Average Error: 1.7 → 0.7
Time: 11.6s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.572611148022768 \cdot 10^{-182}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 8.927747126101457 \cdot 10^{+27}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \left(z \cdot x\right) \cdot \frac{1}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.572611148022768 \cdot 10^{-182}:\\
\;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\

\mathbf{elif}\;x \le 8.927747126101457 \cdot 10^{+27}:\\
\;\;\;\;\left|\frac{4 + x}{y} - \left(z \cdot x\right) \cdot \frac{1}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r536661 = x;
        double r536662 = 4.0;
        double r536663 = r536661 + r536662;
        double r536664 = y;
        double r536665 = r536663 / r536664;
        double r536666 = r536661 / r536664;
        double r536667 = z;
        double r536668 = r536666 * r536667;
        double r536669 = r536665 - r536668;
        double r536670 = fabs(r536669);
        return r536670;
}
↓
double f(double x, double y, double z) {
        double r536671 = x;
        double r536672 = -3.572611148022768e-182;
        bool r536673 = r536671 <= r536672;
        double r536674 = 4.0;
        double r536675 = r536674 + r536671;
        double r536676 = y;
        double r536677 = r536675 / r536676;
        double r536678 = z;
        double r536679 = r536678 / r536676;
        double r536680 = r536671 * r536679;
        double r536681 = r536677 - r536680;
        double r536682 = fabs(r536681);
        double r536683 = 8.927747126101457e+27;
        bool r536684 = r536671 <= r536683;
        double r536685 = r536678 * r536671;
        double r536686 = 1.0;
        double r536687 = r536686 / r536676;
        double r536688 = r536685 * r536687;
        double r536689 = r536677 - r536688;
        double r536690 = fabs(r536689);
        double r536691 = r536684 ? r536690 : r536682;
        double r536692 = r536673 ? r536682 : r536691;
        return r536692;
}

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 < -3.572611148022768e-182 or 8.927747126101457e+27 < x

    1. Initial program 1.0

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

    3. Applied div-inv1.0

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

    4. Applied associate-*l*1.3

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

    5. Simplified1.3

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

    if -3.572611148022768e-182 < x < 8.927747126101457e+27

    1. Initial program 2.6

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

    3. Applied div-inv2.6

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

    4. Applied associate-*l*5.7

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

    5. Simplified5.7

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

    7. Applied div-inv5.7

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

    8. Applied associate-*r*0.1

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\left(x \cdot z\right) \cdot \frac{1}{y}}\right|\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.572611148022768 \cdot 10^{-182}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 8.927747126101457 \cdot 10^{+27}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \left(z \cdot x\right) \cdot \frac{1}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))