Average Error: 6.9 → 6.5
Time: 2.1m
Precision: 64
\[\frac{\frac{1.0}{x}}{y \cdot \left(1.0 + z \cdot z\right)}\]
↓
\[\frac{\frac{-\frac{\sqrt{1.0}}{x}}{\left|\sqrt[3]{1.0 + z \cdot z}\right|} \cdot \frac{\sqrt{1.0}}{y}}{-\sqrt{1.0 + z \cdot z} \cdot \sqrt{\sqrt[3]{1.0 + z \cdot z}}}\]
\frac{\frac{1.0}{x}}{y \cdot \left(1.0 + z \cdot z\right)}
↓
\frac{\frac{-\frac{\sqrt{1.0}}{x}}{\left|\sqrt[3]{1.0 + z \cdot z}\right|} \cdot \frac{\sqrt{1.0}}{y}}{-\sqrt{1.0 + z \cdot z} \cdot \sqrt{\sqrt[3]{1.0 + z \cdot z}}}
double f(double x, double y, double z) {
        double r21599931 = 1.0;
        double r21599932 = x;
        double r21599933 = r21599931 / r21599932;
        double r21599934 = y;
        double r21599935 = z;
        double r21599936 = r21599935 * r21599935;
        double r21599937 = r21599931 + r21599936;
        double r21599938 = r21599934 * r21599937;
        double r21599939 = r21599933 / r21599938;
        return r21599939;
}
↓
double f(double x, double y, double z) {
        double r21599940 = 1.0;
        double r21599941 = sqrt(r21599940);
        double r21599942 = x;
        double r21599943 = r21599941 / r21599942;
        double r21599944 = -r21599943;
        double r21599945 = z;
        double r21599946 = r21599945 * r21599945;
        double r21599947 = r21599940 + r21599946;
        double r21599948 = cbrt(r21599947);
        double r21599949 = fabs(r21599948);
        double r21599950 = r21599944 / r21599949;
        double r21599951 = y;
        double r21599952 = r21599941 / r21599951;
        double r21599953 = r21599950 * r21599952;
        double r21599954 = sqrt(r21599947);
        double r21599955 = sqrt(r21599948);
        double r21599956 = r21599954 * r21599955;
        double r21599957 = -r21599956;
        double r21599958 = r21599953 / r21599957;
        return r21599958;
}

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
Target6.2
Herbie6.5
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1.0 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1.0}{y}}{\left(1.0 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1.0 + z \cdot z\right) \lt 8.680743250567252 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{1.0}{x}}{\left(1.0 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1.0}{y}}{\left(1.0 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.9

    \[\frac{\frac{1.0}{x}}{y \cdot \left(1.0 + z \cdot z\right)}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt6.9

    \[\leadsto \frac{\frac{1.0}{x}}{y \cdot \color{blue}{\left(\sqrt{1.0 + z \cdot z} \cdot \sqrt{1.0 + z \cdot z}\right)}}\]

  4. Applied associate-*r*6.9

    \[\leadsto \frac{\frac{1.0}{x}}{\color{blue}{\left(y \cdot \sqrt{1.0 + z \cdot z}\right) \cdot \sqrt{1.0 + z \cdot z}}}\]

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

    \[\leadsto \frac{\frac{1.0}{\color{blue}{1 \cdot x}}}{\left(y \cdot \sqrt{1.0 + z \cdot z}\right) \cdot \sqrt{1.0 + z \cdot z}}\]

  6. Applied add-sqr-sqrt6.9

    \[\leadsto \frac{\frac{\color{blue}{\sqrt{1.0} \cdot \sqrt{1.0}}}{1 \cdot x}}{\left(y \cdot \sqrt{1.0 + z \cdot z}\right) \cdot \sqrt{1.0 + z \cdot z}}\]

  7. Applied times-frac6.9

    \[\leadsto \frac{\color{blue}{\frac{\sqrt{1.0}}{1} \cdot \frac{\sqrt{1.0}}{x}}}{\left(y \cdot \sqrt{1.0 + z \cdot z}\right) \cdot \sqrt{1.0 + z \cdot z}}\]

  8. Applied times-frac6.6

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{1.0}}{1}}{y \cdot \sqrt{1.0 + z \cdot z}} \cdot \frac{\frac{\sqrt{1.0}}{x}}{\sqrt{1.0 + z \cdot z}}}\]

  9. Simplified6.6

    \[\leadsto \color{blue}{\frac{\frac{\sqrt{1.0}}{y}}{\sqrt{1.0 + z \cdot z}}} \cdot \frac{\frac{\sqrt{1.0}}{x}}{\sqrt{1.0 + z \cdot z}}\]
  10. Using strategy rm

  11. Applied add-cube-cbrt6.6

    \[\leadsto \frac{\frac{\sqrt{1.0}}{y}}{\sqrt{1.0 + z \cdot z}} \cdot \frac{\frac{\sqrt{1.0}}{x}}{\sqrt{\color{blue}{\left(\sqrt[3]{1.0 + z \cdot z} \cdot \sqrt[3]{1.0 + z \cdot z}\right) \cdot \sqrt[3]{1.0 + z \cdot z}}}}\]

  12. Applied sqrt-prod6.6

    \[\leadsto \frac{\frac{\sqrt{1.0}}{y}}{\sqrt{1.0 + z \cdot z}} \cdot \frac{\frac{\sqrt{1.0}}{x}}{\color{blue}{\sqrt{\sqrt[3]{1.0 + z \cdot z} \cdot \sqrt[3]{1.0 + z \cdot z}} \cdot \sqrt{\sqrt[3]{1.0 + z \cdot z}}}}\]

  13. Applied associate-/r*6.6

    \[\leadsto \frac{\frac{\sqrt{1.0}}{y}}{\sqrt{1.0 + z \cdot z}} \cdot \color{blue}{\frac{\frac{\frac{\sqrt{1.0}}{x}}{\sqrt{\sqrt[3]{1.0 + z \cdot z} \cdot \sqrt[3]{1.0 + z \cdot z}}}}{\sqrt{\sqrt[3]{1.0 + z \cdot z}}}}\]

  14. Applied frac-2neg6.6

    \[\leadsto \color{blue}{\frac{-\frac{\sqrt{1.0}}{y}}{-\sqrt{1.0 + z \cdot z}}} \cdot \frac{\frac{\frac{\sqrt{1.0}}{x}}{\sqrt{\sqrt[3]{1.0 + z \cdot z} \cdot \sqrt[3]{1.0 + z \cdot z}}}}{\sqrt{\sqrt[3]{1.0 + z \cdot z}}}\]

  15. Applied frac-times6.5

    \[\leadsto \color{blue}{\frac{\left(-\frac{\sqrt{1.0}}{y}\right) \cdot \frac{\frac{\sqrt{1.0}}{x}}{\sqrt{\sqrt[3]{1.0 + z \cdot z} \cdot \sqrt[3]{1.0 + z \cdot z}}}}{\left(-\sqrt{1.0 + z \cdot z}\right) \cdot \sqrt{\sqrt[3]{1.0 + z \cdot z}}}}\]

  16. Simplified6.5

    \[\leadsto \frac{\color{blue}{\left(-\frac{\sqrt{1.0}}{y}\right) \cdot \frac{\frac{\sqrt{1.0}}{x}}{\left|\sqrt[3]{1.0 + z \cdot z}\right|}}}{\left(-\sqrt{1.0 + z \cdot z}\right) \cdot \sqrt{\sqrt[3]{1.0 + z \cdot z}}}\]

  17. Final simplification6.5

    \[\leadsto \frac{\frac{-\frac{\sqrt{1.0}}{x}}{\left|\sqrt[3]{1.0 + z \cdot z}\right|} \cdot \frac{\sqrt{1.0}}{y}}{-\sqrt{1.0 + z \cdot z} \cdot \sqrt{\sqrt[3]{1.0 + z \cdot z}}}\]

Reproduce

herbie shell --seed 2019158 
(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)))))