Disney BSSRDF, PDF of scattering profile

Average Error: 0.44% → 0.5%

Time: 13.4s

Precision: binary32

Cost: 13696

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

Math

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

↓

\[\frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{s \cdot \left(\pi \cdot 6\right)} \cdot \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r} \]

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))
  (* (/ 0.75 (* s (* PI 6.0))) (/ (exp (/ (* r -0.3333333333333333) 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 ((0.25f / (s * (2.0f * ((float) M_PI)))) * (expf((-r / s)) / r)) + ((0.75f / (s * (((float) M_PI) * 6.0f))) * (expf(((r * -0.3333333333333333f) / 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(Float32(0.25) / Float32(s * Float32(Float32(2.0) * Float32(pi)))) * Float32(exp(Float32(Float32(-r) / s)) / r)) + Float32(Float32(Float32(0.75) / Float32(s * Float32(Float32(pi) * Float32(6.0)))) * Float32(exp(Float32(Float32(r * Float32(-0.3333333333333333)) / 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(0.25) / (s * (single(2.0) * single(pi)))) * (exp((-r / s)) / r)) + ((single(0.75) / (s * (single(pi) * single(6.0)))) * (exp(((r * single(-0.3333333333333333)) / s)) / r));
end

Derivation

1. Initial program 0.44

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

2. Simplified 0.46

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

   [Start] 0.44	\[ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   times-frac [=>] 0.46	\[ \color{blue}{\frac{0.25}{\left(2 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{s}}}{r}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   *-commutative [=>] 0.46	\[ \frac{0.25}{\color{blue}{s \cdot \left(2 \cdot \pi\right)}} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   distribute-frac-neg [=>] 0.46	\[ \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\color{blue}{-\frac{r}{s}}}}{r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   times-frac [=>] 0.46	\[ \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + \color{blue}{\frac{0.75}{\left(6 \cdot \pi\right) \cdot s} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r}} \]
   *-commutative [=>] 0.46	\[ \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + \frac{0.75}{\color{blue}{s \cdot \left(6 \cdot \pi\right)}} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r} \]
   *-commutative [=>] 0.46	\[ \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + \frac{0.75}{s \cdot \color{blue}{\left(\pi \cdot 6\right)}} \cdot \frac{e^{\frac{-r}{3 \cdot s}}}{r} \]
   *-commutative [=>] 0.46	\[ \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + \frac{0.75}{s \cdot \left(\pi \cdot 6\right)} \cdot \frac{e^{\frac{-r}{\color{blue}{s \cdot 3}}}}{r} \]

3. Taylor expanded in r around 0 0.5

   \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + \frac{0.75}{s \cdot \left(\pi \cdot 6\right)} \cdot \frac{e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{r} \]

4. Simplified 0.5

   \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + \frac{0.75}{s \cdot \left(\pi \cdot 6\right)} \cdot \frac{e^{\color{blue}{\frac{-0.3333333333333333 \cdot r}{s}}}}{r} \]
   Proof

   [Start] 0.5	\[ \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + \frac{0.75}{s \cdot \left(\pi \cdot 6\right)} \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}}}{r} \]
   associate-*r/ [=>] 0.5	\[ \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{-\frac{r}{s}}}{r} + \frac{0.75}{s \cdot \left(\pi \cdot 6\right)} \cdot \frac{e^{\color{blue}{\frac{-0.3333333333333333 \cdot r}{s}}}}{r} \]

5. Final simplification 0.5

   \[\leadsto \frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{s \cdot \left(\pi \cdot 6\right)} \cdot \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r} \]

Alternatives

Alternative 1
Error	0.71%
Cost	13472

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

Alternative 2
Error	0.53%
Cost	10208

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

Alternative 3
Error	2.58%
Cost	10144

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

Alternative 4
Error	0.51%
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 5
Error	56.42%
Cost	9792

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

Alternative 6
Error	90.71%
Cost	6816

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

Alternative 7
Error	91.19%
Cost	6624

\[{\left(\pi \cdot \frac{r}{\frac{0.25}{s}}\right)}^{-1} \]

Alternative 8
Error	91.19%
Cost	3392

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

Alternative 9
Error	91.19%
Cost	3392

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

Alternative 10
Error	91.18%
Cost	3392

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

Reproduce

herbie shell --seed 2023121 
(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))))