Average Error: 0.1 → 0.1
Time: 8.1s
Precision: binary32
\[\left(0 \leq s \land s \leq 256\right) \land \left(10^{-6} < r \land r < 1000000\right)\]
\[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
↓
\[\begin{array}{l} t_0 := \sqrt{2 \cdot \left(s \cdot \left(r \cdot \pi\right)\right)}\\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{t_0 \cdot t_0} + \frac{0.75 \cdot e^{\frac{-r}{s \cdot 3}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \end{array} \]
\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{2 \cdot \left(s \cdot \left(r \cdot \pi\right)\right)}\\
\frac{0.25 \cdot e^{\frac{-r}{s}}}{t_0 \cdot t_0} + \frac{0.75 \cdot e^{\frac{-r}{s \cdot 3}}}{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 (sqrt (* 2.0 (* s (* r PI))))))
   (+
    (/ (* 0.25 (exp (/ (- r) s))) (* t_0 t_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) {
	float t_0 = sqrtf(2.0f * (s * (r * ((float) M_PI))));
	return ((0.25f * expf(-r / s)) / (t_0 * t_0)) + ((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. Applied expm1-log1p-u_binary320.1

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

  3. Applied add-sqr-sqrt_binary320.1

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

  4. Simplified0.1

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

  5. Simplified0.1

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

  6. Final simplification0.1

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

Reproduce

herbie shell --seed 2021280 
(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))))