Average Error: 6.9 → 6.4
Time: 4.1s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\left|\sqrt[3]{1 + z \cdot z}\right| \cdot y} \cdot \frac{\frac{\frac{1}{\sqrt[3]{x}}}{\sqrt{1 + z \cdot z}}}{\sqrt{\sqrt[3]{1 + z \cdot z}}}\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\frac{1}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}{\left|\sqrt[3]{1 + z \cdot z}\right| \cdot y} \cdot \frac{\frac{\frac{1}{\sqrt[3]{x}}}{\sqrt{1 + z \cdot z}}}{\sqrt{\sqrt[3]{1 + z \cdot z}}}
double f(double x, double y, double z) {
        double r327305 = 1.0;
        double r327306 = x;
        double r327307 = r327305 / r327306;
        double r327308 = y;
        double r327309 = z;
        double r327310 = r327309 * r327309;
        double r327311 = r327305 + r327310;
        double r327312 = r327308 * r327311;
        double r327313 = r327307 / r327312;
        return r327313;
}
↓
double f(double x, double y, double z) {
        double r327314 = 1.0;
        double r327315 = x;
        double r327316 = cbrt(r327315);
        double r327317 = r327316 * r327316;
        double r327318 = r327314 / r327317;
        double r327319 = 1.0;
        double r327320 = z;
        double r327321 = r327320 * r327320;
        double r327322 = r327319 + r327321;
        double r327323 = cbrt(r327322);
        double r327324 = fabs(r327323);
        double r327325 = y;
        double r327326 = r327324 * r327325;
        double r327327 = r327318 / r327326;
        double r327328 = r327319 / r327316;
        double r327329 = sqrt(r327322);
        double r327330 = r327328 / r327329;
        double r327331 = sqrt(r327323);
        double r327332 = r327330 / r327331;
        double r327333 = r327327 * r327332;
        return r327333;
}

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

Original6.9
Target6.0
Herbie6.4
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.9

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

  3. Applied add-sqr-sqrt6.9

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

  4. Applied associate-*r*6.9

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

  6. Applied associate-/r*6.3

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

  8. Applied add-cube-cbrt6.4

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

  9. Applied sqrt-prod6.4

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

  10. Applied add-cube-cbrt6.9

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

  11. Applied *-un-lft-identity6.9

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

  12. Applied times-frac6.9

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

  13. Applied times-frac6.7

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

  14. Applied times-frac6.6

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

  15. Simplified6.4

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

  16. Final simplification6.4

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

Reproduce

herbie shell --seed 2020018 
(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)))))