Average Error: 6.7 → 6.3
Time: 16.9s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\sqrt{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\frac{\frac{\sqrt{1}}{\sqrt[3]{x}}}{\sqrt[3]{y}}}{1 + z \cdot z}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\sqrt{1}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\frac{\frac{\sqrt{1}}{\sqrt[3]{x}}}{\sqrt[3]{y}}}{1 + z \cdot z}
double f(double x, double y, double z) {
        double r282938 = 1.0;
        double r282939 = x;
        double r282940 = r282938 / r282939;
        double r282941 = y;
        double r282942 = z;
        double r282943 = r282942 * r282942;
        double r282944 = r282938 + r282943;
        double r282945 = r282941 * r282944;
        double r282946 = r282940 / r282945;
        return r282946;
}
↓
double f(double x, double y, double z) {
        double r282947 = 1.0;
        double r282948 = sqrt(r282947);
        double r282949 = x;
        double r282950 = cbrt(r282949);
        double r282951 = r282950 * r282950;
        double r282952 = r282948 / r282951;
        double r282953 = y;
        double r282954 = cbrt(r282953);
        double r282955 = r282954 * r282954;
        double r282956 = r282952 / r282955;
        double r282957 = r282948 / r282950;
        double r282958 = r282957 / r282954;
        double r282959 = z;
        double r282960 = r282959 * r282959;
        double r282961 = r282947 + r282960;
        double r282962 = r282958 / r282961;
        double r282963 = r282956 * r282962;
        return r282963;
}

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

Original6.7
Target6.1
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 6.7

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

  3. Applied add-cube-cbrt7.3

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

  4. Applied times-frac7.1

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

  6. Applied associate-*r/7.4

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

  7. Simplified6.8

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

  9. Applied *-un-lft-identity6.8

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

  10. Applied add-cube-cbrt7.4

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

  11. Applied add-cube-cbrt7.6

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

  12. Applied add-sqr-sqrt7.6

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

  13. Applied times-frac7.6

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

  14. Applied times-frac7.6

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

  15. Applied times-frac6.3

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

  16. Simplified6.3

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

  17. Final simplification6.3

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

Reproduce

herbie shell --seed 2019323 
(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)))))