Disney BSSRDF, sample scattering profile, upper

Average Error: 4.14% → 1.69%

Time: 13.2s

Precision: binary32

Cost: 3488

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

Math

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

↓

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

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 (* -3.0 (log1p (/ (- 0.25 u) 0.75)))))

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 * (-3.0f * log1pf(((0.25f - u) / 0.75f)));
}

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(-3.0) * log1p(Float32(Float32(Float32(0.25) - u) / Float32(0.75)))))
end

Derivation

1. Initial program 4.14

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

2. Simplified 1.69

   \[\leadsto \color{blue}{s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{0.25 + \left(-u\right)}{0.75}\right)\right)} \]
   Proof

   [Start] 4.14	\[ \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
   *-commutative [=>] 4.14	\[ \color{blue}{\left(s \cdot 3\right)} \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
   associate-*l* [=>] 4.13	\[ \color{blue}{s \cdot \left(3 \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right)\right)} \]
   log-rec [=>] 3.32	\[ s \cdot \left(3 \cdot \color{blue}{\left(-\log \left(1 - \frac{u - 0.25}{0.75}\right)\right)}\right) \]
   neg-mul-1 [=>] 3.32	\[ s \cdot \left(3 \cdot \color{blue}{\left(-1 \cdot \log \left(1 - \frac{u - 0.25}{0.75}\right)\right)}\right) \]
   associate-*r* [=>] 3.32	\[ s \cdot \color{blue}{\left(\left(3 \cdot -1\right) \cdot \log \left(1 - \frac{u - 0.25}{0.75}\right)\right)} \]
   metadata-eval [=>] 3.32	\[ s \cdot \left(\color{blue}{-3} \cdot \log \left(1 - \frac{u - 0.25}{0.75}\right)\right) \]
   sub-neg [=>] 3.32	\[ s \cdot \left(-3 \cdot \log \color{blue}{\left(1 + \left(-\frac{u - 0.25}{0.75}\right)\right)}\right) \]
   log1p-def [=>] 1.69	\[ s \cdot \left(-3 \cdot \color{blue}{\mathsf{log1p}\left(-\frac{u - 0.25}{0.75}\right)}\right) \]
   distribute-neg-frac [=>] 1.69	\[ s \cdot \left(-3 \cdot \mathsf{log1p}\left(\color{blue}{\frac{-\left(u - 0.25\right)}{0.75}}\right)\right) \]
   neg-sub0 [=>] 1.69	\[ s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{\color{blue}{0 - \left(u - 0.25\right)}}{0.75}\right)\right) \]
   associate--r- [=>] 1.69	\[ s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{\color{blue}{\left(0 - u\right) + 0.25}}{0.75}\right)\right) \]
   neg-sub0 [<=] 1.69	\[ s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{\color{blue}{\left(-u\right)} + 0.25}{0.75}\right)\right) \]
   +-commutative [<=] 1.69	\[ s \cdot \left(-3 \cdot \mathsf{log1p}\left(\frac{\color{blue}{0.25 + \left(-u\right)}}{0.75}\right)\right) \]

3. Final simplification 1.69

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

Alternatives

Alternative 1
Error	3.86%
Cost	3488

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

Alternative 2
Error	2.12%
Cost	3488

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

Alternative 3
Error	2.09%
Cost	3488

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

Alternative 4
Error	70.07%
Cost	160

\[3 \cdot \left(s \cdot u\right) \]

Alternative 5
Error	70.07%
Cost	160

\[u \cdot \left(s \cdot 3\right) \]

Reproduce

herbie shell --seed 2023102 
(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))))))