Average Error: 1.6 → 1.0
Time: 16.0s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -7.768829658364721501136974287207896432773 \cdot 10^{128} \lor \neg \left(y \le 5.145815002744266464401609035602130947601 \cdot 10^{-163}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;y \le -7.768829658364721501136974287207896432773 \cdot 10^{128} \lor \neg \left(y \le 5.145815002744266464401609035602130947601 \cdot 10^{-163}\right):\\
\;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r32720 = x;
        double r32721 = 4.0;
        double r32722 = r32720 + r32721;
        double r32723 = y;
        double r32724 = r32722 / r32723;
        double r32725 = r32720 / r32723;
        double r32726 = z;
        double r32727 = r32725 * r32726;
        double r32728 = r32724 - r32727;
        double r32729 = fabs(r32728);
        return r32729;
}
↓
double f(double x, double y, double z) {
        double r32730 = y;
        double r32731 = -7.768829658364722e+128;
        bool r32732 = r32730 <= r32731;
        double r32733 = 5.1458150027442665e-163;
        bool r32734 = r32730 <= r32733;
        double r32735 = !r32734;
        bool r32736 = r32732 || r32735;
        double r32737 = x;
        double r32738 = 4.0;
        double r32739 = r32737 + r32738;
        double r32740 = r32739 / r32730;
        double r32741 = z;
        double r32742 = r32741 / r32730;
        double r32743 = r32737 * r32742;
        double r32744 = r32740 - r32743;
        double r32745 = fabs(r32744);
        double r32746 = r32737 * r32741;
        double r32747 = r32746 / r32730;
        double r32748 = r32740 - r32747;
        double r32749 = fabs(r32748);
        double r32750 = r32736 ? r32745 : r32749;
        return r32750;
}

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 y < -7.768829658364722e+128 or 5.1458150027442665e-163 < y

    1. Initial program 2.5

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

    3. Applied div-inv2.5

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

    4. Applied associate-*l*1.1

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

    5. Simplified1.1

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

    if -7.768829658364722e+128 < y < 5.1458150027442665e-163

    1. Initial program 0.1

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

    3. Applied associate-*l/0.9

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -7.768829658364721501136974287207896432773 \cdot 10^{128} \lor \neg \left(y \le 5.145815002744266464401609035602130947601 \cdot 10^{-163}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))