Average Error: 0.1 → 0.1
Time: 19.2s
Precision: binary64
\[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]
↓
\[\begin{array}{l} t_0 := \log \left(\sqrt{z}\right)\\ \mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(y, t_0 + \left(t_0 - z\right), y\right)\right) \end{array} \]
x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)
↓
\begin{array}{l}
t_0 := \log \left(\sqrt{z}\right)\\
\mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(y, t_0 + \left(t_0 - z\right), y\right)\right)
\end{array}

(FPCore (x y z) :precision binary64 (+ (* x 0.5) (* y (+ (- 1.0 z) (log z)))))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (log (sqrt z)))) (fma x 0.5 (fma y (+ t_0 (- t_0 z)) y))))
double code(double x, double y, double z) {
	return (x * 0.5) + (y * ((1.0 - z) + log(z)));
}
↓
double code(double x, double y, double z) {
	double t_0 = log(sqrt(z));
	return fma(x, 0.5, fma(y, (t_0 + (t_0 - z)), y));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.1
Target0.1
Herbie0.1
\[\left(y + 0.5 \cdot x\right) - y \cdot \left(z - \log z\right) \]

Derivation

  1. Initial program 0.1

    \[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right) \]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(y, \log z - z, y\right)\right)} \]

  3. Applied add-sqr-sqrt_binary640.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(y, \log \color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right)} - z, y\right)\right) \]

  4. Applied log-prod_binary640.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(y, \color{blue}{\left(\log \left(\sqrt{z}\right) + \log \left(\sqrt{z}\right)\right)} - z, y\right)\right) \]

  5. Applied associate--l+_binary640.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(y, \color{blue}{\log \left(\sqrt{z}\right) + \left(\log \left(\sqrt{z}\right) - z\right)}, y\right)\right) \]

  6. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, \mathsf{fma}\left(y, \log \left(\sqrt{z}\right) + \left(\log \left(\sqrt{z}\right) - z\right), y\right)\right) \]

Reproduce

herbie shell --seed 2021280 
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"
  :precision binary64

  :herbie-target
  (- (+ y (* 0.5 x)) (* y (- z (log z))))

  (+ (* x 0.5) (* y (+ (- 1.0 z) (log z)))))