Average Error: 14.2 → 0.5
Time: 5.2s
Precision: binary32
\[0.0001 \leq \alpha \land \alpha \leq 1 \land 2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\]
\[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
↓
\[\begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9647257328033447:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\alpha \cdot \left(\alpha \cdot \left(u0 + \left(u0 \cdot \left(u0 \cdot \left(\log \left(e^{u0 \cdot 0.3333333333333333}\right) + 0.5\right)\right) + 0.25 \cdot {u0}^{4}\right)\right)\right)\\ \end{array} \]
\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)
↓
\begin{array}{l}
\mathbf{if}\;1 - u0 \leq 0.9647257328033447:\\
\;\;\;\;\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\alpha \cdot \left(\alpha \cdot \left(u0 + \left(u0 \cdot \left(u0 \cdot \left(\log \left(e^{u0 \cdot 0.3333333333333333}\right) + 0.5\right)\right) + 0.25 \cdot {u0}^{4}\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) 0.9647257328033447)
   (* (- alpha) (* alpha (log (- 1.0 u0))))
   (*
    alpha
    (*
     alpha
     (+
      u0
      (+
       (* u0 (* u0 (+ (log (exp (* u0 0.3333333333333333))) 0.5)))
       (* 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) <= 0.9647257328033447f) {
		tmp = -alpha * (alpha * logf(1.0f - u0));
	} else {
		tmp = alpha * (alpha * (u0 + ((u0 * (u0 * (logf(expf(u0 * 0.3333333333333333f)) + 0.5f))) + (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) < 0.964725733

    1. Initial program 1.1

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

    3. Applied associate-*l*_binary321.1

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

    4. Simplified1.1

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

    if 0.964725733 < (-.f32 1 u0)

    1. Initial program 16.5

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

    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. Simplified0.3

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

    5. Applied associate-*l*_binary320.3

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

    6. Simplified0.3

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

    8. Applied add-log-exp_binary320.3

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u0 \leq 0.9647257328033447:\\ \;\;\;\;\left(-\alpha\right) \cdot \left(\alpha \cdot \log \left(1 - u0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\alpha \cdot \left(\alpha \cdot \left(u0 + \left(u0 \cdot \left(u0 \cdot \left(\log \left(e^{u0 \cdot 0.3333333333333333}\right) + 0.5\right)\right) + 0.25 \cdot {u0}^{4}\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021198 
(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))))