Average Error: 0.2 → 0.2
Time: 14.7s
Precision: binary32
\[\left(0 \leq s \land s \leq 256\right) \land \left(10^{-6} < r \land r < 1000000\right)\]
\[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\left(2 \cdot \pi\right) \cdot \left(r \cdot s\right)}\\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{t_0 \cdot \left(t_0 \cdot t_0\right)} + \frac{0.75 \cdot e^{\frac{\frac{-r}{3}}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \end{array} \]
\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]{\left(2 \cdot \pi\right) \cdot \left(r \cdot s\right)}\\
\frac{0.25 \cdot e^{\frac{-r}{s}}}{t_0 \cdot \left(t_0 \cdot t_0\right)} + \frac{0.75 \cdot e^{\frac{\frac{-r}{3}}{s}}}{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
 (let* ((t_0 (cbrt (* (* 2.0 PI) (* r s)))))
   (+
    (/ (* 0.25 (exp (/ (- r) s))) (* t_0 (* t_0 t_0)))
    (/ (* 0.75 (exp (/ (/ (- r) 3.0) s))) (* 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) {
	float t_0 = cbrtf((2.0f * ((float) M_PI)) * (r * s));
	return ((0.25f * expf(-r / s)) / (t_0 * (t_0 * t_0))) + ((0.75f * expf((-r / 3.0f) / s)) / (r * (s * (((float) M_PI) * 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 0.2

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

  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 associate-*l*_binary320.1

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

  4. Applied add-cube-cbrt_binary320.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2021275 
(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))))