Disney BSSRDF, PDF of scattering profile

Average Accuracy: 99.6% → 99.5%

Time: 24.6s

Precision: binary32

Cost: 13792

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

Math

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

↓

\[\frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \left(\frac{1}{\pi \cdot \left(s \cdot 6\right)} \cdot 0.75\right) \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

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
 (+
  (* (/ 0.25 (* s (* 2.0 PI))) (/ (exp (/ (- r) s)) r))
  (* (* (/ 1.0 (* PI (* s 6.0))) 0.75) (/ (exp (/ (- r) (* s 3.0))) 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 ((0.25f / (s * (2.0f * ((float) M_PI)))) * (expf((-r / s)) / r)) + (((1.0f / (((float) M_PI) * (s * 6.0f))) * 0.75f) * (expf((-r / (s * 3.0f))) / 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 Float32(Float32(Float32(Float32(0.25) / Float32(s * Float32(Float32(2.0) * Float32(pi)))) * Float32(exp(Float32(Float32(-r) / s)) / r)) + Float32(Float32(Float32(Float32(1.0) / Float32(Float32(pi) * Float32(s * Float32(6.0)))) * Float32(0.75)) * Float32(exp(Float32(Float32(-r) / Float32(s * Float32(3.0)))) / r)))
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(0.25) / (s * (single(2.0) * single(pi)))) * (exp((-r / s)) / r)) + (((single(1.0) / (single(pi) * (s * single(6.0)))) * single(0.75)) * (exp((-r / (s * single(3.0)))) / 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 99.6%

   \[\leadsto \color{blue}{\frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{s \cdot \left(\pi \cdot 6\right)} \cdot \frac{e^{-\frac{r}{s \cdot 3}}}{r}} \]
   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	\[ \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   *-commutative [=>] 99.6	\[ \frac{0.25}{\color{blue}{s \cdot \left(2 \cdot \pi\right)}} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   times-frac [=>] 99.6	\[ \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\frac{0.75}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}} \]
   *-commutative [=>] 99.6	\[ \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{\color{blue}{s \cdot \left(6 \cdot \pi\right)}} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r} \]
   *-commutative [=>] 99.6	\[ \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{s \cdot \color{blue}{\left(\pi \cdot 6\right)}} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r} \]
   distribute-frac-neg [=>] 99.6	\[ \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{s \cdot \left(\pi \cdot 6\right)} \cdot \frac{e^{\color{blue}{-\frac{r}{3 \cdot s}}}}{r} \]
   *-commutative [=>] 99.6	\[ \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{s \cdot \left(\pi \cdot 6\right)} \cdot \frac{e^{-\frac{r}{\color{blue}{s \cdot 3}}}}{r} \]

3. Applied egg-rr 99.5%

   \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\left(\frac{1}{\left(s \cdot 6\right) \cdot \pi} \cdot 0.75\right)} \cdot \frac{e^{-\frac{r}{s \cdot 3}}}{r} \]
   Proof

   [Start] 99.6	\[ \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{s \cdot \left(\pi \cdot 6\right)} \cdot \frac{e^{-\frac{r}{s \cdot 3}}}{r} \]
   clear-num [=>] 99.6	\[ \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\frac{1}{\frac{s \cdot \left(\pi \cdot 6\right)}{0.75}}} \cdot \frac{e^{-\frac{r}{s \cdot 3}}}{r} \]
   associate-/r/ [=>] 99.6	\[ \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \color{blue}{\left(\frac{1}{s \cdot \left(\pi \cdot 6\right)} \cdot 0.75\right)} \cdot \frac{e^{-\frac{r}{s \cdot 3}}}{r} \]
   *-commutative [=>] 99.6	\[ \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \left(\frac{1}{s \cdot \color{blue}{\left(6 \cdot \pi\right)}} \cdot 0.75\right) \cdot \frac{e^{-\frac{r}{s \cdot 3}}}{r} \]
   associate-*r* [=>] 99.5	\[ \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \left(\frac{1}{\color{blue}{\left(s \cdot 6\right) \cdot \pi}} \cdot 0.75\right) \cdot \frac{e^{-\frac{r}{s \cdot 3}}}{r} \]

4. Final simplification 99.5%

   \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \left(\frac{1}{\pi \cdot \left(s \cdot 6\right)} \cdot 0.75\right) \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

Alternatives

Alternative 1
Accuracy	99.5%
Cost	13664

\[\frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{e^{\frac{-r}{s \cdot 3}}}{r} \cdot \frac{\frac{0.125}{s}}{\pi} \]

Alternative 2
Accuracy	99.5%
Cost	13600

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

Alternative 3
Accuracy	99.5%
Cost	10208

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

Alternative 4
Accuracy	99.5%
Cost	10208

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

Alternative 5
Accuracy	99.5%
Cost	10144

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

Alternative 6
Accuracy	11.6%
Cost	9792

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

Alternative 7
Accuracy	42.7%
Cost	9792

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

Alternative 8
Accuracy	9.3%
Cost	6912

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

Alternative 9
Accuracy	9.3%
Cost	6880

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

Alternative 10
Accuracy	8.8%
Cost	3456

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

Alternative 11
Accuracy	6.9%
Cost	3392

\[\frac{0.125}{r \cdot \left(s \cdot \pi\right)} \]

Alternative 12
Accuracy	8.8%
Cost	3392

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

Alternative 13
Accuracy	8.8%
Cost	3392

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

Alternative 14
Accuracy	8.8%
Cost	3392

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

Reproduce

herbie shell --seed 2023147 
(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))))