Average Error: 15.3 → 2.5
Time: 3.6s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\frac{\frac{x}{z} \cdot \frac{y}{z + 1}}{z} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\frac{\frac{x}{z} \cdot \frac{y}{z + 1}}{z} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)
double f(double x, double y, double z) {
        double r359763 = x;
        double r359764 = y;
        double r359765 = r359763 * r359764;
        double r359766 = z;
        double r359767 = r359766 * r359766;
        double r359768 = 1.0;
        double r359769 = r359766 + r359768;
        double r359770 = r359767 * r359769;
        double r359771 = r359765 / r359770;
        return r359771;
}
↓
double f(double x, double y, double z) {
        double r359772 = x;
        double r359773 = z;
        double r359774 = r359772 / r359773;
        double r359775 = y;
        double r359776 = 1.0;
        double r359777 = r359773 + r359776;
        double r359778 = r359775 / r359777;
        double r359779 = r359774 * r359778;
        double r359780 = r359779 / r359773;
        double r359781 = 1.0;
        double r359782 = cbrt(r359781);
        double r359783 = r359782 * r359782;
        double r359784 = r359780 * r359783;
        return r359784;
}

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

Original15.3
Target4.1
Herbie2.5
\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307077115547144785523414612:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Initial program 15.3

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

  3. Applied times-frac11.0

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

  5. Applied *-un-lft-identity11.0

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

  6. Applied times-frac5.8

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

  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 *-un-lft-identity2.6

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

  10. Applied add-cube-cbrt2.6

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

  11. Applied times-frac2.6

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

  12. Applied associate-*l*2.6

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

  13. Simplified2.5

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

  14. Final simplification2.5

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

Reproduce

herbie shell --seed 2020001 
(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))))