blender/intern/cycles/util/util_math_fast.h

/*
 * 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;
}

/*
 * 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/substracting 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 = 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 = 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, float* sine, float* cosine)
{
	/* Same argument reduction as fast_sin. */
	int q = fast_rint(x * M_1_PI_F);
	float qf = 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 = 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 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/nans. */
	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 (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);
}

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__ */