Average Error: 6.4 → 6.7
Time: 18.7s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\frac{\frac{1}{x}}{\sqrt{1 + z \cdot z}}}{\sqrt{1 + z \cdot z}}}{y}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\frac{\frac{1}{x}}{\sqrt{1 + z \cdot z}}}{\sqrt{1 + z \cdot z}}}{y}
double f(double x, double y, double z) {
        double r258822 = 1.0;
        double r258823 = x;
        double r258824 = r258822 / r258823;
        double r258825 = y;
        double r258826 = z;
        double r258827 = r258826 * r258826;
        double r258828 = r258822 + r258827;
        double r258829 = r258825 * r258828;
        double r258830 = r258824 / r258829;
        return r258830;
}
↓
double f(double x, double y, double z) {
        double r258831 = 1.0;
        double r258832 = x;
        double r258833 = r258831 / r258832;
        double r258834 = z;
        double r258835 = r258834 * r258834;
        double r258836 = r258831 + r258835;
        double r258837 = sqrt(r258836);
        double r258838 = r258833 / r258837;
        double r258839 = r258838 / r258837;
        double r258840 = y;
        double r258841 = r258839 / r258840;
        return r258841;
}

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
Target5.8
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.4

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

  3. Applied *-un-lft-identity6.4

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

  4. Applied *-un-lft-identity6.4

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

  5. Applied times-frac6.4

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

  6. Applied times-frac6.7

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

  7. Simplified6.7

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

  9. Applied associate-*l/6.6

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

  10. Simplified6.6

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

  12. Applied add-sqr-sqrt6.7

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

  13. Applied associate-/r*6.7

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

  14. Final simplification6.7

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

Reproduce

herbie shell --seed 2019208 
(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)))))