Disney BSSRDF, sample scattering profile, upper

Average Error: 1.3 → 0.6

Time: 14.4s

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

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

↓

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

FPCore

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

↓

(FPCore (s u) :precision binary32 (* -3.0 (* s (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 -3.0f * (s * 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(Float32(-3.0) * Float32(s * log1p(Float32(Float32(Float32(0.25) - u) / Float32(0.75)))))
end

Derivation

1. Initial program 1.3

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

2. Simplified 0.6

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

   [Start] 1.3	\[ \left(3 \cdot s\right) \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right) \]
   associate-*l* [=>] 1.3	\[ \color{blue}{3 \cdot \left(s \cdot \log \left(\frac{1}{1 - \frac{u - 0.25}{0.75}}\right)\right)} \]
   log-rec [=>] 1.1	\[ 3 \cdot \left(s \cdot \color{blue}{\left(-\log \left(1 - \frac{u - 0.25}{0.75}\right)\right)}\right) \]
   sub-neg [=>] 1.1	\[ 3 \cdot \left(s \cdot \left(-\log \color{blue}{\left(1 + \left(-\frac{u - 0.25}{0.75}\right)\right)}\right)\right) \]
   log1p-def [=>] 0.6	\[ 3 \cdot \left(s \cdot \left(-\color{blue}{\mathsf{log1p}\left(-\frac{u - 0.25}{0.75}\right)}\right)\right) \]
   distribute-neg-frac [=>] 0.6	\[ 3 \cdot \left(s \cdot \left(-\mathsf{log1p}\left(\color{blue}{\frac{-\left(u - 0.25\right)}{0.75}}\right)\right)\right) \]
   sub-neg [=>] 0.6	\[ 3 \cdot \left(s \cdot \left(-\mathsf{log1p}\left(\frac{-\color{blue}{\left(u + \left(-0.25\right)\right)}}{0.75}\right)\right)\right) \]
   metadata-eval [=>] 0.6	\[ 3 \cdot \left(s \cdot \left(-\mathsf{log1p}\left(\frac{-\left(u + \color{blue}{-0.25}\right)}{0.75}\right)\right)\right) \]

3. Final simplification 0.6

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

Alternatives

Alternative 1
Error	1.2
Cost	3488

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

Alternative 2
Error	0.7
Cost	3488

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

Alternative 3
Error	0.7
Cost	3488

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

Alternative 4
Error	0.7
Cost	3488

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

Alternative 5
Error	0.6
Cost	3488

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

Alternative 6
Error	26.0
Cost	3424

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

Alternative 7
Error	23.8
Cost	3424

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

Alternative 8
Error	23.8
Cost	3424

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

Alternative 9
Error	29.6
Cost	3296

\[s \cdot \log 0.421875 \]

Reproduce

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