Average Error: 6.2 → 6.3
Time: 13.6s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\sqrt[3]{1}}{y}}{\mathsf{fma}\left(z, z, 1\right) \cdot x} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\sqrt[3]{1}}{y}}{\mathsf{fma}\left(z, z, 1\right) \cdot x} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)
double f(double x, double y, double z) {
        double r170798 = 1.0;
        double r170799 = x;
        double r170800 = r170798 / r170799;
        double r170801 = y;
        double r170802 = z;
        double r170803 = r170802 * r170802;
        double r170804 = r170798 + r170803;
        double r170805 = r170801 * r170804;
        double r170806 = r170800 / r170805;
        return r170806;
}
↓
double f(double x, double y, double z) {
        double r170807 = 1.0;
        double r170808 = cbrt(r170807);
        double r170809 = y;
        double r170810 = r170808 / r170809;
        double r170811 = z;
        double r170812 = fma(r170811, r170811, r170807);
        double r170813 = x;
        double r170814 = r170812 * r170813;
        double r170815 = r170810 / r170814;
        double r170816 = r170808 * r170808;
        double r170817 = r170815 * r170816;
        return r170817;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original6.2
Target5.5
Herbie6.3
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.2

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

  3. Applied div-inv6.2

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

  4. Applied times-frac6.2

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

  5. Simplified6.2

    \[\leadsto \frac{1}{y} \cdot \color{blue}{\frac{\frac{1}{x}}{\mathsf{fma}\left(z, z, 1\right)}}\]
  6. Using strategy rm

  7. Applied *-un-lft-identity6.2

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

  8. Applied add-cube-cbrt6.2

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

  9. Applied times-frac6.2

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

  10. Applied associate-*l*6.2

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

  11. Simplified6.3

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

  12. Final simplification6.3

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

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(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)))))