Disney BSSRDF, PDF of scattering profile

Average Error: 0.1 → 0.1

Time: 12.9s

Precision: binary32

Cost: 13728

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

Math

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

↓

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

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 (exp (/ (- r) s))) (* r (* s (* 2.0 PI))))
  (/ (* 0.75 (exp (/ (- r) (* s 3.0)))) (* (* s (* r PI)) 6.0))))

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 ((0.25f * expf((-r / s))) / (r * (s * (2.0f * ((float) M_PI))))) + ((0.75f * expf((-r / (s * 3.0f)))) / ((s * (r * ((float) M_PI))) * 6.0f));
}

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(0.25) * exp(Float32(Float32(-r) / s))) / Float32(r * Float32(s * Float32(Float32(2.0) * Float32(pi))))) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(s * Float32(3.0))))) / Float32(Float32(s * Float32(r * Float32(pi))) * Float32(6.0))))
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 = ((single(0.25) * exp((-r / s))) / (r * (s * (single(2.0) * single(pi))))) + ((single(0.75) * exp((-r / (s * single(3.0))))) / ((s * (r * single(pi))) * single(6.0)));
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. Taylor expanded in s around 0 0.1

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

3. Simplified 0.1

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

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

4. Final simplification 0.1

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

Alternatives

Alternative 1
Error	0.1
Cost	13728

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

Alternative 2
Error	0.1
Cost	13664

\[\frac{0.125}{s \cdot \pi} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{s \cdot \left(\pi \cdot 6\right)} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]

Alternative 3
Error	0.2
Cost	13440

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

Alternative 4
Error	0.2
Cost	10272

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

Alternative 5
Error	0.1
Cost	10208

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

Alternative 6
Error	0.8
Cost	10144

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

Alternative 7
Error	0.8
Cost	10144

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

Alternative 8
Error	28.2
Cost	9856

\[2 \cdot \frac{0.125}{\mathsf{log1p}\left(\mathsf{expm1}\left(s \cdot \left(r \cdot \pi\right)\right)\right)} \]

Alternative 9
Error	18.4
Cost	9856

\[2 \cdot \frac{0.125}{s \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \pi\right)\right)} \]

Alternative 10
Error	29.1
Cost	7200

\[\frac{0.125}{r \cdot \left(s \cdot \pi\right)} \cdot \left(e^{\frac{-r}{s}} + \left(1 + \frac{r}{s} \cdot \left(-0.3333333333333333 + \frac{r}{s} \cdot 0.05555555555555555\right)\right)\right) \]

Alternative 11
Error	28.9
Cost	6912

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

Alternative 12
Error	28.9
Cost	6816

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

Alternative 13
Error	28.9
Cost	6816

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

Alternative 14
Error	28.9
Cost	6816

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

Alternative 15
Error	28.9
Cost	6816

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

Alternative 16
Error	29.0
Cost	3456

\[2 \cdot \frac{\frac{0.125}{r \cdot s}}{\pi} \]

Alternative 17
Error	29.0
Cost	3456

\[2 \cdot \frac{0.125}{s \cdot \left(r \cdot \pi\right)} \]

Alternative 18
Error	29.0
Cost	3456

\[2 \cdot \frac{0.125}{\pi \cdot \left(r \cdot s\right)} \]

Alternative 19
Error	29.0
Cost	3456

\[2 \cdot \frac{\frac{0.125}{\pi}}{r \cdot s} \]

Reproduce

herbie shell --seed 2023066 
(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))))