Average Error: 14.7 → 1.5
Time: 4.5s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \left(\frac{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}{z} \cdot \frac{y}{z + 1}\right)\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \left(\frac{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}{z} \cdot \frac{y}{z + 1}\right)
double f(double x, double y, double z) {
        double r299630 = x;
        double r299631 = y;
        double r299632 = r299630 * r299631;
        double r299633 = z;
        double r299634 = r299633 * r299633;
        double r299635 = 1.0;
        double r299636 = r299633 + r299635;
        double r299637 = r299634 * r299636;
        double r299638 = r299632 / r299637;
        return r299638;
}
↓
double f(double x, double y, double z) {
        double r299639 = x;
        double r299640 = cbrt(r299639);
        double r299641 = r299640 * r299640;
        double r299642 = z;
        double r299643 = r299641 / r299642;
        double r299644 = cbrt(r299640);
        double r299645 = r299644 * r299644;
        double r299646 = r299645 * r299644;
        double r299647 = r299646 / r299642;
        double r299648 = y;
        double r299649 = 1.0;
        double r299650 = r299642 + r299649;
        double r299651 = r299648 / r299650;
        double r299652 = r299647 * r299651;
        double r299653 = r299643 * r299652;
        return r299653;
}

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

Original14.7
Target4.2
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;z \lt 249.618281453230708:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 14.7

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

  3. Applied times-frac11.2

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

  5. Applied add-cube-cbrt11.6

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

  6. Applied times-frac6.7

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

  7. Applied associate-*l*1.3

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

  9. Applied add-cube-cbrt1.5

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

  10. Final simplification1.5

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

Reproduce

herbie shell --seed 2020056 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z 249.6182814532307) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1 z)) x) z))

  (/ (* x y) (* (* z z) (+ z 1))))