Average Error: 12.3 → 0.4
Time: 11.0s
Precision: binary32
\[0 \leq s \land s \leq 256 \land 2.328306437 \cdot 10^{-10} \leq u \land u \leq 0.25\]
\[s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;4 \cdot u \leq 0.03059847094118595:\\ \;\;\;\;u \cdot \left(4 \cdot s + s \cdot \left(u \cdot 8\right)\right) + s \cdot \left(21.333333333333332 \cdot {u}^{3} + 64 \cdot {u}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(-\sqrt[3]{{\log \left(u \cdot -4 + 1\right)}^{3}}\right)\\ \end{array}\]
s \cdot \log \left(\frac{1}{1 - 4 \cdot u}\right)
↓
\begin{array}{l}
\mathbf{if}\;4 \cdot u \leq 0.03059847094118595:\\
\;\;\;\;u \cdot \left(4 \cdot s + s \cdot \left(u \cdot 8\right)\right) + s \cdot \left(21.333333333333332 \cdot {u}^{3} + 64 \cdot {u}^{4}\right)\\

\mathbf{else}:\\
\;\;\;\;s \cdot \left(-\sqrt[3]{{\log \left(u \cdot -4 + 1\right)}^{3}}\right)\\

\end{array}
(FPCore (s u) :precision binary32 (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))
↓
(FPCore (s u)
 :precision binary32
 (if (<= (* 4.0 u) 0.03059847094118595)
   (+
    (* u (+ (* 4.0 s) (* s (* u 8.0))))
    (* s (+ (* 21.333333333333332 (pow u 3.0)) (* 64.0 (pow u 4.0)))))
   (* s (- (cbrt (pow (log (+ (* u -4.0) 1.0)) 3.0))))))
float code(float s, float u) {
	return s * logf(1.0f / (1.0f - (4.0f * u)));
}
↓
float code(float s, float u) {
	float tmp;
	if ((4.0f * u) <= 0.03059847094118595f) {
		tmp = (u * ((4.0f * s) + (s * (u * 8.0f)))) + (s * ((21.333333333333332f * powf(u, 3.0f)) + (64.0f * powf(u, 4.0f))));
	} else {
		tmp = s * -cbrtf(powf(logf((u * -4.0f) + 1.0f), 3.0f));
	}
	return tmp;
}

Error

Bits error versus s

Bits error versus u

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (*.f32 4 u) < 0.0305984709

    1. Initial program 14.4

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

    2. Simplified13.5

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

    3. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{21.333333333333332 \cdot \left(s \cdot {u}^{3}\right) + \left(4 \cdot \left(s \cdot u\right) + \left(8 \cdot \left(s \cdot {u}^{2}\right) + 64 \cdot \left(s \cdot {u}^{4}\right)\right)\right)}\]

    4. Simplified0.3

      \[\leadsto \color{blue}{s \cdot \left(u \cdot \left(4 + u \cdot 8\right) + \left(21.333333333333332 \cdot {u}^{3} + 64 \cdot {u}^{4}\right)\right)}\]
    5. Using strategy rm

    6. Applied distribute-rgt-in_binary320.3

      \[\leadsto \color{blue}{\left(u \cdot \left(4 + u \cdot 8\right)\right) \cdot s + \left(21.333333333333332 \cdot {u}^{3} + 64 \cdot {u}^{4}\right) \cdot s}\]

    7. Simplified0.3

      \[\leadsto \color{blue}{u \cdot \left(s \cdot \left(4 + u \cdot 8\right)\right)} + \left(21.333333333333332 \cdot {u}^{3} + 64 \cdot {u}^{4}\right) \cdot s\]
    8. Using strategy rm

    9. Applied distribute-lft-in_binary320.2

      \[\leadsto u \cdot \color{blue}{\left(s \cdot 4 + s \cdot \left(u \cdot 8\right)\right)} + \left(21.333333333333332 \cdot {u}^{3} + 64 \cdot {u}^{4}\right) \cdot s\]

    if 0.0305984709 < (*.f32 4 u)

    1. Initial program 1.4

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

    2. Simplified1.0

      \[\leadsto \color{blue}{s \cdot \left(-\log \left(1 - 4 \cdot u\right)\right)}\]
    3. Using strategy rm

    4. Applied add-cbrt-cube_binary321.0

      \[\leadsto s \cdot \left(-\color{blue}{\sqrt[3]{\left(\log \left(1 - 4 \cdot u\right) \cdot \log \left(1 - 4 \cdot u\right)\right) \cdot \log \left(1 - 4 \cdot u\right)}}\right)\]

    5. Simplified1.0

      \[\leadsto s \cdot \left(-\sqrt[3]{\color{blue}{{\log \left(-4 \cdot u + 1\right)}^{3}}}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;4 \cdot u \leq 0.03059847094118595:\\ \;\;\;\;u \cdot \left(4 \cdot s + s \cdot \left(u \cdot 8\right)\right) + s \cdot \left(21.333333333333332 \cdot {u}^{3} + 64 \cdot {u}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;s \cdot \left(-\sqrt[3]{{\log \left(u \cdot -4 + 1\right)}^{3}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021176 
(FPCore (s u)
  :name "Disney BSSRDF, sample scattering profile, lower"
  :precision binary32
  :pre (and (<= 0.0 s 256.0) (<= 2.328306437e-10 u 0.25))
  (* s (log (/ 1.0 (- 1.0 (* 4.0 u))))))