Average Error: 15.0 → 1.4
Time: 11.1s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \frac{\frac{\sqrt[3]{x}}{\frac{z}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}}{\frac{1 + z}{\sqrt[3]{y}}}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \frac{\frac{\sqrt[3]{x}}{\frac{z}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}}{\frac{1 + z}{\sqrt[3]{y}}}
double f(double x, double y, double z) {
        double r307671 = x;
        double r307672 = y;
        double r307673 = r307671 * r307672;
        double r307674 = z;
        double r307675 = r307674 * r307674;
        double r307676 = 1.0;
        double r307677 = r307674 + r307676;
        double r307678 = r307675 * r307677;
        double r307679 = r307673 / r307678;
        return r307679;
}
↓
double f(double x, double y, double z) {
        double r307680 = x;
        double r307681 = cbrt(r307680);
        double r307682 = r307681 * r307681;
        double r307683 = z;
        double r307684 = r307682 / r307683;
        double r307685 = y;
        double r307686 = cbrt(r307685);
        double r307687 = r307686 * r307686;
        double r307688 = r307683 / r307687;
        double r307689 = r307681 / r307688;
        double r307690 = 1.0;
        double r307691 = r307690 + r307683;
        double r307692 = r307691 / r307686;
        double r307693 = r307689 / r307692;
        double r307694 = r307684 * r307693;
        return r307694;
}

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

Original15.0
Target4.1
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;z \lt 249.618281453230708:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 15.0

    \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
  2. Using strategy rm

  3. Applied associate-/l*13.2

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

  4. Simplified11.6

    \[\leadsto \frac{x}{\color{blue}{{z}^{2} \cdot \frac{z + 1}{y}}}\]
  5. Using strategy rm

  6. Applied unpow211.6

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

  7. Applied associate-*l*7.6

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

  8. Simplified8.1

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

  10. Applied add-cube-cbrt8.5

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

  11. Applied times-frac4.6

    \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \frac{\sqrt[3]{x}}{\frac{z \cdot \left(1 + z\right)}{y}}}\]
  12. Using strategy rm

  13. Applied add-cube-cbrt4.7

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

  14. Applied times-frac2.9

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

  15. Applied associate-/r*1.4

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

  16. Final simplification1.4

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

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

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

  (/ (* x y) (* (* z z) (+ z 1))))