Average Error: 6.4 → 3.0
Time: 3.0s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -4.87704907241193947 \cdot 10^{34}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot 1 + \left(y \cdot z\right) \cdot z}\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;y \le -4.87704907241193947 \cdot 10^{34}:\\
\;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{1 + z \cdot z}\\

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

\end{array}
double f(double x, double y, double z) {
        double r369851 = 1.0;
        double r369852 = x;
        double r369853 = r369851 / r369852;
        double r369854 = y;
        double r369855 = z;
        double r369856 = r369855 * r369855;
        double r369857 = r369851 + r369856;
        double r369858 = r369854 * r369857;
        double r369859 = r369853 / r369858;
        return r369859;
}
↓
double f(double x, double y, double z) {
        double r369860 = y;
        double r369861 = -4.8770490724119395e+34;
        bool r369862 = r369860 <= r369861;
        double r369863 = 1.0;
        double r369864 = r369863 / r369860;
        double r369865 = x;
        double r369866 = r369864 / r369865;
        double r369867 = z;
        double r369868 = r369867 * r369867;
        double r369869 = r369863 + r369868;
        double r369870 = r369866 / r369869;
        double r369871 = r369863 / r369865;
        double r369872 = r369860 * r369863;
        double r369873 = r369860 * r369867;
        double r369874 = r369873 * r369867;
        double r369875 = r369872 + r369874;
        double r369876 = r369871 / r369875;
        double r369877 = r369862 ? r369870 : r369876;
        return r369877;
}

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
Target5.6
Herbie3.0
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if y < -4.8770490724119395e+34

    1. Initial program 3.9

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

    3. Applied associate-/r*1.2

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

    4. Simplified1.2

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

    if -4.8770490724119395e+34 < y

    1. Initial program 7.2

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

    3. Applied distribute-lft-in7.2

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

    5. Applied associate-*r*3.6

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

  4. Final simplification3.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -4.87704907241193947 \cdot 10^{34}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{y \cdot 1 + \left(y \cdot z\right) \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(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)))))