Average Error: 1.4 → 1.7
Time: 12.1s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.968897763889820057096245510304041678837 \cdot 10^{89}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z}{y} \cdot x\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le 1.968897763889820057096245510304041678837 \cdot 10^{89}:\\
\;\;\;\;\left|\frac{4 + x}{y} - \frac{z \cdot x}{y}\right|\\

\mathbf{else}:\\
\;\;\;\;\left|\frac{4 + x}{y} - \frac{z}{y} \cdot x\right|\\

\end{array}
double f(double x, double y, double z) {
        double r21807 = x;
        double r21808 = 4.0;
        double r21809 = r21807 + r21808;
        double r21810 = y;
        double r21811 = r21809 / r21810;
        double r21812 = r21807 / r21810;
        double r21813 = z;
        double r21814 = r21812 * r21813;
        double r21815 = r21811 - r21814;
        double r21816 = fabs(r21815);
        return r21816;
}
↓
double f(double x, double y, double z) {
        double r21817 = x;
        double r21818 = 1.96889776388982e+89;
        bool r21819 = r21817 <= r21818;
        double r21820 = 4.0;
        double r21821 = r21820 + r21817;
        double r21822 = y;
        double r21823 = r21821 / r21822;
        double r21824 = z;
        double r21825 = r21824 * r21817;
        double r21826 = r21825 / r21822;
        double r21827 = r21823 - r21826;
        double r21828 = fabs(r21827);
        double r21829 = r21824 / r21822;
        double r21830 = r21829 * r21817;
        double r21831 = r21823 - r21830;
        double r21832 = fabs(r21831);
        double r21833 = r21819 ? r21828 : r21832;
        return r21833;
}

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

Derivation

  1. Split input into 2 regimes

  2. if x < 1.96889776388982e+89

    1. Initial program 1.6

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    2. Using strategy rm

    3. Applied associate-*l/1.9

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{x \cdot z}{y}}\right|\]

    4. Simplified1.9

      \[\leadsto \left|\frac{x + 4}{y} - \frac{\color{blue}{z \cdot x}}{y}\right|\]

    if 1.96889776388982e+89 < x

    1. Initial program 0.1

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    2. Using strategy rm

    3. Applied *-un-lft-identity0.1

      \[\leadsto \left|\color{blue}{1 \cdot \left(\frac{x + 4}{y} - \frac{x}{y} \cdot z\right)}\right|\]
    4. Using strategy rm

    5. Applied div-inv0.1

      \[\leadsto \left|1 \cdot \left(\frac{x + 4}{y} - \color{blue}{\left(x \cdot \frac{1}{y}\right)} \cdot z\right)\right|\]

    6. Applied associate-*l*0.1

      \[\leadsto \left|1 \cdot \left(\frac{x + 4}{y} - \color{blue}{x \cdot \left(\frac{1}{y} \cdot z\right)}\right)\right|\]

    7. Simplified0.1

      \[\leadsto \left|1 \cdot \left(\frac{x + 4}{y} - x \cdot \color{blue}{\frac{z}{y}}\right)\right|\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.968897763889820057096245510304041678837 \cdot 10^{89}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z}{y} \cdot x\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))