Average Error: 13.9 → 0.3
Time: 4.5s
Precision: binary32
\[\left(0.0001 \leq \alpha \land \alpha \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u0 \land u0 \leq 1\right)\]
\[\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right) \]
↓
\[\mathsf{log1p}\left(-u0\right) \cdot \left(\alpha \cdot \left(-\alpha\right)\right) \]
\left(\left(-\alpha\right) \cdot \alpha\right) \cdot \log \left(1 - u0\right)
↓
\mathsf{log1p}\left(-u0\right) \cdot \left(\alpha \cdot \left(-\alpha\right)\right)

(FPCore (alpha u0)
 :precision binary32
 (* (* (- alpha) alpha) (log (- 1.0 u0))))
↓
(FPCore (alpha u0) :precision binary32 (* (log1p (- u0)) (* alpha (- alpha))))
float code(float alpha, float u0) {
	return (-alpha * alpha) * logf(1.0f - u0);
}
↓
float code(float alpha, float u0) {
	return log1pf(-u0) * (alpha * -alpha);
}

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. Initial program 13.9

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

  2. Simplified0.3

    \[\leadsto \color{blue}{\left(\alpha \cdot \left(-\alpha\right)\right) \cdot \mathsf{log1p}\left(-u0\right)} \]

  3. Applied associate-*l*_binary320.3

    \[\leadsto \color{blue}{\alpha \cdot \left(\left(-\alpha\right) \cdot \mathsf{log1p}\left(-u0\right)\right)} \]

  4. Taylor expanded in alpha around 0 13.9

    \[\leadsto \color{blue}{-1 \cdot \left(\log \left(1 - u0\right) \cdot {\alpha}^{2}\right)} \]

  5. Simplified0.3

    \[\leadsto \color{blue}{\mathsf{log1p}\left(-u0\right) \cdot \left(\alpha \cdot \left(-\alpha\right)\right)} \]

  6. Final simplification0.3

    \[\leadsto \mathsf{log1p}\left(-u0\right) \cdot \left(\alpha \cdot \left(-\alpha\right)\right) \]

Reproduce

herbie shell --seed 2021280 
(FPCore (alpha u0)
  :name "Beckmann Distribution sample, tan2theta, alphax == alphay"
  :precision binary32
  :pre (and (and (<= 0.0001 alpha) (<= alpha 1.0)) (and (<= 2.328306437e-10 u0) (<= u0 1.0)))
  (* (* (- alpha) alpha) (log (- 1.0 u0))))