Average Error: 0.1 → 0.2
Time: 3.9s
Precision: binary32
\[\left(0 \leq s \land s \leq 256\right) \land \left(10^{-6} < r \land r < 1000000\right)\]
\[\frac{0.25 \cdot e^{\frac{-r}{s}}}{\left(\left(2 \cdot \pi\right) \cdot s\right) \cdot r} + \frac{0.75 \cdot e^{\frac{-r}{3 \cdot s}}}{\left(\left(6 \cdot \pi\right) \cdot s\right) \cdot r} \]
↓
\[\frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot {\left(\sqrt[3]{\pi \cdot \left(s \cdot 2\right)}\right)}^{3}} + \frac{0.75 \cdot e^{\frac{-r}{s \cdot 3}}}{6 \cdot \left(s \cdot \left(r \cdot \pi\right)\right)} \]
(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))) (* r (pow (cbrt (* PI (* s 2.0))) 3.0)))
  (/ (* 0.75 (exp (/ (- r) (* s 3.0)))) (* 6.0 (* s (* r PI))))))
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))) / (r * powf(cbrtf((((float) M_PI) * (s * 2.0f))), 3.0f))) + ((0.75f * expf((-r / (s * 3.0f)))) / (6.0f * (s * (r * ((float) M_PI)))));
}

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)
	return Float32(Float32(Float32(Float32(0.25) * exp(Float32(Float32(-r) / s))) / Float32(r * (cbrt(Float32(Float32(pi) * Float32(s * Float32(2.0)))) ^ Float32(3.0)))) + Float32(Float32(Float32(0.75) * exp(Float32(Float32(-r) / Float32(s * Float32(3.0))))) / Float32(Float32(6.0) * Float32(s * Float32(r * Float32(pi))))))
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}
↓
\frac{0.25 \cdot e^{\frac{-r}{s}}}{r \cdot {\left(\sqrt[3]{\pi \cdot \left(s \cdot 2\right)}\right)}^{3}} + \frac{0.75 \cdot e^{\frac{-r}{s \cdot 3}}}{6 \cdot \left(s \cdot \left(r \cdot \pi\right)\right)}

Error

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

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

  3. Taylor expanded in s around 0 0.2

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

  4. Final simplification0.2

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

Reproduce

herbie shell --seed 2022211 
(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))))