Disney BSSRDF, PDF of scattering profile

Average Error: 0.1 → 0.2

Time: 11.8s

Precision: binary32

Cost: 13344

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

Math

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

↓

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

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
  (/
   (+ (pow E (* (/ r s) -0.3333333333333333)) (exp (/ (- r) s)))
   (* 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 * ((powf(((float) M_E), ((r / s) * -0.3333333333333333f)) + expf((-r / s))) / (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(0.125) * Float32(Float32((Float32(exp(1)) ^ Float32(Float32(r / s) * Float32(-0.3333333333333333))) + exp(Float32(Float32(-r) / s))) / 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(2.71828182845904523536) ^ ((r / s) * single(-0.3333333333333333))) + exp((-r / s))) / (s * (r * single(pi))));
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 1.8

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

   [Start] 0.1	\[ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   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* [<=] 0.7	\[ \frac{0.75}{\color{blue}{\left(6 \cdot \pi\right) \cdot \left(s \cdot r\right)}} \cdot e^{\frac{-r}{s}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
   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	\[ \color{blue}{\frac{0.75}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \cdot \left(e^{\frac{-r}{s}} + e^{\frac{-r}{3 \cdot s}}\right)} \]

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. Applied egg-rr 0.2

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

5. Simplified 0.2

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

   [Start] 0.2	\[ 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + {\left(e^{1}\right)}^{\left(-0.3333333333333333 \cdot \frac{r}{s}\right)}}{s \cdot \left(r \cdot \pi\right)} \]
   exp-1-e [=>] 0.2	\[ 0.125 \cdot \frac{e^{-1 \cdot \frac{r}{s}} + {\color{blue}{e}}^{\left(-0.3333333333333333 \cdot \frac{r}{s}\right)}}{s \cdot \left(r \cdot \pi\right)} \]

6. Final simplification 0.2

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

Alternatives

Alternative 1
Error	0.7
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 2
Error	0.2
Cost	10144

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

Alternative 3
Error	0.2
Cost	10144

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

Alternative 4
Error	17.7
Cost	9792

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

Alternative 5
Error	29.0
Cost	6880

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

Alternative 6
Error	29.0
Cost	6816

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

Alternative 7
Error	29.1
Cost	3456

\[\frac{\frac{\frac{1}{r}}{\pi}}{s \cdot 4} \]

Alternative 8
Error	29.1
Cost	3392

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

Alternative 9
Error	29.1
Cost	3392

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

Reproduce

herbie shell --seed 2023017 
(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))))