Average Error: 6.6 → 6.1
Time: 15.4s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{1}{x}}{\sqrt{1 + z \cdot z}} \cdot \frac{\frac{1}{y}}{\sqrt{1 + z \cdot z}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{1}{x}}{\sqrt{1 + z \cdot z}} \cdot \frac{\frac{1}{y}}{\sqrt{1 + z \cdot z}}
double f(double x, double y, double z) {
        double r302291 = 1.0;
        double r302292 = x;
        double r302293 = r302291 / r302292;
        double r302294 = y;
        double r302295 = z;
        double r302296 = r302295 * r302295;
        double r302297 = r302291 + r302296;
        double r302298 = r302294 * r302297;
        double r302299 = r302293 / r302298;
        return r302299;
}
↓
double f(double x, double y, double z) {
        double r302300 = 1.0;
        double r302301 = x;
        double r302302 = r302300 / r302301;
        double r302303 = 1.0;
        double r302304 = z;
        double r302305 = r302304 * r302304;
        double r302306 = r302303 + r302305;
        double r302307 = sqrt(r302306);
        double r302308 = r302302 / r302307;
        double r302309 = y;
        double r302310 = r302303 / r302309;
        double r302311 = r302310 / r302307;
        double r302312 = r302308 * r302311;
        return r302312;
}

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
Target6.0
Herbie6.1
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.6

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

  3. Applied div-inv6.6

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

  4. Applied times-frac6.5

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

  6. Applied add-sqr-sqrt6.5

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

  7. Applied *-un-lft-identity6.5

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

  8. Applied add-cube-cbrt6.5

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

  9. Applied times-frac6.5

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

  10. Applied times-frac6.6

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

  11. Applied associate-*r*6.1

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

  12. Simplified6.1

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

  13. Final simplification6.1

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

Reproduce

herbie shell --seed 2019306 
(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)))))