Average Error: 6.5 → 6.3
Time: 3.5s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{1}{\left(1 + z \cdot z\right) \cdot x}}{y}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{1}{\left(1 + z \cdot z\right) \cdot x}}{y}
double f(double x, double y, double z) {
        double r404234 = 1.0;
        double r404235 = x;
        double r404236 = r404234 / r404235;
        double r404237 = y;
        double r404238 = z;
        double r404239 = r404238 * r404238;
        double r404240 = r404234 + r404239;
        double r404241 = r404237 * r404240;
        double r404242 = r404236 / r404241;
        return r404242;
}
↓
double f(double x, double y, double z) {
        double r404243 = 1.0;
        double r404244 = z;
        double r404245 = r404244 * r404244;
        double r404246 = r404243 + r404245;
        double r404247 = x;
        double r404248 = r404246 * r404247;
        double r404249 = r404243 / r404248;
        double r404250 = y;
        double r404251 = r404249 / r404250;
        return r404251;
}

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

Original6.5
Target5.9
Herbie6.3
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.5

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

  3. Applied div-inv6.5

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

  4. Applied times-frac6.3

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

  6. Applied associate-*l/6.2

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

  7. Simplified6.3

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

  8. Final simplification6.3

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

Reproduce

herbie shell --seed 2020089 
(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)))))