Average Error: 6.8 → 6.6
Time: 2.9s
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 r359041 = 1.0;
        double r359042 = x;
        double r359043 = r359041 / r359042;
        double r359044 = y;
        double r359045 = z;
        double r359046 = r359045 * r359045;
        double r359047 = r359041 + r359046;
        double r359048 = r359044 * r359047;
        double r359049 = r359043 / r359048;
        return r359049;
}
↓
double f(double x, double y, double z) {
        double r359050 = 1.0;
        double r359051 = y;
        double r359052 = r359050 / r359051;
        double r359053 = z;
        double r359054 = r359053 * r359053;
        double r359055 = r359050 + r359054;
        double r359056 = x;
        double r359057 = r359055 * r359056;
        double r359058 = r359052 / r359057;
        return r359058;
}

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)))))