Average Error: 1.6 → 1.0
Time: 35.5s
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 r45684 = x;
        double r45685 = 4.0;
        double r45686 = r45684 + r45685;
        double r45687 = y;
        double r45688 = r45686 / r45687;
        double r45689 = r45684 / r45687;
        double r45690 = z;
        double r45691 = r45689 * r45690;
        double r45692 = r45688 - r45691;
        double r45693 = fabs(r45692);
        return r45693;
}
↓
double f(double x, double y, double z) {
        double r45694 = y;
        double r45695 = -7.768829658364722e+128;
        bool r45696 = r45694 <= r45695;
        double r45697 = 5.1458150027442665e-163;
        bool r45698 = r45694 <= r45697;
        double r45699 = !r45698;
        bool r45700 = r45696 || r45699;
        double r45701 = x;
        double r45702 = 4.0;
        double r45703 = r45701 + r45702;
        double r45704 = r45703 / r45694;
        double r45705 = z;
        double r45706 = r45705 / r45694;
        double r45707 = r45701 * r45706;
        double r45708 = r45704 - r45707;
        double r45709 = fabs(r45708);
        double r45710 = r45701 * r45705;
        double r45711 = r45703 - r45710;
        double r45712 = r45711 / r45694;
        double r45713 = fabs(r45712);
        double r45714 = r45700 ? r45709 : r45713;
        return r45714;
}

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