Average Error: 6.1 → 6.0
Time: 5.2s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\left(\frac{1}{y} \cdot \frac{\frac{1}{x}}{\sqrt{1 + z \cdot z}}\right) \cdot \frac{1}{\sqrt{1 + z \cdot z}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\left(\frac{1}{y} \cdot \frac{\frac{1}{x}}{\sqrt{1 + z \cdot z}}\right) \cdot \frac{1}{\sqrt{1 + z \cdot z}}
double f(double x, double y, double z) {
        double r355820 = 1.0;
        double r355821 = x;
        double r355822 = r355820 / r355821;
        double r355823 = y;
        double r355824 = z;
        double r355825 = r355824 * r355824;
        double r355826 = r355820 + r355825;
        double r355827 = r355823 * r355826;
        double r355828 = r355822 / r355827;
        return r355828;
}
↓
double f(double x, double y, double z) {
        double r355829 = 1.0;
        double r355830 = y;
        double r355831 = r355829 / r355830;
        double r355832 = 1.0;
        double r355833 = x;
        double r355834 = r355832 / r355833;
        double r355835 = z;
        double r355836 = r355835 * r355835;
        double r355837 = r355829 + r355836;
        double r355838 = sqrt(r355837);
        double r355839 = r355834 / r355838;
        double r355840 = r355831 * r355839;
        double r355841 = r355832 / r355838;
        double r355842 = r355840 * r355841;
        return r355842;
}

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

Original6.1
Target5.5
Herbie6.0
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.1

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

  3. Applied div-inv6.1

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

  4. Applied times-frac6.4

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

  6. Applied add-sqr-sqrt6.4

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

  7. Applied associate-/r*6.4

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

  9. Applied div-inv6.4

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

  10. Applied associate-*r*6.0

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

  11. Final simplification6.0

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

Reproduce

herbie shell --seed 2020047 
(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)))))