Average Error: 6.2 → 6.0
Time: 6.8s
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 r305656 = 1.0;
        double r305657 = x;
        double r305658 = r305656 / r305657;
        double r305659 = y;
        double r305660 = z;
        double r305661 = r305660 * r305660;
        double r305662 = r305656 + r305661;
        double r305663 = r305659 * r305662;
        double r305664 = r305658 / r305663;
        return r305664;
}
↓
double f(double x, double y, double z) {
        double r305665 = 1.0;
        double r305666 = y;
        double r305667 = r305665 / r305666;
        double r305668 = cbrt(r305667);
        double r305669 = z;
        double r305670 = r305669 * r305669;
        double r305671 = r305665 + r305670;
        double r305672 = sqrt(r305671);
        double r305673 = r305672 / r305668;
        double r305674 = r305668 / r305673;
        double r305675 = x;
        double r305676 = r305668 / r305675;
        double r305677 = r305676 / r305672;
        double r305678 = r305674 * r305677;
        return r305678;
}

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

Original6.2
Target5.6
Herbie6.0
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.2

    \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
  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-frac6.0

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

  11. Simplified6.0

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

  12. Final simplification6.0

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

Reproduce

herbie shell --seed 2020024 
(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)))))