Average Error: 0.1 → 0.1
Time: 19.3s
Precision: 64
\[x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)\]
↓
\[x \cdot 0.5 + \left(y \cdot \mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), 1 - z\right) + y \cdot \log \left(\sqrt[3]{{z}^{\frac{2}{3}}} \cdot \sqrt[3]{\sqrt[3]{z}}\right)\right)\]
x \cdot 0.5 + y \cdot \left(\left(1 - z\right) + \log z\right)
↓
x \cdot 0.5 + \left(y \cdot \mathsf{fma}\left(2, \log \left(\sqrt[3]{z}\right), 1 - z\right) + y \cdot \log \left(\sqrt[3]{{z}^{\frac{2}{3}}} \cdot \sqrt[3]{\sqrt[3]{z}}\right)\right)
double f(double x, double y, double z) {
        double r249105 = x;
        double r249106 = 0.5;
        double r249107 = r249105 * r249106;
        double r249108 = y;
        double r249109 = 1.0;
        double r249110 = z;
        double r249111 = r249109 - r249110;
        double r249112 = log(r249110);
        double r249113 = r249111 + r249112;
        double r249114 = r249108 * r249113;
        double r249115 = r249107 + r249114;
        return r249115;
}
↓
double f(double x, double y, double z) {
        double r249116 = x;
        double r249117 = 0.5;
        double r249118 = r249116 * r249117;
        double r249119 = y;
        double r249120 = 2.0;
        double r249121 = z;
        double r249122 = cbrt(r249121);
        double r249123 = log(r249122);
        double r249124 = 1.0;
        double r249125 = r249124 - r249121;
        double r249126 = fma(r249120, r249123, r249125);
        double r249127 = r249119 * r249126;
        double r249128 = 0.6666666666666666;
        double r249129 = pow(r249121, r249128);
        double r249130 = cbrt(r249129);
        double r249131 = cbrt(r249122);
        double r249132 = r249130 * r249131;
        double r249133 = log(r249132);
        double r249134 = r249119 * r249133;
        double r249135 = r249127 + r249134;
        double r249136 = r249118 + r249135;
        return r249136;
}

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. Simplified0.1

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

  6. Applied add-cube-cbrt0.1

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

  7. Applied log-prod0.1

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

  8. Applied distribute-lft-in0.1

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

  9. Applied associate-+r+0.1

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

  10. Simplified0.1

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

  12. Applied add-cube-cbrt0.1

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

  13. Applied cbrt-prod0.1

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

  14. Simplified0.1

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

  15. Final simplification0.1

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

Reproduce

herbie shell --seed 2019208 +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)))))