Average Error: 6.5 → 6.1
Time: 8.1s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\sqrt{1}}{\sqrt{1 + z \cdot z} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \frac{\frac{\frac{\sqrt{1}}{y}}{\sqrt[3]{x}}}{\sqrt{1 + z \cdot z}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\sqrt{1}}{\sqrt{1 + z \cdot z} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \frac{\frac{\frac{\sqrt{1}}{y}}{\sqrt[3]{x}}}{\sqrt{1 + z \cdot z}}
double f(double x, double y, double z) {
        double r378382 = 1.0;
        double r378383 = x;
        double r378384 = r378382 / r378383;
        double r378385 = y;
        double r378386 = z;
        double r378387 = r378386 * r378386;
        double r378388 = r378382 + r378387;
        double r378389 = r378385 * r378388;
        double r378390 = r378384 / r378389;
        return r378390;
}
↓
double f(double x, double y, double z) {
        double r378391 = 1.0;
        double r378392 = sqrt(r378391);
        double r378393 = z;
        double r378394 = r378393 * r378393;
        double r378395 = r378391 + r378394;
        double r378396 = sqrt(r378395);
        double r378397 = x;
        double r378398 = cbrt(r378397);
        double r378399 = r378398 * r378398;
        double r378400 = r378396 * r378399;
        double r378401 = r378392 / r378400;
        double r378402 = y;
        double r378403 = r378392 / r378402;
        double r378404 = r378403 / r378398;
        double r378405 = r378404 / r378396;
        double r378406 = r378401 * r378405;
        return r378406;
}

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
Target5.7
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 6.5

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

  3. Applied associate-/r*6.5

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

  4. Simplified6.5

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

  6. Applied add-sqr-sqrt6.5

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

  7. Applied add-cube-cbrt7.1

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

  8. Applied *-un-lft-identity7.1

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

  9. Applied add-sqr-sqrt7.1

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

  10. Applied times-frac7.1

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

  11. Applied times-frac7.1

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

  12. Applied times-frac6.1

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

  13. Simplified6.1

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

  14. Final simplification6.1

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

Reproduce

herbie shell --seed 2019354 
(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)))))