Average Error: 0.2 → 0.2
Time: 11.8s
Precision: binary32
\[10^{-05} \leq u \land u \leq 1 \land 0 \leq v \land v \leq 109.746574\]
\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
↓
\[\begin{array}{l} t_0 := \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\\ 1 + \sqrt[3]{\left(v \cdot \left(v \cdot v\right)\right) \cdot \left(t_0 \cdot \left(t_0 \cdot t_0\right)\right)} \end{array} \]
1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)
↓
\begin{array}{l}
t_0 := \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\\
1 + \sqrt[3]{\left(v \cdot \left(v \cdot v\right)\right) \cdot \left(t_0 \cdot \left(t_0 \cdot t_0\right)\right)}
\end{array}

(FPCore (u v)
 :precision binary32
 (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))
↓
(FPCore (u v)
 :precision binary32
 (let* ((t_0 (log (+ u (* (- 1.0 u) (exp (/ -2.0 v)))))))
   (+ 1.0 (cbrt (* (* v (* v v)) (* t_0 (* t_0 t_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) {
	float t_0 = logf(u + ((1.0f - u) * expf(-2.0f / v)));
	return 1.0f + cbrtf((v * (v * v)) * (t_0 * (t_0 * t_0)));
}

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. Applied add-cbrt-cube_binary320.2

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

  5. Applied cbrt-unprod_binary320.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2021190 
(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))))))))