Average Error: 6.4 → 3.4
Time: 12.7s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \cdot z \le 4.652915945381896807258263887763215141468 \cdot 10^{233}:\\ \;\;\;\;\frac{1 \cdot \frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{z \cdot \left(y \cdot z\right)} - \frac{\frac{1}{x}}{y \cdot {z}^{4}}\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;z \cdot z \le 4.652915945381896807258263887763215141468 \cdot 10^{233}:\\
\;\;\;\;\frac{1 \cdot \frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{x}}{z \cdot \left(y \cdot z\right)} - \frac{\frac{1}{x}}{y \cdot {z}^{4}}\\

\end{array}
double f(double x, double y, double z) {
        double r239858 = 1.0;
        double r239859 = x;
        double r239860 = r239858 / r239859;
        double r239861 = y;
        double r239862 = z;
        double r239863 = r239862 * r239862;
        double r239864 = r239858 + r239863;
        double r239865 = r239861 * r239864;
        double r239866 = r239860 / r239865;
        return r239866;
}
↓
double f(double x, double y, double z) {
        double r239867 = z;
        double r239868 = r239867 * r239867;
        double r239869 = 4.652915945381897e+233;
        bool r239870 = r239868 <= r239869;
        double r239871 = 1.0;
        double r239872 = 1.0;
        double r239873 = y;
        double r239874 = r239872 / r239873;
        double r239875 = r239871 * r239874;
        double r239876 = r239871 + r239868;
        double r239877 = x;
        double r239878 = r239876 * r239877;
        double r239879 = r239875 / r239878;
        double r239880 = r239871 / r239877;
        double r239881 = r239873 * r239867;
        double r239882 = r239867 * r239881;
        double r239883 = r239880 / r239882;
        double r239884 = 4.0;
        double r239885 = pow(r239867, r239884);
        double r239886 = r239873 * r239885;
        double r239887 = r239880 / r239886;
        double r239888 = r239883 - r239887;
        double r239889 = r239870 ? r239879 : r239888;
        return r239889;
}

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

Derivation

  1. Split input into 2 regimes

  2. if (* z z) < 4.652915945381897e+233

    1. Initial program 1.6

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

    3. Applied clear-num2.0

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

    4. Simplified1.9

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

    6. Applied associate-/r*1.6

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

    7. Simplified1.6

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

    if 4.652915945381897e+233 < (* z z)

    1. Initial program 16.3

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

    3. Applied clear-num16.4

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

    4. Simplified16.6

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

    5. Taylor expanded around inf 16.3

      \[\leadsto \color{blue}{1 \cdot \frac{1}{x \cdot \left({z}^{2} \cdot y\right)} - 1 \cdot \frac{1}{x \cdot \left({z}^{4} \cdot y\right)}}\]

    6. Simplified7.2

      \[\leadsto \color{blue}{\frac{\frac{1}{x}}{z \cdot \left(z \cdot y\right)} - \frac{\frac{1}{x}}{y \cdot {z}^{4}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot z \le 4.652915945381896807258263887763215141468 \cdot 10^{233}:\\ \;\;\;\;\frac{1 \cdot \frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{z \cdot \left(y \cdot z\right)} - \frac{\frac{1}{x}}{y \cdot {z}^{4}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(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)))))