Average Error: 6.5 → 5.8
Time: 10.5s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{y} \cdot \frac{\sqrt[3]{\frac{\sqrt[3]{1}}{x}} \cdot \sqrt[3]{\frac{\sqrt[3]{1}}{x}}}{\sqrt[3]{1 + z \cdot z} \cdot \sqrt[3]{1 + z \cdot z}}\right) \cdot \frac{\sqrt[3]{\frac{\sqrt[3]{1}}{x}}}{\sqrt[3]{1 + z \cdot z}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\left(\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{y} \cdot \frac{\sqrt[3]{\frac{\sqrt[3]{1}}{x}} \cdot \sqrt[3]{\frac{\sqrt[3]{1}}{x}}}{\sqrt[3]{1 + z \cdot z} \cdot \sqrt[3]{1 + z \cdot z}}\right) \cdot \frac{\sqrt[3]{\frac{\sqrt[3]{1}}{x}}}{\sqrt[3]{1 + z \cdot z}}
double f(double x, double y, double z) {
        double r1637 = 1.0;
        double r1638 = x;
        double r1639 = r1637 / r1638;
        double r1640 = y;
        double r1641 = z;
        double r1642 = r1641 * r1641;
        double r1643 = r1637 + r1642;
        double r1644 = r1640 * r1643;
        double r1645 = r1639 / r1644;
        return r1645;
}
↓
double f(double x, double y, double z) {
        double r1646 = 1.0;
        double r1647 = cbrt(r1646);
        double r1648 = r1647 * r1647;
        double r1649 = y;
        double r1650 = r1648 / r1649;
        double r1651 = x;
        double r1652 = r1647 / r1651;
        double r1653 = cbrt(r1652);
        double r1654 = r1653 * r1653;
        double r1655 = z;
        double r1656 = r1655 * r1655;
        double r1657 = r1646 + r1656;
        double r1658 = cbrt(r1657);
        double r1659 = r1658 * r1658;
        double r1660 = r1654 / r1659;
        double r1661 = r1650 * r1660;
        double r1662 = r1653 / r1658;
        double r1663 = r1661 * r1662;
        return r1663;
}

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

Original6.5
Target5.7
Herbie5.8
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.5

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

  3. Applied *-un-lft-identity6.5

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

  4. Applied add-cube-cbrt6.5

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

  5. Applied times-frac6.5

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

  6. Applied times-frac6.5

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

  7. Simplified6.5

    \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{y}} \cdot \frac{\frac{\sqrt[3]{1}}{x}}{1 + z \cdot z}\]
  8. Using strategy rm

  9. Applied add-cube-cbrt6.7

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

  10. Applied add-cube-cbrt7.1

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

  11. Applied times-frac7.1

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

  12. Applied associate-*r*5.8

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

  13. Final simplification5.8

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

Reproduce

herbie shell --seed 2020025 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< (* y (+ 1 (* z z))) #f) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 8.680743250567252e+305) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x))))

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