Average Error: 0.1 → 0.2
Time: 6.5s
Precision: binary32
\[\left(0 \leq s \land s \leq 256\right) \land \left(10^{-6} < r \land r < 1000000\right)\]
\[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{r \cdot \pi}\\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot e^{\frac{\frac{-r}{3}}{s}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(s \cdot \left({t_0}^{2.5} \cdot \sqrt{t_0}\right)\right) \cdot 6\right)\right)} \end{array} \]
\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]{r \cdot \pi}\\
\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{\frac{-r}{3}}{s}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(s \cdot \left({t_0}^{2.5} \cdot \sqrt{t_0}\right)\right) \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
 (let* ((t_0 (cbrt (* r PI))))
   (+
    (/ (* 0.25 (exp (/ (- r) s))) (* r (* s (* 2.0 PI))))
    (/
     (* 0.75 (exp (/ (/ (- r) 3.0) s)))
     (expm1 (log1p (* (* s (* (pow t_0 2.5) (sqrt t_0))) 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) {
	float t_0 = cbrtf(r * ((float) M_PI));
	return ((0.25f * expf(-r / s)) / (r * (s * (2.0f * ((float) M_PI))))) + ((0.75f * expf((-r / 3.0f) / s)) / expm1f(log1pf((s * (powf(t_0, 2.5f) * sqrtf(t_0))) * 6.0f)));
}

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 associate-/r*_binary320.1

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

  3. Applied expm1-log1p-u_binary320.2

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

  4. Simplified0.2

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

  5. Applied add-cube-cbrt_binary320.2

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

  6. Applied add-sqr-sqrt_binary320.2

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

  7. Applied associate-*r*_binary320.2

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

  8. Simplified0.2

    \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{\frac{-r}{3}}{s}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(s \cdot \left(\color{blue}{{\left(\sqrt[3]{r \cdot \pi}\right)}^{2.5}} \cdot \sqrt{\sqrt[3]{\pi \cdot r}}\right)\right) \cdot 6\right)\right)} \]

  9. Final simplification0.2

    \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(s \cdot \left(2 \cdot \pi\right)\right)} + \frac{0.75 \cdot e^{\frac{\frac{-r}{3}}{s}}}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(s \cdot \left({\left(\sqrt[3]{r \cdot \pi}\right)}^{2.5} \cdot \sqrt{\sqrt[3]{r \cdot \pi}}\right)\right) \cdot 6\right)\right)} \]

Reproduce

herbie shell --seed 2022039 
(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))))