Average Error: 14.9 → 2.0
Time: 15.1s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{\frac{\sqrt[3]{x}}{\sqrt[3]{z}}}{1 + z} \cdot \left(\left(\frac{\sqrt[3]{x}}{\sqrt[3]{z}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{z}}\right) \cdot \frac{y}{z}\right)\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{\frac{\sqrt[3]{x}}{\sqrt[3]{z}}}{1 + z} \cdot \left(\left(\frac{\sqrt[3]{x}}{\sqrt[3]{z}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{z}}\right) \cdot \frac{y}{z}\right)
double f(double x, double y, double z) {
        double r276906 = x;
        double r276907 = y;
        double r276908 = r276906 * r276907;
        double r276909 = z;
        double r276910 = r276909 * r276909;
        double r276911 = 1.0;
        double r276912 = r276909 + r276911;
        double r276913 = r276910 * r276912;
        double r276914 = r276908 / r276913;
        return r276914;
}
↓
double f(double x, double y, double z) {
        double r276915 = x;
        double r276916 = cbrt(r276915);
        double r276917 = z;
        double r276918 = cbrt(r276917);
        double r276919 = r276916 / r276918;
        double r276920 = 1.0;
        double r276921 = r276920 + r276917;
        double r276922 = r276919 / r276921;
        double r276923 = r276919 * r276919;
        double r276924 = y;
        double r276925 = r276924 / r276917;
        double r276926 = r276923 * r276925;
        double r276927 = r276922 * r276926;
        return r276927;
}

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

Derivation

  1. Initial program 14.9

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

  2. Simplified3.2

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

  4. Applied *-un-lft-identity3.2

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

  5. Applied add-cube-cbrt3.7

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

  6. Applied add-cube-cbrt3.9

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

  7. Applied times-frac3.9

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

  8. Applied times-frac3.9

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

  9. Applied associate-*r*2.0

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

  10. Simplified2.0

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

  11. Final simplification2.0

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

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))))