Average Error: 6.1 → 6.4
Time: 13.9s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\sqrt{1}}{y} \cdot \frac{\frac{\sqrt{1}}{\sqrt{1 + z \cdot z} \cdot x}}{\sqrt{1 + z \cdot z}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\sqrt{1}}{y} \cdot \frac{\frac{\sqrt{1}}{\sqrt{1 + z \cdot z} \cdot x}}{\sqrt{1 + z \cdot z}}
double f(double x, double y, double z) {
        double r398805 = 1.0;
        double r398806 = x;
        double r398807 = r398805 / r398806;
        double r398808 = y;
        double r398809 = z;
        double r398810 = r398809 * r398809;
        double r398811 = r398805 + r398810;
        double r398812 = r398808 * r398811;
        double r398813 = r398807 / r398812;
        return r398813;
}
↓
double f(double x, double y, double z) {
        double r398814 = 1.0;
        double r398815 = sqrt(r398814);
        double r398816 = y;
        double r398817 = r398815 / r398816;
        double r398818 = z;
        double r398819 = r398818 * r398818;
        double r398820 = r398814 + r398819;
        double r398821 = sqrt(r398820);
        double r398822 = x;
        double r398823 = r398821 * r398822;
        double r398824 = r398815 / r398823;
        double r398825 = r398824 / r398821;
        double r398826 = r398817 * r398825;
        return r398826;
}

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
Herbie6.4
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.1

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

  3. Applied *-un-lft-identity6.1

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

  4. Applied add-sqr-sqrt6.1

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

  5. Applied times-frac6.1

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

  6. Applied times-frac6.4

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

  7. Simplified6.4

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

  9. Applied add-sqr-sqrt6.4

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

  10. Applied associate-/r*6.4

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

  12. Applied associate-/l/6.4

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

  13. Final simplification6.4

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

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)))))