Average Error: 14.2 → 1.2
Time: 18.4s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1}\right)\]
\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{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1}\right)
double f(double x, double y, double z) {
        double r201765 = x;
        double r201766 = y;
        double r201767 = r201765 * r201766;
        double r201768 = z;
        double r201769 = r201768 * r201768;
        double r201770 = 1.0;
        double r201771 = r201768 + r201770;
        double r201772 = r201769 * r201771;
        double r201773 = r201767 / r201772;
        return r201773;
}
↓
double f(double x, double y, double z) {
        double r201774 = x;
        double r201775 = cbrt(r201774);
        double r201776 = r201775 * r201775;
        double r201777 = z;
        double r201778 = r201776 / r201777;
        double r201779 = r201775 / r201777;
        double r201780 = y;
        double r201781 = 1.0;
        double r201782 = r201777 + r201781;
        double r201783 = r201780 / r201782;
        double r201784 = r201779 * r201783;
        double r201785 = r201778 * r201784;
        return r201785;
}

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

Original14.2
Target3.8
Herbie1.2
\[\begin{array}{l} \mathbf{if}\;z \lt 249.618281453230708:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 14.2

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

  3. Applied times-frac10.9

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

  4. Simplified10.9

    \[\leadsto \color{blue}{\frac{x}{{z}^{2}}} \cdot \frac{y}{z + 1}\]
  5. Using strategy rm

  6. Applied unpow210.9

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

  7. Applied add-cube-cbrt11.3

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

  8. Applied times-frac6.5

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

  9. Applied associate-*l*1.2

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

  10. Final simplification1.2

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

Reproduce

herbie shell --seed 2019198 +o rules:numerics
(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))))