Disney BSSRDF, sample scattering profile, upper

Average Accuracy: 95.9% → 98.2%

Time: 15.2s

Precision: binary32

Cost: 10304

\[\left(0 \leq s \land s \leq 256\right) \land \left(0.25 \leq u \land u \leq 1\right)\]

Math

\[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

↓

\[s \cdot \left(\left(\mathsf{log1p}\left(2.3703703703703702 \cdot {\left(0.25 - u\right)}^{3}\right) - \mathsf{log1p}\left(\left(0.25 - u\right) \cdot \left(\left(0.25 - u\right) \cdot 1.7777777777777777 - 1.3333333333333333\right)\right)\right) \cdot -3\right) \]

FPCore

(FPCore (s u)
 :precision binary32
 (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))

↓

(FPCore (s u)
 :precision binary32
 (*
  s
  (*
   (-
    (log1p (* 2.3703703703703702 (pow (- 0.25 u) 3.0)))
    (log1p
     (* (- 0.25 u) (- (* (- 0.25 u) 1.7777777777777777) 1.3333333333333333))))
   -3.0)))

C

float code(float s, float u) {
	return (3.0f * s) * logf((1.0f / (1.0f - ((u - 0.25f) / 0.75f))));
}

↓

float code(float s, float u) {
	return s * ((log1pf((2.3703703703703702f * powf((0.25f - u), 3.0f))) - log1pf(((0.25f - u) * (((0.25f - u) * 1.7777777777777777f) - 1.3333333333333333f)))) * -3.0f);
}

Julia

function code(s, u)
	return Float32(Float32(Float32(3.0) * s) * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(u - Float32(0.25)) / Float32(0.75))))))
end

↓

function code(s, u)
	return Float32(s * Float32(Float32(log1p(Float32(Float32(2.3703703703703702) * (Float32(Float32(0.25) - u) ^ Float32(3.0)))) - log1p(Float32(Float32(Float32(0.25) - u) * Float32(Float32(Float32(Float32(0.25) - u) * Float32(1.7777777777777777)) - Float32(1.3333333333333333))))) * Float32(-3.0)))
end

Derivation

1. Initial program 95.8%

   \[\left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]

2. Simplified 98.3%

   \[\leadsto \color{blue}{s \cdot \left(\mathsf{log1p}\left(\frac{0.25 - u}{0.75}\right) \cdot -3\right)} \]
   Step-by-step derivation

   [Start] 95.8	\[ \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
   associate-*l* [=>] 95.7	\[ \color{blue}{3 \cdot \left(s \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right)\right)} \]
   *-commutative [=>] 95.7	\[ \color{blue}{\left(s \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right)\right) \cdot 3} \]
   associate-*l* [=>] 95.8	\[ \color{blue}{s \cdot \left(\log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \cdot 3\right)} \]
   log-rec [=>] 96.6	\[ s \cdot \left(\color{blue}{\left(-\log \left(1 - \frac{u - 0.25}{0.75}\right)\right)} \cdot 3\right) \]
   distribute-lft-neg-out [=>] 96.6	\[ s \cdot \color{blue}{\left(-\log \left(1 - \frac{u - 0.25}{0.75}\right) \cdot 3\right)} \]
   distribute-rgt-neg-in [=>] 96.6	\[ s \cdot \color{blue}{\left(\log \left(1 - \frac{u - 0.25}{0.75}\right) \cdot \left(-3\right)\right)} \]
   sub-neg [=>] 96.6	\[ s \cdot \left(\log \color{blue}{\left(1 + \left(-\frac{u - 0.25}{0.75}\right)\right)} \cdot \left(-3\right)\right) \]
   log1p-def [=>] 98.3	\[ s \cdot \left(\color{blue}{\mathsf{log1p}\left(-\frac{u - 0.25}{0.75}\right)} \cdot \left(-3\right)\right) \]
   distribute-neg-frac [=>] 98.3	\[ s \cdot \left(\mathsf{log1p}\left(\color{blue}{\frac{-\left(u - 0.25\right)}{0.75}}\right) \cdot \left(-3\right)\right) \]
   sub-neg [=>] 98.3	\[ s \cdot \left(\mathsf{log1p}\left(\frac{-\color{blue}{\left(u + \left(-0.25\right)\right)}}{0.75}\right) \cdot \left(-3\right)\right) \]
   +-commutative [=>] 98.3	\[ s \cdot \left(\mathsf{log1p}\left(\frac{-\color{blue}{\left(\left(-0.25\right) + u\right)}}{0.75}\right) \cdot \left(-3\right)\right) \]
   distribute-neg-in [=>] 98.3	\[ s \cdot \left(\mathsf{log1p}\left(\frac{\color{blue}{\left(-\left(-0.25\right)\right) + \left(-u\right)}}{0.75}\right) \cdot \left(-3\right)\right) \]
   metadata-eval [=>] 98.3	\[ s \cdot \left(\mathsf{log1p}\left(\frac{\left(-\color{blue}{-0.25}\right) + \left(-u\right)}{0.75}\right) \cdot \left(-3\right)\right) \]
   metadata-eval [=>] 98.3	\[ s \cdot \left(\mathsf{log1p}\left(\frac{\color{blue}{0.25} + \left(-u\right)}{0.75}\right) \cdot \left(-3\right)\right) \]
   unsub-neg [=>] 98.3	\[ s \cdot \left(\mathsf{log1p}\left(\frac{\color{blue}{0.25 - u}}{0.75}\right) \cdot \left(-3\right)\right) \]
   metadata-eval [=>] 98.3	\[ s \cdot \left(\mathsf{log1p}\left(\frac{0.25 - u}{0.75}\right) \cdot \color{blue}{-3}\right) \]

3. Applied egg-rr 95.6%

   \[\leadsto s \cdot \left(\color{blue}{\left(\mathsf{log1p}\left({\left(0.3333333333333333 - u \cdot 1.3333333333333333\right)}^{3}\right) - \log \left(1 + \left({\left(0.25 - u\right)}^{2} \cdot 1.7777777777777777 - \left(0.3333333333333333 - u \cdot 1.3333333333333333\right)\right)\right)\right)} \cdot -3\right) \]
   Step-by-step derivation

   [Start] 98.3	\[ s \cdot \left(\mathsf{log1p}\left(\frac{0.25 - u}{0.75}\right) \cdot -3\right) \]
   log1p-udef [=>] 96.6	\[ s \cdot \left(\color{blue}{\log \left(1 + \frac{0.25 - u}{0.75}\right)} \cdot -3\right) \]
   flip3-+ [=>] 95.8	\[ s \cdot \left(\log \color{blue}{\left(\frac{{1}^{3} + {\left(\frac{0.25 - u}{0.75}\right)}^{3}}{1 \cdot 1 + \left(\frac{0.25 - u}{0.75} \cdot \frac{0.25 - u}{0.75} - 1 \cdot \frac{0.25 - u}{0.75}\right)}\right)} \cdot -3\right) \]
   log-div [=>] 95.9	\[ s \cdot \left(\color{blue}{\left(\log \left({1}^{3} + {\left(\frac{0.25 - u}{0.75}\right)}^{3}\right) - \log \left(1 \cdot 1 + \left(\frac{0.25 - u}{0.75} \cdot \frac{0.25 - u}{0.75} - 1 \cdot \frac{0.25 - u}{0.75}\right)\right)\right)} \cdot -3\right) \]
   metadata-eval [=>] 95.9	\[ s \cdot \left(\left(\log \left(\color{blue}{1} + {\left(\frac{0.25 - u}{0.75}\right)}^{3}\right) - \log \left(1 \cdot 1 + \left(\frac{0.25 - u}{0.75} \cdot \frac{0.25 - u}{0.75} - 1 \cdot \frac{0.25 - u}{0.75}\right)\right)\right) \cdot -3\right) \]
   pow3 [<=] 95.9	\[ s \cdot \left(\left(\log \left(1 + \color{blue}{\left(\frac{0.25 - u}{0.75} \cdot \frac{0.25 - u}{0.75}\right) \cdot \frac{0.25 - u}{0.75}}\right) - \log \left(1 \cdot 1 + \left(\frac{0.25 - u}{0.75} \cdot \frac{0.25 - u}{0.75} - 1 \cdot \frac{0.25 - u}{0.75}\right)\right)\right) \cdot -3\right) \]
   log1p-udef [<=] 95.9	\[ s \cdot \left(\left(\color{blue}{\mathsf{log1p}\left(\left(\frac{0.25 - u}{0.75} \cdot \frac{0.25 - u}{0.75}\right) \cdot \frac{0.25 - u}{0.75}\right)} - \log \left(1 \cdot 1 + \left(\frac{0.25 - u}{0.75} \cdot \frac{0.25 - u}{0.75} - 1 \cdot \frac{0.25 - u}{0.75}\right)\right)\right) \cdot -3\right) \]
   pow3 [=>] 95.9	\[ s \cdot \left(\left(\mathsf{log1p}\left(\color{blue}{{\left(\frac{0.25 - u}{0.75}\right)}^{3}}\right) - \log \left(1 \cdot 1 + \left(\frac{0.25 - u}{0.75} \cdot \frac{0.25 - u}{0.75} - 1 \cdot \frac{0.25 - u}{0.75}\right)\right)\right) \cdot -3\right) \]
   div-sub [=>] 95.7	\[ s \cdot \left(\left(\mathsf{log1p}\left({\color{blue}{\left(\frac{0.25}{0.75} - \frac{u}{0.75}\right)}}^{3}\right) - \log \left(1 \cdot 1 + \left(\frac{0.25 - u}{0.75} \cdot \frac{0.25 - u}{0.75} - 1 \cdot \frac{0.25 - u}{0.75}\right)\right)\right) \cdot -3\right) \]
   metadata-eval [=>] 95.7	\[ s \cdot \left(\left(\mathsf{log1p}\left({\left(\color{blue}{0.3333333333333333} - \frac{u}{0.75}\right)}^{3}\right) - \log \left(1 \cdot 1 + \left(\frac{0.25 - u}{0.75} \cdot \frac{0.25 - u}{0.75} - 1 \cdot \frac{0.25 - u}{0.75}\right)\right)\right) \cdot -3\right) \]
   div-inv [=>] 95.9	\[ s \cdot \left(\left(\mathsf{log1p}\left({\left(0.3333333333333333 - \color{blue}{u \cdot \frac{1}{0.75}}\right)}^{3}\right) - \log \left(1 \cdot 1 + \left(\frac{0.25 - u}{0.75} \cdot \frac{0.25 - u}{0.75} - 1 \cdot \frac{0.25 - u}{0.75}\right)\right)\right) \cdot -3\right) \]
   metadata-eval [=>] 95.9	\[ s \cdot \left(\left(\mathsf{log1p}\left({\left(0.3333333333333333 - u \cdot \color{blue}{1.3333333333333333}\right)}^{3}\right) - \log \left(1 \cdot 1 + \left(\frac{0.25 - u}{0.75} \cdot \frac{0.25 - u}{0.75} - 1 \cdot \frac{0.25 - u}{0.75}\right)\right)\right) \cdot -3\right) \]
   metadata-eval [=>] 95.9	\[ s \cdot \left(\left(\mathsf{log1p}\left({\left(0.3333333333333333 - u \cdot 1.3333333333333333\right)}^{3}\right) - \log \left(\color{blue}{1} + \left(\frac{0.25 - u}{0.75} \cdot \frac{0.25 - u}{0.75} - 1 \cdot \frac{0.25 - u}{0.75}\right)\right)\right) \cdot -3\right) \]

4. Simplified 98.3%

   \[\leadsto s \cdot \left(\color{blue}{\left(\mathsf{log1p}\left(2.3703703703703702 \cdot {\left(0.25 - u\right)}^{3}\right) - \mathsf{log1p}\left(\left(0.25 - u\right) \cdot \left(\left(0.25 - u\right) \cdot 1.7777777777777777 - 1.3333333333333333\right)\right)\right)} \cdot -3\right) \]
   Step-by-step derivation

   [Start] 95.6	\[ s \cdot \left(\left(\mathsf{log1p}\left({\left(0.3333333333333333 - u \cdot 1.3333333333333333\right)}^{3}\right) - \log \left(1 + \left({\left(0.25 - u\right)}^{2} \cdot 1.7777777777777777 - \left(0.3333333333333333 - u \cdot 1.3333333333333333\right)\right)\right)\right) \cdot -3\right) \]
   cancel-sign-sub-inv [=>] 95.6	\[ s \cdot \left(\left(\mathsf{log1p}\left({\color{blue}{\left(0.3333333333333333 + \left(-u\right) \cdot 1.3333333333333333\right)}}^{3}\right) - \log \left(1 + \left({\left(0.25 - u\right)}^{2} \cdot 1.7777777777777777 - \left(0.3333333333333333 - u \cdot 1.3333333333333333\right)\right)\right)\right) \cdot -3\right) \]
   metadata-eval [<=] 95.6	\[ s \cdot \left(\left(\mathsf{log1p}\left({\left(\color{blue}{0.25 \cdot 1.3333333333333333} + \left(-u\right) \cdot 1.3333333333333333\right)}^{3}\right) - \log \left(1 + \left({\left(0.25 - u\right)}^{2} \cdot 1.7777777777777777 - \left(0.3333333333333333 - u \cdot 1.3333333333333333\right)\right)\right)\right) \cdot -3\right) \]
   mul-1-neg [<=] 95.6	\[ s \cdot \left(\left(\mathsf{log1p}\left({\left(0.25 \cdot 1.3333333333333333 + \color{blue}{\left(-1 \cdot u\right)} \cdot 1.3333333333333333\right)}^{3}\right) - \log \left(1 + \left({\left(0.25 - u\right)}^{2} \cdot 1.7777777777777777 - \left(0.3333333333333333 - u \cdot 1.3333333333333333\right)\right)\right)\right) \cdot -3\right) \]
   distribute-rgt-in [<=] 95.8	\[ s \cdot \left(\left(\mathsf{log1p}\left({\color{blue}{\left(1.3333333333333333 \cdot \left(0.25 + -1 \cdot u\right)\right)}}^{3}\right) - \log \left(1 + \left({\left(0.25 - u\right)}^{2} \cdot 1.7777777777777777 - \left(0.3333333333333333 - u \cdot 1.3333333333333333\right)\right)\right)\right) \cdot -3\right) \]
   mul-1-neg [=>] 95.8	\[ s \cdot \left(\left(\mathsf{log1p}\left({\left(1.3333333333333333 \cdot \left(0.25 + \color{blue}{\left(-u\right)}\right)\right)}^{3}\right) - \log \left(1 + \left({\left(0.25 - u\right)}^{2} \cdot 1.7777777777777777 - \left(0.3333333333333333 - u \cdot 1.3333333333333333\right)\right)\right)\right) \cdot -3\right) \]
   sub-neg [<=] 95.8	\[ s \cdot \left(\left(\mathsf{log1p}\left({\left(1.3333333333333333 \cdot \color{blue}{\left(0.25 - u\right)}\right)}^{3}\right) - \log \left(1 + \left({\left(0.25 - u\right)}^{2} \cdot 1.7777777777777777 - \left(0.3333333333333333 - u \cdot 1.3333333333333333\right)\right)\right)\right) \cdot -3\right) \]
   cube-prod [=>] 95.7	\[ s \cdot \left(\left(\mathsf{log1p}\left(\color{blue}{{1.3333333333333333}^{3} \cdot {\left(0.25 - u\right)}^{3}}\right) - \log \left(1 + \left({\left(0.25 - u\right)}^{2} \cdot 1.7777777777777777 - \left(0.3333333333333333 - u \cdot 1.3333333333333333\right)\right)\right)\right) \cdot -3\right) \]
   metadata-eval [=>] 96.0	\[ s \cdot \left(\left(\mathsf{log1p}\left(\color{blue}{2.3703703703703702} \cdot {\left(0.25 - u\right)}^{3}\right) - \log \left(1 + \left({\left(0.25 - u\right)}^{2} \cdot 1.7777777777777777 - \left(0.3333333333333333 - u \cdot 1.3333333333333333\right)\right)\right)\right) \cdot -3\right) \]
   log1p-def [=>] 97.5	\[ s \cdot \left(\left(\mathsf{log1p}\left(2.3703703703703702 \cdot {\left(0.25 - u\right)}^{3}\right) - \color{blue}{\mathsf{log1p}\left({\left(0.25 - u\right)}^{2} \cdot 1.7777777777777777 - \left(0.3333333333333333 - u \cdot 1.3333333333333333\right)\right)}\right) \cdot -3\right) \]
   unpow2 [=>] 97.5	\[ s \cdot \left(\left(\mathsf{log1p}\left(2.3703703703703702 \cdot {\left(0.25 - u\right)}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\left(\left(0.25 - u\right) \cdot \left(0.25 - u\right)\right)} \cdot 1.7777777777777777 - \left(0.3333333333333333 - u \cdot 1.3333333333333333\right)\right)\right) \cdot -3\right) \]
   associate-*l* [=>] 97.5	\[ s \cdot \left(\left(\mathsf{log1p}\left(2.3703703703703702 \cdot {\left(0.25 - u\right)}^{3}\right) - \mathsf{log1p}\left(\color{blue}{\left(0.25 - u\right) \cdot \left(\left(0.25 - u\right) \cdot 1.7777777777777777\right)} - \left(0.3333333333333333 - u \cdot 1.3333333333333333\right)\right)\right) \cdot -3\right) \]
   cancel-sign-sub-inv [=>] 97.5	\[ s \cdot \left(\left(\mathsf{log1p}\left(2.3703703703703702 \cdot {\left(0.25 - u\right)}^{3}\right) - \mathsf{log1p}\left(\left(0.25 - u\right) \cdot \left(\left(0.25 - u\right) \cdot 1.7777777777777777\right) - \color{blue}{\left(0.3333333333333333 + \left(-u\right) \cdot 1.3333333333333333\right)}\right)\right) \cdot -3\right) \]
   metadata-eval [<=] 97.5	\[ s \cdot \left(\left(\mathsf{log1p}\left(2.3703703703703702 \cdot {\left(0.25 - u\right)}^{3}\right) - \mathsf{log1p}\left(\left(0.25 - u\right) \cdot \left(\left(0.25 - u\right) \cdot 1.7777777777777777\right) - \left(\color{blue}{0.25 \cdot 1.3333333333333333} + \left(-u\right) \cdot 1.3333333333333333\right)\right)\right) \cdot -3\right) \]
   mul-1-neg [<=] 97.5	\[ s \cdot \left(\left(\mathsf{log1p}\left(2.3703703703703702 \cdot {\left(0.25 - u\right)}^{3}\right) - \mathsf{log1p}\left(\left(0.25 - u\right) \cdot \left(\left(0.25 - u\right) \cdot 1.7777777777777777\right) - \left(0.25 \cdot 1.3333333333333333 + \color{blue}{\left(-1 \cdot u\right)} \cdot 1.3333333333333333\right)\right)\right) \cdot -3\right) \]
   distribute-rgt-in [<=] 98.3	\[ s \cdot \left(\left(\mathsf{log1p}\left(2.3703703703703702 \cdot {\left(0.25 - u\right)}^{3}\right) - \mathsf{log1p}\left(\left(0.25 - u\right) \cdot \left(\left(0.25 - u\right) \cdot 1.7777777777777777\right) - \color{blue}{1.3333333333333333 \cdot \left(0.25 + -1 \cdot u\right)}\right)\right) \cdot -3\right) \]
   mul-1-neg [=>] 98.3	\[ s \cdot \left(\left(\mathsf{log1p}\left(2.3703703703703702 \cdot {\left(0.25 - u\right)}^{3}\right) - \mathsf{log1p}\left(\left(0.25 - u\right) \cdot \left(\left(0.25 - u\right) \cdot 1.7777777777777777\right) - 1.3333333333333333 \cdot \left(0.25 + \color{blue}{\left(-u\right)}\right)\right)\right) \cdot -3\right) \]
   sub-neg [<=] 98.3	\[ s \cdot \left(\left(\mathsf{log1p}\left(2.3703703703703702 \cdot {\left(0.25 - u\right)}^{3}\right) - \mathsf{log1p}\left(\left(0.25 - u\right) \cdot \left(\left(0.25 - u\right) \cdot 1.7777777777777777\right) - 1.3333333333333333 \cdot \color{blue}{\left(0.25 - u\right)}\right)\right) \cdot -3\right) \]
   *-commutative [=>] 98.3	\[ s \cdot \left(\left(\mathsf{log1p}\left(2.3703703703703702 \cdot {\left(0.25 - u\right)}^{3}\right) - \mathsf{log1p}\left(\left(0.25 - u\right) \cdot \left(\left(0.25 - u\right) \cdot 1.7777777777777777\right) - \color{blue}{\left(0.25 - u\right) \cdot 1.3333333333333333}\right)\right) \cdot -3\right) \]

5. Final simplification 98.3%

   \[\leadsto s \cdot \left(\left(\mathsf{log1p}\left(2.3703703703703702 \cdot {\left(0.25 - u\right)}^{3}\right) - \mathsf{log1p}\left(\left(0.25 - u\right) \cdot \left(\left(0.25 - u\right) \cdot 1.7777777777777777 - 1.3333333333333333\right)\right)\right) \cdot -3\right) \]

Alternatives

Alternative 1
Accuracy	98.3%
Cost	3552

\[\left(s \cdot 3\right) \cdot \left(-\mathsf{log1p}\left(\frac{\left(-u\right) - -0.25}{0.75}\right)\right) \]

Alternative 2
Accuracy	96.1%
Cost	3488

\[-3 \cdot \left(s \cdot \log \left(1.3333333333333333 + u \cdot -1.3333333333333333\right)\right) \]

Alternative 3
Accuracy	97.9%
Cost	3488

\[-3 \cdot \left(s \cdot \mathsf{log1p}\left(\left(0.25 - u\right) \cdot 1.3333333333333333\right)\right) \]

Alternative 4
Accuracy	97.9%
Cost	3488

\[s \cdot \left(-3 \cdot \mathsf{log1p}\left(\left(0.25 - u\right) \cdot 1.3333333333333333\right)\right) \]

Alternative 5
Accuracy	98.3%
Cost	3488

\[s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{0.25 - u}{0.75}\right)\right) \]

Alternative 6
Accuracy	18.9%
Cost	3424

\[-3 \cdot \left(s \cdot \mathsf{log1p}\left(u \cdot -1.3333333333333333\right)\right) \]

Alternative 7
Accuracy	25.7%
Cost	3424

\[3 \cdot \left(s \cdot \left(u + \log 0.75\right)\right) \]

Alternative 8
Accuracy	25.7%
Cost	3424

\[s \cdot \left(u \cdot 3 + \log 0.421875\right) \]

Alternative 9
Accuracy	7.4%
Cost	3296

\[s \cdot \log 0.421875 \]

Reproduce

herbie shell --seed 2023159 
(FPCore (s u)
  :name "Disney BSSRDF, sample scattering profile, upper"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 0.25 u) (<= u 1.0)))
  (* (* 3.0 s) (log (/ 1.0 (- 1.0 (/ (- u 0.25) 0.75))))))