Disney BSSRDF, PDF of scattering profile

Average Error: 0.1 → 0.2

Time: 21.9s

Precision: binary32

Cost: 10208

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

Math

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

↓

\[\frac{\frac{\frac{e^{\frac{r}{-s}} + e^{\frac{r \cdot \frac{0.5}{s}}{-1.5}}}{\pi}}{r}}{s \cdot 8} \]

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
 (/
  (/ (/ (+ (exp (/ r (- s))) (exp (/ (* r (/ 0.5 s)) -1.5))) PI) r)
  (* s 8.0)))

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 (((expf((r / -s)) + expf(((r * (0.5f / s)) / -1.5f))) / ((float) M_PI)) / r) / (s * 8.0f);
}

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(exp(Float32(r / Float32(-s))) + exp(Float32(Float32(r * Float32(Float32(0.5) / s)) / Float32(-1.5)))) / Float32(pi)) / r) / Float32(s * Float32(8.0)))
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 = (((exp((r / -s)) + exp(((r * (single(0.5) / s)) / single(-1.5)))) / single(pi)) / r) / (s * single(8.0));
end

Derivation

1. Initial program 0.1

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

2. Simplified 0.8

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

   [Start] 0.1	\[ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   rational.json-simplify-67 [=>] 0.8	\[ \color{blue}{\left(0.25 \cdot e^{\frac{-r}{s}} + 0.25 \cdot e^{\frac{-r}{s}}\right) \cdot \frac{0.5}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   rational.json-simplify-41 [=>] 0.1	\[ \color{blue}{\frac{0.25 \cdot e^{\frac{-r}{s}} + 0.25 \cdot e^{\frac{-r}{s}}}{\frac{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}{0.5}}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   rational.json-simplify-67 [=>] 0.8	\[ \color{blue}{\left(\left(0.25 \cdot e^{\frac{-r}{s}} + 0.25 \cdot e^{\frac{-r}{s}}\right) + \left(0.25 \cdot e^{\frac{-r}{s}} + 0.25 \cdot e^{\frac{-r}{s}}\right)\right) \cdot \frac{0.5}{\frac{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}{0.5}}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   rational.json-simplify-41 [<=] 0.8	\[ \left(\left(0.25 \cdot e^{\frac{-r}{s}} + 0.25 \cdot e^{\frac{-r}{s}}\right) + \left(0.25 \cdot e^{\frac{-r}{s}} + 0.25 \cdot e^{\frac{-r}{s}}\right)\right) \cdot \color{blue}{\left(0.5 \cdot \frac{0.5}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right)} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   metadata-eval [<=] 0.8	\[ \left(\left(0.25 \cdot e^{\frac{-r}{s}} + 0.25 \cdot e^{\frac{-r}{s}}\right) + \left(0.25 \cdot e^{\frac{-r}{s}} + 0.25 \cdot e^{\frac{-r}{s}}\right)\right) \cdot \left(\color{blue}{\left(0.25 + 0.25\right)} \cdot \frac{0.5}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}\right) + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   rational.json-simplify-67 [<=] 0.8	\[ \left(\left(0.25 \cdot e^{\frac{-r}{s}} + 0.25 \cdot e^{\frac{-r}{s}}\right) + \left(0.25 \cdot e^{\frac{-r}{s}} + 0.25 \cdot e^{\frac{-r}{s}}\right)\right) \cdot \color{blue}{\frac{0.25}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   rational.json-simplify-43 [=>] 0.8	\[ \left(\left(0.25 \cdot e^{\frac{-r}{s}} + 0.25 \cdot e^{\frac{-r}{s}}\right) + \left(0.25 \cdot e^{\frac{-r}{s}} + 0.25 \cdot e^{\frac{-r}{s}}\right)\right) \cdot \frac{0.25}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{e^{\frac{-r}{3 \cdot s}} \cdot \frac{0.75}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r}} \]
   rational.json-simplify-50 [=>] 0.8	\[ \left(\left(0.25 \cdot e^{\frac{-r}{s}} + 0.25 \cdot e^{\frac{-r}{s}}\right) + \left(0.25 \cdot e^{\frac{-r}{s}} + 0.25 \cdot e^{\frac{-r}{s}}\right)\right) \cdot \frac{0.25}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \color{blue}{\frac{0.75}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \cdot e^{\frac{-r}{3 \cdot s}}} \]

3. Applied egg-rr 0.9

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

4. Applied egg-rr 0.9

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

5. Applied egg-rr 0.2

   \[\leadsto \color{blue}{\frac{\frac{\frac{e^{\frac{r}{-s}} + e^{r \cdot \frac{-0.3333333333333333}{s}}}{\pi}}{r}}{s \cdot 8}} \]

6. Applied egg-rr 0.2

   \[\leadsto \frac{\frac{\frac{e^{\frac{r}{-s}} + e^{\color{blue}{\frac{r \cdot \frac{0.5}{s}}{-1.5}}}}{\pi}}{r}}{s \cdot 8} \]

7. Final simplification 0.2

   \[\leadsto \frac{\frac{\frac{e^{\frac{r}{-s}} + e^{\frac{r \cdot \frac{0.5}{s}}{-1.5}}}{\pi}}{r}}{s \cdot 8} \]

Alternatives

Alternative 1
Error	0.2
Cost	10144

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

Alternative 2
Error	0.2
Cost	10144

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

Alternative 3
Error	29.1
Cost	6912

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

Alternative 4
Error	29.1
Cost	6912

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

Alternative 5
Error	29.1
Cost	6848

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

Alternative 6
Error	29.2
Cost	3744

\[\frac{0.25}{\frac{\frac{r}{\frac{0.25}{s}} + \left(r \cdot s\right) \cdot -5}{-\frac{1}{\pi}}} \]

Alternative 7
Error	29.2
Cost	3680

\[\frac{0.25}{\left(\frac{r}{\frac{0.25}{s}} - r \cdot \left(5 \cdot s\right)\right) \cdot \left(-\pi\right)} \]

Alternative 8
Error	29.2
Cost	3456

\[\frac{\frac{0.5}{\pi}}{r \cdot s} \cdot 0.5 \]

Alternative 9
Error	29.2
Cost	3456

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

Alternative 10
Error	29.2
Cost	3392

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

Alternative 11
Error	29.2
Cost	3392

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

Alternative 12
Error	29.2
Cost	3392

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

Alternative 13
Error	32.0
Cost	160

\[\frac{0.25}{s - s} \]

Reproduce

herbie shell --seed 2023073 
(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))))