Average Error: 14.9 → 3.3
Time: 3.3s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \le 9.1025958135814896 \cdot 10^{-267} \lor \neg \left(\left(z \cdot z\right) \cdot \left(z + 1\right) \le 9.98869995522394163 \cdot 10^{297}\right):\\ \;\;\;\;\left(\sqrt[3]{\frac{1}{z}} \cdot \sqrt[3]{\frac{1}{z}}\right) \cdot \left(\sqrt[3]{\frac{1}{z}} \cdot \left(\frac{x}{z} \cdot \frac{y}{z + 1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{\left(z \cdot z\right) \cdot \left(z + 1\right)}{y}}\\ \end{array}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\begin{array}{l}
\mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \le 9.1025958135814896 \cdot 10^{-267} \lor \neg \left(\left(z \cdot z\right) \cdot \left(z + 1\right) \le 9.98869995522394163 \cdot 10^{297}\right):\\
\;\;\;\;\left(\sqrt[3]{\frac{1}{z}} \cdot \sqrt[3]{\frac{1}{z}}\right) \cdot \left(\sqrt[3]{\frac{1}{z}} \cdot \left(\frac{x}{z} \cdot \frac{y}{z + 1}\right)\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{\frac{\left(z \cdot z\right) \cdot \left(z + 1\right)}{y}}\\

\end{array}
double f(double x, double y, double z) {
        double r399594 = x;
        double r399595 = y;
        double r399596 = r399594 * r399595;
        double r399597 = z;
        double r399598 = r399597 * r399597;
        double r399599 = 1.0;
        double r399600 = r399597 + r399599;
        double r399601 = r399598 * r399600;
        double r399602 = r399596 / r399601;
        return r399602;
}
↓
double f(double x, double y, double z) {
        double r399603 = z;
        double r399604 = r399603 * r399603;
        double r399605 = 1.0;
        double r399606 = r399603 + r399605;
        double r399607 = r399604 * r399606;
        double r399608 = 9.10259581358149e-267;
        bool r399609 = r399607 <= r399608;
        double r399610 = 9.988699955223942e+297;
        bool r399611 = r399607 <= r399610;
        double r399612 = !r399611;
        bool r399613 = r399609 || r399612;
        double r399614 = 1.0;
        double r399615 = r399614 / r399603;
        double r399616 = cbrt(r399615);
        double r399617 = r399616 * r399616;
        double r399618 = x;
        double r399619 = r399618 / r399603;
        double r399620 = y;
        double r399621 = r399620 / r399606;
        double r399622 = r399619 * r399621;
        double r399623 = r399616 * r399622;
        double r399624 = r399617 * r399623;
        double r399625 = r399607 / r399620;
        double r399626 = r399618 / r399625;
        double r399627 = r399613 ? r399624 : r399626;
        return r399627;
}

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
Target4.1
Herbie3.3
\[\begin{array}{l} \mathbf{if}\;z \lt 249.618281453230708:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (* (* z z) (+ z 1.0)) < 9.10259581358149e-267 or 9.988699955223942e+297 < (* (* z z) (+ z 1.0))

    1. Initial program 19.0

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

    3. Applied times-frac13.6

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

    5. Applied *-un-lft-identity13.6

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

    6. Applied times-frac5.9

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

    7. Applied associate-*l*1.8

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

    9. Applied add-cube-cbrt2.1

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

    10. Applied associate-*l*2.1

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

    if 9.10259581358149e-267 < (* (* z z) (+ z 1.0)) < 9.988699955223942e+297

    1. Initial program 6.2

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

    3. Applied associate-/l*5.7

      \[\leadsto \color{blue}{\frac{x}{\frac{\left(z \cdot z\right) \cdot \left(z + 1\right)}{y}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left(z \cdot z\right) \cdot \left(z + 1\right) \le 9.1025958135814896 \cdot 10^{-267} \lor \neg \left(\left(z \cdot z\right) \cdot \left(z + 1\right) \le 9.98869995522394163 \cdot 10^{297}\right):\\ \;\;\;\;\left(\sqrt[3]{\frac{1}{z}} \cdot \sqrt[3]{\frac{1}{z}}\right) \cdot \left(\sqrt[3]{\frac{1}{z}} \cdot \left(\frac{x}{z} \cdot \frac{y}{z + 1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{\left(z \cdot z\right) \cdot \left(z + 1\right)}{y}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(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))))