Disney BSSRDF, sample scattering profile, lower

Average Error: 12.4 → 0.2

Time: 14.6s

Precision: binary32

Cost: 3392

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

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 Float32(log1p(Float32(u * Float32(-4.0))) * Float32(-s))
end

Derivation

1. Initial program 12.4

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

2. Simplified 0.2

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

   [Start] 12.4	\[ s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
   log-rec [=>] 11.6	\[ s \cdot \color{blue}{\left(-\log \left(1 - 4 \cdot u\right)\right)} \]
   distribute-rgt-neg-out [=>] 11.6	\[ \color{blue}{-s \cdot \log \left(1 - 4 \cdot u\right)} \]
   distribute-lft-neg-in [=>] 11.6	\[ \color{blue}{\left(-s\right) \cdot \log \left(1 - 4 \cdot u\right)} \]
   *-commutative [<=] 11.6	\[ \color{blue}{\log \left(1 - 4 \cdot u\right) \cdot \left(-s\right)} \]
   sub-neg [=>] 11.6	\[ \log \color{blue}{\left(1 + \left(-4 \cdot u\right)\right)} \cdot \left(-s\right) \]
   log1p-def [=>] 0.2	\[ \color{blue}{\mathsf{log1p}\left(-4 \cdot u\right)} \cdot \left(-s\right) \]
   distribute-lft-neg-in [=>] 0.2	\[ \mathsf{log1p}\left(\color{blue}{\left(-4\right) \cdot u}\right) \cdot \left(-s\right) \]
   *-commutative [=>] 0.2	\[ \mathsf{log1p}\left(\color{blue}{u \cdot \left(-4\right)}\right) \cdot \left(-s\right) \]
   metadata-eval [=>] 0.2	\[ \mathsf{log1p}\left(u \cdot \color{blue}{-4}\right) \cdot \left(-s\right) \]

3. Final simplification 0.2

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

Alternatives

Alternative 1
Error	4.2
Cost	352

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

Alternative 2
Error	3.6
Cost	352

\[\frac{u}{\frac{0.25}{s} + -0.5 \cdot \frac{u}{s}} \]

Alternative 3
Error	3.6
Cost	352

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

Alternative 4
Error	4.2
Cost	288

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

Alternative 5
Error	8.3
Cost	160

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

Alternative 6
Error	8.3
Cost	160

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

Alternative 7
Error	26.6
Cost	32

\[0 \]

Reproduce

herbie shell --seed 2023027 
(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))))))