HairBSDF, sample_f, cosTheta

Average Accuracy: 99.4% → 99.4%

Time: 18.1s

Precision: binary32

Cost: 13216

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

Math

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

↓

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

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 (* 2.0 (log (sqrt (fma (- 1.0 u) (exp (/ -2.0 v)) u)))))))

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 * (2.0f * logf(sqrtf(fmaf((1.0f - u), expf((-2.0f / v)), u)))));
}

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 * Float32(Float32(2.0) * log(sqrt(fma(Float32(Float32(1.0) - u), exp(Float32(Float32(-2.0) / v)), u))))))
end

Derivation

1. Initial program 99.4%

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

2. Applied egg-rr 99.4%

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

   [Start] 99.4	\[ 1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
   add-sqr-sqrt [=>] 99.3	\[ 1 + v \cdot \log \color{blue}{\left(\sqrt{u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}} \cdot \sqrt{u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)} \]
   log-prod [=>] 99.4	\[ 1 + v \cdot \color{blue}{\left(\log \left(\sqrt{u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right) + \log \left(\sqrt{u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)\right)} \]
   +-commutative [=>] 99.4	\[ 1 + v \cdot \left(\log \left(\sqrt{\color{blue}{\left(1 - u\right) \cdot e^{\frac{-2}{v}} + u}}\right) + \log \left(\sqrt{u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)\right) \]
   fma-def [=>] 99.4	\[ 1 + v \cdot \left(\log \left(\sqrt{\color{blue}{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}}\right) + \log \left(\sqrt{u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}}\right)\right) \]
   +-commutative [=>] 99.4	\[ 1 + v \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right) + \log \left(\sqrt{\color{blue}{\left(1 - u\right) \cdot e^{\frac{-2}{v}} + u}}\right)\right) \]
   fma-def [=>] 99.4	\[ 1 + v \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right) + \log \left(\sqrt{\color{blue}{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}}\right)\right) \]

3. Simplified 99.4%

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

   [Start] 99.4	\[ 1 + v \cdot \left(\log \left(\sqrt{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right) + \log \left(\sqrt{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right)\right) \]
   count-2 [=>] 99.4	\[ 1 + v \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt{\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)}\right)\right)} \]

4. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	9952

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

Alternative 2
Accuracy	99.4%
Cost	6816

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

Alternative 3
Accuracy	95.8%
Cost	6688

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

Alternative 4
Accuracy	94.2%
Cost	3360

\[1 + v \cdot \log u \]

Alternative 5
Accuracy	90.9%
Cost	932

\[\begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot u + \frac{u}{v \cdot v} \cdot \left(1.3333333333333333 + u \cdot -4\right)\right) - \frac{u \cdot \left(-2 + 2 \cdot u\right)}{v}\right) + -1\\ \end{array} \]

Alternative 6
Accuracy	90.4%
Cost	548

\[\begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + v \cdot \left(u \cdot \frac{1 + v}{v \cdot \left(v \cdot 0.5\right)}\right)\\ \end{array} \]

Alternative 7
Accuracy	90.5%
Cost	548

\[\begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(2 \cdot u - \frac{u \cdot \left(-2 + 2 \cdot u\right)}{v}\right)\\ \end{array} \]

Alternative 8
Accuracy	90.4%
Cost	516

\[\begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + v \cdot \left(u \cdot \frac{-2 - \frac{2}{v}}{-v}\right)\\ \end{array} \]

Alternative 9
Accuracy	90.4%
Cost	484

\[\begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + v \cdot \left(\frac{u}{v} \cdot \left(2 + \frac{2}{v}\right)\right)\\ \end{array} \]

Alternative 10
Accuracy	90.4%
Cost	356

\[\begin{array}{l} \mathbf{if}\;v \leq 0.15000000596046448:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + 2 \cdot \left(u + \frac{u}{v}\right)\\ \end{array} \]

Alternative 11
Accuracy	5.9%
Cost	32

\[-1 \]

Alternative 12
Accuracy	86.5%
Cost	32

\[1 \]

Reproduce

herbie shell --seed 2023153 
(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))))))))