Average Error: 0.1 → 0.1
Time: 29.9s
Precision: 64
\[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)\]
↓
\[\mathsf{fma}\left(x, 0.5, \left(1 - \left(\mathsf{fma}\left(-2, \log \left(\sqrt[3]{z}\right), z\right) - \log \left(\sqrt[3]{z}\right)\right)\right) \cdot y\right)\]
x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)
↓
\mathsf{fma}\left(x, 0.5, \left(1 - \left(\mathsf{fma}\left(-2, \log \left(\sqrt[3]{z}\right), z\right) - \log \left(\sqrt[3]{z}\right)\right)\right) \cdot y\right)
double f(double x, double y, double z) {
        double r12569895 = x;
        double r12569896 = 0.5;
        double r12569897 = r12569895 * r12569896;
        double r12569898 = y;
        double r12569899 = 1.0;
        double r12569900 = z;
        double r12569901 = r12569899 - r12569900;
        double r12569902 = log(r12569900);
        double r12569903 = r12569901 + r12569902;
        double r12569904 = r12569898 * r12569903;
        double r12569905 = r12569897 + r12569904;
        return r12569905;
}
↓
double f(double x, double y, double z) {
        double r12569906 = x;
        double r12569907 = 0.5;
        double r12569908 = 1.0;
        double r12569909 = -2.0;
        double r12569910 = z;
        double r12569911 = cbrt(r12569910);
        double r12569912 = log(r12569911);
        double r12569913 = fma(r12569909, r12569912, r12569910);
        double r12569914 = r12569913 - r12569912;
        double r12569915 = r12569908 - r12569914;
        double r12569916 = y;
        double r12569917 = r12569915 * r12569916;
        double r12569918 = fma(r12569906, r12569907, r12569917);
        return r12569918;
}

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

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.5, \left(1 - \left(z - \log z\right)\right) \cdot y\right)}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, \left(1 - \left(z - \log \color{blue}{\left(\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}\right)}\right)\right) \cdot y\right)\]

  5. Applied log-prod0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, \left(1 - \left(z - \color{blue}{\left(\log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) + \log \left(\sqrt[3]{z}\right)\right)}\right)\right) \cdot y\right)\]

  6. Applied associate--r+0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, \left(1 - \color{blue}{\left(\left(z - \log \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) - \log \left(\sqrt[3]{z}\right)\right)}\right) \cdot y\right)\]

  7. Simplified0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, \left(1 - \left(\color{blue}{\mathsf{fma}\left(-2, \log \left(\sqrt[3]{z}\right), z\right)} - \log \left(\sqrt[3]{z}\right)\right)\right) \cdot y\right)\]

  8. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x, 0.5, \left(1 - \left(\mathsf{fma}\left(-2, \log \left(\sqrt[3]{z}\right), z\right) - \log \left(\sqrt[3]{z}\right)\right)\right) \cdot y\right)\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x y z)
  :name "System.Random.MWC.Distributions:gamma from mwc-random-0.13.3.2"

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

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