Disney BSSRDF, sample scattering profile, lower

?

Percentage Accurate: 61.1% → 99.4%
Time: 12.0s
Precision: binary32
Cost: 3392

?

\[\left(0 \leq s \land s \leq 256\right) \land \left(2.328306437 \cdot 10^{-10} \leq u \land u \leq 0.25\right)\]
\[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right) \]
↓
\[\mathsf{log1p}\left(u \cdot -4\right) \cdot \left(-s\right) \]
(FPCore (s u) :precision binary32 (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))
↓
(FPCore (s u) :precision binary32 (* (log1p (* u -4.0)) (- s)))
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;
}

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

s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)
↓
\mathsf{log1p}\left(u \cdot -4\right) \cdot \left(-s\right)

Local Percentage Accuracy?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 57.0%

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

  2. Simplified99.4%

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

    [Start]57.0

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

    *-commutative [=>]57.0

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

    log-rec [=>]59.7

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

    distribute-lft-neg-out [=>]59.7

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

    distribute-rgt-neg-in [=>]59.7

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

    sub-neg [=>]59.7

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

    log1p-def [=>]99.4

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

    *-commutative [=>]99.4

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

    distribute-rgt-neg-in [=>]99.4

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

    metadata-eval [=>]99.4

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

  3. Final simplification99.4%

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

Alternatives

Alternative 1
Accuracy87.1%
Cost352
\[s \cdot \left(u \cdot \left(u \cdot 8\right) + u \cdot 4\right) \]
Alternative 2
Accuracy87.1%
Cost352
\[u \cdot \left(s \cdot 4 + s \cdot \left(u \cdot 8\right)\right) \]
Alternative 3
Accuracy86.9%
Cost288
\[s \cdot \left(u \cdot \left(4 + u \cdot 8\right)\right) \]
Alternative 4
Accuracy73.9%
Cost160
\[4 \cdot \left(u \cdot s\right) \]
Alternative 5
Accuracy74.1%
Cost160
\[s \cdot \left(u \cdot 4\right) \]

Error

Reproduce?

herbie shell --seed 2023160 
(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))))))