Average Error: 6.2 → 6.0
Time: 6.0s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\sqrt[3]{\frac{1}{y}}}{\frac{\sqrt{1 + z \cdot z}}{\sqrt[3]{\frac{1}{y}}}} \cdot \frac{\frac{\sqrt[3]{\frac{1}{y}}}{x}}{\sqrt{1 + z \cdot z}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\sqrt[3]{\frac{1}{y}}}{\frac{\sqrt{1 + z \cdot z}}{\sqrt[3]{\frac{1}{y}}}} \cdot \frac{\frac{\sqrt[3]{\frac{1}{y}}}{x}}{\sqrt{1 + z \cdot z}}
double f(double x, double y, double z) {
        double r304177 = 1.0;
        double r304178 = x;
        double r304179 = r304177 / r304178;
        double r304180 = y;
        double r304181 = z;
        double r304182 = r304181 * r304181;
        double r304183 = r304177 + r304182;
        double r304184 = r304180 * r304183;
        double r304185 = r304179 / r304184;
        return r304185;
}
↓
double f(double x, double y, double z) {
        double r304186 = 1.0;
        double r304187 = y;
        double r304188 = r304186 / r304187;
        double r304189 = cbrt(r304188);
        double r304190 = z;
        double r304191 = r304190 * r304190;
        double r304192 = r304186 + r304191;
        double r304193 = sqrt(r304192);
        double r304194 = r304193 / r304189;
        double r304195 = r304189 / r304194;
        double r304196 = x;
        double r304197 = r304189 / r304196;
        double r304198 = r304197 / r304193;
        double r304199 = r304195 * r304198;
        return r304199;
}

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
Target5.6
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 6.2

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

  3. Applied associate-/r*6.6

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

  4. Simplified6.6

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

  6. Applied add-sqr-sqrt6.6

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

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

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

  8. Applied add-cube-cbrt7.2

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

  9. Applied times-frac7.2

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

  10. Applied times-frac6.0

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

  11. Simplified6.0

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

  12. Final simplification6.0

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

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(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)))))