Average Error: 1.7 → 0.1
Time: 3.5s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1097352739.18932986 \lor \neg \left(x \le 1.1704247077394178 \cdot 10^{-39}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -1097352739.18932986 \lor \neg \left(x \le 1.1704247077394178 \cdot 10^{-39}\right):\\
\;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\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 r29684 = x;
        double r29685 = 4.0;
        double r29686 = r29684 + r29685;
        double r29687 = y;
        double r29688 = r29686 / r29687;
        double r29689 = r29684 / r29687;
        double r29690 = z;
        double r29691 = r29689 * r29690;
        double r29692 = r29688 - r29691;
        double r29693 = fabs(r29692);
        return r29693;
}
↓
double f(double x, double y, double z) {
        double r29694 = x;
        double r29695 = -1097352739.1893299;
        bool r29696 = r29694 <= r29695;
        double r29697 = 1.1704247077394178e-39;
        bool r29698 = r29694 <= r29697;
        double r29699 = !r29698;
        bool r29700 = r29696 || r29699;
        double r29701 = 4.0;
        double r29702 = r29694 + r29701;
        double r29703 = y;
        double r29704 = r29702 / r29703;
        double r29705 = r29694 / r29703;
        double r29706 = z;
        double r29707 = r29705 * r29706;
        double r29708 = r29704 - r29707;
        double r29709 = fabs(r29708);
        double r29710 = r29694 * r29706;
        double r29711 = r29702 - r29710;
        double r29712 = r29711 / r29703;
        double r29713 = fabs(r29712);
        double r29714 = r29700 ? r29709 : r29713;
        return r29714;
}

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 < -1097352739.1893299 or 1.1704247077394178e-39 < x

    1. Initial program 0.2

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

    if -1097352739.1893299 < x < 1.1704247077394178e-39

    1. Initial program 2.8

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

    3. Applied associate-*l/0.1

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

    4. Applied sub-div0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1097352739.18932986 \lor \neg \left(x \le 1.1704247077394178 \cdot 10^{-39}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))