Average Error: 6.5 → 6.7
Time: 19.3s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\sqrt[3]{\frac{1}{x}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{x}}}{y} \cdot \frac{\sqrt[3]{\frac{1}{x}}}{1 + z \cdot z}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\sqrt[3]{\frac{1}{x}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{x}}}{y} \cdot \frac{\sqrt[3]{\frac{1}{x}}}{1 + z \cdot z}
double f(double x, double y, double z) {
        double r249244 = 1.0;
        double r249245 = x;
        double r249246 = r249244 / r249245;
        double r249247 = y;
        double r249248 = z;
        double r249249 = r249248 * r249248;
        double r249250 = r249244 + r249249;
        double r249251 = r249247 * r249250;
        double r249252 = r249246 / r249251;
        return r249252;
}
↓
double f(double x, double y, double z) {
        double r249253 = 1.0;
        double r249254 = x;
        double r249255 = r249253 / r249254;
        double r249256 = cbrt(r249255);
        double r249257 = cbrt(r249253);
        double r249258 = cbrt(r249254);
        double r249259 = r249257 / r249258;
        double r249260 = r249256 * r249259;
        double r249261 = y;
        double r249262 = r249260 / r249261;
        double r249263 = z;
        double r249264 = r249263 * r249263;
        double r249265 = r249253 + r249264;
        double r249266 = r249256 / r249265;
        double r249267 = r249262 * r249266;
        return r249267;
}

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

Original6.5
Target5.9
Herbie6.7
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.5

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

  3. Applied add-cube-cbrt7.1

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

  4. Applied times-frac6.8

    \[\leadsto \color{blue}{\frac{\sqrt[3]{\frac{1}{x}} \cdot \sqrt[3]{\frac{1}{x}}}{y} \cdot \frac{\sqrt[3]{\frac{1}{x}}}{1 + z \cdot z}}\]
  5. Using strategy rm

  6. Applied cbrt-div6.7

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

  7. Final simplification6.7

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

Reproduce

herbie shell --seed 2019297 
(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)))))