Disney BSSRDF, sample scattering profile, lower

Percentage Accurate: 61.2% → 99.3%

Time: 13.5s

Precision: binary32

Cost: 9984

\[\left(0 \leq s \land s \leq 256\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 0.25\right)\]

Math

\[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]

↓

\[\mathsf{fma}\left(\mathsf{log1p}\left(u \cdot 4\right), s, s \cdot \left(-\mathsf{log1p}\left(u \cdot \left(u \cdot -16\right)\right)\right)\right) \]

FPCore

(FPCore (s u) :precision binary32 (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))

↓

(FPCore (s u)
 :precision binary32
 (fma (log1p (* u 4.0)) s (* s (- (log1p (* u (* u -16.0)))))))

C

float code(float s, float u) {
	return s * logf((1.0f / (1.0f - (4.0f * u))));
}

↓

float code(float s, float u) {
	return fmaf(log1pf((u * 4.0f)), s, (s * -log1pf((u * (u * -16.0f)))));
}

Julia

function code(s, u)
	return Float32(s * log(Float32(Float32(1.0) / Float32(Float32(1.0) - Float32(Float32(4.0) * u)))))
end

↓

function code(s, u)
	return fma(log1p(Float32(u * Float32(4.0))), s, Float32(s * Float32(-log1p(Float32(u * Float32(u * Float32(-16.0)))))))
end

Derivation

1. Initial program 60.6%

   \[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]

2. Simplified 99.2%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(u \cdot -4\right) \cdot \left(-s\right)} \]
   Step-by-step derivation

   [Start] 60.6	\[ s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   *-commutative [=>] 60.6	\[ \color{blue}{\log \left(\frac{1}{1 - 4 \cdot u}\right) \cdot s} \]
   log-rec [=>] 63.6	\[ \color{blue}{\left(-\log \left(1 - 4 \cdot u\right)\right)} \cdot s \]
   distribute-lft-neg-out [=>] 63.6	\[ \color{blue}{-\log \left(1 - 4 \cdot u\right) \cdot s} \]
   distribute-rgt-neg-in [=>] 63.6	\[ \color{blue}{\log \left(1 - 4 \cdot u\right) \cdot \left(-s\right)} \]
   sub-neg [=>] 63.6	\[ \log \color{blue}{\left(1 + \left(-4 \cdot u\right)\right)} \cdot \left(-s\right) \]
   log1p-def [=>] 99.2	\[ \color{blue}{\mathsf{log1p}\left(-4 \cdot u\right)} \cdot \left(-s\right) \]
   *-commutative [=>] 99.2	\[ \mathsf{log1p}\left(-\color{blue}{u \cdot 4}\right) \cdot \left(-s\right) \]
   distribute-rgt-neg-in [=>] 99.2	\[ \mathsf{log1p}\left(\color{blue}{u \cdot \left(-4\right)}\right) \cdot \left(-s\right) \]
   metadata-eval [=>] 99.2	\[ \mathsf{log1p}\left(u \cdot \color{blue}{-4}\right) \cdot \left(-s\right) \]

3. Applied egg-rr 97.7%

   \[\leadsto \color{blue}{\left({\left(\sqrt[3]{\mathsf{log1p}\left(u \cdot -4\right)}\right)}^{2} \cdot \sqrt[3]{\mathsf{log1p}\left(u \cdot -4\right)}\right)} \cdot \left(-s\right) \]
   Step-by-step derivation

   [Start] 99.2	\[ \mathsf{log1p}\left(u \cdot -4\right) \cdot \left(-s\right) \]
   add-cube-cbrt [=>] 97.8	\[ \color{blue}{\left(\left(\sqrt[3]{\mathsf{log1p}\left(u \cdot -4\right)} \cdot \sqrt[3]{\mathsf{log1p}\left(u \cdot -4\right)}\right) \cdot \sqrt[3]{\mathsf{log1p}\left(u \cdot -4\right)}\right)} \cdot \left(-s\right) \]
   pow2 [=>] 97.7	\[ \left(\color{blue}{{\left(\sqrt[3]{\mathsf{log1p}\left(u \cdot -4\right)}\right)}^{2}} \cdot \sqrt[3]{\mathsf{log1p}\left(u \cdot -4\right)}\right) \cdot \left(-s\right) \]

4. Applied egg-rr 60.4%

   \[\leadsto \color{blue}{\left(\log \left(1 - 16 \cdot \left(u \cdot u\right)\right) - \log \left(1 - u \cdot -4\right)\right)} \cdot \left(-s\right) \]
   Step-by-step derivation

   [Start] 97.7	\[ \left({\left(\sqrt[3]{\mathsf{log1p}\left(u \cdot -4\right)}\right)}^{2} \cdot \sqrt[3]{\mathsf{log1p}\left(u \cdot -4\right)}\right) \cdot \left(-s\right) \]
   unpow2 [=>] 97.8	\[ \left(\color{blue}{\left(\sqrt[3]{\mathsf{log1p}\left(u \cdot -4\right)} \cdot \sqrt[3]{\mathsf{log1p}\left(u \cdot -4\right)}\right)} \cdot \sqrt[3]{\mathsf{log1p}\left(u \cdot -4\right)}\right) \cdot \left(-s\right) \]
   add-cube-cbrt [<=] 99.2	\[ \color{blue}{\mathsf{log1p}\left(u \cdot -4\right)} \cdot \left(-s\right) \]
   log1p-udef [=>] 63.6	\[ \color{blue}{\log \left(1 + u \cdot -4\right)} \cdot \left(-s\right) \]
   flip-+ [=>] 60.4	\[ \log \color{blue}{\left(\frac{1 \cdot 1 - \left(u \cdot -4\right) \cdot \left(u \cdot -4\right)}{1 - u \cdot -4}\right)} \cdot \left(-s\right) \]
   log-div [=>] 60.4	\[ \color{blue}{\left(\log \left(1 \cdot 1 - \left(u \cdot -4\right) \cdot \left(u \cdot -4\right)\right) - \log \left(1 - u \cdot -4\right)\right)} \cdot \left(-s\right) \]
   metadata-eval [=>] 60.4	\[ \left(\log \left(\color{blue}{1} - \left(u \cdot -4\right) \cdot \left(u \cdot -4\right)\right) - \log \left(1 - u \cdot -4\right)\right) \cdot \left(-s\right) \]
   *-commutative [=>] 60.4	\[ \left(\log \left(1 - \color{blue}{\left(-4 \cdot u\right)} \cdot \left(u \cdot -4\right)\right) - \log \left(1 - u \cdot -4\right)\right) \cdot \left(-s\right) \]
   *-commutative [=>] 60.4	\[ \left(\log \left(1 - \left(-4 \cdot u\right) \cdot \color{blue}{\left(-4 \cdot u\right)}\right) - \log \left(1 - u \cdot -4\right)\right) \cdot \left(-s\right) \]
   swap-sqr [=>] 60.4	\[ \left(\log \left(1 - \color{blue}{\left(-4 \cdot -4\right) \cdot \left(u \cdot u\right)}\right) - \log \left(1 - u \cdot -4\right)\right) \cdot \left(-s\right) \]
   metadata-eval [=>] 60.4	\[ \left(\log \left(1 - \color{blue}{16} \cdot \left(u \cdot u\right)\right) - \log \left(1 - u \cdot -4\right)\right) \cdot \left(-s\right) \]

5. Simplified 99.0%

   \[\leadsto \color{blue}{\left(\mathsf{log1p}\left(\left(u \cdot u\right) \cdot -16\right) - \mathsf{log1p}\left(u \cdot 4\right)\right)} \cdot \left(-s\right) \]
   Step-by-step derivation

   [Start] 60.4	\[ \left(\log \left(1 - 16 \cdot \left(u \cdot u\right)\right) - \log \left(1 - u \cdot -4\right)\right) \cdot \left(-s\right) \]
   sub-neg [=>] 60.4	\[ \left(\log \color{blue}{\left(1 + \left(-16 \cdot \left(u \cdot u\right)\right)\right)} - \log \left(1 - u \cdot -4\right)\right) \cdot \left(-s\right) \]
   log1p-def [=>] 60.9	\[ \left(\color{blue}{\mathsf{log1p}\left(-16 \cdot \left(u \cdot u\right)\right)} - \log \left(1 - u \cdot -4\right)\right) \cdot \left(-s\right) \]
   unpow2 [<=] 60.9	\[ \left(\mathsf{log1p}\left(-16 \cdot \color{blue}{{u}^{2}}\right) - \log \left(1 - u \cdot -4\right)\right) \cdot \left(-s\right) \]
   *-commutative [=>] 60.9	\[ \left(\mathsf{log1p}\left(-\color{blue}{{u}^{2} \cdot 16}\right) - \log \left(1 - u \cdot -4\right)\right) \cdot \left(-s\right) \]
   distribute-rgt-neg-in [=>] 60.9	\[ \left(\mathsf{log1p}\left(\color{blue}{{u}^{2} \cdot \left(-16\right)}\right) - \log \left(1 - u \cdot -4\right)\right) \cdot \left(-s\right) \]
   unpow2 [=>] 60.9	\[ \left(\mathsf{log1p}\left(\color{blue}{\left(u \cdot u\right)} \cdot \left(-16\right)\right) - \log \left(1 - u \cdot -4\right)\right) \cdot \left(-s\right) \]
   metadata-eval [=>] 60.9	\[ \left(\mathsf{log1p}\left(\left(u \cdot u\right) \cdot \color{blue}{-16}\right) - \log \left(1 - u \cdot -4\right)\right) \cdot \left(-s\right) \]
   sub-neg [=>] 60.9	\[ \left(\mathsf{log1p}\left(\left(u \cdot u\right) \cdot -16\right) - \log \color{blue}{\left(1 + \left(-u \cdot -4\right)\right)}\right) \cdot \left(-s\right) \]
   log1p-def [=>] 99.0	\[ \left(\mathsf{log1p}\left(\left(u \cdot u\right) \cdot -16\right) - \color{blue}{\mathsf{log1p}\left(-u \cdot -4\right)}\right) \cdot \left(-s\right) \]
   distribute-rgt-neg-in [=>] 99.0	\[ \left(\mathsf{log1p}\left(\left(u \cdot u\right) \cdot -16\right) - \mathsf{log1p}\left(\color{blue}{u \cdot \left(--4\right)}\right)\right) \cdot \left(-s\right) \]
   metadata-eval [=>] 99.0	\[ \left(\mathsf{log1p}\left(\left(u \cdot u\right) \cdot -16\right) - \mathsf{log1p}\left(u \cdot \color{blue}{4}\right)\right) \cdot \left(-s\right) \]

6. Taylor expanded in s around 0 60.4%

   \[\leadsto \color{blue}{-1 \cdot \left(s \cdot \left(\log \left(1 + -16 \cdot {u}^{2}\right) - \log \left(1 + 4 \cdot u\right)\right)\right)} \]

7. Simplified 99.0%

   \[\leadsto \color{blue}{s \cdot \left(\mathsf{log1p}\left(u \cdot 4\right) - \mathsf{log1p}\left(u \cdot \left(u \cdot -16\right)\right)\right)} \]
   Step-by-step derivation

   [Start] 60.4	\[ -1 \cdot \left(s \cdot \left(\log \left(1 + -16 \cdot {u}^{2}\right) - \log \left(1 + 4 \cdot u\right)\right)\right) \]
   *-commutative [=>] 60.4	\[ \color{blue}{\left(s \cdot \left(\log \left(1 + -16 \cdot {u}^{2}\right) - \log \left(1 + 4 \cdot u\right)\right)\right) \cdot -1} \]
   +-commutative [=>] 60.4	\[ \left(s \cdot \left(\log \color{blue}{\left(-16 \cdot {u}^{2} + 1\right)} - \log \left(1 + 4 \cdot u\right)\right)\right) \cdot -1 \]
   *-commutative [=>] 60.4	\[ \left(s \cdot \left(\log \left(\color{blue}{{u}^{2} \cdot -16} + 1\right) - \log \left(1 + 4 \cdot u\right)\right)\right) \cdot -1 \]
   unpow2 [=>] 60.4	\[ \left(s \cdot \left(\log \left(\color{blue}{\left(u \cdot u\right)} \cdot -16 + 1\right) - \log \left(1 + 4 \cdot u\right)\right)\right) \cdot -1 \]
   associate-*r* [<=] 60.4	\[ \left(s \cdot \left(\log \left(\color{blue}{u \cdot \left(u \cdot -16\right)} + 1\right) - \log \left(1 + 4 \cdot u\right)\right)\right) \cdot -1 \]
   fma-udef [<=] 60.5	\[ \left(s \cdot \left(\log \color{blue}{\left(\mathsf{fma}\left(u, u \cdot -16, 1\right)\right)} - \log \left(1 + 4 \cdot u\right)\right)\right) \cdot -1 \]
   +-commutative [=>] 60.5	\[ \left(s \cdot \left(\log \left(\mathsf{fma}\left(u, u \cdot -16, 1\right)\right) - \log \color{blue}{\left(4 \cdot u + 1\right)}\right)\right) \cdot -1 \]
   *-commutative [=>] 60.5	\[ \left(s \cdot \left(\log \left(\mathsf{fma}\left(u, u \cdot -16, 1\right)\right) - \log \left(\color{blue}{u \cdot 4} + 1\right)\right)\right) \cdot -1 \]
   fma-udef [<=] 60.5	\[ \left(s \cdot \left(\log \left(\mathsf{fma}\left(u, u \cdot -16, 1\right)\right) - \log \color{blue}{\left(\mathsf{fma}\left(u, 4, 1\right)\right)}\right)\right) \cdot -1 \]
   log-div [<=] 60.4	\[ \left(s \cdot \color{blue}{\log \left(\frac{\mathsf{fma}\left(u, u \cdot -16, 1\right)}{\mathsf{fma}\left(u, 4, 1\right)}\right)}\right) \cdot -1 \]
   associate-*l* [=>] 60.4	\[ \color{blue}{s \cdot \left(\log \left(\frac{\mathsf{fma}\left(u, u \cdot -16, 1\right)}{\mathsf{fma}\left(u, 4, 1\right)}\right) \cdot -1\right)} \]
   metadata-eval [<=] 60.4	\[ s \cdot \left(\log \left(\frac{\mathsf{fma}\left(u, u \cdot -16, 1\right)}{\mathsf{fma}\left(u, 4, 1\right)}\right) \cdot \color{blue}{\left(-1\right)}\right) \]
   distribute-rgt-neg-in [<=] 60.4	\[ s \cdot \color{blue}{\left(-\log \left(\frac{\mathsf{fma}\left(u, u \cdot -16, 1\right)}{\mathsf{fma}\left(u, 4, 1\right)}\right) \cdot 1\right)} \]
   *-rgt-identity [=>] 60.4	\[ s \cdot \left(-\color{blue}{\log \left(\frac{\mathsf{fma}\left(u, u \cdot -16, 1\right)}{\mathsf{fma}\left(u, 4, 1\right)}\right)}\right) \]

8. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(u \cdot 4\right), s, \left(-\mathsf{log1p}\left(u \cdot \left(u \cdot -16\right)\right)\right) \cdot s\right)} \]
   Step-by-step derivation

   [Start] 99.0	\[ s \cdot \left(\mathsf{log1p}\left(u \cdot 4\right) - \mathsf{log1p}\left(u \cdot \left(u \cdot -16\right)\right)\right) \]
   sub-neg [=>] 99.0	\[ s \cdot \color{blue}{\left(\mathsf{log1p}\left(u \cdot 4\right) + \left(-\mathsf{log1p}\left(u \cdot \left(u \cdot -16\right)\right)\right)\right)} \]
   distribute-rgt-in [=>] 99.0	\[ \color{blue}{\mathsf{log1p}\left(u \cdot 4\right) \cdot s + \left(-\mathsf{log1p}\left(u \cdot \left(u \cdot -16\right)\right)\right) \cdot s} \]
   fma-def [=>] 99.4	\[ \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(u \cdot 4\right), s, \left(-\mathsf{log1p}\left(u \cdot \left(u \cdot -16\right)\right)\right) \cdot s\right)} \]

9. Final simplification 99.4%

   \[\leadsto \mathsf{fma}\left(\mathsf{log1p}\left(u \cdot 4\right), s, s \cdot \left(-\mathsf{log1p}\left(u \cdot \left(u \cdot -16\right)\right)\right)\right) \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	3392

\[s \cdot \left(-\mathsf{log1p}\left(u \cdot -4\right)\right) \]

Alternative 2
Accuracy	89.1%
Cost	384

\[s \cdot \frac{u \cdot \left(-16\right)}{-4 + u \cdot 8} \]

Alternative 3
Accuracy	88.3%
Cost	352

\[-16 \cdot \frac{u \cdot s}{u \cdot 8 - 4} \]

Alternative 4
Accuracy	87.1%
Cost	288

\[s \cdot \left(u \cdot \left(4 + u \cdot 8\right)\right) \]

Alternative 5
Accuracy	87.1%
Cost	288

\[u \cdot \left(s \cdot \left(4 - u \cdot -8\right)\right) \]

Alternative 6
Accuracy	74.0%
Cost	160

\[4 \cdot \left(u \cdot s\right) \]

Alternative 7
Accuracy	74.3%
Cost	160

\[u \cdot \left(4 \cdot s\right) \]

Alternative 8
Accuracy	8.2%
Cost	96

\[s \cdot -2 \]

Reproduce

herbie shell --seed 2023159 
(FPCore (s u)
  :name "Disney BSSRDF, sample scattering profile, lower"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (<= 2.328306437e-10 u) (<= u 0.25)))
  (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))