Disney BSSRDF, PDF of scattering profile

Average Error: 0.2 → 0.2

Time: 12.8s

Precision: binary32

Cost: 10208

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

Math

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

↓

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

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
 (*
  (/ 0.125 (* s PI))
  (+ (/ (exp (/ r (* s -3.0))) r) (/ (exp (/ (- r) s)) 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 (0.125f / (s * ((float) M_PI))) * ((expf((r / (s * -3.0f))) / r) + (expf((-r / s)) / 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(0.125) / Float32(s * Float32(pi))) * Float32(Float32(exp(Float32(r / Float32(s * Float32(-3.0)))) / r) + Float32(exp(Float32(Float32(-r) / s)) / 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(0.125) / (s * single(pi))) * ((exp((r / (s * single(-3.0)))) / r) + (exp((-r / s)) / r));
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 0.2

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

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

3. Applied egg-rr 0.2

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

4. Taylor expanded in s around 0 0.2

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

5. Simplified 0.2

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

   [Start] 0.2	\[ \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r} + \frac{e^{-\frac{r}{s}}}{r}\right) \]
   metadata-eval [<=] 0.2	\[ \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\color{blue}{\frac{1}{-3}} \cdot \frac{r}{s}}}{r} + \frac{e^{-\frac{r}{s}}}{r}\right) \]
   times-frac [<=] 0.2	\[ \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\color{blue}{\frac{1 \cdot r}{-3 \cdot s}}}}{r} + \frac{e^{-\frac{r}{s}}}{r}\right) \]
   *-lft-identity [=>] 0.2	\[ \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{\color{blue}{r}}{-3 \cdot s}}}{r} + \frac{e^{-\frac{r}{s}}}{r}\right) \]
   *-commutative [=>] 0.2	\[ \frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{r}{\color{blue}{s \cdot -3}}}}{r} + \frac{e^{-\frac{r}{s}}}{r}\right) \]

6. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.2
Cost	10208

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

Alternative 2
Error	0.8
Cost	10144

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

Alternative 3
Error	0.2
Cost	10144

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

Alternative 4
Error	0.2
Cost	10144

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

Alternative 5
Error	0.2
Cost	10144

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

Alternative 6
Error	17.7
Cost	9792

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

Alternative 7
Error	28.9
Cost	6880

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

Alternative 8
Error	28.9
Cost	6816

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

Alternative 9
Error	28.9
Cost	6816

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

Alternative 10
Error	29.0
Cost	3456

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

Alternative 11
Error	29.0
Cost	3392

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

Alternative 12
Error	29.0
Cost	3392

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

Reproduce

herbie shell --seed 2022354 
(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))))