Inf, or divide by zero.
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*/
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/* Round to nearest integer, returning as an int. */
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ccl_device_inline int fast_rint(float x)
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{
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/* used by sin/cos/tan range reduction. */
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#ifdef __KERNEL_SSE4__
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/* Single roundps instruction on SSE4.1+ (for gcc/clang at least). */
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return float_to_int(rintf(x));
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#else
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/* emulate rounding by adding/substracting 0.5. */
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return float_to_int(x + copysignf(0.5f, x));
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#endif
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}
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ccl_device float fast_sinf(float x)
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{
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/* Very accurate argument reduction from SLEEF,
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* starts failing around x=262000
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*
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* Results on: [-2pi,2pi].
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*
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* Examined 2173837240 values of sin: 0.00662760244 avg ulp diff, 2 max ulp,
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* 1.19209e-07 max error
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*/
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int q = fast_rint(x * M_1_PI_F);
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float qf = q;
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x = madd(qf, -0.78515625f*4, x);
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x = madd(qf, -0.00024187564849853515625f*4, x);
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x = madd(qf, -3.7747668102383613586e-08f*4, x);
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x = madd(qf, -1.2816720341285448015e-12f*4, x);
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x = M_PI_2_F - (M_PI_2_F - x); /* Crush denormals */
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float s = x * x;
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if((q & 1) != 0) x = -x;
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/* This polynomial approximation has very low error on [-pi/2,+pi/2]
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* 1.19209e-07 max error in total over [-2pi,+2pi]. */
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float u = 2.6083159809786593541503e-06f;
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u = madd(u, s, -0.0001981069071916863322258f);
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u = madd(u, s, +0.00833307858556509017944336f);
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u = madd(u, s, -0.166666597127914428710938f);
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u = madd(s, u * x, x);
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/* For large x, the argument reduction can fail and the polynomial can be
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* evaluated with arguments outside the valid internal. Just clamp the bad
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* values away (setting to 0.0f means no branches need to be generated). */
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if(fabsf(u) > 1.0f) {
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u = 0.0f;
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}
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return u;
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}
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ccl_device float fast_cosf(float x)
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{
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/* Same argument reduction as fast_sinf(). */
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int q = fast_rint(x * M_1_PI_F);
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float qf = q;
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x = madd(qf, -0.78515625f*4, x);
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x = madd(qf, -0.00024187564849853515625f*4, x);
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x = madd(qf, -3.7747668102383613586e-08f*4, x);
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x = madd(qf, -1.2816720341285448015e-12f*4, x);
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x = M_PI_2_F - (M_PI_2_F - x); /* Crush denormals. */
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float s = x * x;
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/* Polynomial from SLEEF's sincosf, max error is
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* 4.33127e-07 over [-2pi,2pi] (98% of values are "exact"). */
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float u = -2.71811842367242206819355e-07f;
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u = madd(u, s, +2.47990446951007470488548e-05f);
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u = madd(u, s, -0.00138888787478208541870117f);
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u = madd(u, s, +0.0416666641831398010253906f);
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u = madd(u, s, -0.5f);
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u = madd(u, s, +1.0f);
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if((q & 1) != 0) {
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u = -u;
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}
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if(fabsf(u) > 1.0f) {
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u = 0.0f;
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}
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return u;
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}
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ccl_device void fast_sincosf(float x, float* sine, float* cosine)
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{
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/* Same argument reduction as fast_sin. */
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int q = fast_rint(x * float(M_1_PI));
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float qf = q;
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x = madd(qf, -0.78515625f*4, x);
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x = madd(qf, -0.00024187564849853515625f*4, x);
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x = madd(qf, -3.7747668102383613586e-08f*4, x);
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x = madd(qf, -1.2816720341285448015e-12f*4, x);
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x = M_PI_2_F - (M_PI_2_F - x); // crush denormals
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float s = x * x;
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/* NOTE: same exact polynomials as fast_sinf() and fast_cosf() above. */
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if((q & 1) != 0) {
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x = -x;
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}
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float su = 2.6083159809786593541503e-06f;
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su = madd(su, s, -0.0001981069071916863322258f);
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su = madd(su, s, +0.00833307858556509017944336f);
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su = madd(su, s, -0.166666597127914428710938f);
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su = madd(s, su * x, x);
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float cu = -2.71811842367242206819355e-07f;
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cu = madd(cu, s, +2.47990446951007470488548e-05f);
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cu = madd(cu, s, -0.00138888787478208541870117f);
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cu = madd(cu, s, +0.0416666641831398010253906f);
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cu = madd(cu, s, -0.5f);
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cu = madd(cu, s, +1.0f);
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if((q & 1) != 0) {
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cu = -cu;
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}
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if(fabsf(su) > 1.0f) {
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su = 0.0f;
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}
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if(fabsf(cu) > 1.0f) {
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cu = 0.0f;
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}
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*sine = su;
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*cosine = cu;
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}
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/* NOTE: this approximation is only valid on [-8192.0,+8192.0], it starts
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* becoming really poor outside of this range because the reciprocal amplifies
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* errors.
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*/
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ccl_device float fast_tanf(float x)
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{
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/* Derived from SLEEF implementation.
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*
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* Note that we cannot apply the "denormal crush" trick everywhere because
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* we sometimes need to take the reciprocal of the polynomial
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*/
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int q = fast_rint(x * 2.0f * M_1_PI_F);
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float qf = q;
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x = madd(qf, -0.78515625f*2, x);
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x = madd(qf, -0.00024187564849853515625f*2, x);
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x = madd(qf, -3.7747668102383613586e-08f*2, x);
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x = madd(qf, -1.2816720341285448015e-12f*2, x);
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if((q & 1) == 0) {
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/* Crush denormals (only if we aren't inverting the result later). */
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x = M_PI_4_F - (M_PI_4_F - x);
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}
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float s = x * x;
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float u = 0.00927245803177356719970703f;
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u = madd(u, s, 0.00331984995864331722259521f);
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u = madd(u, s, 0.0242998078465461730957031f);
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u = madd(u, s, 0.0534495301544666290283203f);
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u = madd(u, s, 0.133383005857467651367188f);
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u = madd(u, s, 0.333331853151321411132812f);
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u = madd(s, u * x, x);
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if((q & 1) != 0) {
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u = -1.0f / u;
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}
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return u;
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}
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/* Fast, approximate sin(x*M_PI) with maximum absolute error of 0.000918954611.
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*
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* Adapted from http://devmaster.net/posts/9648/fast-and-accurate-sine-cosine#comment-76773
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*/
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ccl_device float fast_sinpif(float x)
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{
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/* Fast trick to strip the integral part off, so our domain is [-1, 1]. */
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const float z = x - ((x + 25165824.0f) - 25165824.0f);
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const float y = z - z * fabsf(z);
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const float Q = 3.10396624f;
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const float P = 3.584135056f; /* P = 16-4*Q */
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return y * (Q + P * fabsf(y));
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/* The original article used used inferior constants for Q and P and
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* so had max error 1.091e-3.
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*
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* The optimal value for Q was determined by exhaustive search, minimizing
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* the absolute numerical error relative to float(std::sin(double(phi*M_PI)))
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* over the interval [0,2] (which is where most of the invocations happen).
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*
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* The basic idea of this approximation starts with the coarse approximation:
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* sin(pi*x) ~= f(x) = 4 * (x - x * abs(x))
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*
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* This approximation always _over_ estimates the target. On the otherhand,
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* the curve:
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* sin(pi*x) ~= f(x) * abs(f(x)) / 4
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*
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* always lies _under_ the target. Thus we can simply numerically search for
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* the optimal constant to LERP these curves into a more precise
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* approximation.
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*
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* After folding the constants together and simplifying the resulting math,
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* we end up with the compact implementation above.
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*
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* NOTE: this function actually computes sin(x * pi) which avoids one or two
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* mults in many cases and guarantees exact values at integer periods.
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*/
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}
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/* Fast approximate cos(x*M_PI) with ~0.1% absolute error. */
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ccl_device_inline float fast_cospif(float x)
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{
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return fast_sinpif(x+0.5f);
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}
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ccl_device float fast_acosf(float x)
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{
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const float f = fabsf(x);
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/* clamp and crush denormals. */
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const float m = (f < 1.0f) ? 1.0f - (1.0f - f) : 1.0f;
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/* Based on http://www.pouet.net/topic.php?which=9132&page=2
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* 85% accurate (ulp 0)
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* Examined 2130706434 values of acos: 15.2000597 avg ulp diff, 4492 max ulp, 4.51803e-05 max error // without "denormal crush"
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* Examined 2130706434 values of acos: 15.2007108 avg ulp diff, 4492 max ulp, 4.51803e-05 max error // with "denormal crush"
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*/
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const float a = sqrtf(1.0f - m) *
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(1.5707963267f + m * (-0.213300989f + m *
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(0.077980478f + m * -0.02164095f)));
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return x < 0 ? M_PI_F - a : a;
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}
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ccl_device float fast_asinf(float x)
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{
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/* Based on acosf approximation above.
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* Max error is 4.51133e-05 (ulps are higher because we are consistently off
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* by a little amount).
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*/
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const float f = fabsf(x);
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/* Clamp and crush denormals. */
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const float m = (f < 1.0f) ? 1.0f - (1.0f - f) : 1.0f;
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const float a = M_PI_2_F - sqrtf(1.0f - m) *
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(1.5707963267f + m * (-0.213300989f + m *
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(0.077980478f + m * -0.02164095f)));
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return copysignf(a, x);
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}
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ccl_device float fast_atanf(float x)
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{
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const float a = fabsf(x);
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const float k = a > 1.0f ? 1 / a : a;
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const float s = 1.0f - (1.0f - k); /* Crush denormals. */
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const float t = s * s;
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/* http://mathforum.org/library/drmath/view/62672.html
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* Examined 4278190080 values of atan: 2.36864877 avg ulp diff, 302 max ulp, 6.55651e-06 max error // (with denormals)
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* Examined 4278190080 values of atan: 171160502 avg ulp diff, 855638016 max ulp, 6.55651e-06 max error // (crush denormals)
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*/
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float r = s * madd(0.43157974f, t, 1.0f) /
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madd(madd(0.05831938f, t, 0.76443945f), t, 1.0f);
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if(a > 1.0f) {
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/* TODO(sergey): Is it M_PI_2_F? */
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r = 1.570796326794896557998982f - r;
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}
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return copysignf(r, x);
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}
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ccl_device float fast_atan2f(float y, float x)
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{
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/* Based on atan approximation above.
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*
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* The special cases around 0 and infinity were tested explicitly.
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*
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* The only case not handled correctly is x=NaN,y=0 which returns 0 instead
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* of nan.
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*/
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const float a = fabsf(x);
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const float b = fabsf(y);
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const float k = (b == 0) ? 0.0f : ((a == b) ? 1.0f : (b > a ? a / b : b / a));
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const float s = 1.0f - (1.0f - k); /* Crush denormals */
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const float t = s * s;
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float r = s * madd(0.43157974f, t, 1.0f) /
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madd(madd(0.05831938f, t, 0.76443945f), t, 1.0f);
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if(b > a) {
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/* Account for arg reduction. */
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/* TODO(sergey): Is it M_PI_2_F? */
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r = 1.570796326794896557998982f - r;
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}
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/* Test sign bit of x. */
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if(__float_as_uint(x) & 0x80000000u) {
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r = M_PI_F - r;
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}
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return copysignf(r, y);
|
||||
}
|
||||
|
||||
/* Based on:
|
||||
*
|
||||
* https://github.com/LiraNuna/glsl-sse2/blob/master/source/vec4.h
|
||||
*
|
||||
*/
|
||||
ccl_device float fast_log2f(float x)
|
||||
{
|
||||
/* NOTE: clamp to avoid special cases and make result "safe" from large
|
||||
* negative values/nans. */
|
||||
clamp(x, FLT_MIN, FLT_MAX);
|
||||
unsigned bits = __float_as_uint(x);
|
||||
int exponent = int(bits >> 23) - 127;
|
||||
float f = __uint_as_float((bits & 0x007FFFFF) | 0x3f800000) - 1.0f;
|
||||
/* Examined 2130706432 values of log2 on [1.17549435e-38,3.40282347e+38]:
|
||||
* 0.0797524457 avg ulp diff, 3713596 max ulp, 7.62939e-06 max error.
|
||||
* ulp histogram:
|
||||
* 0 = 97.46%
|
||||
* 1 = 2.29%
|
||||
* 2 = 0.11%
|
||||
*/
|
||||
float f2 = f * f;
|
||||
float f4 = f2 * f2;
|
||||
float hi = madd(f, -0.00931049621349f, 0.05206469089414f);
|
||||
float lo = madd(f, 0.47868480909345f, -0.72116591947498f);
|
||||
hi = madd(f, hi, -0.13753123777116f);
|
||||
hi = madd(f, hi, 0.24187369696082f);
|
||||
hi = madd(f, hi, -0.34730547155299f);
|
||||
lo = madd(f, lo, 1.442689881667200f);
|
||||
return ((f4 * hi) + (f * lo)) + exponent;
|
||||
}
|
||||
|
||||
ccl_device_inline float fast_logf(float x)
|
||||
{
|
||||
/* Examined 2130706432 values of logf on [1.17549435e-38,3.40282347e+38]:
|
||||
* 0.313865375 avg ulp diff, 5148137 max ulp, 7.62939e-06 max error.
|
||||
*/
|
||||
return fast_log2f(x) * float(M_LN2);
|
||||
}
|
||||
|
||||
ccl_device_inline float fast_log10(float x)
|
||||
{
|
||||
/* Examined 2130706432 values of log10f on [1.17549435e-38,3.40282347e+38]:
|
||||
* 0.631237033 avg ulp diff, 4471615 max ulp, 3.8147e-06 max error.
|
||||
*/
|
||||
return fast_log2f(x) * float(M_LN2 / M_LN10);
|
||||
}
|
||||
|
||||
ccl_device float fast_logb(float x)
|
||||
{
|
||||
/* Don't bother with denormals. */
|
||||
x = fabsf(x);
|
||||
clamp(x, FLT_MIN, FLT_MAX);
|
||||
unsigned bits = __float_as_uint(x);
|
||||
return int(bits >> 23) - 127;
|
||||
}
|
||||
|
||||
ccl_device float fast_exp2f(float x)
|
||||
{
|
||||
/* Clamp to safe range for final addition. */
|
||||
clamp(x, -126.0f, 126.0f);
|
||||
/* Range reduction. */
|
||||
int m = int(x); x -= m;
|
||||
x = 1.0f - (1.0f - x); /* Crush denormals (does not affect max ulps!). */
|
||||
/* 5th degree polynomial generated with sollya
|
||||
* Examined 2247622658 values of exp2 on [-126,126]: 2.75764912 avg ulp diff,
|
||||
* 232 max ulp.
|
||||
*
|
||||
* ulp histogram:
|
||||
* 0 = 87.81%
|
||||
* 1 = 4.18%
|
||||
*/
|
||||
float r = 1.33336498402e-3f;
|
||||
r = madd(x, r, 9.810352697968e-3f);
|
||||
r = madd(x, r, 5.551834031939e-2f);
|
||||
r = madd(x, r, 0.2401793301105f);
|
||||
r = madd(x, r, 0.693144857883f);
|
||||
r = madd(x, r, 1.0f);
|
||||
/* Multiply by 2 ^ m by adding in the exponent. */
|
||||
/* NOTE: left-shift of negative number is undefined behavior. */
|
||||
return __uint_as_float(__float_as_uint(r) + (unsigned(m) << 23));
|
||||
}
|
||||
|
||||
ccl_device_inline float fast_expf(float x)
|
||||
{
|
||||
/* Examined 2237485550 values of exp on [-87.3300018,87.3300018]:
|
||||
* 2.6666452 avg ulp diff, 230 max ulp.
|
||||
*/
|
||||
return fast_exp2f(x * float(1.0 / M_LN2));
|
||||
}
|
||||
|
||||
ccl_device_inline float fast_exp10(float x)
|
||||
{
|
||||
/* Examined 2217701018 values of exp10 on [-37.9290009,37.9290009]:
|
||||
* 2.71732409 avg ulp diff, 232 max ulp.
|
||||
*/
|
||||
return fast_exp2f(x * float(M_LN10 / M_LN2));
|
||||
}
|
||||
|
||||
ccl_device_inline float fast_expm1f(float x)
|
||||
{
|
||||
if(fabsf(x) < 1e-5f) {
|
||||
x = 1.0f - (1.0f - x); /* Crush denormals. */
|
||||
return madd(0.5f, x * x, x);
|
||||
}
|
||||
else {
|
||||
return fast_expf(x) - 1.0f;
|
||||
}
|
||||
}
|
||||
|
||||
ccl_device float fast_sinhf(float x)
|
||||
{
|
||||
float a = fabsf(x);
|
||||
if(a > 1.0f) {
|
||||
/* Examined 53389559 values of sinh on [1,87.3300018]:
|
||||
* 33.6886442 avg ulp diff, 178 max ulp. */
|
||||
float e = fast_expf(a);
|
||||
return copysignf(0.5f * e - 0.5f / e, x);
|
||||
}
|
||||
else {
|
||||
a = 1.0f - (1.0f - a); /* Crush denorms. */
|
||||
float a2 = a * a;
|
||||
/* Degree 7 polynomial generated with sollya. */
|
||||
/* Examined 2130706434 values of sinh on [-1,1]: 1.19209e-07 max error. */
|
||||
float r = 2.03945513931e-4f;
|
||||
r = madd(r, a2, 8.32990277558e-3f);
|
||||
r = madd(r, a2, 0.1666673421859f);
|
||||
r = madd(r * a, a2, a);
|
||||
return copysignf(r, x);
|
||||
}
|
||||
}
|
||||
|
||||
ccl_device_inline float fast_coshf(float x)
|
||||
{
|
||||
/* Examined 2237485550 values of cosh on [-87.3300018,87.3300018]:
|
||||
* 1.78256726 avg ulp diff, 178 max ulp.
|
||||
*/
|
||||
float e = fast_expf(fabsf(x));
|
||||
return 0.5f * e + 0.5f / e;
|
||||
}
|
||||
|
||||
ccl_device_inline float fast_tanhf(float x)
|
||||
{
|
||||
/* Examined 4278190080 values of tanh on [-3.40282347e+38,3.40282347e+38]:
|
||||
* 3.12924e-06 max error.
|
||||
*/
|
||||
/* NOTE: ulp error is high because of sub-optimal handling around the origin. */
|
||||
float e = fast_expf(2.0f * fabsf(x));
|
||||
return copysignf(1.0f - 2.0f / (1.0f + e), x);
|
||||
}
|
||||
|
||||
ccl_device float fast_safe_powf(float x, float y)
|
||||
{
|
||||
if(y == 0) return 1.0f; /* x^1=1 */
|
||||
if(x == 0) return 0.0f; /* 0^y=0 */
|
||||
float sign = 1.0f;
|
||||
if(x < 0.0f) {
|
||||
/* if x is negative, only deal with integer powers
|
||||
* powf returns NaN for non-integers, we will return 0 instead.
|
||||
*/
|
||||
int ybits = __float_as_int(y) & 0x7fffffff;
|
||||
if(ybits >= 0x4b800000) {
|
||||
// always even int, keep positive
|
||||
}
|
||||
else if(ybits >= 0x3f800000) {
|
||||
/* Bigger than 1, check. */
|
||||
int k = (ybits >> 23) - 127; /* Get exponent. */
|
||||
int j = ybits >> (23 - k); /* Shift out possible fractional bits. */
|
||||
if((j << (23 - k)) == ybits) { /* rebuild number and check for a match. */
|
||||
/* +1 for even, -1 for odd. */
|
||||
sign = __int_as_float(0x3f800000 | (j << 31));
|
||||
}
|
||||
else {
|
||||
/* Not an integer. */
|
||||
return 0.0f;
|
||||
}
|
||||
}
|
||||
else {
|
||||
/* Not an integer. */
|
||||
return 0.0f;
|
||||
}
|
||||
}
|
||||
return sign * fast_exp2f(y * fast_log2f(fabsf(x)));
|
||||
}
|
||||
|
||||
/* TODO(sergey): Check speed with our erf functions implementation from
|
||||
* bsdf_microfaset.h.
|
||||
*/
|
||||
|
||||
ccl_device float fast_erff(float x)
|
||||
{
|
||||
/* Examined 1082130433 values of erff on [0,4]: 1.93715e-06 max error. */
|
||||
/* Abramowitz and Stegun, 7.1.28. */
|
||||
const float a1 = 0.0705230784f;
|
||||
const float a2 = 0.0422820123f;
|
||||
const float a3 = 0.0092705272f;
|
||||
const float a4 = 0.0001520143f;
|
||||
const float a5 = 0.0002765672f;
|
||||
const float a6 = 0.0000430638f;
|
||||
const float a = fabsf(x);
|
||||
const float b = 1.0f - (1.0f - a); /* Crush denormals. */
|
||||
const float r = madd(madd(madd(madd(madd(madd(a6, b, a5), b, a4), b, a3), b, a2), b, a1), b, 1.0f);
|
||||
const float s = r * r; /* ^2 */
|
||||
const float t = s * s; /* ^4 */
|
||||
const float u = t * t; /* ^8 */
|
||||
const float v = u * u; /* ^16 */
|
||||
return copysignf(1.0f - 1.0f / v, x);
|
||||
}
|
||||
|
||||
ccl_device_inline float fast_erfcf(float x)
|
||||
{
|
||||
/* Examined 2164260866 values of erfcf on [-4,4]: 1.90735e-06 max error.
|
||||
*
|
||||
* ulp histogram:
|
||||
*
|
||||
* 0 = 80.30%
|
||||
*/
|
||||
return 1.0f - fast_erff(x);
|
||||
}
|
||||
|
||||
ccl_device float fast_ierff(float x)
|
||||
{
|
||||
/* From: Approximating the erfinv function by Mike Giles. */
|
||||
/* To avoid trouble at the limit, clamp input to 1-eps. */
|
||||
float a = fabsf(x);
|
||||
if(a > 0.99999994f) {
|
||||
a = 0.99999994f;
|
||||
}
|
||||
float w = -fast_logf((1.0f - a) * (1.0f + a)), p;
|
||||
if(w < 5.0f) {
|
||||
w = w - 2.5f;
|
||||
p = 2.81022636e-08f;
|
||||
p = madd(p, w, 3.43273939e-07f);
|
||||
p = madd(p, w, -3.5233877e-06f);
|
||||
p = madd(p, w, -4.39150654e-06f);
|
||||
p = madd(p, w, 0.00021858087f);
|
||||
p = madd(p, w, -0.00125372503f);
|
||||
p = madd(p, w, -0.00417768164f);
|
||||
p = madd(p, w, 0.246640727f);
|
||||
p = madd(p, w, 1.50140941f);
|
||||
}
|
||||
else {
|
||||
w = sqrtf(w) - 3.0f;
|
||||
p = -0.000200214257f;
|
||||
p = madd(p, w, 0.000100950558f);
|
||||
p = madd(p, w, 0.00134934322f);
|
||||
p = madd(p, w, -0.00367342844f);
|
||||
p = madd(p, w, 0.00573950773f);
|
||||
p = madd(p, w, -0.0076224613f);
|
||||
p = madd(p, w, 0.00943887047f);
|
||||
p = madd(p, w, 1.00167406f);
|
||||
p = madd(p, w, 2.83297682f);
|
||||
}
|
||||
return p * x;
|
||||
}
|
||||
|
||||
CCL_NAMESPACE_END
|
||||
|
||||
#endif /* __UTIL_FAST_MATH__ */
|
||||
Loading…
Reference in New Issue