Average Error: 6.6 → 6.5
Time: 14.1s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{y} \cdot \frac{\frac{\sqrt[3]{1}}{x}}{1 + z \cdot z}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{y} \cdot \frac{\frac{\sqrt[3]{1}}{x}}{1 + z \cdot z}
double f(double x, double y, double z) {
        double r294124 = 1.0;
        double r294125 = x;
        double r294126 = r294124 / r294125;
        double r294127 = y;
        double r294128 = z;
        double r294129 = r294128 * r294128;
        double r294130 = r294124 + r294129;
        double r294131 = r294127 * r294130;
        double r294132 = r294126 / r294131;
        return r294132;
}
↓
double f(double x, double y, double z) {
        double r294133 = 1.0;
        double r294134 = cbrt(r294133);
        double r294135 = r294134 * r294134;
        double r294136 = y;
        double r294137 = r294135 / r294136;
        double r294138 = x;
        double r294139 = r294134 / r294138;
        double r294140 = z;
        double r294141 = r294140 * r294140;
        double r294142 = r294133 + r294141;
        double r294143 = r294139 / r294142;
        double r294144 = r294137 * r294143;
        return r294144;
}

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

Original6.6
Target5.9
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.6

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

  3. Applied *-un-lft-identity6.6

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

  4. Applied add-cube-cbrt6.6

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

  5. Applied times-frac6.6

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

  6. Applied times-frac6.5

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

  7. Simplified6.5

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

  8. Final simplification6.5

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

Reproduce

herbie shell --seed 2019303 
(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))) -inf.bf) (/ (/ 1 y) (* (+ 1 (* z z)) x)) (if (< (* y (+ 1 (* z z))) 8.68074325056725162e305) (/ (/ 1 x) (* (+ 1 (* z z)) y)) (/ (/ 1 y) (* (+ 1 (* z z)) x))))

  (/ (/ 1 x) (* y (+ 1 (* z z)))))