Average Error: 14.7 → 3.0
Time: 3.9s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{1}{z} \cdot \left(\frac{\frac{x}{z}}{\sqrt[3]{z + 1} \cdot \sqrt[3]{z + 1}} \cdot \frac{y}{\sqrt[3]{z + 1}}\right)\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{1}{z} \cdot \left(\frac{\frac{x}{z}}{\sqrt[3]{z + 1} \cdot \sqrt[3]{z + 1}} \cdot \frac{y}{\sqrt[3]{z + 1}}\right)
double f(double x, double y, double z) {
        double r291741 = x;
        double r291742 = y;
        double r291743 = r291741 * r291742;
        double r291744 = z;
        double r291745 = r291744 * r291744;
        double r291746 = 1.0;
        double r291747 = r291744 + r291746;
        double r291748 = r291745 * r291747;
        double r291749 = r291743 / r291748;
        return r291749;
}
↓
double f(double x, double y, double z) {
        double r291750 = 1.0;
        double r291751 = z;
        double r291752 = r291750 / r291751;
        double r291753 = x;
        double r291754 = r291753 / r291751;
        double r291755 = 1.0;
        double r291756 = r291751 + r291755;
        double r291757 = cbrt(r291756);
        double r291758 = r291757 * r291757;
        double r291759 = r291754 / r291758;
        double r291760 = y;
        double r291761 = r291760 / r291757;
        double r291762 = r291759 * r291761;
        double r291763 = r291752 * r291762;
        return r291763;
}

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
Target3.9
Herbie3.0
\[\begin{array}{l} \mathbf{if}\;z \lt 249.618281453230708:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

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-frac10.5

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

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

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

  6. Applied times-frac6.1

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

  7. Applied associate-*l*2.6

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

  9. Applied add-cube-cbrt2.8

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

  10. Applied *-un-lft-identity2.8

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

  11. Applied times-frac2.8

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

  12. Applied associate-*r*3.0

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

  13. Simplified3.0

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

  14. Final simplification3.0

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

Reproduce

herbie shell --seed 2020100 +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))))