Average Error: 0.1 → 0.1
Time: 6.6s
Precision: binary32
\[0 \leq s \land s \leq 256 \land 10^{-6} < r \land r < 1000000\]
\[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
↓
\[\frac{0.25 \cdot e^{\frac{-r}{s}}}{{\left(\sqrt[3]{2 \cdot \left(s \cdot \left(r \cdot \pi\right)\right)}\right)}^{3}} + \frac{0.75 \cdot e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]
\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{0.25 \cdot e^{\frac{-r}{s}}}{{\left(\sqrt[3]{2 \cdot \left(s \cdot \left(r \cdot \pi\right)\right)}\right)}^{3}} + \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
 (+
  (/ (* 0.25 (exp (/ (- r) s))) (pow (cbrt (* 2.0 (* s (* r PI)))) 3.0))
  (/ (* 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 ((0.25f * expf(-r / s)) / powf(cbrtf(2.0f * (s * (r * ((float) M_PI)))), 3.0f)) + ((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 add-cube-cbrt_binary320.1

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

  4. Simplified0.1

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

  5. Simplified0.1

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

  7. Applied pow1_binary320.1

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

  8. Applied pow1_binary320.1

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

  9. Applied pow1_binary320.1

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

  10. Applied pow-sqr_binary320.1

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

  11. Applied pow-prod-up_binary320.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2021204 
(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))))