Disney BSSRDF, PDF of scattering profile

Average Accuracy: 99.6% → 99.5%

Time: 14.4s

Precision: binary32

Cost: 10144

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

Math

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

↓

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

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) -0.3333333333333333)) (exp (- (/ r s))))
   (* 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) * -0.3333333333333333f)) + expf(-(r / s))) / (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(0.125) * Float32(Float32(exp(Float32(Float32(r / s) * Float32(-0.3333333333333333))) + exp(Float32(-Float32(r / s)))) / 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) * single(-0.3333333333333333))) + exp(-(r / s))) / (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 95.7%

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

   [Start] 99.6	\[ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   associate-*l/ [<=] 97.6	\[ \color{blue}{\frac{0.25}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \cdot e^{\frac{-r}{s}}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   associate-*l* [=>] 97.6	\[ \frac{0.25}{\color{blue}{\left(2 \cdot \pi\right) \cdot \left(s \cdot r\right)}} \cdot e^{\frac{-r}{s}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   associate-*l* [=>] 97.6	\[ \frac{0.25}{\color{blue}{2 \cdot \left(\pi \cdot \left(s \cdot r\right)\right)}} \cdot e^{\frac{-r}{s}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   associate-/r* [=>] 97.6	\[ \color{blue}{\frac{\frac{0.25}{2}}{\pi \cdot \left(s \cdot r\right)}} \cdot e^{\frac{-r}{s}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   metadata-eval [=>] 97.6	\[ \frac{\color{blue}{0.125}}{\pi \cdot \left(s \cdot r\right)} \cdot e^{\frac{-r}{s}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   metadata-eval [<=] 97.6	\[ \frac{\color{blue}{\frac{0.75}{6}}}{\pi \cdot \left(s \cdot r\right)} \cdot e^{\frac{-r}{s}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   associate-/r* [<=] 97.6	\[ \color{blue}{\frac{0.75}{6 \cdot \left(\pi \cdot \left(s \cdot r\right)\right)}} \cdot e^{\frac{-r}{s}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   associate-*l* [<=] 97.6	\[ \frac{0.75}{\color{blue}{\left(6 \cdot \pi\right) \cdot \left(s \cdot r\right)}} \cdot e^{\frac{-r}{s}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   associate-*l* [<=] 97.6	\[ \frac{0.75}{\color{blue}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} \cdot e^{\frac{-r}{s}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   associate-*l/ [<=] 97.6	\[ \frac{0.75}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \cdot e^{\frac{-r}{s}} + \color{blue}{\frac{0.75}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \cdot e^{\frac{-r}{3 \cdot s}}} \]
   distribute-lft-out [=>] 97.6	\[ \color{blue}{\frac{0.75}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \cdot \left(e^{\frac{-r}{s}} + e^{\frac{-r}{3 \cdot s}}\right)} \]

3. Taylor expanded in r around inf 99.5%

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

4. Final simplification 99.5%

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

Alternatives

Alternative 1
Accuracy	97.5%
Cost	10144

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

Alternative 2
Accuracy	97.5%
Cost	10144

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

Alternative 3
Accuracy	99.5%
Cost	10144

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

Alternative 4
Accuracy	11.8%
Cost	9792

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

Alternative 5
Accuracy	44.0%
Cost	9792

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

Alternative 6
Accuracy	9.4%
Cost	6816

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

Alternative 7
Accuracy	9.4%
Cost	6816

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

Alternative 8
Accuracy	9.4%
Cost	6816

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

Alternative 9
Accuracy	8.9%
Cost	3456

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

Alternative 10
Accuracy	8.9%
Cost	3392

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

Alternative 11
Accuracy	8.9%
Cost	3392

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

Reproduce

herbie shell --seed 2023139 
(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))))