Disney BSSRDF, PDF of scattering profile

?

Percentage Accurate: 99.6% → 99.6%
Time: 11.3s
Precision: binary32
Cost: 13696

?

\[\left(0 \leq s \land s \leq 256\right) \land \left(10^{-6} < r \land r < 1000000\right)\]
\[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
↓
\[\begin{array}{l} \\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot e^{\frac{-0.3333333333333333}{\frac{s}{r}}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \end{array} \]
(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 (/ -0.3333333333333333 (/ s r)))) (* r (* s (* PI 6.0))))))
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((-0.3333333333333333f / (s / r)))) / (r * (s * (((float) M_PI) * 6.0f))));
}

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(-0.3333333333333333) / Float32(s / r)))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))))
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) * exp((-r / s))) / (r * (s * (single(2.0) * single(pi))))) + ((single(0.75) * exp((single(-0.3333333333333333) / (s / r)))) / (r * (s * (single(pi) * single(6.0)))));
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}
↓
\begin{array}{l}

\\
\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{-0.3333333333333333}{\frac{s}{r}}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)}
\end{array}

Local Percentage Accuracy vs ?

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.

Herbie found 14 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Bogosity?

Bogosity

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 99.3%

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

  2. Taylor expanded in r around 0 99.3%

    \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

  3. Simplified99.4%

    \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{\frac{-0.3333333333333333}{\frac{s}{r}}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
    Step-by-step derivation

    [Start]99.3%

    \[ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{-0.3333333333333333 \cdot \frac{r}{s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    associate-*r/ [=>]99.3%

    \[ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{\frac{-0.3333333333333333 \cdot r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

    associate-/l* [=>]99.4%

    \[ \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\color{blue}{\frac{-0.3333333333333333}{\frac{s}{r}}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

  4. Final simplification99.4%

    \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot e^{\frac{-0.3333333333333333}{\frac{s}{r}}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]

Alternatives

Alternative 1
Accuracy99.6%
Cost13696
\[\frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot e^{\frac{-0.3333333333333333}{\frac{s}{r}}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]
Alternative 2
Accuracy99.5%
Cost10208
\[\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{-r}{s}}}{r} + \frac{e^{\frac{r \cdot -0.3333333333333333}{s}}}{r}\right) \]
Alternative 3
Accuracy97.5%
Cost10144
\[\frac{\frac{0.125}{r \cdot \pi}}{s} \cdot \left(e^{\frac{-r}{s}} + e^{-0.3333333333333333 \cdot \frac{r}{s}}\right) \]
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
Accuracy99.5%
Cost10144
\[0.125 \cdot \frac{e^{\frac{-r}{s}} + e^{\frac{r \cdot -0.3333333333333333}{s}}}{s \cdot \left(r \cdot \pi\right)} \]
Alternative 6
Accuracy44.2%
Cost9792
\[\frac{0.25}{s \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(r \cdot \pi\right)\right)} \]
Alternative 7
Accuracy10.4%
Cost7264
\[\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{-r}{s}}}{r} + \left(\frac{1}{r} + \left(\frac{r \cdot 0.05555555555555555}{s \cdot s} - \frac{0.3333333333333333}{s}\right)\right)\right) \]
Alternative 8
Accuracy9.3%
Cost6880
\[\frac{0.125}{s \cdot \pi} \cdot \left(\frac{e^{\frac{-r}{s}}}{r} + \frac{1}{r}\right) \]
Alternative 9
Accuracy9.3%
Cost6880
\[\frac{0.125 \cdot \left(\frac{e^{\frac{-r}{s}}}{r} + \frac{1}{r}\right)}{s \cdot \pi} \]
Alternative 10
Accuracy9.3%
Cost6816
\[0.125 \cdot \frac{e^{\frac{-r}{s}} + 1}{\pi \cdot \left(r \cdot s\right)} \]
Alternative 11
Accuracy9.3%
Cost6816
\[\frac{0.125}{r \cdot \pi} \cdot \frac{e^{\frac{-r}{s}} + 1}{s} \]
Alternative 12
Accuracy8.9%
Cost3392
\[\frac{0.25}{r \cdot \left(s \cdot \pi\right)} \]
Alternative 13
Accuracy8.9%
Cost3392
\[\frac{0.25}{s \cdot \left(r \cdot \pi\right)} \]
Alternative 14
Accuracy8.9%
Cost3392
\[\frac{\frac{0.25}{s}}{r \cdot \pi} \]

Reproduce?

herbie shell --seed 2023167 
(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))))