Average Error: 6.6 → 6.5
Time: 14.0s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{\frac{\sqrt[3]{1}}{y}}{\left(1 + z \cdot z\right) \cdot x}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \frac{\frac{\sqrt[3]{1}}{y}}{\left(1 + z \cdot z\right) \cdot x}
double f(double x, double y, double z) {
        double r308387 = 1.0;
        double r308388 = x;
        double r308389 = r308387 / r308388;
        double r308390 = y;
        double r308391 = z;
        double r308392 = r308391 * r308391;
        double r308393 = r308387 + r308392;
        double r308394 = r308390 * r308393;
        double r308395 = r308389 / r308394;
        return r308395;
}
↓
double f(double x, double y, double z) {
        double r308396 = 1.0;
        double r308397 = cbrt(r308396);
        double r308398 = r308397 * r308397;
        double r308399 = y;
        double r308400 = r308397 / r308399;
        double r308401 = z;
        double r308402 = r308401 * r308401;
        double r308403 = r308396 + r308402;
        double r308404 = x;
        double r308405 = r308403 * r308404;
        double r308406 = r308400 / r308405;
        double r308407 = r308398 * r308406;
        return r308407;
}

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

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 *-un-lft-identity6.5

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

  7. Applied add-cube-cbrt6.5

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

  8. Applied times-frac6.5

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

  9. Applied associate-*l*6.5

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

  10. Simplified6.5

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

  11. Final simplification6.5

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

Reproduce

herbie shell --seed 2019303 
(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)))))