Average Error: 6.5 → 5.8
Time: 17.2s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{1}{x}}{\sqrt[3]{y} \cdot \sqrt{1 + z \cdot z}} \cdot \frac{\frac{\frac{1}{\sqrt[3]{y}}}{\sqrt[3]{y}}}{\sqrt{1 + z \cdot z}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{1}{x}}{\sqrt[3]{y} \cdot \sqrt{1 + z \cdot z}} \cdot \frac{\frac{\frac{1}{\sqrt[3]{y}}}{\sqrt[3]{y}}}{\sqrt{1 + z \cdot z}}
double f(double x, double y, double z) {
        double r309996 = 1.0;
        double r309997 = x;
        double r309998 = r309996 / r309997;
        double r309999 = y;
        double r310000 = z;
        double r310001 = r310000 * r310000;
        double r310002 = r309996 + r310001;
        double r310003 = r309999 * r310002;
        double r310004 = r309998 / r310003;
        return r310004;
}
↓
double f(double x, double y, double z) {
        double r310005 = 1.0;
        double r310006 = x;
        double r310007 = r310005 / r310006;
        double r310008 = y;
        double r310009 = cbrt(r310008);
        double r310010 = 1.0;
        double r310011 = z;
        double r310012 = r310011 * r310011;
        double r310013 = r310010 + r310012;
        double r310014 = sqrt(r310013);
        double r310015 = r310009 * r310014;
        double r310016 = r310007 / r310015;
        double r310017 = r310010 / r310009;
        double r310018 = r310017 / r310009;
        double r310019 = r310018 / r310014;
        double r310020 = r310016 * r310019;
        return r310020;
}

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
Herbie5.8
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.680743250567251617010582226806563373013 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.5

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

  2. Simplified6.7

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

  4. Applied add-sqr-sqrt6.7

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

  5. Applied add-cube-cbrt7.2

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

  6. Applied div-inv7.2

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

  7. Applied times-frac7.3

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

  8. Applied times-frac6.1

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

  9. Simplified6.1

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

  10. Simplified5.8

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

  11. Final simplification5.8

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

Reproduce

herbie shell --seed 2019174 
(FPCore (x y z)
  :name "Statistics.Distribution.CauchyLorentz:$cdensity from math-functions-0.1.5.2"

  :herbie-target
  (if (< (* y (+ 1.0 (* z z))) -inf.0) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x)) (if (< (* y (+ 1.0 (* z z))) 8.680743250567252e+305) (/ (/ 1.0 x) (* (+ 1.0 (* z z)) y)) (/ (/ 1.0 y) (* (+ 1.0 (* z z)) x))))

  (/ (/ 1.0 x) (* y (+ 1.0 (* z z)))))