Average Error: 14.4 → 1.1
Time: 18.1s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1.0\right)}\]
↓
\[\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \left(\frac{y}{z + 1.0} \cdot \frac{\sqrt[3]{x}}{z}\right)\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1.0\right)}
↓
\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \left(\frac{y}{z + 1.0} \cdot \frac{\sqrt[3]{x}}{z}\right)
double f(double x, double y, double z) {
        double r12167833 = x;
        double r12167834 = y;
        double r12167835 = r12167833 * r12167834;
        double r12167836 = z;
        double r12167837 = r12167836 * r12167836;
        double r12167838 = 1.0;
        double r12167839 = r12167836 + r12167838;
        double r12167840 = r12167837 * r12167839;
        double r12167841 = r12167835 / r12167840;
        return r12167841;
}
↓
double f(double x, double y, double z) {
        double r12167842 = x;
        double r12167843 = cbrt(r12167842);
        double r12167844 = r12167843 * r12167843;
        double r12167845 = z;
        double r12167846 = r12167844 / r12167845;
        double r12167847 = y;
        double r12167848 = 1.0;
        double r12167849 = r12167845 + r12167848;
        double r12167850 = r12167847 / r12167849;
        double r12167851 = r12167843 / r12167845;
        double r12167852 = r12167850 * r12167851;
        double r12167853 = r12167846 * r12167852;
        return r12167853;
}

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

Original14.4
Target4.1
Herbie1.1
\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 14.4

    \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1.0\right)}\]
  2. Using strategy rm

  3. Applied times-frac10.3

    \[\leadsto \color{blue}{\frac{x}{z \cdot z} \cdot \frac{y}{z + 1.0}}\]
  4. Using strategy rm

  5. Applied add-cube-cbrt10.6

    \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{z \cdot z} \cdot \frac{y}{z + 1.0}\]

  6. Applied times-frac6.0

    \[\leadsto \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \frac{\sqrt[3]{x}}{z}\right)} \cdot \frac{y}{z + 1.0}\]

  7. Applied associate-*l*1.1

    \[\leadsto \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \left(\frac{\sqrt[3]{x}}{z} \cdot \frac{y}{z + 1.0}\right)}\]

  8. Final simplification1.1

    \[\leadsto \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{z} \cdot \left(\frac{y}{z + 1.0} \cdot \frac{\sqrt[3]{x}}{z}\right)\]

Reproduce

herbie shell --seed 2019168 +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 z)) x) z))

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