Average Error: 14.0 → 0.5
Time: 6.8s
Precision: binary32
\[0.0001 \leq \alpha \land \alpha \leq 1 \land 2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\]
\[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9620054960250854:\\ \;\;\;\;\begin{array}{l} t_0 := \sqrt[3]{\log \left(1 - u0\right)}\\ t_0 \cdot \left(\left(t_0 \cdot t_0\right) \cdot \left(\alpha \cdot \left(-\alpha\right)\right)\right) \end{array}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_1 := \sqrt{u0 \cdot \left(u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)}\\ \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(t_1 \cdot t_1 + 0.25 \cdot {u0}^{4}\right)\right) \end{array}\\ \end{array} \]
\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)
↓
\begin{array}{l}
\mathbf{if}\;1 - u0 \leq 0.9620054960250854:\\
\;\;\;\;\begin{array}{l}
t_0 := \sqrt[3]{\log \left(1 - u0\right)}\\
t_0 \cdot \left(\left(t_0 \cdot t_0\right) \cdot \left(\alpha \cdot \left(-\alpha\right)\right)\right)
\end{array}\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_1 := \sqrt{u0 \cdot \left(u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)}\\
\left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(t_1 \cdot t_1 + 0.25 \cdot {u0}^{4}\right)\right)
\end{array}\\


\end{array}

(FPCore (alpha u0)
 :precision binary32
 (* (* (- alpha) alpha) (log (- 1.0 u0))))
↓
(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9620054960250854)
   (let* ((t_0 (cbrt (log (- 1.0 u0)))))
     (* t_0 (* (* t_0 t_0) (* alpha (- alpha)))))
   (let* ((t_1 (sqrt (* u0 (* u0 (+ 0.5 (* u0 0.3333333333333333)))))))
     (* (* alpha alpha) (+ u0 (+ (* t_1 t_1) (* 0.25 (pow u0 4.0))))))))
float code(float alpha, float u0) {
	return (-alpha * alpha) * logf(1.0f - u0);
}
↓
float code(float alpha, float u0) {
	float tmp;
	if ((1.0f - u0) <= 0.9620054960250854f) {
		float t_0_1 = cbrtf(logf(1.0f - u0));
		tmp = t_0_1 * ((t_0_1 * t_0_1) * (alpha * -alpha));
	} else {
		float t_1 = sqrtf(u0 * (u0 * (0.5f + (u0 * 0.3333333333333333f))));
		tmp = (alpha * alpha) * (u0 + ((t_1 * t_1) + (0.25f * powf(u0, 4.0f))));
	}
	return tmp;
}

Error

Bits error versus alpha

Bits error versus u0

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (-.f32 1 u0) < 0.962005496

    1. Initial program 1.1

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
    2. Using strategy rm

    3. Applied add-cube-cbrt_binary321.3

      \[\leadsto \left(\left(-\alpha\right) \cdot \alpha\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\log \left(1 - u0\right)} \cdot \sqrt[3]{\log \left(1 - u0\right)}\right) \cdot \sqrt[3]{\log \left(1 - u0\right)}\right)} \]

    4. Applied associate-*r*_binary321.3

      \[\leadsto \color{blue}{\left(\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \left(\sqrt[3]{\log \left(1 - u0\right)} \cdot \sqrt[3]{\log \left(1 - u0\right)}\right)\right) \cdot \sqrt[3]{\log \left(1 - u0\right)}} \]

    5. Simplified1.3

      \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\log \left(1 - u0\right)} \cdot \sqrt[3]{\log \left(1 - u0\right)}\right) \cdot \left(\alpha \cdot \left(-\alpha\right)\right)\right)} \cdot \sqrt[3]{\log \left(1 - u0\right)} \]

    if 0.962005496 < (-.f32 1 u0)

    1. Initial program 16.3

      \[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]

    2. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left({u0}^{3} \cdot {\alpha}^{2}\right) + \left(0.5 \cdot \left({u0}^{2} \cdot {\alpha}^{2}\right) + \left(0.25 \cdot \left({u0}^{4} \cdot {\alpha}^{2}\right) + u0 \cdot {\alpha}^{2}\right)\right)} \]

    3. Simplified0.4

      \[\leadsto \color{blue}{\left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(\left(u0 \cdot u0\right) \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right) + 0.25 \cdot {u0}^{4}\right)\right)} \]
    4. Using strategy rm

    5. Applied add-sqr-sqrt_binary320.4

      \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(\color{blue}{\sqrt{\left(u0 \cdot u0\right) \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)} \cdot \sqrt{\left(u0 \cdot u0\right) \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)}} + 0.25 \cdot {u0}^{4}\right)\right) \]

    6. Simplified0.4

      \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(\color{blue}{\sqrt{u0 \cdot \left(u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)}} \cdot \sqrt{\left(u0 \cdot u0\right) \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)} + 0.25 \cdot {u0}^{4}\right)\right) \]

    7. Simplified0.4

      \[\leadsto \left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(\sqrt{u0 \cdot \left(u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)} \cdot \color{blue}{\sqrt{u0 \cdot \left(u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)}} + 0.25 \cdot {u0}^{4}\right)\right) \]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9620054960250854:\\ \;\;\;\;\sqrt[3]{\log \left(1 - u0\right)} \cdot \left(\left(\sqrt[3]{\log \left(1 - u0\right)} \cdot \sqrt[3]{\log \left(1 - u0\right)}\right) \cdot \left(\alpha \cdot \left(-\alpha\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\alpha \cdot \alpha\right) \cdot \left(u0 + \left(\sqrt{u0 \cdot \left(u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)} \cdot \sqrt{u0 \cdot \left(u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)} + 0.25 \cdot {u0}^{4}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021204 
(FPCore (alpha u0)
  :name "Beckmann Distribution sample, tan2theta, alphax == alphay"
  :precision binary32
  :pre (and (<= 0.0001 alpha 1.0) (<= 2.328306437e-10 u0 1.0))
  (* (* (- alpha) alpha) (log (- 1.0 u0))))