Average Error: 15.0 → 3.0
Time: 18.0s
Precision: 64
\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;z \le 2.145140342720879107448244448209964264379 \cdot 10^{-92}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{x}{z}}{\frac{z + 1}{y}}\\ \mathbf{elif}\;z \le 1.657287012418471255626513760736376625879 \cdot 10^{87}:\\ \;\;\;\;\frac{\frac{\frac{x \cdot y}{z}}{z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\frac{y}{z}}{z + 1}\\ \end{array}\]
\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}
↓
\begin{array}{l}
\mathbf{if}\;z \le 2.145140342720879107448244448209964264379 \cdot 10^{-92}:\\
\;\;\;\;\frac{1}{z} \cdot \frac{\frac{x}{z}}{\frac{z + 1}{y}}\\

\mathbf{elif}\;z \le 1.657287012418471255626513760736376625879 \cdot 10^{87}:\\
\;\;\;\;\frac{\frac{\frac{x \cdot y}{z}}{z}}{z + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{z} \cdot \frac{\frac{y}{z}}{z + 1}\\

\end{array}
double f(double x, double y, double z) {
        double r242761 = x;
        double r242762 = y;
        double r242763 = r242761 * r242762;
        double r242764 = z;
        double r242765 = r242764 * r242764;
        double r242766 = 1.0;
        double r242767 = r242764 + r242766;
        double r242768 = r242765 * r242767;
        double r242769 = r242763 / r242768;
        return r242769;
}
↓
double f(double x, double y, double z) {
        double r242770 = z;
        double r242771 = 2.145140342720879e-92;
        bool r242772 = r242770 <= r242771;
        double r242773 = 1.0;
        double r242774 = r242773 / r242770;
        double r242775 = x;
        double r242776 = r242775 / r242770;
        double r242777 = 1.0;
        double r242778 = r242770 + r242777;
        double r242779 = y;
        double r242780 = r242778 / r242779;
        double r242781 = r242776 / r242780;
        double r242782 = r242774 * r242781;
        double r242783 = 1.6572870124184713e+87;
        bool r242784 = r242770 <= r242783;
        double r242785 = r242775 * r242779;
        double r242786 = r242785 / r242770;
        double r242787 = r242786 / r242770;
        double r242788 = r242787 / r242778;
        double r242789 = r242779 / r242770;
        double r242790 = r242789 / r242778;
        double r242791 = r242776 * r242790;
        double r242792 = r242784 ? r242788 : r242791;
        double r242793 = r242772 ? r242782 : r242792;
        return r242793;
}

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

Original15.0
Target3.9
Herbie3.0
\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307077115547144785523414612:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if z < 2.145140342720879e-92

    1. Initial program 18.3

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

    3. Applied associate-/r*16.7

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

    5. Applied *-un-lft-identity16.7

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

    6. Applied times-frac3.1

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

    7. Applied times-frac3.7

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

    8. Simplified3.7

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

    10. Applied *-un-lft-identity3.7

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

    11. Applied *-un-lft-identity3.7

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

    12. Applied times-frac3.7

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

    13. Applied associate-/l*3.9

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

    15. Applied associate-/r/3.9

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

    16. Applied *-un-lft-identity3.9

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

    17. Applied add-cube-cbrt3.9

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

    18. Applied times-frac3.9

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

    19. Applied times-frac3.8

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

    20. Applied associate-*r*3.0

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

    21. Simplified3.0

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

    if 2.145140342720879e-92 < z < 1.6572870124184713e+87

    1. Initial program 5.3

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

    3. Applied associate-/r*5.4

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

    5. Applied associate-/r*5.4

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

    if 1.6572870124184713e+87 < z

    1. Initial program 13.2

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

    3. Applied associate-/r*9.4

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

    5. Applied *-un-lft-identity9.4

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

    6. Applied times-frac0.6

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

    7. Applied times-frac1.5

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

    8. Simplified1.5

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

  4. Final simplification3.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 2.145140342720879107448244448209964264379 \cdot 10^{-92}:\\ \;\;\;\;\frac{1}{z} \cdot \frac{\frac{x}{z}}{\frac{z + 1}{y}}\\ \mathbf{elif}\;z \le 1.657287012418471255626513760736376625879 \cdot 10^{87}:\\ \;\;\;\;\frac{\frac{\frac{x \cdot y}{z}}{z}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\frac{y}{z}}{z + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z 249.618281453230708) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1 z)) x) z))

  (/ (* x y) (* (* z z) (+ z 1))))