HairBSDF, sample_f, cosTheta

Average Accuracy: 99.5% → 99.5%

Time: 19.2s

Precision: binary32

Cost: 10016

\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]

Math

\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]

↓

\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {e}^{\left(\frac{-2}{v}\right)}\right) \]

FPCore

(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))

↓

(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (pow E (/ -2.0 v))))))))

C

float code(float u, float v) {
	return 1.0f + (v * logf((u + ((1.0f - u) * expf((-2.0f / v))))));
}

↓

float code(float u, float v) {
	return 1.0f + (v * logf((u + ((1.0f - u) * powf(((float) M_E), (-2.0f / v))))));
}

Julia

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * exp(Float32(Float32(-2.0) / v)))))))
end

↓

function code(u, v)
	return Float32(Float32(1.0) + Float32(v * log(Float32(u + Float32(Float32(Float32(1.0) - u) * (Float32(exp(1)) ^ Float32(Float32(-2.0) / v)))))))
end

MATLAB

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * exp((single(-2.0) / v))))));
end

↓

function tmp = code(u, v)
	tmp = single(1.0) + (v * log((u + ((single(1.0) - u) * (single(2.71828182845904523536) ^ (single(-2.0) / v))))));
end

Derivation

1. Initial program 99.5%

   \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]

2. Applied egg-rr 99.5%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{{e}^{\left(\frac{-2}{v}\right)}}\right) \]
   Proof

   [Start] 99.5	\[ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   *-un-lft-identity [=>] 99.5	\[ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\color{blue}{1 \cdot \frac{-2}{v}}}\right) \]
   exp-prod [=>] 99.5	\[ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot \color{blue}{{\left(e^{1}\right)}^{\left(\frac{-2}{v}\right)}}\right) \]
   exp-1-e [=>] 99.5	\[ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {\color{blue}{e}}^{\left(\frac{-2}{v}\right)}\right) \]

3. Final simplification 99.5%

   \[\leadsto 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot {e}^{\left(\frac{-2}{v}\right)}\right) \]

Alternatives

Alternative 1
Accuracy	99.5%
Cost	6816

\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]

Alternative 2
Accuracy	90.3%
Cost	4676

\[\begin{array}{l} t_0 := \left(1 - u\right) \cdot -2\\ \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{1 + {t_0}^{3}}{t_0 \cdot t_0 + \left(1 + -2 \cdot \left(u + -1\right)\right)} + \frac{0.5}{v} \cdot \left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	90.3%
Cost	3748

\[\begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{u \cdot u}{v}, 2 \cdot \left(u + \frac{u}{v}\right)\right) + -1\\ \end{array} \]

Alternative 4
Accuracy	90.1%
Cost	740

\[\begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 - v \cdot \left(2 \cdot \frac{1}{v} + u \cdot \left(\frac{-2}{v \cdot v} + \frac{-2}{v}\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	90.3%
Cost	740

\[\begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{v} \cdot \left(\left(1 - u\right) \cdot \left(\left(1 - u\right) \cdot -4 + 4\right)\right) + \left(-1 + u \cdot 2\right)\\ \end{array} \]

Alternative 6
Accuracy	90.1%
Cost	356

\[\begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(u + \frac{u}{v}\right) + -1\\ \end{array} \]

Alternative 7
Accuracy	89.5%
Cost	228

\[\begin{array}{l} \mathbf{if}\;v \leq 0.20000000298023224:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + u \cdot 2\\ \end{array} \]

Alternative 8
Accuracy	5.9%
Cost	32

\[-1 \]

Alternative 9
Accuracy	86.3%
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023130 
(FPCore (u v)
  :name "HairBSDF, sample_f, cosTheta"
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0)) (and (<= 0.0 v) (<= v 109.746574)))
  (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))