Average Error: 0.1 → 0.1
Time: 15.1s
Precision: 64
\[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)\]
↓
\[x \cdot 0.5 + \left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{\sqrt{z}} \cdot \sqrt[3]{\sqrt{z}}\right), 1 - z\right) \cdot y + \log \left({\left(\frac{1}{z}\right)}^{\frac{-1}{3}}\right) \cdot y\right)\]
x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)
↓
x \cdot 0.5 + \left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{\sqrt{z}} \cdot \sqrt[3]{\sqrt{z}}\right), 1 - z\right) \cdot y + \log \left({\left(\frac{1}{z}\right)}^{\frac{-1}{3}}\right) \cdot y\right)
double f(double x, double y, double z) {
        double r159842 = x;
        double r159843 = 0.5;
        double r159844 = r159842 * r159843;
        double r159845 = y;
        double r159846 = 1.0;
        double r159847 = z;
        double r159848 = r159846 - r159847;
        double r159849 = log(r159847);
        double r159850 = r159848 + r159849;
        double r159851 = r159845 * r159850;
        double r159852 = r159844 + r159851;
        return r159852;
}
↓
double f(double x, double y, double z) {
        double r159853 = x;
        double r159854 = 0.5;
        double r159855 = r159853 * r159854;
        double r159856 = 2.0;
        double r159857 = z;
        double r159858 = sqrt(r159857);
        double r159859 = cbrt(r159858);
        double r159860 = r159859 * r159859;
        double r159861 = log(r159860);
        double r159862 = 1.0;
        double r159863 = r159862 - r159857;
        double r159864 = fma(r159856, r159861, r159863);
        double r159865 = y;
        double r159866 = r159864 * r159865;
        double r159867 = 1.0;
        double r159868 = r159867 / r159857;
        double r159869 = -0.3333333333333333;
        double r159870 = pow(r159868, r159869);
        double r159871 = log(r159870);
        double r159872 = r159871 * r159865;
        double r159873 = r159866 + r159872;
        double r159874 = r159855 + r159873;
        return r159874;
}

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. Using strategy rm

  3. Applied distribute-lft-in0.1

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

  5. Applied add-cube-cbrt0.1

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

  6. Applied log-prod0.1

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

  7. Applied distribute-rgt-in0.1

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

  8. Applied associate-+r+0.1

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

  9. Simplified0.1

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

  10. Taylor expanded around inf 0.1

    \[\leadsto x \cdot 0.5 + \left(\mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), 1 - z\right) \cdot y + \log \color{blue}{\left({\left(\frac{1}{z}\right)}^{\frac{-1}{3}}\right)} \cdot y\right)\]
  11. Using strategy rm

  12. Applied add-sqr-sqrt0.1

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

  13. Applied cbrt-prod0.1

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

  14. Final simplification0.1

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

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(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 z) (log z)))))