Average Error: 6.6 → 6.5
Time: 18.6s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x} \cdot {\left(\sqrt[3]{1}\right)}^{3}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x} \cdot {\left(\sqrt[3]{1}\right)}^{3}
double f(double x, double y, double z) {
        double r418737 = 1.0;
        double r418738 = x;
        double r418739 = r418737 / r418738;
        double r418740 = y;
        double r418741 = z;
        double r418742 = r418741 * r418741;
        double r418743 = r418737 + r418742;
        double r418744 = r418740 * r418743;
        double r418745 = r418739 / r418744;
        return r418745;
}
↓
double f(double x, double y, double z) {
        double r418746 = 1.0;
        double r418747 = y;
        double r418748 = r418746 / r418747;
        double r418749 = 1.0;
        double r418750 = z;
        double r418751 = r418750 * r418750;
        double r418752 = r418749 + r418751;
        double r418753 = x;
        double r418754 = r418752 * r418753;
        double r418755 = r418748 / r418754;
        double r418756 = cbrt(r418749);
        double r418757 = 3.0;
        double r418758 = pow(r418756, r418757);
        double r418759 = r418755 * r418758;
        return r418759;
}

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

Original6.6
Target5.9
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.6

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

  3. Applied *-un-lft-identity6.6

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

  4. Applied *-un-lft-identity6.6

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

  5. Applied times-frac6.6

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

  6. Applied times-frac6.5

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

  7. Simplified6.5

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

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

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

  10. Applied add-cube-cbrt6.5

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

  11. Applied times-frac6.5

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

  12. Applied associate-/l*6.6

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

  13. Final simplification6.5

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

Reproduce

herbie shell --seed 2019303 
(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))) -inf.bf) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 8.68074325056725162e305) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x))))

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