Average Error: 1.5 → 2.4
Time: 3.5s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le 8.04631218254802011 \cdot 10^{144}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;y \le 8.04631218254802011 \cdot 10^{144}:\\
\;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r31675 = x;
        double r31676 = 4.0;
        double r31677 = r31675 + r31676;
        double r31678 = y;
        double r31679 = r31677 / r31678;
        double r31680 = r31675 / r31678;
        double r31681 = z;
        double r31682 = r31680 * r31681;
        double r31683 = r31679 - r31682;
        double r31684 = fabs(r31683);
        return r31684;
}
↓
double f(double x, double y, double z) {
        double r31685 = y;
        double r31686 = 8.04631218254802e+144;
        bool r31687 = r31685 <= r31686;
        double r31688 = x;
        double r31689 = 4.0;
        double r31690 = r31688 + r31689;
        double r31691 = r31690 / r31685;
        double r31692 = z;
        double r31693 = r31688 * r31692;
        double r31694 = r31693 / r31685;
        double r31695 = r31691 - r31694;
        double r31696 = fabs(r31695);
        double r31697 = r31692 / r31685;
        double r31698 = r31688 * r31697;
        double r31699 = r31691 - r31698;
        double r31700 = fabs(r31699);
        double r31701 = r31687 ? r31696 : r31700;
        return r31701;
}

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 < 8.04631218254802e+144

    1. Initial program 0.9

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

    3. Applied associate-*l/2.8

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

    if 8.04631218254802e+144 < y

    1. Initial program 4.4

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

    3. Applied div-inv4.4

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

    4. Applied associate-*l*0.1

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

    5. Simplified0.1

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

  4. Final simplification2.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 8.04631218254802011 \cdot 10^{144}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))