Average Error: 0.1 → 0.2
Time: 3.8s
Precision: binary32
\[\left(0 \leq s \land s \leq 256\right) \land \left(10^{-6} < r \land r < 1000000\right)\]
\[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{s \cdot \pi}\\ \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(t_0 \cdot \left(2 \cdot {t_0}^{2}\right)\right)} + \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \end{array} \]
(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 (* s PI))))
   (+
    (/ (* 0.25 (exp (/ (- r) s))) (* r (* t_0 (* 2.0 (pow t_0 2.0)))))
    (/
     (* 0.75 (exp (* r (/ -0.3333333333333333 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((s * ((float) M_PI)));
	return ((0.25f * expf((-r / s))) / (r * (t_0 * (2.0f * powf(t_0, 2.0f))))) + ((0.75f * expf((r * (-0.3333333333333333f / s)))) / (r * (s * (((float) M_PI) * 6.0f))));
}

function code(s, r)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(Float32(Float32(Float32(2.0) * Float32(pi)) * s) * r)) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(Float32(3.0) * s)))) / Float32(Float32(Float32(Float32(6.0) * Float32(pi)) * s) * r)))
end
↓
function code(s, r)
	t_0 = cbrt(Float32(s * Float32(pi)))
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(r * Float32(t_0 * Float32(Float32(2.0) * (t_0 ^ Float32(2.0)))))) + Float32(Float32(Float32(0.75) * exp(Float32(r * Float32(Float32(-0.3333333333333333) / s)))) / Float32(r * Float32(s * Float32(Float32(pi) * Float32(6.0))))))
end

\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]{s \cdot \pi}\\
\frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(t_0 \cdot \left(2 \cdot {t_0}^{2}\right)\right)} + \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)}
\end{array}

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 egg-rr0.2

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

  3. Taylor expanded in r around 0 0.2

    \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{{\left(\sqrt[3]{s \cdot \left(\pi \cdot 2\right)}\right)}^{3} \cdot r} + \frac{0.75 \cdot e^{\color{blue}{-0.3333333333333333 \cdot \frac{r}{s}}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

  4. Simplified0.2

    \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{{\left(\sqrt[3]{s \cdot \left(\pi \cdot 2\right)}\right)}^{3} \cdot r} + \frac{0.75 \cdot e^{\color{blue}{\frac{-0.3333333333333333}{s} \cdot r}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

  5. Applied egg-rr0.2

    \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\color{blue}{e^{\log \left(2 \cdot \left(s \cdot \pi\right)\right)}} \cdot r} + \frac{0.75 \cdot e^{\frac{-0.3333333333333333}{s} \cdot r}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

  6. Applied egg-rr0.2

    \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{\color{blue}{\left(\left(2 \cdot {\left(\sqrt[3]{s \cdot \pi}\right)}^{2}\right) \cdot \sqrt[3]{s \cdot \pi}\right)} \cdot r} + \frac{0.75 \cdot e^{\frac{-0.3333333333333333}{s} \cdot r}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]

  7. Final simplification0.2

    \[\leadsto \frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot \left(\sqrt[3]{s \cdot \pi} \cdot \left(2 \cdot {\left(\sqrt[3]{s \cdot \pi}\right)}^{2}\right)\right)} + \frac{0.75 \cdot e^{r \cdot \frac{-0.3333333333333333}{s}}}{r \cdot \left(s \cdot \left(\pi \cdot 6\right)\right)} \]

Reproduce

herbie shell --seed 2022166 
(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))))