Disney BSSRDF, PDF of scattering profile

Average Accuracy: 99.5% → 99.3%

Time: 26.0s

Precision: binary32

Cost: 19904

\[\left(0 \leq s \land s \leq 256\right) \land \left(10^{-6} < r \land r < 1000000\right)\]

Math

\[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

↓

\[\mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{-r}{s}}}{r}, e^{-0.3333333333333333 \cdot \frac{r}{s} - \log \left(r \cdot \frac{\pi}{\frac{0.125}{s}}\right)}\right) \]

FPCore

(FPCore (s r)
 :precision binary32
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r))
  (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r))))

↓

(FPCore (s r)
 :precision binary32
 (fma
  (/ 0.125 (* s PI))
  (/ (exp (/ (- r) s)) r)
  (exp (- (* -0.3333333333333333 (/ r s)) (log (* r (/ PI (/ 0.125 s))))))))

C

float code(float s, float r) {
	return ((0.25f * expf((-r / s))) / (((2.0f * ((float) M_PI)) * s) * r)) + ((0.75f * expf((-r / (3.0f * s)))) / (((6.0f * ((float) M_PI)) * s) * r));
}

↓

float code(float s, float r) {
	return fmaf((0.125f / (s * ((float) M_PI))), (expf((-r / s)) / r), expf(((-0.3333333333333333f * (r / s)) - logf((r * (((float) M_PI) / (0.125f / s)))))));
}

Julia

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(Float32(3.0) * s)))) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end

↓

function code(s, r)
	return fma(Float32(Float32(0.125) / Float32(s * Float32(pi))), Float32(exp(Float32(Float32(-r) / s)) / r), exp(Float32(Float32(Float32(-0.3333333333333333) * Float32(r / s)) - log(Float32(r * Float32(Float32(pi) / Float32(Float32(0.125) / s)))))))
end

Derivation

1. Initial program 99.5%

   \[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

2. Simplified 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{-r}{s}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}\right)} \]
   Proof

   [Start] 99.5	\[ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   times-frac [=>] 99.5	\[ \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   fma-def [=>] 99.5	\[ \color{blue}{\mathsf{fma}\left(\frac{0.25}{\left(2 \cdot \pi\right) \cdot s}, \frac{e^{\frac{-r}{s}}}{r}, \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}\right)} \]
   associate-*l* [=>] 99.5	\[ \mathsf{fma}\left(\frac{0.25}{\color{blue}{2 \cdot \left(\pi \cdot s\right)}}, \frac{e^{\frac{-r}{s}}}{r}, \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]
   associate-/r* [=>] 99.5	\[ \mathsf{fma}\left(\color{blue}{\frac{\frac{0.25}{2}}{\pi \cdot s}}, \frac{e^{\frac{-r}{s}}}{r}, \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]
   metadata-eval [=>] 99.5	\[ \mathsf{fma}\left(\frac{\color{blue}{0.125}}{\pi \cdot s}, \frac{e^{\frac{-r}{s}}}{r}, \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]
   *-commutative [=>] 99.5	\[ \mathsf{fma}\left(\frac{0.125}{\color{blue}{s \cdot \pi}}, \frac{e^{\frac{-r}{s}}}{r}, \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}\right) \]
   times-frac [=>] 99.6	\[ \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{-r}{s}}}{r}, \color{blue}{\frac{0.75}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}}\right) \]
   associate-*l* [=>] 99.5	\[ \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{-r}{s}}}{r}, \frac{0.75}{\color{blue}{6 \cdot \left(\pi \cdot s\right)}} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}\right) \]
   associate-/r* [=>] 99.5	\[ \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{-r}{s}}}{r}, \color{blue}{\frac{\frac{0.75}{6}}{\pi \cdot s}} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}\right) \]
   metadata-eval [=>] 99.5	\[ \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{-r}{s}}}{r}, \frac{\color{blue}{0.125}}{\pi \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}\right) \]
   *-commutative [=>] 99.5	\[ \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{-r}{s}}}{r}, \frac{0.125}{\color{blue}{s \cdot \pi}} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}\right) \]

3. Applied egg-rr 99.3%

   \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{-r}{s}}}{r}, \color{blue}{e^{-0.3333333333333333 \cdot \frac{r}{s} - \log \left(r \cdot \frac{\pi}{\frac{0.125}{s}}\right)}}\right) \]
   Proof

   [Start] 99.5	\[ \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{-r}{s}}}{r}, \frac{0.125}{s \cdot \pi} \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}\right) \]
   add-exp-log [=>] 99.3	\[ \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{-r}{s}}}{r}, \color{blue}{e^{\log \left(\frac{0.125}{s \cdot \pi} \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}\right)}}\right) \]
   *-commutative [=>] 99.3	\[ \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{-r}{s}}}{r}, e^{\log \color{blue}{\left(\frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r} \cdot \frac{0.125}{s \cdot \pi}\right)}}\right) \]
   clear-num [=>] 99.3	\[ \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{-r}{s}}}{r}, e^{\log \left(\frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r} \cdot \color{blue}{\frac{1}{\frac{s \cdot \pi}{0.125}}}\right)}\right) \]
   frac-times [=>] 99.3	\[ \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{-r}{s}}}{r}, e^{\log \color{blue}{\left(\frac{e^{-0.3333333333333333 \cdot \frac{r}{s}} \cdot 1}{r \cdot \frac{s \cdot \pi}{0.125}}\right)}}\right) \]
   *-commutative [<=] 99.3	\[ \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{-r}{s}}}{r}, e^{\log \left(\frac{\color{blue}{1 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}}{r \cdot \frac{s \cdot \pi}{0.125}}\right)}\right) \]
   *-un-lft-identity [<=] 99.3	\[ \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{-r}{s}}}{r}, e^{\log \left(\frac{\color{blue}{e^{-0.3333333333333333 \cdot \frac{r}{s}}}}{r \cdot \frac{s \cdot \pi}{0.125}}\right)}\right) \]
   log-div [=>] 99.3	\[ \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{-r}{s}}}{r}, e^{\color{blue}{\log \left(e^{-0.3333333333333333 \cdot \frac{r}{s}}\right) - \log \left(r \cdot \frac{s \cdot \pi}{0.125}\right)}}\right) \]
   add-log-exp [<=] 99.3	\[ \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{-r}{s}}}{r}, e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}} - \log \left(r \cdot \frac{s \cdot \pi}{0.125}\right)}\right) \]
   *-commutative [=>] 99.3	\[ \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{-r}{s}}}{r}, e^{-0.3333333333333333 \cdot \frac{r}{s} - \log \left(r \cdot \frac{\color{blue}{\pi \cdot s}}{0.125}\right)}\right) \]
   associate-/l* [=>] 99.3	\[ \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{-r}{s}}}{r}, e^{-0.3333333333333333 \cdot \frac{r}{s} - \log \left(r \cdot \color{blue}{\frac{\pi}{\frac{0.125}{s}}}\right)}\right) \]

4. Final simplification 99.3%

   \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{\frac{-r}{s}}}{r}, e^{-0.3333333333333333 \cdot \frac{r}{s} - \log \left(r \cdot \frac{\pi}{\frac{0.125}{s}}\right)}\right) \]

Alternatives

Alternative 1
Accuracy	99.5%
Cost	13664

\[\frac{e^{\frac{-r}{s}}}{r} \cdot \frac{\frac{0.125}{s}}{\pi} + \frac{0.75}{s \cdot \left(\pi \cdot 6\right)} \cdot \frac{e^{\left(-r\right) \cdot \frac{0.3333333333333333}{s}}}{r} \]

Alternative 2
Accuracy	99.5%
Cost	13600

\[\frac{e^{\frac{-r}{s}}}{r} \cdot \frac{\frac{0.125}{s}}{\pi} + \frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

Alternative 3
Accuracy	99.5%
Cost	10144

\[0.125 \cdot \frac{e^{\frac{-r}{s}} + e^{\frac{r \cdot -0.3333333333333333}{s}}}{\left(s \cdot \pi\right) \cdot r} \]

Alternative 4
Accuracy	99.5%
Cost	10144

\[0.125 \cdot \frac{e^{\frac{-r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{s \cdot \left(\pi \cdot r\right)} \]

Alternative 5
Accuracy	11.9%
Cost	9792

\[\frac{0.25}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(s \cdot \pi\right) \cdot r\right)\right)} \]

Alternative 6
Accuracy	44.4%
Cost	9792

\[\frac{0.25}{s \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot r\right)\right)} \]

Alternative 7
Accuracy	9.5%
Cost	6816

\[\frac{0.125}{s} \cdot \frac{e^{\frac{-r}{s}} + 1}{\pi \cdot r} \]

Alternative 8
Accuracy	9.0%
Cost	3456

\[\frac{\frac{\frac{1}{\pi}}{r}}{s \cdot 4} \]

Alternative 9
Accuracy	9.0%
Cost	3392

\[\frac{0.25}{\left(s \cdot \pi\right) \cdot r} \]

Alternative 10
Accuracy	9.0%
Cost	3392

\[\frac{0.25}{s \cdot \left(\pi \cdot r\right)} \]

Alternative 11
Accuracy	9.0%
Cost	3392

\[\frac{\frac{0.25}{s}}{\pi \cdot r} \]

Alternative 12
Accuracy	9.0%
Cost	3392

\[\frac{\frac{0.25}{s \cdot \pi}}{r} \]

Reproduce

herbie shell --seed 2023151 
(FPCore (s r)
  :name "Disney BSSRDF, PDF of scattering profile"
  :precision binary32
  :pre (and (and (<= 0.0 s) (<= s 256.0)) (and (< 1e-6 r) (< r 1000000.0)))
  (+ (/ (* 0.25 (exp (/ (- r) s))) (* (* (* 2.0 PI) s) r)) (/ (* 0.75 (exp (/ (- r) (* 3.0 s)))) (* (* (* 6.0 PI) s) r))))