Disney BSSRDF, PDF of scattering profile

Average Error: 0.2 → 0.2

Time: 12.5s

Precision: binary32

Cost: 10176

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

Math

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

↓

\[\frac{0.125 \cdot \left(\frac{1}{e^{\frac{r}{s}}} + e^{\frac{r}{s \cdot -3}}\right)}{s \cdot \left(r \cdot \pi\right)} \]

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 (+ (/ 1.0 (exp (/ r s))) (exp (/ r (* s -3.0))))) (* 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 * ((1.0f / expf((r / s))) + expf((r / (s * -3.0f))))) / (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(Float32(Float32(1.0) / exp(Float32(r / s))) + exp(Float32(r / Float32(s * Float32(-3.0)))))) / 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) * ((single(1.0) / exp((r / s))) + exp((r / (s * single(-3.0)))))) / (s * (r * single(pi)));
end

Derivation

1. Initial program 0.2

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

2. Simplified 1.3

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

   [Start] 0.2	\[ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   associate-*l/ [<=] 0.7	\[ \color{blue}{\frac{0.25}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} \cdot e^{\frac{-r}{s}}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   associate-*l* [=>] 0.7	\[ \frac{0.25}{\color{blue}{\left(2 \cdot \pi\right) \cdot \left(s \cdot r\right)}} \cdot e^{\frac{-r}{s}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   associate-*l* [=>] 0.7	\[ \frac{0.25}{\color{blue}{2 \cdot \left(\pi \cdot \left(s \cdot r\right)\right)}} \cdot e^{\frac{-r}{s}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   associate-/r* [=>] 0.7	\[ \color{blue}{\frac{\frac{0.25}{2}}{\pi \cdot \left(s \cdot r\right)}} \cdot e^{\frac{-r}{s}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   metadata-eval [=>] 0.7	\[ \frac{\color{blue}{0.125}}{\pi \cdot \left(s \cdot r\right)} \cdot e^{\frac{-r}{s}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   metadata-eval [<=] 0.7	\[ \frac{\color{blue}{\frac{0.75}{6}}}{\pi \cdot \left(s \cdot r\right)} \cdot e^{\frac{-r}{s}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   associate-/r* [<=] 0.7	\[ \color{blue}{\frac{0.75}{6 \cdot \left(\pi \cdot \left(s \cdot r\right)\right)}} \cdot e^{\frac{-r}{s}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   associate-*l* [<=] 0.6	\[ \frac{0.75}{\color{blue}{\left(6 \cdot \pi\right) \cdot \left(s \cdot r\right)}} \cdot e^{\frac{-r}{s}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   associate-*l* [<=] 0.7	\[ \frac{0.75}{\color{blue}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} \cdot e^{\frac{-r}{s}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   associate-*l/ [<=] 0.7	\[ \frac{0.75}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \cdot e^{\frac{-r}{s}} + \color{blue}{\frac{0.75}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \cdot e^{\frac{-r}{3 \cdot s}}} \]
   distribute-lft-out [=>] 0.7	\[ \color{blue}{\frac{0.75}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \cdot \left(e^{\frac{-r}{s}} + e^{\frac{-r}{3 \cdot s}}\right)} \]
   associate-/r* [=>] 0.7	\[ \color{blue}{\frac{\frac{0.75}{\left(6 \cdot \pi\right) \cdot s}}{r}} \cdot \left(e^{\frac{-r}{s}} + e^{\frac{-r}{3 \cdot s}}\right) \]
   associate-*l* [=>] 0.7	\[ \frac{\frac{0.75}{\color{blue}{6 \cdot \left(\pi \cdot s\right)}}}{r} \cdot \left(e^{\frac{-r}{s}} + e^{\frac{-r}{3 \cdot s}}\right) \]
   associate-/r* [=>] 0.7	\[ \frac{\color{blue}{\frac{\frac{0.75}{6}}{\pi \cdot s}}}{r} \cdot \left(e^{\frac{-r}{s}} + e^{\frac{-r}{3 \cdot s}}\right) \]
   associate-/l/ [=>] 0.7	\[ \color{blue}{\frac{\frac{0.75}{6}}{r \cdot \left(\pi \cdot s\right)}} \cdot \left(e^{\frac{-r}{s}} + e^{\frac{-r}{3 \cdot s}}\right) \]
   metadata-eval [=>] 0.7	\[ \frac{\color{blue}{0.125}}{r \cdot \left(\pi \cdot s\right)} \cdot \left(e^{\frac{-r}{s}} + e^{\frac{-r}{3 \cdot s}}\right) \]
   *-commutative [=>] 0.7	\[ \frac{0.125}{r \cdot \color{blue}{\left(s \cdot \pi\right)}} \cdot \left(e^{\frac{-r}{s}} + e^{\frac{-r}{3 \cdot s}}\right) \]
   neg-mul-1 [=>] 0.7	\[ \frac{0.125}{r \cdot \left(s \cdot \pi\right)} \cdot \left(e^{\frac{-r}{s}} + e^{\frac{\color{blue}{-1 \cdot r}}{3 \cdot s}}\right) \]
   associate-*l/ [<=] 0.7	\[ \frac{0.125}{r \cdot \left(s \cdot \pi\right)} \cdot \left(e^{\frac{-r}{s}} + e^{\color{blue}{\frac{-1}{3 \cdot s} \cdot r}}\right) \]
   *-commutative [=>] 0.7	\[ \frac{0.125}{r \cdot \left(s \cdot \pi\right)} \cdot \left(e^{\frac{-r}{s}} + e^{\color{blue}{r \cdot \frac{-1}{3 \cdot s}}}\right) \]
   exp-prod [=>] 1.3	\[ \frac{0.125}{r \cdot \left(s \cdot \pi\right)} \cdot \left(e^{\frac{-r}{s}} + \color{blue}{{\left(e^{r}\right)}^{\left(\frac{-1}{3 \cdot s}\right)}}\right) \]
   associate-/r* [=>] 1.3	\[ \frac{0.125}{r \cdot \left(s \cdot \pi\right)} \cdot \left(e^{\frac{-r}{s}} + {\left(e^{r}\right)}^{\color{blue}{\left(\frac{\frac{-1}{3}}{s}\right)}}\right) \]
   metadata-eval [=>] 1.3	\[ \frac{0.125}{r \cdot \left(s \cdot \pi\right)} \cdot \left(e^{\frac{-r}{s}} + {\left(e^{r}\right)}^{\left(\frac{\color{blue}{-0.3333333333333333}}{s}\right)}\right) \]

3. Taylor expanded in r around inf 0.2

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

4. Simplified 0.2

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

   [Start] 0.2	\[ 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{s \cdot \left(r \cdot \pi\right)} \]
   *-commutative [=>] 0.2	\[ 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\color{blue}{\left(r \cdot \pi\right) \cdot s}} \]
   associate-*r/ [=>] 0.2	\[ \color{blue}{\frac{0.125 \cdot \left(e^{-1 \cdot \frac{r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}\right)}{\left(r \cdot \pi\right) \cdot s}} \]
   mul-1-neg [=>] 0.2	\[ \frac{0.125 \cdot \left(e^{\color{blue}{-\frac{r}{s}}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}\right)}{\left(r \cdot \pi\right) \cdot s} \]
   distribute-neg-frac [=>] 0.2	\[ \frac{0.125 \cdot \left(e^{\color{blue}{\frac{-r}{s}}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}\right)}{\left(r \cdot \pi\right) \cdot s} \]
   *-commutative [=>] 0.2	\[ \frac{0.125 \cdot \left(e^{\frac{-r}{s}} + e^{\color{blue}{\frac{r}{s} \cdot -0.3333333333333333}}\right)}{\left(r \cdot \pi\right) \cdot s} \]
   associate-*l/ [=>] 0.2	\[ \frac{0.125 \cdot \left(e^{\frac{-r}{s}} + e^{\color{blue}{\frac{r \cdot -0.3333333333333333}{s}}}\right)}{\left(r \cdot \pi\right) \cdot s} \]
   associate-/l* [=>] 0.2	\[ \frac{0.125 \cdot \left(e^{\frac{-r}{s}} + e^{\color{blue}{\frac{r}{\frac{s}{-0.3333333333333333}}}}\right)}{\left(r \cdot \pi\right) \cdot s} \]
   *-commutative [<=] 0.2	\[ \frac{0.125 \cdot \left(e^{\frac{-r}{s}} + e^{\frac{r}{\frac{s}{-0.3333333333333333}}}\right)}{\color{blue}{s \cdot \left(r \cdot \pi\right)}} \]

5. Applied egg-rr 0.2

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

6. Applied egg-rr 0.2

   \[\leadsto \frac{0.125 \cdot \left(\color{blue}{\frac{1}{e^{\frac{r}{s}}}} + e^{\frac{r}{s \cdot -3}}\right)}{s \cdot \left(r \cdot \pi\right)} \]

7. Final simplification 0.2

   \[\leadsto \frac{0.125 \cdot \left(\frac{1}{e^{\frac{r}{s}}} + e^{\frac{r}{s \cdot -3}}\right)}{s \cdot \left(r \cdot \pi\right)} \]

Alternatives

Alternative 1
Error	0.7
Cost	10144

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

Alternative 2
Error	0.7
Cost	10144

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

Alternative 3
Error	0.2
Cost	10144

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

Alternative 4
Error	0.2
Cost	10144

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

Alternative 5
Error	18.0
Cost	9792

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

Alternative 6
Error	28.9
Cost	6816

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

Alternative 7
Error	29.1
Cost	3392

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

Alternative 8
Error	29.1
Cost	3392

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

Alternative 9
Error	29.1
Cost	3392

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

Alternative 10
Error	29.1
Cost	3392

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

Reproduce

herbie shell --seed 2022364 
(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))))