Average Error: 1.6 → 1.0
Time: 15.3s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -7.768829658364721501136974287207896432773 \cdot 10^{128} \lor \neg \left(y \le 5.145815002744266464401609035602130947601 \cdot 10^{-163}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]
\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{\left(x + 4\right) - x \cdot z}{y}\right|\\

\end{array}
double f(double x, double y, double z) {
        double r38629 = x;
        double r38630 = 4.0;
        double r38631 = r38629 + r38630;
        double r38632 = y;
        double r38633 = r38631 / r38632;
        double r38634 = r38629 / r38632;
        double r38635 = z;
        double r38636 = r38634 * r38635;
        double r38637 = r38633 - r38636;
        double r38638 = fabs(r38637);
        return r38638;
}
↓
double f(double x, double y, double z) {
        double r38639 = y;
        double r38640 = -7.768829658364722e+128;
        bool r38641 = r38639 <= r38640;
        double r38642 = 5.1458150027442665e-163;
        bool r38643 = r38639 <= r38642;
        double r38644 = !r38643;
        bool r38645 = r38641 || r38644;
        double r38646 = x;
        double r38647 = 4.0;
        double r38648 = r38646 + r38647;
        double r38649 = r38648 / r38639;
        double r38650 = z;
        double r38651 = r38650 / r38639;
        double r38652 = r38646 * r38651;
        double r38653 = r38649 - r38652;
        double r38654 = fabs(r38653);
        double r38655 = r38646 * r38650;
        double r38656 = r38648 - r38655;
        double r38657 = r38656 / r38639;
        double r38658 = fabs(r38657);
        double r38659 = r38645 ? r38654 : r38658;
        return r38659;
}

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 div-inv0.2

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

    4. Applied associate-*l*7.9

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

    5. Simplified7.9

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

    7. Applied associate-*r/0.9

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

    8. Applied sub-div0.9

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

  4. Final simplification1.0

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

Reproduce

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