Average Error: 6.3 → 6.1
Time: 6.6s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\sqrt[3]{\frac{1}{y}}}{\frac{\sqrt{1 + z \cdot z}}{\sqrt[3]{\frac{1}{y}}}} \cdot \frac{\frac{\sqrt[3]{\frac{1}{y}}}{x}}{\sqrt{1 + z \cdot z}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\sqrt[3]{\frac{1}{y}}}{\frac{\sqrt{1 + z \cdot z}}{\sqrt[3]{\frac{1}{y}}}} \cdot \frac{\frac{\sqrt[3]{\frac{1}{y}}}{x}}{\sqrt{1 + z \cdot z}}
double f(double x, double y, double z) {
        double r431035 = 1.0;
        double r431036 = x;
        double r431037 = r431035 / r431036;
        double r431038 = y;
        double r431039 = z;
        double r431040 = r431039 * r431039;
        double r431041 = r431035 + r431040;
        double r431042 = r431038 * r431041;
        double r431043 = r431037 / r431042;
        return r431043;
}
↓
double f(double x, double y, double z) {
        double r431044 = 1.0;
        double r431045 = y;
        double r431046 = r431044 / r431045;
        double r431047 = cbrt(r431046);
        double r431048 = z;
        double r431049 = r431048 * r431048;
        double r431050 = r431044 + r431049;
        double r431051 = sqrt(r431050);
        double r431052 = r431051 / r431047;
        double r431053 = r431047 / r431052;
        double r431054 = x;
        double r431055 = r431047 / r431054;
        double r431056 = r431055 / r431051;
        double r431057 = r431053 * r431056;
        return r431057;
}

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

Original6.3
Target5.7
Herbie6.1
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.3

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

  3. Applied associate-/r*6.6

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

  4. Simplified6.6

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

  6. Applied add-sqr-sqrt6.6

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

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

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

  8. Applied add-cube-cbrt7.2

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

  9. Applied times-frac7.2

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

  10. Applied times-frac6.1

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

  11. Simplified6.1

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

  12. Final simplification6.1

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

Reproduce

herbie shell --seed 2019353 
(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)))))