Average Error: 6.4 → 3.8
Time: 2.8m
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 8.76578475530872594577852830720147433255 \cdot 10^{297}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{\sqrt[3]{y}}}{z \cdot z + 1} \cdot \frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{z \cdot \left(y \cdot z\right)}\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;z \cdot z \le 8.76578475530872594577852830720147433255 \cdot 10^{297}:\\
\;\;\;\;\frac{\frac{\frac{1}{x}}{\sqrt[3]{y}}}{z \cdot z + 1} \cdot \frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{x}}{z \cdot \left(y \cdot z\right)}\\

\end{array}
double f(double x, double y, double z) {
        double r18358901 = 1.0;
        double r18358902 = x;
        double r18358903 = r18358901 / r18358902;
        double r18358904 = y;
        double r18358905 = z;
        double r18358906 = r18358905 * r18358905;
        double r18358907 = r18358901 + r18358906;
        double r18358908 = r18358904 * r18358907;
        double r18358909 = r18358903 / r18358908;
        return r18358909;
}
↓
double f(double x, double y, double z) {
        double r18358910 = z;
        double r18358911 = r18358910 * r18358910;
        double r18358912 = 8.765784755308726e+297;
        bool r18358913 = r18358911 <= r18358912;
        double r18358914 = 1.0;
        double r18358915 = x;
        double r18358916 = r18358914 / r18358915;
        double r18358917 = y;
        double r18358918 = cbrt(r18358917);
        double r18358919 = r18358916 / r18358918;
        double r18358920 = 1.0;
        double r18358921 = r18358911 + r18358920;
        double r18358922 = r18358919 / r18358921;
        double r18358923 = r18358918 * r18358918;
        double r18358924 = r18358920 / r18358923;
        double r18358925 = r18358922 * r18358924;
        double r18358926 = r18358920 / r18358915;
        double r18358927 = r18358917 * r18358910;
        double r18358928 = r18358910 * r18358927;
        double r18358929 = r18358926 / r18358928;
        double r18358930 = r18358913 ? r18358925 : r18358929;
        return r18358930;
}

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

Original6.4
Target5.7
Herbie3.8
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* z z) < 8.765784755308726e+297

    1. Initial program 2.1

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

    3. Applied associate-/r*2.0

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

    5. Applied *-un-lft-identity2.0

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

    6. Applied add-cube-cbrt2.8

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

    7. Applied div-inv2.8

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

    8. Applied times-frac2.8

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

    9. Applied times-frac2.3

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

    10. Simplified2.3

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

    if 8.765784755308726e+297 < (* z z)

    1. Initial program 17.7

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

    3. Applied associate-/r*18.1

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

    4. Taylor expanded around inf 17.7

      \[\leadsto \color{blue}{\frac{1}{x \cdot \left({z}^{2} \cdot y\right)}}\]

    5. Simplified7.6

      \[\leadsto \color{blue}{\frac{\frac{1}{x}}{\left(y \cdot z\right) \cdot z}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 8.76578475530872594577852830720147433255 \cdot 10^{297}:\\ \;\;\;\;\frac{\frac{\frac{1}{x}}{\sqrt[3]{y}}}{z \cdot z + 1} \cdot \frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{z \cdot \left(y \cdot z\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"

  :herbie-target
  (if (< (* y (+ 1.0 (* z z))) -inf.0) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x)) (if (< (* y (+ 1.0 (* z z))) 8.680743250567252e+305) (/ (/ 1.0 x) (* (+ 1.0 (* z z)) y)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x))))

  (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))