Average Error: 6.2 → 5.8
Time: 5.8s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{y}}{\sqrt{1 + z \cdot z}} \cdot \frac{\frac{\sqrt[3]{1}}{x}}{\sqrt{1 + z \cdot z}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{y}}{\sqrt{1 + z \cdot z}} \cdot \frac{\frac{\sqrt[3]{1}}{x}}{\sqrt{1 + z \cdot z}}
double f(double x, double y, double z) {
        double r271814 = 1.0;
        double r271815 = x;
        double r271816 = r271814 / r271815;
        double r271817 = y;
        double r271818 = z;
        double r271819 = r271818 * r271818;
        double r271820 = r271814 + r271819;
        double r271821 = r271817 * r271820;
        double r271822 = r271816 / r271821;
        return r271822;
}
↓
double f(double x, double y, double z) {
        double r271823 = 1.0;
        double r271824 = cbrt(r271823);
        double r271825 = r271824 * r271824;
        double r271826 = y;
        double r271827 = r271825 / r271826;
        double r271828 = z;
        double r271829 = r271828 * r271828;
        double r271830 = r271823 + r271829;
        double r271831 = sqrt(r271830);
        double r271832 = r271827 / r271831;
        double r271833 = x;
        double r271834 = r271824 / r271833;
        double r271835 = r271834 / r271831;
        double r271836 = r271832 * r271835;
        return r271836;
}

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

Original6.2
Target5.5
Herbie5.8
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.2

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

  3. Applied add-sqr-sqrt6.2

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

  4. Applied associate-*r*6.2

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

  6. Applied *-un-lft-identity6.2

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

  7. Applied add-cube-cbrt6.2

    \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{1 \cdot x}}{\left(y \cdot \sqrt{1 + z \cdot z}\right) \cdot \sqrt{1 + z \cdot z}}\]

  8. Applied times-frac6.2

    \[\leadsto \frac{\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1} \cdot \frac{\sqrt[3]{1}}{x}}}{\left(y \cdot \sqrt{1 + z \cdot z}\right) \cdot \sqrt{1 + z \cdot z}}\]

  9. Applied times-frac5.9

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{1}}{y \cdot \sqrt{1 + z \cdot z}} \cdot \frac{\frac{\sqrt[3]{1}}{x}}{\sqrt{1 + z \cdot z}}}\]

  10. Simplified5.8

    \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{y}}{\sqrt{1 + z \cdot z}}} \cdot \frac{\frac{\sqrt[3]{1}}{x}}{\sqrt{1 + z \cdot z}}\]

  11. Final simplification5.8

    \[\leadsto \frac{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{y}}{\sqrt{1 + z \cdot z}} \cdot \frac{\frac{\sqrt[3]{1}}{x}}{\sqrt{1 + z \cdot z}}\]

Reproduce

herbie shell --seed 2019322 
(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)))))