Average Error: 6.3 → 6.1
Time: 6.4s
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 r369144 = 1.0;
        double r369145 = x;
        double r369146 = r369144 / r369145;
        double r369147 = y;
        double r369148 = z;
        double r369149 = r369148 * r369148;
        double r369150 = r369144 + r369149;
        double r369151 = r369147 * r369150;
        double r369152 = r369146 / r369151;
        return r369152;
}
↓
double f(double x, double y, double z) {
        double r369153 = 1.0;
        double r369154 = y;
        double r369155 = r369153 / r369154;
        double r369156 = cbrt(r369155);
        double r369157 = z;
        double r369158 = r369157 * r369157;
        double r369159 = r369153 + r369158;
        double r369160 = sqrt(r369159);
        double r369161 = r369160 / r369156;
        double r369162 = r369156 / r369161;
        double r369163 = x;
        double r369164 = r369156 / r369163;
        double r369165 = r369164 / r369160;
        double r369166 = r369162 * r369165;
        return r369166;
}

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)))))