Disney BSSRDF, PDF of scattering profile

Average Accuracy: 99.6% → 99.6%

Time: 15.8s

Precision: binary32

Cost: 13728

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

Math

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

↓

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

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)))) (* r (* s (* 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)))) / (r * (s * (((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(r * Float32(s * Float32(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))))) / (r * (s * (single(pi) * single(6.0)))));
end

Derivation

1. Initial program 99.6%

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

2. Final simplification 99.6%

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

Alternatives

Alternative 1
Accuracy	99.6%
Cost	13728

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

Alternative 2
Accuracy	99.6%
Cost	13728

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

Alternative 3
Accuracy	99.6%
Cost	13664

\[\frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{e^{\frac{-r}{s \cdot 3}}}{r} \cdot \frac{0.125}{s \cdot \pi} \]

Alternative 4
Accuracy	99.6%
Cost	13664

\[\frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{e^{\frac{-r}{s \cdot 3}}}{r} \cdot \frac{\frac{0.125}{s}}{\pi} \]

Alternative 5
Accuracy	99.6%
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
Accuracy	49.9%
Cost	9892

\[\begin{array}{l} \mathbf{if}\;r \leq 0.5:\\ \;\;\;\;\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{-r}{s}}}{r} + \frac{\frac{r}{s} \cdot -0.3333333333333333 + 1}{r}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \left(-\pi\right)\right)\right)}\\ \end{array} \]

Alternative 7
Accuracy	44.7%
Cost	9792

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

Alternative 8
Accuracy	9.3%
Cost	6944

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

Alternative 9
Accuracy	9.3%
Cost	6880

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

Alternative 10
Accuracy	9.3%
Cost	6816

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

Alternative 11
Accuracy	8.8%
Cost	3456

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

Alternative 12
Accuracy	8.8%
Cost	3456

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

Alternative 13
Accuracy	8.8%
Cost	3392

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

Alternative 14
Accuracy	8.8%
Cost	3392

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

Alternative 15
Accuracy	8.8%
Cost	3392

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

Alternative 16
Accuracy	8.8%
Cost	3392

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

Alternative 17
Accuracy	8.8%
Cost	3392

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

Alternative 18
Accuracy	8.8%
Cost	3392

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

Alternative 19
Accuracy	8.8%
Cost	3392

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

Reproduce

herbie shell --seed 2023142 
(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))))