Average Error: 6.8 → 6.6
Time: 3.2s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}
double f(double x, double y, double z) {
        double r266845 = 1.0;
        double r266846 = x;
        double r266847 = r266845 / r266846;
        double r266848 = y;
        double r266849 = z;
        double r266850 = r266849 * r266849;
        double r266851 = r266845 + r266850;
        double r266852 = r266848 * r266851;
        double r266853 = r266847 / r266852;
        return r266853;
}
↓
double f(double x, double y, double z) {
        double r266854 = 1.0;
        double r266855 = y;
        double r266856 = r266854 / r266855;
        double r266857 = z;
        double r266858 = r266857 * r266857;
        double r266859 = r266854 + r266858;
        double r266860 = x;
        double r266861 = r266859 * r266860;
        double r266862 = r266856 / r266861;
        return r266862;
}

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

Original6.8
Target6.0
Herbie6.6
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.8

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

  3. Applied associate-/r*6.9

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

  4. Simplified6.9

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

  6. Applied div-inv7.0

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

  7. Applied associate-/l*6.7

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

  8. Simplified6.6

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

  9. Final simplification6.6

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

Reproduce

herbie shell --seed 2020036 
(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)))))