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Diffstat (limited to 'intern/cycles/util/math_fast.h')
| -rw-r--r-- | intern/cycles/util/math_fast.h | 652 |
1 files changed, 652 insertions, 0 deletions
diff --git a/intern/cycles/util/math_fast.h b/intern/cycles/util/math_fast.h new file mode 100644 index 00000000000..cc924f36a71 --- /dev/null +++ b/intern/cycles/util/math_fast.h @@ -0,0 +1,652 @@ +/* + * Adapted from OpenImageIO library with this license: + * + * Copyright 2008-2014 Larry Gritz and the other authors and contributors. + * All Rights Reserved. + + * Redistribution and use in source and binary forms, with or without + * modification, are permitted provided that the following conditions are + * met: + * * Redistributions of source code must retain the above copyright + * notice, this list of conditions and the following disclaimer. + * * Redistributions in binary form must reproduce the above copyright + * notice, this list of conditions and the following disclaimer in the + * documentation and/or other materials provided with the distribution. + * * Neither the name of the software's owners nor the names of its + * contributors may be used to endorse or promote products derived from + * this software without specific prior written permission. + * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS + * "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT + * LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR + * A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT + * OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, + * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT + * LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, + * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY + * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT + * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE + * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + * + * (This is the Modified BSD License) + * + * A few bits here are based upon code from NVIDIA that was also released + * under the same modified BSD license, and marked as: + * Copyright 2004 NVIDIA Corporation. All Rights Reserved. + * + * Some parts of this file were first open-sourced in Open Shading Language, + * then later moved here. The original copyright notice was: + * Copyright (c) 2009-2014 Sony Pictures Imageworks Inc., et al. + * + * Many of the math functions were copied from or inspired by other + * public domain sources or open source packages with compatible licenses. + * The individual functions give references were applicable. + */ + +#ifndef __UTIL_FAST_MATH__ +#define __UTIL_FAST_MATH__ + +CCL_NAMESPACE_BEGIN + +ccl_device_inline float madd(const float a, const float b, const float c) +{ + /* NOTE: In the future we may want to explicitly ask for a fused + * multiply-add in a specialized version for float. + * + * NOTE: GCC/ICC will turn this (for float) into a FMA unless + * explicitly asked not to, clang seems to leave the code alone. + */ + return a * b + c; +} + +ccl_device_inline float4 madd4(const float4 a, const float4 b, const float4 c) +{ + return a * b + c; +} + +/* + * FAST & APPROXIMATE MATH + * + * The functions named "fast_*" provide a set of replacements to libm that + * are much faster at the expense of some accuracy and robust handling of + * extreme values. One design goal for these approximation was to avoid + * branches as much as possible and operate on single precision values only + * so that SIMD versions should be straightforward ports We also try to + * implement "safe" semantics (ie: clamp to valid range where possible) + * natively since wrapping these inline calls in another layer would be + * wasteful. + * + * Some functions are fast_safe_*, which is both a faster approximation as + * well as clamped input domain to ensure no NaN, Inf, or divide by zero. + */ + +/* Round to nearest integer, returning as an int. */ +ccl_device_inline int fast_rint(float x) +{ + /* used by sin/cos/tan range reduction. */ +#ifdef __KERNEL_SSE4__ + /* Single `roundps` instruction on SSE4.1+ (for gcc/clang at least). */ + return float_to_int(rintf(x)); +#else + /* emulate rounding by adding/subtracting 0.5. */ + return float_to_int(x + copysignf(0.5f, x)); +#endif +} + +ccl_device float fast_sinf(float x) +{ + /* Very accurate argument reduction from SLEEF, + * starts failing around x=262000 + * + * Results on: [-2pi,2pi]. + * + * Examined 2173837240 values of sin: 0.00662760244 avg ulp diff, 2 max ulp, + * 1.19209e-07 max error + */ + int q = fast_rint(x * M_1_PI_F); + float qf = (float)q; + x = madd(qf, -0.78515625f * 4, x); + x = madd(qf, -0.00024187564849853515625f * 4, x); + x = madd(qf, -3.7747668102383613586e-08f * 4, x); + x = madd(qf, -1.2816720341285448015e-12f * 4, x); + x = M_PI_2_F - (M_PI_2_F - x); /* Crush denormals */ + float s = x * x; + if ((q & 1) != 0) + x = -x; + /* This polynomial approximation has very low error on [-pi/2,+pi/2] + * 1.19209e-07 max error in total over [-2pi,+2pi]. */ + float u = 2.6083159809786593541503e-06f; + u = madd(u, s, -0.0001981069071916863322258f); + u = madd(u, s, +0.00833307858556509017944336f); + u = madd(u, s, -0.166666597127914428710938f); + u = madd(s, u * x, x); + /* For large x, the argument reduction can fail and the polynomial can be + * evaluated with arguments outside the valid internal. Just clamp the bad + * values away (setting to 0.0f means no branches need to be generated). */ + if (fabsf(u) > 1.0f) { + u = 0.0f; + } + return u; +} + +ccl_device float fast_cosf(float x) +{ + /* Same argument reduction as fast_sinf(). */ + int q = fast_rint(x * M_1_PI_F); + float qf = (float)q; + x = madd(qf, -0.78515625f * 4, x); + x = madd(qf, -0.00024187564849853515625f * 4, x); + x = madd(qf, -3.7747668102383613586e-08f * 4, x); + x = madd(qf, -1.2816720341285448015e-12f * 4, x); + x = M_PI_2_F - (M_PI_2_F - x); /* Crush denormals. */ + float s = x * x; + /* Polynomial from SLEEF's sincosf, max error is + * 4.33127e-07 over [-2pi,2pi] (98% of values are "exact"). */ + float u = -2.71811842367242206819355e-07f; + u = madd(u, s, +2.47990446951007470488548e-05f); + u = madd(u, s, -0.00138888787478208541870117f); + u = madd(u, s, +0.0416666641831398010253906f); + u = madd(u, s, -0.5f); + u = madd(u, s, +1.0f); + if ((q & 1) != 0) { + u = -u; + } + if (fabsf(u) > 1.0f) { + u = 0.0f; + } + return u; +} + +ccl_device void fast_sincosf(float x, ccl_private float *sine, ccl_private float *cosine) +{ + /* Same argument reduction as fast_sin. */ + int q = fast_rint(x * M_1_PI_F); + float qf = (float)q; + x = madd(qf, -0.78515625f * 4, x); + x = madd(qf, -0.00024187564849853515625f * 4, x); + x = madd(qf, -3.7747668102383613586e-08f * 4, x); + x = madd(qf, -1.2816720341285448015e-12f * 4, x); + x = M_PI_2_F - (M_PI_2_F - x); // crush denormals + float s = x * x; + /* NOTE: same exact polynomials as fast_sinf() and fast_cosf() above. */ + if ((q & 1) != 0) { + x = -x; + } + float su = 2.6083159809786593541503e-06f; + su = madd(su, s, -0.0001981069071916863322258f); + su = madd(su, s, +0.00833307858556509017944336f); + su = madd(su, s, -0.166666597127914428710938f); + su = madd(s, su * x, x); + float cu = -2.71811842367242206819355e-07f; + cu = madd(cu, s, +2.47990446951007470488548e-05f); + cu = madd(cu, s, -0.00138888787478208541870117f); + cu = madd(cu, s, +0.0416666641831398010253906f); + cu = madd(cu, s, -0.5f); + cu = madd(cu, s, +1.0f); + if ((q & 1) != 0) { + cu = -cu; + } + if (fabsf(su) > 1.0f) { + su = 0.0f; + } + if (fabsf(cu) > 1.0f) { + cu = 0.0f; + } + *sine = su; + *cosine = cu; +} + +/* NOTE: this approximation is only valid on [-8192.0,+8192.0], it starts + * becoming really poor outside of this range because the reciprocal amplifies + * errors. + */ +ccl_device float fast_tanf(float x) +{ + /* Derived from SLEEF implementation. + * + * Note that we cannot apply the "denormal crush" trick everywhere because + * we sometimes need to take the reciprocal of the polynomial + */ + int q = fast_rint(x * 2.0f * M_1_PI_F); + float qf = (float)q; + x = madd(qf, -0.78515625f * 2, x); + x = madd(qf, -0.00024187564849853515625f * 2, x); + x = madd(qf, -3.7747668102383613586e-08f * 2, x); + x = madd(qf, -1.2816720341285448015e-12f * 2, x); + if ((q & 1) == 0) { + /* Crush denormals (only if we aren't inverting the result later). */ + x = M_PI_4_F - (M_PI_4_F - x); + } + float s = x * x; + float u = 0.00927245803177356719970703f; + u = madd(u, s, 0.00331984995864331722259521f); + u = madd(u, s, 0.0242998078465461730957031f); + u = madd(u, s, 0.0534495301544666290283203f); + u = madd(u, s, 0.133383005857467651367188f); + u = madd(u, s, 0.333331853151321411132812f); + u = madd(s, u * x, x); + if ((q & 1) != 0) { + u = -1.0f / u; + } + return u; +} + +/* Fast, approximate sin(x*M_PI) with maximum absolute error of 0.000918954611. + * + * Adapted from http://devmaster.net/posts/9648/fast-and-accurate-sine-cosine#comment-76773 + */ +ccl_device float fast_sinpif(float x) +{ + /* Fast trick to strip the integral part off, so our domain is [-1, 1]. */ + const float z = x - ((x + 25165824.0f) - 25165824.0f); + const float y = z - z * fabsf(z); + const float Q = 3.10396624f; + const float P = 3.584135056f; /* P = 16-4*Q */ + return y * (Q + P * fabsf(y)); + + /* The original article used inferior constants for Q and P and + * so had max error 1.091e-3. + * + * The optimal value for Q was determined by exhaustive search, minimizing + * the absolute numerical error relative to float(std::sin(double(phi*M_PI))) + * over the interval [0,2] (which is where most of the invocations happen). + * + * The basic idea of this approximation starts with the coarse approximation: + * sin(pi*x) ~= f(x) = 4 * (x - x * abs(x)) + * + * This approximation always _over_ estimates the target. On the other hand, + * the curve: + * sin(pi*x) ~= f(x) * abs(f(x)) / 4 + * + * always lies _under_ the target. Thus we can simply numerically search for + * the optimal constant to LERP these curves into a more precise + * approximation. + * + * After folding the constants together and simplifying the resulting math, + * we end up with the compact implementation above. + * + * NOTE: this function actually computes sin(x * pi) which avoids one or two + * mults in many cases and guarantees exact values at integer periods. + */ +} + +/* Fast approximate cos(x*M_PI) with ~0.1% absolute error. */ +ccl_device_inline float fast_cospif(float x) +{ + return fast_sinpif(x + 0.5f); +} + +ccl_device float fast_acosf(float x) +{ + const float f = fabsf(x); + /* clamp and crush denormals. */ + const float m = (f < 1.0f) ? 1.0f - (1.0f - f) : 1.0f; + /* Based on http://www.pouet.net/topic.php?which=9132&page=2 + * 85% accurate (ulp 0) + * Examined 2130706434 values of acos: + * 15.2000597 avg ulp diff, 4492 max ulp, 4.51803e-05 max error // without "denormal crush" + * Examined 2130706434 values of acos: + * 15.2007108 avg ulp diff, 4492 max ulp, 4.51803e-05 max error // with "denormal crush" + */ + const float a = sqrtf(1.0f - m) * + (1.5707963267f + m * (-0.213300989f + m * (0.077980478f + m * -0.02164095f))); + return x < 0 ? M_PI_F - a : a; +} + +ccl_device float fast_asinf(float x) +{ + /* Based on acosf approximation above. + * Max error is 4.51133e-05 (ulps are higher because we are consistently off + * by a little amount). + */ + const float f = fabsf(x); + /* Clamp and crush denormals. */ + const float m = (f < 1.0f) ? 1.0f - (1.0f - f) : 1.0f; + const float a = M_PI_2_F - + sqrtf(1.0f - m) * (1.5707963267f + + m * (-0.213300989f + m * (0.077980478f + m * -0.02164095f))); + return copysignf(a, x); +} + +ccl_device float fast_atanf(float x) +{ + const float a = fabsf(x); + const float k = a > 1.0f ? 1 / a : a; + const float s = 1.0f - (1.0f - k); /* Crush denormals. */ + const float t = s * s; + /* http://mathforum.org/library/drmath/view/62672.html + * Examined 4278190080 values of atan: + * 2.36864877 avg ulp diff, 302 max ulp, 6.55651e-06 max error // (with denormals) + * Examined 4278190080 values of atan: + * 171160502 avg ulp diff, 855638016 max ulp, 6.55651e-06 max error // (crush denormals) + */ + float r = s * madd(0.43157974f, t, 1.0f) / madd(madd(0.05831938f, t, 0.76443945f), t, 1.0f); + if (a > 1.0f) { + r = M_PI_2_F - r; + } + return copysignf(r, x); +} + +ccl_device float fast_atan2f(float y, float x) +{ + /* Based on atan approximation above. + * + * The special cases around 0 and infinity were tested explicitly. + * + * The only case not handled correctly is x=NaN,y=0 which returns 0 instead + * of nan. + */ + const float a = fabsf(x); + const float b = fabsf(y); + + const float k = (b == 0) ? 0.0f : ((a == b) ? 1.0f : (b > a ? a / b : b / a)); + const float s = 1.0f - (1.0f - k); /* Crush denormals */ + const float t = s * s; + + float r = s * madd(0.43157974f, t, 1.0f) / madd(madd(0.05831938f, t, 0.76443945f), t, 1.0f); + + if (b > a) { + /* Account for arg reduction. */ + r = M_PI_2_F - r; + } + /* Test sign bit of x. */ + if (__float_as_uint(x) & 0x80000000u) { + r = M_PI_F - r; + } + 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/NAN's. */ + x = 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) * M_LN2_F; +} + +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) * M_LN2_F / M_LN10_F; +} + +ccl_device float fast_logb(float x) +{ + /* Don't bother with denormals. */ + x = fabsf(x); + x = clamp(x, FLT_MIN, FLT_MAX); + unsigned bits = __float_as_uint(x); + return (float)((int)(bits >> 23) - 127); +} + +ccl_device float fast_exp2f(float x) +{ + /* Clamp to safe range for final addition. */ + x = 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 / M_LN2_F); +} + +#if defined(__KERNEL_CPU__) && !defined(_MSC_VER) +/* MSVC seems to have a code-gen bug here in at least SSE41/AVX, see + * T78047 and T78869 for details. Just disable for now, it only makes + * a small difference in denoising performance. */ +ccl_device float4 fast_exp2f4(float4 x) +{ + const float4 one = make_float4(1.0f); + const float4 limit = make_float4(126.0f); + x = clamp(x, -limit, limit); + int4 m = make_int4(x); + x = one - (one - (x - make_float4(m))); + float4 r = make_float4(1.33336498402e-3f); + r = madd4(x, r, make_float4(9.810352697968e-3f)); + r = madd4(x, r, make_float4(5.551834031939e-2f)); + r = madd4(x, r, make_float4(0.2401793301105f)); + r = madd4(x, r, make_float4(0.693144857883f)); + r = madd4(x, r, make_float4(1.0f)); + return __int4_as_float4(__float4_as_int4(r) + (m << 23)); +} + +ccl_device_inline float4 fast_expf4(float4 x) +{ + return fast_exp2f4(x / M_LN2_F); +} +#else +ccl_device_inline float4 fast_expf4(float4 x) +{ + return make_float4(fast_expf(x.x), fast_expf(x.y), fast_expf(x.z), fast_expf(x.w)); +} +#endif + +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 * M_LN10_F / M_LN2_F); +} + +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_microfacet.h. + */ + +ccl_device_inline 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); + if (a >= 12.3f) { + return copysignf(1.0f, 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_inline 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__ */ |