Disney BSSRDF, sample scattering profile, lower

Average Accuracy: 61.7% → 99.3%

Time: 22.9s

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

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

↓

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

FPCore

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

↓

(FPCore (s u)
 :precision binary32
 (fma (log1p (* 4.0 u)) 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((4.0f * u)), 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(Float32(4.0) * u)), s, Float32(s * Float32(-log1p(Float32(u * Float32(u * Float32(-16.0)))))))
end

Derivation

1. Initial program 61.7%

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

2. Applied egg-rr 87.7%

   \[\leadsto s \cdot \color{blue}{\left(\log \left(\frac{1}{1 - {\left(4 \cdot u\right)}^{2}}\right) + \mathsf{log1p}\left(4 \cdot u\right)\right)} \]
   Proof

   [Start] 61.7	\[ s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   flip-- [=>] 61.1	\[ s \cdot \log \left(\frac{1}{\color{blue}{\frac{1 \cdot 1 - \left(4 \cdot u\right) \cdot \left(4 \cdot u\right)}{1 + 4 \cdot u}}}\right) \]
   associate-/r/ [=>] 60.8	\[ s \cdot \log \color{blue}{\left(\frac{1}{1 \cdot 1 - \left(4 \cdot u\right) \cdot \left(4 \cdot u\right)} \cdot \left(1 + 4 \cdot u\right)\right)} \]
   log-prod [=>] 60.9	\[ s \cdot \color{blue}{\left(\log \left(\frac{1}{1 \cdot 1 - \left(4 \cdot u\right) \cdot \left(4 \cdot u\right)}\right) + \log \left(1 + 4 \cdot u\right)\right)} \]
   metadata-eval [=>] 60.9	\[ s \cdot \left(\log \left(\frac{1}{\color{blue}{1} - \left(4 \cdot u\right) \cdot \left(4 \cdot u\right)}\right) + \log \left(1 + 4 \cdot u\right)\right) \]
   pow2 [=>] 60.9	\[ s \cdot \left(\log \left(\frac{1}{1 - \color{blue}{{\left(4 \cdot u\right)}^{2}}}\right) + \log \left(1 + 4 \cdot u\right)\right) \]
   log1p-udef [<=] 87.7	\[ s \cdot \left(\log \left(\frac{1}{1 - {\left(4 \cdot u\right)}^{2}}\right) + \color{blue}{\mathsf{log1p}\left(4 \cdot u\right)}\right) \]

3. Simplified 99.1%

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

   [Start] 87.7	\[ s \cdot \left(\log \left(\frac{1}{1 - {\left(4 \cdot u\right)}^{2}}\right) + \mathsf{log1p}\left(4 \cdot u\right)\right) \]
   +-commutative [=>] 87.7	\[ s \cdot \color{blue}{\left(\mathsf{log1p}\left(4 \cdot u\right) + \log \left(\frac{1}{1 - {\left(4 \cdot u\right)}^{2}}\right)\right)} \]
   log-rec [=>] 88.9	\[ s \cdot \left(\mathsf{log1p}\left(4 \cdot u\right) + \color{blue}{\left(-\log \left(1 - {\left(4 \cdot u\right)}^{2}\right)\right)}\right) \]
   unsub-neg [=>] 88.9	\[ s \cdot \color{blue}{\left(\mathsf{log1p}\left(4 \cdot u\right) - \log \left(1 - {\left(4 \cdot u\right)}^{2}\right)\right)} \]
   sub-neg [=>] 88.9	\[ s \cdot \left(\mathsf{log1p}\left(4 \cdot u\right) - \log \color{blue}{\left(1 + \left(-{\left(4 \cdot u\right)}^{2}\right)\right)}\right) \]
   log1p-def [=>] 99.1	\[ s \cdot \left(\mathsf{log1p}\left(4 \cdot u\right) - \color{blue}{\mathsf{log1p}\left(-{\left(4 \cdot u\right)}^{2}\right)}\right) \]
   unpow2 [=>] 99.1	\[ s \cdot \left(\mathsf{log1p}\left(4 \cdot u\right) - \mathsf{log1p}\left(-\color{blue}{\left(4 \cdot u\right) \cdot \left(4 \cdot u\right)}\right)\right) \]
   swap-sqr [=>] 99.1	\[ s \cdot \left(\mathsf{log1p}\left(4 \cdot u\right) - \mathsf{log1p}\left(-\color{blue}{\left(4 \cdot 4\right) \cdot \left(u \cdot u\right)}\right)\right) \]
   metadata-eval [=>] 99.1	\[ s \cdot \left(\mathsf{log1p}\left(4 \cdot u\right) - \mathsf{log1p}\left(-\color{blue}{16} \cdot \left(u \cdot u\right)\right)\right) \]
   unpow2 [<=] 99.1	\[ s \cdot \left(\mathsf{log1p}\left(4 \cdot u\right) - \mathsf{log1p}\left(-16 \cdot \color{blue}{{u}^{2}}\right)\right) \]
   *-commutative [=>] 99.1	\[ s \cdot \left(\mathsf{log1p}\left(4 \cdot u\right) - \mathsf{log1p}\left(-\color{blue}{{u}^{2} \cdot 16}\right)\right) \]
   distribute-rgt-neg-in [=>] 99.1	\[ s \cdot \left(\mathsf{log1p}\left(4 \cdot u\right) - \mathsf{log1p}\left(\color{blue}{{u}^{2} \cdot \left(-16\right)}\right)\right) \]
   unpow2 [=>] 99.1	\[ s \cdot \left(\mathsf{log1p}\left(4 \cdot u\right) - \mathsf{log1p}\left(\color{blue}{\left(u \cdot u\right)} \cdot \left(-16\right)\right)\right) \]
   metadata-eval [=>] 99.1	\[ s \cdot \left(\mathsf{log1p}\left(4 \cdot u\right) - \mathsf{log1p}\left(\left(u \cdot u\right) \cdot \color{blue}{-16}\right)\right) \]

4. Applied egg-rr 99.3%

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

   [Start] 99.1	\[ s \cdot \left(\mathsf{log1p}\left(4 \cdot u\right) - \mathsf{log1p}\left(\left(u \cdot u\right) \cdot -16\right)\right) \]
   sub-neg [=>] 99.1	\[ s \cdot \color{blue}{\left(\mathsf{log1p}\left(4 \cdot u\right) + \left(-\mathsf{log1p}\left(\left(u \cdot u\right) \cdot -16\right)\right)\right)} \]
   distribute-rgt-in [=>] 99.1	\[ \color{blue}{\mathsf{log1p}\left(4 \cdot u\right) \cdot s + \left(-\mathsf{log1p}\left(\left(u \cdot u\right) \cdot -16\right)\right) \cdot s} \]
   fma-def [=>] 99.3	\[ \color{blue}{\mathsf{fma}\left(\mathsf{log1p}\left(4 \cdot u\right), s, \left(-\mathsf{log1p}\left(\left(u \cdot u\right) \cdot -16\right)\right) \cdot s\right)} \]
   associate-*l* [=>] 99.3	\[ \mathsf{fma}\left(\mathsf{log1p}\left(4 \cdot u\right), s, \left(-\mathsf{log1p}\left(\color{blue}{u \cdot \left(u \cdot -16\right)}\right)\right) \cdot s\right) \]

5. Final simplification 99.3%

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

Alternatives

Alternative 1
Accuracy	99.4%
Cost	3392

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

Alternative 2
Accuracy	91.3%
Cost	480

\[s \cdot \left(4 \cdot u + \left(u \cdot u\right) \cdot \left(u \cdot 21.333333333333332 + 8\right)\right) \]

Alternative 3
Accuracy	91.1%
Cost	448

\[s \cdot \left(\left(-4 + u \cdot \left(u \cdot -21.333333333333332 + -8\right)\right) \cdot \left(-u\right)\right) \]

Alternative 4
Accuracy	86.9%
Cost	352

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

Alternative 5
Accuracy	86.7%
Cost	288

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

Alternative 6
Accuracy	86.7%
Cost	288

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

Alternative 7
Accuracy	73.4%
Cost	160

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

Alternative 8
Accuracy	73.6%
Cost	160

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

Alternative 9
Accuracy	16.5%
Cost	96

\[s \cdot 0 \]

Reproduce

herbie shell --seed 2023146 
(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))))))