Average Error: 14.9 → 2.7
Time: 32.8s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\left(\frac{1}{\frac{\sqrt[3]{1 + z}}{\frac{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{\sqrt[3]{1 + z}}}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt[3]{1 + z}}{\sqrt[3]{\sqrt[3]{y}}}}\right) \cdot \left(\frac{\sqrt[3]{x}}{\frac{z}{\sqrt[3]{y}}} \cdot \frac{\sqrt[3]{x}}{\frac{z}{\sqrt[3]{y}}}\right)\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\left(\frac{1}{\frac{\sqrt[3]{1 + z}}{\frac{\sqrt[3]{\sqrt[3]{y} \cdot \sqrt[3]{y}}}{\sqrt[3]{1 + z}}}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt[3]{1 + z}}{\sqrt[3]{\sqrt[3]{y}}}}\right) \cdot \left(\frac{\sqrt[3]{x}}{\frac{z}{\sqrt[3]{y}}} \cdot \frac{\sqrt[3]{x}}{\frac{z}{\sqrt[3]{y}}}\right)
double f(double x, double y, double z) {
        double r17939705 = x;
        double r17939706 = y;
        double r17939707 = r17939705 * r17939706;
        double r17939708 = z;
        double r17939709 = r17939708 * r17939708;
        double r17939710 = 1.0;
        double r17939711 = r17939708 + r17939710;
        double r17939712 = r17939709 * r17939711;
        double r17939713 = r17939707 / r17939712;
        return r17939713;
}
↓
double f(double x, double y, double z) {
        double r17939714 = 1.0;
        double r17939715 = 1.0;
        double r17939716 = z;
        double r17939717 = r17939715 + r17939716;
        double r17939718 = cbrt(r17939717);
        double r17939719 = y;
        double r17939720 = cbrt(r17939719);
        double r17939721 = r17939720 * r17939720;
        double r17939722 = cbrt(r17939721);
        double r17939723 = r17939722 / r17939718;
        double r17939724 = r17939718 / r17939723;
        double r17939725 = r17939714 / r17939724;
        double r17939726 = x;
        double r17939727 = cbrt(r17939726);
        double r17939728 = cbrt(r17939720);
        double r17939729 = r17939718 / r17939728;
        double r17939730 = r17939727 / r17939729;
        double r17939731 = r17939725 * r17939730;
        double r17939732 = r17939716 / r17939720;
        double r17939733 = r17939727 / r17939732;
        double r17939734 = r17939733 * r17939733;
        double r17939735 = r17939731 * r17939734;
        return r17939735;
}

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

Original14.9
Target3.9
Herbie2.7
\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307077115547144785523414612:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 14.9

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

  3. Applied associate-/l*13.6

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

  5. Applied add-cube-cbrt14.0

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

  6. Applied times-frac12.1

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

  7. Applied add-cube-cbrt12.2

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

  8. Applied times-frac9.8

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

  9. Simplified2.6

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

  11. Applied add-cube-cbrt2.6

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

  12. Applied cbrt-prod2.7

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

  13. Applied add-cube-cbrt2.7

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

  14. Applied times-frac2.7

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

  15. Applied *-un-lft-identity2.7

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

  16. Applied cbrt-prod2.7

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

  17. Applied times-frac2.7

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

  18. Simplified2.7

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

  19. Final simplification2.7

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

Reproduce

herbie shell --seed 2019179 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1.0 z)) x) z))

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