Average Error: 0.1 → 0.1
Time: 7.2s
Precision: binary32
\[\left(0 \leq s \land s \leq 256\right) \land \left(10^{-6} < r \land r < 1000000\right)\]
\[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
↓
\[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\sqrt{r} \cdot \left(\left(2 \cdot \left(s \cdot \pi\right)\right) \cdot \sqrt{r}\right)} + \frac{0.75 \cdot e^{\frac{-r}{s \cdot 3}}}{\left(\pi \cdot 6\right) \cdot \left(r \cdot s\right)} \]
\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 \cdot e^{\frac{-r}{s}}}{\sqrt{r} \cdot \left(\left(2 \cdot \left(s \cdot \pi\right)\right) \cdot \sqrt{r}\right)} + \frac{0.75 \cdot e^{\frac{-r}{s \cdot 3}}}{\left(\pi \cdot 6\right) \cdot \left(r \cdot s\right)}

(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))) (* (sqrt r) (* (* 2.0 (* s PI)) (sqrt r))))
  (/ (* 0.75 (exp (/ (- r) (* s 3.0)))) (* (* PI 6.0) (* r s)))))
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)) / (sqrtf(r) * ((2.0f * (s * ((float) M_PI))) * sqrtf(r)))) + ((0.75f * expf(-r / (s * 3.0f))) / ((((float) M_PI) * 6.0f) * (r * s)));
}

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-*l*_binary320.1

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

  3. Applied add-exp-log_binary320.2

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

  4. Applied add-sqr-sqrt_binary320.2

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

  5. Applied associate-*r*_binary320.2

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

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2022082 
(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))))