Average Error: 6.9 → 6.4
Time: 4.5s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\left|\sqrt[3]{1 + z \cdot z}\right| \cdot y} \cdot \frac{\frac{\frac{\sqrt[3]{1}}{\sqrt[3]{x}}}{\sqrt{1 + z \cdot z}}}{\sqrt{\sqrt[3]{1 + z \cdot z}}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\left|\sqrt[3]{1 + z \cdot z}\right| \cdot y} \cdot \frac{\frac{\frac{\sqrt[3]{1}}{\sqrt[3]{x}}}{\sqrt{1 + z \cdot z}}}{\sqrt{\sqrt[3]{1 + z \cdot z}}}
double f(double x, double y, double z) {
        double r317863 = 1.0;
        double r317864 = x;
        double r317865 = r317863 / r317864;
        double r317866 = y;
        double r317867 = z;
        double r317868 = r317867 * r317867;
        double r317869 = r317863 + r317868;
        double r317870 = r317866 * r317869;
        double r317871 = r317865 / r317870;
        return r317871;
}
↓
double f(double x, double y, double z) {
        double r317872 = 1.0;
        double r317873 = cbrt(r317872);
        double r317874 = r317873 * r317873;
        double r317875 = x;
        double r317876 = cbrt(r317875);
        double r317877 = r317876 * r317876;
        double r317878 = r317874 / r317877;
        double r317879 = z;
        double r317880 = r317879 * r317879;
        double r317881 = r317872 + r317880;
        double r317882 = cbrt(r317881);
        double r317883 = fabs(r317882);
        double r317884 = y;
        double r317885 = r317883 * r317884;
        double r317886 = r317878 / r317885;
        double r317887 = r317873 / r317876;
        double r317888 = sqrt(r317881);
        double r317889 = r317887 / r317888;
        double r317890 = sqrt(r317882);
        double r317891 = r317889 / r317890;
        double r317892 = r317886 * r317891;
        return r317892;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

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Results

Enter valid numbers for all inputs

Target

Original6.9
Target6.0
Herbie6.4
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.9

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

  3. Applied add-sqr-sqrt6.9

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

  4. Applied associate-*r*6.9

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

  6. Applied associate-/r*6.3

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

  8. Applied add-cube-cbrt6.4

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

  9. Applied sqrt-prod6.4

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

  10. Applied add-cube-cbrt6.9

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

  11. Applied add-cube-cbrt6.9

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

  12. Applied times-frac6.9

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

  13. Applied times-frac6.7

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

  14. Applied times-frac6.6

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

  15. Simplified6.4

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

  16. Final simplification6.4

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

Reproduce

herbie shell --seed 2020018 +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)))))