Average Error: 6.2 → 6.0
Time: 6.2s
Precision: 64
\[\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}\]
↓
\[\frac{\sqrt{1}}{y} \cdot \left(\frac{\left|\sqrt[3]{1}\right|}{\sqrt{1 + z \cdot z}} \cdot \frac{\frac{\sqrt{\sqrt[3]{1}}}{x}}{\sqrt{1 + z \cdot z}}\right)\]
\frac{\frac{1}{x}}{y \cdot \left(1 + z \cdot z\right)}
↓
\frac{\sqrt{1}}{y} \cdot \left(\frac{\left|\sqrt[3]{1}\right|}{\sqrt{1 + z \cdot z}} \cdot \frac{\frac{\sqrt{\sqrt[3]{1}}}{x}}{\sqrt{1 + z \cdot z}}\right)
double f(double x, double y, double z) {
        double r386479 = 1.0;
        double r386480 = x;
        double r386481 = r386479 / r386480;
        double r386482 = y;
        double r386483 = z;
        double r386484 = r386483 * r386483;
        double r386485 = r386479 + r386484;
        double r386486 = r386482 * r386485;
        double r386487 = r386481 / r386486;
        return r386487;
}
↓
double f(double x, double y, double z) {
        double r386488 = 1.0;
        double r386489 = sqrt(r386488);
        double r386490 = y;
        double r386491 = r386489 / r386490;
        double r386492 = cbrt(r386488);
        double r386493 = fabs(r386492);
        double r386494 = z;
        double r386495 = r386494 * r386494;
        double r386496 = r386488 + r386495;
        double r386497 = sqrt(r386496);
        double r386498 = r386493 / r386497;
        double r386499 = sqrt(r386492);
        double r386500 = x;
        double r386501 = r386499 / r386500;
        double r386502 = r386501 / r386497;
        double r386503 = r386498 * r386502;
        double r386504 = r386491 * r386503;
        return r386504;
}

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

Original6.2
Target5.5
Herbie6.0
\[\begin{array}{l} \mathbf{if}\;y \cdot \left(1 + z \cdot z\right) \lt -\infty:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \mathbf{elif}\;y \cdot \left(1 + z \cdot z\right) \lt 8.68074325056725162 \cdot 10^{305}:\\ \;\;\;\;\frac{\frac{1}{x}}{\left(1 + z \cdot z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{y}}{\left(1 + z \cdot z\right) \cdot x}\\ \end{array}\]

Derivation

  1. Initial program 6.2

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

  3. Applied *-un-lft-identity6.2

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

  4. Applied add-sqr-sqrt6.2

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

  5. Applied times-frac6.2

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

  6. Applied times-frac6.0

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

  7. Simplified6.0

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

  9. Applied add-sqr-sqrt6.0

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

  10. Applied *-un-lft-identity6.0

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

  11. Applied add-cube-cbrt6.0

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

  12. Applied sqrt-prod6.0

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

  13. Applied times-frac6.0

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

  14. Applied times-frac6.0

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

  15. Simplified6.0

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

  16. Final simplification6.0

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

Reproduce

herbie shell --seed 2020027 
(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)))))