Average Error: 0.1 → 0.1
Time: 6.0s
Precision: binary32
\[\left(0 \leq s \land s \leq 256\right) \land \left(10^{-6} < r \land r < 1000000\right)\]
\[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\sqrt{r}}\\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot e^{\frac{-r}{s \cdot 3}}}{t_0 \cdot \left(\left(\left(\pi \cdot 6\right) \cdot \left(s \cdot \sqrt[3]{r \cdot r}\right)\right) \cdot t_0\right)} \end{array} \]
\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}
t_0 := \sqrt[3]{\sqrt{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}}}{t_0 \cdot \left(\left(\left(\pi \cdot 6\right) \cdot \left(s \cdot \sqrt[3]{r \cdot r}\right)\right) \cdot t_0\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
 (let* ((t_0 (cbrt (sqrt r))))
   (+
    (/ (* 0.25 (exp (/ (- r) s))) (* r (* s (* 2.0 PI))))
    (/
     (* 0.75 (exp (/ (- r) (* s 3.0))))
     (* t_0 (* (* (* PI 6.0) (* s (cbrt (* r r)))) t_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) {
	float t_0 = cbrtf(sqrtf(r));
	return ((0.25f * expf((-r / s))) / (r * (s * (2.0f * ((float) M_PI))))) + ((0.75f * expf((-r / (s * 3.0f)))) / (t_0 * (((((float) M_PI) * 6.0f) * (s * cbrtf((r * r)))) * t_0)));
}

Error

Bits error versus s

Bits error versus r

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Applied add-cube-cbrt_binary320.2

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

  3. Applied associate-*r*_binary320.1

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

  4. Applied add-sqr-sqrt_binary320.1

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

  5. Applied cbrt-prod_binary320.1

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

  6. Applied associate-*r*_binary320.1

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

  7. Applied associate-*l*_binary320.1

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

  8. Simplified0.1

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

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2022125 
(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))))