Average Error: 14.8 → 2.7
Time: 18.1s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\left(\frac{x}{z} \cdot \left(\frac{1}{\sqrt[3]{z + 1} \cdot \sqrt[3]{z + 1}} \cdot \frac{y}{\sqrt[3]{z + 1}}\right)\right) \cdot \frac{1}{z}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\left(\frac{x}{z} \cdot \left(\frac{1}{\sqrt[3]{z + 1} \cdot \sqrt[3]{z + 1}} \cdot \frac{y}{\sqrt[3]{z + 1}}\right)\right) \cdot \frac{1}{z}
double f(double x, double y, double z) {
        double r258767 = x;
        double r258768 = y;
        double r258769 = r258767 * r258768;
        double r258770 = z;
        double r258771 = r258770 * r258770;
        double r258772 = 1.0;
        double r258773 = r258770 + r258772;
        double r258774 = r258771 * r258773;
        double r258775 = r258769 / r258774;
        return r258775;
}
↓
double f(double x, double y, double z) {
        double r258776 = x;
        double r258777 = z;
        double r258778 = r258776 / r258777;
        double r258779 = 1.0;
        double r258780 = 1.0;
        double r258781 = r258777 + r258780;
        double r258782 = cbrt(r258781);
        double r258783 = r258782 * r258782;
        double r258784 = r258779 / r258783;
        double r258785 = y;
        double r258786 = r258785 / r258782;
        double r258787 = r258784 * r258786;
        double r258788 = r258778 * r258787;
        double r258789 = r258779 / r258777;
        double r258790 = r258788 * r258789;
        return r258790;
}

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

Original14.8
Target4.1
Herbie2.7
\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307077115547144785523414612:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 14.8

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

  3. Applied times-frac10.9

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

  4. Simplified10.9

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

  6. Applied sqr-pow10.9

    \[\leadsto \frac{x}{\color{blue}{{z}^{\left(\frac{2}{2}\right)} \cdot {z}^{\left(\frac{2}{2}\right)}}} \cdot \frac{y}{z + 1}\]

  7. Applied *-un-lft-identity10.9

    \[\leadsto \frac{\color{blue}{1 \cdot x}}{{z}^{\left(\frac{2}{2}\right)} \cdot {z}^{\left(\frac{2}{2}\right)}} \cdot \frac{y}{z + 1}\]

  8. Applied times-frac6.1

    \[\leadsto \color{blue}{\left(\frac{1}{{z}^{\left(\frac{2}{2}\right)}} \cdot \frac{x}{{z}^{\left(\frac{2}{2}\right)}}\right)} \cdot \frac{y}{z + 1}\]

  9. Applied associate-*l*2.5

    \[\leadsto \color{blue}{\frac{1}{{z}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{x}{{z}^{\left(\frac{2}{2}\right)}} \cdot \frac{y}{z + 1}\right)}\]

  10. Simplified2.5

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

  12. Applied add-cube-cbrt2.7

    \[\leadsto \frac{1}{{z}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{x}{z} \cdot \frac{y}{\color{blue}{\left(\sqrt[3]{z + 1} \cdot \sqrt[3]{z + 1}\right) \cdot \sqrt[3]{z + 1}}}\right)\]

  13. Applied *-un-lft-identity2.7

    \[\leadsto \frac{1}{{z}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{x}{z} \cdot \frac{\color{blue}{1 \cdot y}}{\left(\sqrt[3]{z + 1} \cdot \sqrt[3]{z + 1}\right) \cdot \sqrt[3]{z + 1}}\right)\]

  14. Applied times-frac2.7

    \[\leadsto \frac{1}{{z}^{\left(\frac{2}{2}\right)}} \cdot \left(\frac{x}{z} \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{z + 1} \cdot \sqrt[3]{z + 1}} \cdot \frac{y}{\sqrt[3]{z + 1}}\right)}\right)\]

  15. Final simplification2.7

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

Reproduce

herbie shell --seed 2019209 +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 249.618281453230708) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1 z)) x) z))

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