Disney BSSRDF, PDF of scattering profile

?

Percentage Accurate: 99.5% → 99.5%
Time: 16.5s
Precision: binary32
Cost: 13728

?

\[\left(0 \leq s \land s \leq 256\right) \land \left(10^{-6} < r \land r < 1000000\right)\]
\[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
↓
\[\frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{0.75}{s \cdot \left(\pi \cdot 6\right)} \cdot \frac{e^{\frac{-r}{s \cdot 3}}}{r} \]
(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 (* s (* 2.0 PI))) (/ (exp (/ (- r) s)) r))
  (* (/ 0.75 (* s (* PI 6.0))) (/ (exp (/ (- r) (* s 3.0))) r))))
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 / (s * (2.0f * ((float) M_PI)))) * (expf((-r / s)) / r)) + ((0.75f / (s * (((float) M_PI) * 6.0f))) * (expf((-r / (s * 3.0f))) / r));
}

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) / Float32(s * Float32(Float32(2.0) * Float32(pi)))) * Float32(exp(Float32(Float32(-r) / s)) / r)) + Float32(Float32(Float32(0.75) / Float32(s * Float32(Float32(pi) * Float32(6.0)))) * Float32(exp(Float32(Float32(-r) / Float32(s * Float32(3.0)))) / r)))
end

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) / (s * (single(2.0) * single(pi)))) * (exp((-r / s)) / r)) + ((single(0.75) / (s * (single(pi) * single(6.0)))) * (exp((-r / (s * single(3.0)))) / r));
end

\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}{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}

Local Percentage Accuracy?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

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Results

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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. Simplified99.6%

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

    [Start]99.6

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

    times-frac [=>]99.7

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

    *-commutative [=>]99.7

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

    times-frac [=>]99.6

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

    *-commutative [=>]99.6

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

    *-commutative [=>]99.6

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

    distribute-frac-neg [=>]99.6

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

    *-commutative [=>]99.6

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

  3. Final simplification99.6%

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

Alternatives

Alternative 1
Accuracy99.5%
Cost13728
\[\frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]
Alternative 2
Accuracy99.5%
Cost13664
\[\frac{0.25}{s \cdot \left(2 \cdot \pi\right)} \cdot \frac{e^{\frac{-r}{s}}}{r} + \frac{e^{\frac{-r}{s \cdot 3}}}{r} \cdot \frac{\frac{0.125}{s}}{\pi} \]
Alternative 3
Accuracy99.5%
Cost10208
\[\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{-r}{s}}}{r} + \frac{e^{\frac{-0.3333333333333333}{\frac{s}{r}}}}{r}\right) \]
Alternative 4
Accuracy99.5%
Cost10144
\[0.125 \cdot \frac{e^{\frac{-r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}}{s \cdot \left(r \cdot \pi\right)} \]
Alternative 5
Accuracy50.6%
Cost9892
\[\begin{array}{l} \mathbf{if}\;r \leq 1.5:\\ \;\;\;\;0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}} + \left(1 + \frac{r}{s} \cdot \left(-1 + \frac{r}{s} \cdot 0.5\right)\right)}{s \cdot \left(r \cdot \pi\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.25}{s \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \left(-\pi\right)\right)\right)}\\ \end{array} \]
Alternative 6
Accuracy44.2%
Cost9792
\[\frac{0.25}{s \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \pi\right)\right)} \]
Alternative 7
Accuracy10.0%
Cost7232
\[0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}} + \left(1 + \frac{r}{s} \cdot \left(-1 + \frac{r}{s} \cdot 0.5\right)\right)}{s \cdot \left(r \cdot \pi\right)} \]
Alternative 8
Accuracy9.7%
Cost7072
\[\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{-r}{s}}}{r} + \frac{-0.3333333333333333 \cdot \frac{r}{s} + 1}{r}\right) \]
Alternative 9
Accuracy9.1%
Cost6976
\[0.125 \cdot \frac{e^{-0.3333333333333333 \cdot \frac{r}{s}} + \left(1 - \frac{r}{s}\right)}{s \cdot \left(r \cdot \pi\right)} \]
Alternative 10
Accuracy9.0%
Cost6880
\[\frac{0.25}{s} \cdot \frac{1}{r \cdot \pi} + \frac{-0.16666666666666666}{s \cdot \left(s \cdot \pi\right)} \]
Alternative 11
Accuracy9.5%
Cost6816
\[0.125 \cdot \frac{e^{\frac{-r}{s}} + 1}{s \cdot \left(r \cdot \pi\right)} \]
Alternative 12
Accuracy9.0%
Cost6816
\[\frac{-0.16666666666666666}{s \cdot \left(s \cdot \pi\right)} + \frac{0.25}{s \cdot \left(r \cdot \pi\right)} \]
Alternative 13
Accuracy9.0%
Cost6816
\[\frac{-0.16666666666666666}{s \cdot \left(s \cdot \pi\right)} + \frac{\frac{0.25}{s}}{r \cdot \pi} \]
Alternative 14
Accuracy9.0%
Cost3392
\[\frac{0.25}{s \cdot \left(r \cdot \pi\right)} \]
Alternative 15
Accuracy9.0%
Cost3392
\[\frac{0.25}{\pi \cdot \left(r \cdot s\right)} \]
Alternative 16
Accuracy9.0%
Cost3392
\[\frac{\frac{\frac{0.25}{r}}{s}}{\pi} \]

Error

Reproduce?

herbie shell --seed 2023160 
(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))))