Average Error: 6.9 → 6.5
Time: 13.6s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{\frac{1}{x}}{1 + z \cdot z}}{y}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{\frac{1}{x}}{1 + z \cdot z}}{y}
double f(double x, double y, double z) {
        double r291551 = 1.0;
        double r291552 = x;
        double r291553 = r291551 / r291552;
        double r291554 = y;
        double r291555 = z;
        double r291556 = r291555 * r291555;
        double r291557 = r291551 + r291556;
        double r291558 = r291554 * r291557;
        double r291559 = r291553 / r291558;
        return r291559;
}
↓
double f(double x, double y, double z) {
        double r291560 = 1.0;
        double r291561 = x;
        double r291562 = r291560 / r291561;
        double r291563 = z;
        double r291564 = r291563 * r291563;
        double r291565 = r291560 + r291564;
        double r291566 = r291562 / r291565;
        double r291567 = y;
        double r291568 = r291566 / r291567;
        return r291568;
}

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

Original6.9
Target6.2
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.9

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

  3. Applied *-un-lft-identity6.9

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

  4. Applied *-un-lft-identity6.9

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

  5. Applied times-frac6.9

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

  6. Applied times-frac6.6

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

  7. Simplified6.6

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

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

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

  10. Applied add-cube-cbrt6.6

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

  11. Applied times-frac6.6

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

  12. Applied associate-*l*6.6

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

  13. Simplified6.5

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

  14. Final simplification6.5

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

Reproduce

herbie shell --seed 2019198 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"

  :herbie-target
  (if (< (* y (+ 1.0 (* z z))) -inf.0) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x)) (if (< (* y (+ 1.0 (* z z))) 8.680743250567252e+305) (/ (/ 1.0 x) (* (+ 1.0 (* z z)) y)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x))))

  (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))