Average Error: 14.2 → 1.6
Time: 18.8s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1.0\right)}\]
↓
\[\frac{\frac{\frac{1}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}}{\frac{z}{\frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}}}}}}{z + 1.0}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1.0\right)}
↓
\frac{\frac{\frac{1}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}}{\frac{z}{\frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}}}}}}{z + 1.0}
double f(double x, double y, double z) {
        double r19009887 = x;
        double r19009888 = y;
        double r19009889 = r19009887 * r19009888;
        double r19009890 = z;
        double r19009891 = r19009890 * r19009890;
        double r19009892 = 1.0;
        double r19009893 = r19009890 + r19009892;
        double r19009894 = r19009891 * r19009893;
        double r19009895 = r19009889 / r19009894;
        return r19009895;
}
↓
double f(double x, double y, double z) {
        double r19009896 = 1.0;
        double r19009897 = z;
        double r19009898 = cbrt(r19009897);
        double r19009899 = r19009898 * r19009898;
        double r19009900 = y;
        double r19009901 = cbrt(r19009900);
        double r19009902 = r19009901 * r19009901;
        double r19009903 = r19009899 / r19009902;
        double r19009904 = r19009896 / r19009903;
        double r19009905 = x;
        double r19009906 = r19009898 / r19009901;
        double r19009907 = r19009905 / r19009906;
        double r19009908 = r19009897 / r19009907;
        double r19009909 = r19009904 / r19009908;
        double r19009910 = 1.0;
        double r19009911 = r19009897 + r19009910;
        double r19009912 = r19009909 / r19009911;
        return r19009912;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.2
Target3.9
Herbie1.6
\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 14.2

    \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1.0\right)}\]
  2. Using strategy rm

  3. Applied times-frac10.4

    \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1.0}}\]
  4. Using strategy rm

  5. Applied *-un-lft-identity10.4

    \[\leadsto \frac{\color{blue}{1 \cdot x}}{z \cdot z} \cdot \frac{y}{z + 1.0}\]

  6. Applied times-frac5.5

    \[\leadsto \color{blue}{\left(\frac{1}{z} \cdot \frac{x}{z}\right)} \cdot \frac{y}{z + 1.0}\]

  7. Applied associate-*l*2.5

    \[\leadsto \color{blue}{\frac{1}{z} \cdot \left(\frac{x}{z} \cdot \frac{y}{z + 1.0}\right)}\]
  8. Using strategy rm

  9. Applied associate-*r/3.0

    \[\leadsto \frac{1}{z} \cdot \color{blue}{\frac{\frac{x}{z} \cdot y}{z + 1.0}}\]

  10. Applied associate-*r/3.0

    \[\leadsto \color{blue}{\frac{\frac{1}{z} \cdot \left(\frac{x}{z} \cdot y\right)}{z + 1.0}}\]

  11. Simplified3.0

    \[\leadsto \frac{\color{blue}{\frac{\frac{x}{\frac{z}{y}}}{z}}}{z + 1.0}\]
  12. Using strategy rm

  13. Applied add-cube-cbrt3.5

    \[\leadsto \frac{\frac{\frac{x}{\frac{z}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}}}{z}}{z + 1.0}\]

  14. Applied add-cube-cbrt3.6

    \[\leadsto \frac{\frac{\frac{x}{\frac{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}}{z}}{z + 1.0}\]

  15. Applied times-frac3.6

    \[\leadsto \frac{\frac{\frac{x}{\color{blue}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{y}}}}}{z}}{z + 1.0}\]

  16. Applied *-un-lft-identity3.6

    \[\leadsto \frac{\frac{\frac{\color{blue}{1 \cdot x}}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{y}}}}{z}}{z + 1.0}\]

  17. Applied times-frac2.6

    \[\leadsto \frac{\frac{\color{blue}{\frac{1}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}}}}}{z}}{z + 1.0}\]

  18. Applied associate-/l*1.6

    \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}}{\frac{z}{\frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}}}}}}}{z + 1.0}\]

  19. Final simplification1.6

    \[\leadsto \frac{\frac{\frac{1}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}}{\frac{z}{\frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}}}}}}{z + 1.0}\]

Reproduce

herbie shell --seed 2019164 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1 z)) x) z))

  (/ (* x y) (* (* z z) (+ z 1.0))))