Average Error: 14.6 → 2.0
Time: 16.9s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\left(\left(\frac{\sqrt[3]{x}}{\sqrt[3]{z}} \cdot \frac{y}{1 + z}\right) \cdot \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{1}{z}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\left(\left(\frac{\sqrt[3]{x}}{\sqrt[3]{z}} \cdot \frac{y}{1 + z}\right) \cdot \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{1}{z}
double f(double x, double y, double z) {
        double r16056782 = x;
        double r16056783 = y;
        double r16056784 = r16056782 * r16056783;
        double r16056785 = z;
        double r16056786 = r16056785 * r16056785;
        double r16056787 = 1.0;
        double r16056788 = r16056785 + r16056787;
        double r16056789 = r16056786 * r16056788;
        double r16056790 = r16056784 / r16056789;
        return r16056790;
}
↓
double f(double x, double y, double z) {
        double r16056791 = x;
        double r16056792 = cbrt(r16056791);
        double r16056793 = z;
        double r16056794 = cbrt(r16056793);
        double r16056795 = r16056792 / r16056794;
        double r16056796 = y;
        double r16056797 = 1.0;
        double r16056798 = r16056797 + r16056793;
        double r16056799 = r16056796 / r16056798;
        double r16056800 = r16056795 * r16056799;
        double r16056801 = r16056792 * r16056792;
        double r16056802 = r16056794 * r16056794;
        double r16056803 = r16056801 / r16056802;
        double r16056804 = r16056800 * r16056803;
        double r16056805 = 1.0;
        double r16056806 = r16056805 / r16056793;
        double r16056807 = r16056804 * r16056806;
        return r16056807;
}

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

Original14.6
Target3.9
Herbie2.0
\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307077115547144785523414612:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 14.6

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

  3. Applied times-frac10.9

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

  5. Applied *-un-lft-identity10.9

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

  6. Applied times-frac5.9

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

  7. Applied associate-*l*2.5

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

  9. Applied add-cube-cbrt3.0

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

  10. Applied add-cube-cbrt3.2

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

  11. Applied times-frac3.2

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

  12. Applied associate-*l*2.0

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

  13. Final simplification2.0

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

Reproduce

herbie shell --seed 2019172 
(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))))