Average Error: 0.2 → 0.2
Time: 9.3s
Precision: binary32
\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]
\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{\sqrt{v}}\\ 1 + \left(t_0 \cdot \left(t_0 \cdot \sqrt[3]{v}\right)\right) \cdot \left(\sqrt[3]{v} \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)\right) \end{array} \]
1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)
↓
\begin{array}{l}
t_0 := \sqrt[3]{\sqrt{v}}\\
1 + \left(t_0 \cdot \left(t_0 \cdot \sqrt[3]{v}\right)\right) \cdot \left(\sqrt[3]{v} \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\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 (cbrt (sqrt v))))
   (+
    1.0
    (*
     (* t_0 (* t_0 (cbrt v)))
     (* (cbrt v) (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))))
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 = cbrtf(sqrtf(v));
	return 1.0f + ((t_0 * (t_0 * cbrtf(v))) * (cbrtf(v) * logf(u + ((1.0f - u) * expf(-2.0f / v)))));
}

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. Applied add-cube-cbrt_binary320.2

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

  3. Applied associate-*l*_binary320.2

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

  4. Applied add-sqr-sqrt_binary320.2

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

  5. Applied cbrt-prod_binary320.2

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

  6. Applied associate-*l*_binary320.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2022061 
(FPCore (u v)
  :name "HairBSDF, sample_f, cosTheta"
  :precision binary32
  :pre (and (and (<= 1e-5 u) (<= u 1.0)) (and (<= 0.0 v) (<= v 109.746574)))
  (+ 1.0 (* v (log (+ u (* (- 1.0 u) (exp (/ -2.0 v))))))))