Average Error: 1.8 → 0.8
Time: 8.7s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.4444603965533054 \cdot 10^{117} \lor \neg \left(x \le 7.43806599639702096 \cdot 10^{-140}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.4444603965533054 \cdot 10^{117} \lor \neg \left(x \le 7.43806599639702096 \cdot 10^{-140}\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 r26717 = x;
        double r26718 = 4.0;
        double r26719 = r26717 + r26718;
        double r26720 = y;
        double r26721 = r26719 / r26720;
        double r26722 = r26717 / r26720;
        double r26723 = z;
        double r26724 = r26722 * r26723;
        double r26725 = r26721 - r26724;
        double r26726 = fabs(r26725);
        return r26726;
}
↓
double f(double x, double y, double z) {
        double r26727 = x;
        double r26728 = -2.4444603965533054e+117;
        bool r26729 = r26727 <= r26728;
        double r26730 = 7.438065996397021e-140;
        bool r26731 = r26727 <= r26730;
        double r26732 = !r26731;
        bool r26733 = r26729 || r26732;
        double r26734 = 4.0;
        double r26735 = r26727 + r26734;
        double r26736 = y;
        double r26737 = r26735 / r26736;
        double r26738 = z;
        double r26739 = r26738 / r26736;
        double r26740 = r26727 * r26739;
        double r26741 = r26737 - r26740;
        double r26742 = fabs(r26741);
        double r26743 = r26727 * r26738;
        double r26744 = r26735 - r26743;
        double r26745 = r26744 / r26736;
        double r26746 = fabs(r26745);
        double r26747 = r26733 ? r26742 : r26746;
        return r26747;
}

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 < -2.4444603965533054e+117 or 7.438065996397021e-140 < x

    1. Initial program 0.8

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

    3. Applied div-inv0.9

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

    4. Applied associate-*l*1.0

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

    5. Simplified1.0

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

    if -2.4444603965533054e+117 < x < 7.438065996397021e-140

    1. Initial program 2.6

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

    3. Applied associate-*l/0.7

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

    4. Applied sub-div0.7

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.4444603965533054 \cdot 10^{117} \lor \neg \left(x \le 7.43806599639702096 \cdot 10^{-140}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))