Average Error: 6.6 → 6.5
Time: 10.0s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\sqrt{1}}{y} \cdot \frac{\frac{\sqrt{1}}{x}}{1 + z \cdot z}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\sqrt{1}}{y} \cdot \frac{\frac{\sqrt{1}}{x}}{1 + z \cdot z}
double f(double x, double y, double z) {
        double r237829 = 1.0;
        double r237830 = x;
        double r237831 = r237829 / r237830;
        double r237832 = y;
        double r237833 = z;
        double r237834 = r237833 * r237833;
        double r237835 = r237829 + r237834;
        double r237836 = r237832 * r237835;
        double r237837 = r237831 / r237836;
        return r237837;
}
↓
double f(double x, double y, double z) {
        double r237838 = 1.0;
        double r237839 = sqrt(r237838);
        double r237840 = y;
        double r237841 = r237839 / r237840;
        double r237842 = x;
        double r237843 = r237839 / r237842;
        double r237844 = z;
        double r237845 = r237844 * r237844;
        double r237846 = r237838 + r237845;
        double r237847 = r237843 / r237846;
        double r237848 = r237841 * r237847;
        return r237848;
}

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

Original6.6
Target5.8
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.6

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

  3. Applied *-un-lft-identity6.6

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

  4. Applied add-sqr-sqrt6.6

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

  5. Applied times-frac6.6

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

  6. Applied times-frac6.5

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

  7. Simplified6.5

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

  8. Final simplification6.5

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

Reproduce

herbie shell --seed 2019212 
(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)))))