Disney BSSRDF, PDF of scattering profile

Average Accuracy: 99.6% → 99.4%

Time: 15.6s

Precision: binary32

Cost: 19904

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

Math

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

↓

\[\mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, e^{\frac{\frac{r}{-3}}{s} - \log r}, \frac{0.125 \cdot \frac{e^{\frac{-r}{s}}}{s \cdot \pi}}{r}\right) \]

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 -3.0) s) (log r)))
  (/ (* 0.125 (/ (exp (/ (- r) s)) (* s PI))) r)))

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 / -3.0f) / s) - logf(r))), ((0.125f * (expf((-r / s)) / (s * ((float) M_PI)))) / r));
}

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))), exp(Float32(Float32(Float32(r / Float32(-3.0)) / s) - log(r))), Float32(Float32(Float32(0.125) * Float32(exp(Float32(Float32(-r) / s)) / Float32(s * Float32(pi)))) / r))
end

Derivation

1. Initial program 99.6%

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

2. Simplified 99.5%

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

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

3. Applied egg-rr 99.4%

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

   [Start] 99.5	\[ \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r}, \frac{0.125 \cdot \frac{e^{\frac{-r}{s}}}{s \cdot \pi}}{r}\right) \]
   add-exp-log [=>] 99.4	\[ \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\color{blue}{e^{\log r}}}, \frac{0.125 \cdot \frac{e^{\frac{-r}{s}}}{s \cdot \pi}}{r}\right) \]
   div-exp [=>] 99.4	\[ \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, \color{blue}{e^{-0.3333333333333333 \cdot \frac{r}{s} - \log r}}, \frac{0.125 \cdot \frac{e^{\frac{-r}{s}}}{s \cdot \pi}}{r}\right) \]

4. Applied egg-rr 99.4%

   \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, e^{\color{blue}{\frac{\frac{r}{-3}}{s}} - \log r}, \frac{0.125 \cdot \frac{e^{\frac{-r}{s}}}{s \cdot \pi}}{r}\right) \]
   Proof

   [Start] 99.4	\[ \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, e^{-0.3333333333333333 \cdot \frac{r}{s} - \log r}, \frac{0.125 \cdot \frac{e^{\frac{-r}{s}}}{s \cdot \pi}}{r}\right) \]
   associate-*r/ [=>] 99.4	\[ \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, e^{\color{blue}{\frac{-0.3333333333333333 \cdot r}{s}} - \log r}, \frac{0.125 \cdot \frac{e^{\frac{-r}{s}}}{s \cdot \pi}}{r}\right) \]
   *-un-lft-identity [=>] 99.4	\[ \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, e^{\frac{-0.3333333333333333 \cdot r}{\color{blue}{1 \cdot s}} - \log r}, \frac{0.125 \cdot \frac{e^{\frac{-r}{s}}}{s \cdot \pi}}{r}\right) \]
   associate-/r* [=>] 99.4	\[ \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, e^{\color{blue}{\frac{\frac{-0.3333333333333333 \cdot r}{1}}{s}} - \log r}, \frac{0.125 \cdot \frac{e^{\frac{-r}{s}}}{s \cdot \pi}}{r}\right) \]
   *-commutative [=>] 99.4	\[ \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, e^{\frac{\frac{\color{blue}{r \cdot -0.3333333333333333}}{1}}{s} - \log r}, \frac{0.125 \cdot \frac{e^{\frac{-r}{s}}}{s \cdot \pi}}{r}\right) \]
   associate-/l* [=>] 99.4	\[ \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, e^{\frac{\color{blue}{\frac{r}{\frac{1}{-0.3333333333333333}}}}{s} - \log r}, \frac{0.125 \cdot \frac{e^{\frac{-r}{s}}}{s \cdot \pi}}{r}\right) \]
   metadata-eval [=>] 99.4	\[ \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, e^{\frac{\frac{r}{\color{blue}{-3}}}{s} - \log r}, \frac{0.125 \cdot \frac{e^{\frac{-r}{s}}}{s \cdot \pi}}{r}\right) \]

5. Final simplification 99.4%

   \[\leadsto \mathsf{fma}\left(\frac{0.125}{s \cdot \pi}, e^{\frac{\frac{r}{-3}}{s} - \log r}, \frac{0.125 \cdot \frac{e^{\frac{-r}{s}}}{s \cdot \pi}}{r}\right) \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	19904

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

Alternative 2
Accuracy	99.6%
Cost	13728

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

Alternative 3
Accuracy	99.5%
Cost	10272

\[\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{r \cdot \frac{1}{\frac{s}{-0.3333333333333333}}}}{r} + \frac{e^{\frac{-r}{s}}}{r}\right) \]

Alternative 4
Accuracy	99.5%
Cost	10208

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

Alternative 5
Accuracy	99.5%
Cost	10208

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

Alternative 6
Accuracy	97.9%
Cost	10144

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

Alternative 7
Accuracy	97.9%
Cost	10144

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

Alternative 8
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 9
Accuracy	43.4%
Cost	9792

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

Alternative 10
Accuracy	9.8%
Cost	6816

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

Alternative 11
Accuracy	9.3%
Cost	3392

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

Reproduce

herbie shell --seed 2023135 
(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))))