Average Error: 6.6 → 6.1
Time: 8.8s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{y}}{\sqrt{1 + z \cdot z}} \cdot \frac{\frac{\sqrt[3]{1}}{x}}{\sqrt{1 + z \cdot z}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{y}}{\sqrt{1 + z \cdot z}} \cdot \frac{\frac{\sqrt[3]{1}}{x}}{\sqrt{1 + z \cdot z}}
double f(double x, double y, double z) {
        double r338842 = 1.0;
        double r338843 = x;
        double r338844 = r338842 / r338843;
        double r338845 = y;
        double r338846 = z;
        double r338847 = r338846 * r338846;
        double r338848 = r338842 + r338847;
        double r338849 = r338845 * r338848;
        double r338850 = r338844 / r338849;
        return r338850;
}
↓
double f(double x, double y, double z) {
        double r338851 = 1.0;
        double r338852 = cbrt(r338851);
        double r338853 = r338852 * r338852;
        double r338854 = y;
        double r338855 = r338853 / r338854;
        double r338856 = z;
        double r338857 = r338856 * r338856;
        double r338858 = r338851 + r338857;
        double r338859 = sqrt(r338858);
        double r338860 = r338855 / r338859;
        double r338861 = x;
        double r338862 = r338852 / r338861;
        double r338863 = r338862 / r338859;
        double r338864 = r338860 * r338863;
        return r338864;
}

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

Original6.6
Target5.9
Herbie6.1
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.6

    \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
  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 add-cube-cbrt6.6

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

  5. Applied times-frac6.6

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

  6. Applied times-frac6.3

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

  7. Simplified6.3

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

  9. Applied add-sqr-sqrt6.3

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

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

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

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

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

  12. Applied times-frac6.3

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

  13. Applied times-frac6.3

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

  14. Applied associate-*r*6.1

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

  15. Simplified6.1

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

  16. Final simplification6.1

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

Reproduce

herbie shell --seed 2020003 
(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)))))