Average Error: 14.2 → 0.5
Time: 9.3s
Precision: binary32
\[0.0001 \leq \alpha \land \alpha \leq 1 \land 2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\]
\[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9502819776535034:\\ \;\;\;\;\begin{array}{l} t_0 := \log \left(1 - u0\right)\\ \sqrt[3]{\left(\left(\alpha \cdot \left(-\alpha\right)\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)\right)\right) \cdot \left(t_0 \cdot \left(t_0 \cdot t_0\right)\right)} \end{array}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(\alpha \cdot \alpha\right) + \alpha \cdot \left(\alpha \cdot \left(0.25 \cdot {u0}^{4} + \left(u0 \cdot u0\right) \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)\right)\\ \end{array} \]
\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)
↓
\begin{array}{l}
\mathbf{if}\;1 - u0 \leq 0.9502819776535034:\\
\;\;\;\;\begin{array}{l}
t_0 := \log \left(1 - u0\right)\\
\sqrt[3]{\left(\left(\alpha \cdot \left(-\alpha\right)\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)\right)\right) \cdot \left(t_0 \cdot \left(t_0 \cdot t_0\right)\right)}
\end{array}\\

\mathbf{else}:\\
\;\;\;\;u0 \cdot \left(\alpha \cdot \alpha\right) + \alpha \cdot \left(\alpha \cdot \left(0.25 \cdot {u0}^{4} + \left(u0 \cdot u0\right) \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)\right)\\


\end{array}

(FPCore (alpha u0)
 :precision binary32
 (* (* (- alpha) alpha) (log (- 1.0 u0))))
↓
(FPCore (alpha u0)
 :precision binary32
 (if (<= (- 1.0 u0) 0.9502819776535034)
   (let* ((t_0 (log (- 1.0 u0))))
     (cbrt
      (*
       (* (* alpha (- alpha)) (* (* alpha alpha) (* alpha alpha)))
       (* t_0 (* t_0 t_0)))))
   (+
    (* u0 (* alpha alpha))
    (*
     alpha
     (*
      alpha
      (+
       (* 0.25 (pow u0 4.0))
       (* (* u0 u0) (+ 0.5 (* u0 0.3333333333333333)))))))))
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) <= 0.9502819776535034f) {
		float t_0_1 = logf(1.0f - u0);
		tmp = cbrtf(((alpha * -alpha) * ((alpha * alpha) * (alpha * alpha))) * (t_0_1 * (t_0_1 * t_0_1)));
	} else {
		tmp = (u0 * (alpha * alpha)) + (alpha * (alpha * ((0.25f * powf(u0, 4.0f)) + ((u0 * u0) * (0.5f + (u0 * 0.3333333333333333f))))));
	}
	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) < 0.950281978

    1. Initial program 1.0

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

    3. Applied add-cbrt-cube_binary321.0

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

    4. Applied add-cbrt-cube_binary321.0

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

    5. Applied cbrt-unprod_binary321.0

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

    if 0.950281978 < (-.f32 1 u0)

    1. Initial program 16.3

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

    2. Taylor expanded in u0 around 0 0.4

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

    3. Simplified0.4

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

    4. Taylor expanded in u0 around 0 0.4

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

    5. Simplified0.4

      \[\leadsto \color{blue}{\alpha \cdot \left(\alpha \cdot \left(u0 + u0 \cdot \left(0.25 \cdot {u0}^{3} + u0 \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)\right)\right)} \]
    6. Using strategy rm

    7. Applied distribute-rgt-in_binary320.4

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

    8. Applied distribute-rgt-in_binary320.4

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

    9. Simplified0.4

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

    10. Simplified0.4

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9502819776535034:\\ \;\;\;\;\sqrt[3]{\left(\left(\alpha \cdot \left(-\alpha\right)\right) \cdot \left(\left(\alpha \cdot \alpha\right) \cdot \left(\alpha \cdot \alpha\right)\right)\right) \cdot \left(\log \left(1 - u0\right) \cdot \left(\log \left(1 - u0\right) \cdot \log \left(1 - u0\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;u0 \cdot \left(\alpha \cdot \alpha\right) + \alpha \cdot \left(\alpha \cdot \left(0.25 \cdot {u0}^{4} + \left(u0 \cdot u0\right) \cdot \left(0.5 + u0 \cdot 0.3333333333333333\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021206 
(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))))