Average Error: 0.2 → 0.2
Time: 7.8s
Precision: binary32
\[\left(10^{-5} \leq u \land u \leq 1\right) \land \left(0 \leq v \land v \leq 109.746574\right)\]
\[1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right) \]
↓
\[\begin{array}{l} t_0 := \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)\\ \mathsf{fma}\left(v, \sqrt[3]{t_0 \cdot t_0} \cdot \sqrt[3]{t_0}, 1\right) \end{array} \]
1 + v \cdot \log \left(u + \left(1 - u\right) \cdot e^{\frac{-2}{v}}\right)
↓
\begin{array}{l}
t_0 := \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right)\\
\mathsf{fma}\left(v, \sqrt[3]{t_0 \cdot t_0} \cdot \sqrt[3]{t_0}, 1\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 (fma (- 1.0 u) (exp (/ -2.0 v)) u))))
   (fma v (* (cbrt (* t_0 t_0)) (cbrt t_0)) 1.0)))
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(fmaf((1.0f - u), expf(-2.0f / v), u));
	return fmaf(v, (cbrtf(t_0 * t_0) * cbrtf(t_0)), 1.0f);
}

Error

Bits error versus u

Bits error versus v

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(v, \log \left(\mathsf{fma}\left(1 - u, e^{\frac{-2}{v}}, u\right)\right), 1\right)} \]

  3. Applied add-cube-cbrt_binary320.2

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

  4. Applied add-cbrt-cube_binary320.2

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

  5. Simplified0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2021329 
(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))))))))