Average Error: 6.7 → 6.2
Time: 4.9s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{1}{y}}{\sqrt{1 + z \cdot z}} \cdot \frac{\frac{1}{x}}{\sqrt{1 + z \cdot z}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{1}{y}}{\sqrt{1 + z \cdot z}} \cdot \frac{\frac{1}{x}}{\sqrt{1 + z \cdot z}}
double f(double x, double y, double z) {
        double r340916 = 1.0;
        double r340917 = x;
        double r340918 = r340916 / r340917;
        double r340919 = y;
        double r340920 = z;
        double r340921 = r340920 * r340920;
        double r340922 = r340916 + r340921;
        double r340923 = r340919 * r340922;
        double r340924 = r340918 / r340923;
        return r340924;
}
↓
double f(double x, double y, double z) {
        double r340925 = 1.0;
        double r340926 = y;
        double r340927 = r340925 / r340926;
        double r340928 = z;
        double r340929 = r340928 * r340928;
        double r340930 = r340925 + r340929;
        double r340931 = sqrt(r340930);
        double r340932 = r340927 / r340931;
        double r340933 = 1.0;
        double r340934 = x;
        double r340935 = r340933 / r340934;
        double r340936 = r340935 / r340931;
        double r340937 = r340932 * r340936;
        return r340937;
}

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

Original6.7
Target6.1
Herbie6.2
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.7

    \[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
  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 add-sqr-sqrt6.9

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

  7. Applied div-inv6.9

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

  8. Applied times-frac6.2

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

  9. Final simplification6.2

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

Reproduce

herbie shell --seed 2020046 
(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)))))