Disney BSSRDF, PDF of scattering profile

Average Error: 0.45% → 0.5%

Time: 12.6s

Precision: binary32

Cost: 20128

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

Math

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

↓

\[\frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot e^{\frac{-r}{s \cdot 3}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(r \cdot \pi\right) \cdot \left(s \cdot 6\right)\right)\right)} \]

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
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (* r (* s (* 2.0 PI))))
  (/
   (* 0.75 (exp (/ (- r) (* s 3.0))))
   (expm1 (log1p (* (* r PI) (* s 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)))) / expm1f(log1pf(((r * ((float) M_PI)) * (s * 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))))) / expm1(log1p(Float32(Float32(r * Float32(pi)) * Float32(s * Float32(6.0)))))))
end

Derivation

1. Initial program 0.45

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

2. Taylor expanded in s around 0 0.47

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

3. Simplified 0.47

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

   [Start] 0.47	\[ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{6 \cdot \left(s \cdot \left(r \cdot \pi\right)\right)} \]
   associate-*r* [=>] 0.46	\[ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\color{blue}{\left(6 \cdot s\right) \cdot \left(r \cdot \pi\right)}} \]
   *-commutative [=>] 0.46	\[ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\color{blue}{\left(s \cdot 6\right)} \cdot \left(r \cdot \pi\right)} \]
   associate-*l* [=>] 0.47	\[ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\color{blue}{s \cdot \left(6 \cdot \left(r \cdot \pi\right)\right)}} \]
   *-commutative [=>] 0.47	\[ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{s \cdot \left(6 \cdot \color{blue}{\left(\pi \cdot r\right)}\right)} \]

4. Applied egg-rr 0.5

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\pi \cdot r\right) \cdot \left(s \cdot 6\right)\right)\right)}} \]

5. Final simplification 0.5

   \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot e^{\frac{-r}{s \cdot 3}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(r \cdot \pi\right) \cdot \left(s \cdot 6\right)\right)\right)} \]

Alternatives

Alternative 1
Error	0.47%
Cost	13728

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

Alternative 2
Error	0.46%
Cost	13728

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

Alternative 3
Error	0.45%
Cost	13728

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

Alternative 4
Error	2.5%
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 5
Error	0.5%
Cost	10144

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

Alternative 6
Error	56%
Cost	9792

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

Alternative 7
Error	90.42%
Cost	6816

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

Alternative 8
Error	90.42%
Cost	6816

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

Alternative 9
Error	90.42%
Cost	6816

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

Alternative 10
Error	90.42%
Cost	6816

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

Alternative 11
Error	90.93%
Cost	3392

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

Alternative 12
Error	90.93%
Cost	3392

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

Alternative 13
Error	90.93%
Cost	3392

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

Alternative 14
Error	90.93%
Cost	3392

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

Alternative 15
Error	90.93%
Cost	3392

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

Reproduce

herbie shell --seed 2023089 
(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))))