Average Error: 0.1 → 0.1
Time: 7.0s
Precision: binary32
\[0 \leq s \land s \leq 256 \land 10^{-6} < r \land r < 1000000\]
\[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
↓
\[\frac{1}{\frac{r \cdot \pi}{e^{\frac{-r}{s}}} \cdot \frac{s}{0.125}} + \frac{0.75 \cdot e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]
\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{1}{\frac{r \cdot \pi}{e^{\frac{-r}{s}}} \cdot \frac{s}{0.125}} + \frac{0.75 \cdot e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\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
 (+
  (/ 1.0 (* (/ (* r PI) (exp (/ (- r) s))) (/ s 0.125)))
  (/ (* 0.75 (exp (/ (- r) (* s 3.0)))) (* 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 (1.0f / (((r * ((float) M_PI)) / expf(-r / s)) * (s / 0.125f))) + ((0.75f * expf(-r / (s * 3.0f))) / (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 0.1

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

  3. Applied frac-2neg_binary320.1

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

  4. Simplified0.1

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

  5. Simplified0.1

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

  7. Applied clear-num_binary320.1

    \[\leadsto \color{blue}{\frac{1}{\frac{\left(r \cdot \pi\right) \cdot \left(s \cdot -2\right)}{e^{\frac{-r}{s}} \cdot -0.25}}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

  8. Simplified0.1

    \[\leadsto \frac{1}{\color{blue}{\frac{r \cdot \pi}{e^{\frac{-r}{s}}} \cdot \frac{s}{0.125}}} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

  9. Final simplification0.1

    \[\leadsto \frac{1}{\frac{r \cdot \pi}{e^{\frac{-r}{s}}} \cdot \frac{s}{0.125}} + \frac{0.75 \cdot e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]

Reproduce

herbie shell --seed 2021197 
(FPCore (s r)
  :name "Disney BSSRDF, PDF of scattering profile"
  :precision binary32
  :pre (and (<= 0.0 s 256.0) (< 1e-6 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))))