Average Error: 0.2 → 0.2
Time: 12.0s
Precision: binary32
\[10^{-05} \leq u \land u \leq 1 \land 0 \leq v \land v \leq 109.746574\]
\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\]
↓
\[1 + v \cdot \sqrt[3]{{\log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}^{3}}\]
1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)
↓
1 + v \cdot \sqrt[3]{{\log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}^{3}}
(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))
↓
(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (cbrt (pow (log (+ u (* (exp (/ -2.0 v)) (- 1.0 u)))) 3.0)))))
float code(float u, float v) {
	return 1.0f + (v * logf(u + ((1.0f - u) * expf(-2.0f / v))));
}
↓
float code(float u, float v) {
	return 1.0f + (v * cbrtf(powf(logf(u + (expf(-2.0f / v) * (1.0f - u))), 3.0f)));
}

Error

Bits error versus u

Bits error versus v

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 0.2

    \[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\]
  2. Using strategy rm

  3. Applied add-cbrt-cube_binary320.2

    \[\leadsto 1 + v \cdot \color{blue}{\sqrt[3]{\left(\log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right) \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)}}\]

  4. Simplified0.2

    \[\leadsto 1 + v \cdot \sqrt[3]{\color{blue}{{\log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}^{3}}}\]

  5. Final simplification0.2

    \[\leadsto 1 + v \cdot \sqrt[3]{{\log \left(u + e^{\frac{-2}{v}} \cdot \left(1 - u\right)\right)}^{3}}\]

Reproduce

herbie shell --seed 2021175 
(FPCore (u v)
  :name "HairBSDF, sample_f, cosTheta"
  :precision binary32
  :pre (and (<= 1e-05 u 1.0) (<= 0.0 v 109.746574))
  (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))