Disney BSSRDF, PDF of scattering profile

Average Accuracy: 99.6% → 99.5%

Time: 13.8s

Precision: binary32

Cost: 10176

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

Math

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

↓

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

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
 (/
  (* 0.125 (+ (exp (/ (- r) s)) (exp (/ 0.3333333333333333 (/ (- s) r)))))
  (* s (* r PI))))

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 (0.125f * (expf((-r / s)) + expf((0.3333333333333333f / (-s / r))))) / (s * (r * ((float) M_PI)));
}

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 Float32(Float32(Float32(0.125) * Float32(exp(Float32(Float32(-r) / s)) + exp(Float32(Float32(0.3333333333333333) / Float32(Float32(-s) / r))))) / Float32(s * Float32(r * Float32(pi))))
end

MATLAB

function tmp = code(s, r)
	tmp = ((single(0.25) * exp((-r / s))) / (((single(2.0) * single(pi)) * s) * r)) + ((single(0.75) * exp((-r / (single(3.0) * s)))) / (((single(6.0) * single(pi)) * s) * r));
end

↓

function tmp = code(s, r)
	tmp = (single(0.125) * (exp((-r / s)) + exp((single(0.3333333333333333) / (-s / r))))) / (s * (r * single(pi)));
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%

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

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

3. Applied egg-rr 99.5%

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

4. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{0.125}{s}}{\pi}\right)} - 1\right)} \cdot \left(\frac{e^{-\frac{-0.3333333333333333}{\frac{-s}{r}}}}{r} + \frac{e^{-\frac{r}{s}}}{r}\right) \]

5. Simplified 99.5%

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

   [Start] 98.5	\[ \left(e^{\mathsf{log1p}\left(\frac{\frac{0.125}{s}}{\pi}\right)} - 1\right) \cdot \left(\frac{e^{-\frac{-0.3333333333333333}{\frac{-s}{r}}}}{r} + \frac{e^{-\frac{r}{s}}}{r}\right) \]
   expm1-def [=>] 99.5	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{0.125}{s}}{\pi}\right)\right)} \cdot \left(\frac{e^{-\frac{-0.3333333333333333}{\frac{-s}{r}}}}{r} + \frac{e^{-\frac{r}{s}}}{r}\right) \]
   expm1-log1p [=>] 99.5	\[ \color{blue}{\frac{\frac{0.125}{s}}{\pi}} \cdot \left(\frac{e^{-\frac{-0.3333333333333333}{\frac{-s}{r}}}}{r} + \frac{e^{-\frac{r}{s}}}{r}\right) \]

6. Taylor expanded in r around inf 99.5%

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

7. Simplified 99.5%

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

   [Start] 99.5	\[ 0.125 \cdot \frac{e^{-\frac{r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{s \cdot \left(r \cdot \pi\right)} \]
   associate-*r/ [=>] 99.5	\[ \color{blue}{\frac{0.125 \cdot \left(e^{-\frac{r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}\right)}{s \cdot \left(r \cdot \pi\right)}} \]
   distribute-neg-frac [=>] 99.5	\[ \frac{0.125 \cdot \left(e^{\color{blue}{\frac{-r}{s}}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}\right)}{s \cdot \left(r \cdot \pi\right)} \]
   distribute-lft-neg-in [=>] 99.5	\[ \frac{0.125 \cdot \left(e^{\frac{-r}{s}} + e^{\color{blue}{\left(-0.3333333333333333\right) \cdot \frac{r}{s}}}\right)}{s \cdot \left(r \cdot \pi\right)} \]
   metadata-eval [=>] 99.5	\[ \frac{0.125 \cdot \left(e^{\frac{-r}{s}} + e^{\color{blue}{-0.3333333333333333} \cdot \frac{r}{s}}\right)}{s \cdot \left(r \cdot \pi\right)} \]
   associate-*r/ [=>] 99.5	\[ \frac{0.125 \cdot \left(e^{\frac{-r}{s}} + e^{\color{blue}{\frac{-0.3333333333333333 \cdot r}{s}}}\right)}{s \cdot \left(r \cdot \pi\right)} \]
   *-commutative [=>] 99.5	\[ \frac{0.125 \cdot \left(e^{\frac{-r}{s}} + e^{\frac{-0.3333333333333333 \cdot r}{s}}\right)}{s \cdot \color{blue}{\left(\pi \cdot r\right)}} \]

8. Applied egg-rr 99.5%

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

9. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	97.5%
Cost	10144

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

Alternative 2
Accuracy	99.5%
Cost	10144

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

Alternative 3
Accuracy	99.5%
Cost	10144

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

Alternative 4
Accuracy	43.6%
Cost	9792

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

Alternative 5
Accuracy	9.3%
Cost	6880

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

Alternative 6
Accuracy	9.3%
Cost	6880

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

Alternative 7
Accuracy	9.2%
Cost	6816

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

Alternative 8
Accuracy	9.3%
Cost	6816

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

Alternative 9
Accuracy	8.8%
Cost	3456

\[\frac{0.25}{\frac{s}{\frac{\frac{1}{\pi}}{r}}} \]

Alternative 10
Accuracy	8.8%
Cost	3392

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

Alternative 11
Accuracy	8.8%
Cost	3392

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

Alternative 12
Accuracy	8.8%
Cost	3392

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

Alternative 13
Accuracy	8.8%
Cost	3392

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

Reproduce

herbie shell --seed 2023129 
(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))))