Average Error: 6.4 → 5.0
Time: 9.3s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -38289380947.143463:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(y \cdot \sqrt{1 + z \cdot z}\right) \cdot \sqrt{1 + z \cdot z}}\\ \mathbf{elif}\;x \le 1.7144269134100035 \cdot 10^{68}:\\ \;\;\;\;\frac{\sqrt{1}}{y} \cdot \frac{\frac{\sqrt{1}}{x}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{1 + z \cdot z}\\ \end{array}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\begin{array}{l}
\mathbf{if}\;x \le -38289380947.143463:\\
\;\;\;\;\frac{\frac{1}{x}}{\left(y \cdot \sqrt{1 + z \cdot z}\right) \cdot \sqrt{1 + z \cdot z}}\\

\mathbf{elif}\;x \le 1.7144269134100035 \cdot 10^{68}:\\
\;\;\;\;\frac{\sqrt{1}}{y} \cdot \frac{\frac{\sqrt{1}}{x}}{1 + z \cdot z}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{1 + z \cdot z}\\

\end{array}
double f(double x, double y, double z) {
        double r232674 = 1.0;
        double r232675 = x;
        double r232676 = r232674 / r232675;
        double r232677 = y;
        double r232678 = z;
        double r232679 = r232678 * r232678;
        double r232680 = r232674 + r232679;
        double r232681 = r232677 * r232680;
        double r232682 = r232676 / r232681;
        return r232682;
}
↓
double f(double x, double y, double z) {
        double r232683 = x;
        double r232684 = -38289380947.14346;
        bool r232685 = r232683 <= r232684;
        double r232686 = 1.0;
        double r232687 = r232686 / r232683;
        double r232688 = y;
        double r232689 = z;
        double r232690 = r232689 * r232689;
        double r232691 = r232686 + r232690;
        double r232692 = sqrt(r232691);
        double r232693 = r232688 * r232692;
        double r232694 = r232693 * r232692;
        double r232695 = r232687 / r232694;
        double r232696 = 1.7144269134100035e+68;
        bool r232697 = r232683 <= r232696;
        double r232698 = sqrt(r232686);
        double r232699 = r232698 / r232688;
        double r232700 = r232698 / r232683;
        double r232701 = r232700 / r232691;
        double r232702 = r232699 * r232701;
        double r232703 = r232686 / r232688;
        double r232704 = r232703 / r232683;
        double r232705 = r232704 / r232691;
        double r232706 = r232697 ? r232702 : r232705;
        double r232707 = r232685 ? r232695 : r232706;
        return r232707;
}

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
Herbie5.0
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -38289380947.14346

    1. Initial program 1.3

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

    3. Applied add-sqr-sqrt1.3

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

    4. Applied associate-*r*1.3

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

    if -38289380947.14346 < x < 1.7144269134100035e+68

    1. Initial program 11.1

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

    3. Applied *-un-lft-identity11.1

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

    4. Applied add-sqr-sqrt11.1

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

    5. Applied times-frac11.1

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

    6. Applied times-frac8.5

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

    7. Simplified8.5

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

    if 1.7144269134100035e+68 < x

    1. Initial program 0.7

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

    3. Applied associate-/r*0.7

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

    4. Simplified0.7

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

  4. Final simplification5.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -38289380947.143463:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(y \cdot \sqrt{1 + z \cdot z}\right) \cdot \sqrt{1 + z \cdot z}}\\ \mathbf{elif}\;x \le 1.7144269134100035 \cdot 10^{68}:\\ \;\;\;\;\frac{\sqrt{1}}{y} \cdot \frac{\frac{\sqrt{1}}{x}}{1 + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{1}{y}}{x}}{1 + z \cdot z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(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)))))