Average Error: 1.7 → 1.8
Time: 15.6s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.644056109949103 \cdot 10^{+33}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{z}{y} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.644056109949103 \cdot 10^{+33}:\\
\;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{z}{y} \cdot x\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r1138712 = x;
        double r1138713 = 4.0;
        double r1138714 = r1138712 + r1138713;
        double r1138715 = y;
        double r1138716 = r1138714 / r1138715;
        double r1138717 = r1138712 / r1138715;
        double r1138718 = z;
        double r1138719 = r1138717 * r1138718;
        double r1138720 = r1138716 - r1138719;
        double r1138721 = fabs(r1138720);
        return r1138721;
}
↓
double f(double x, double y, double z) {
        double r1138722 = x;
        double r1138723 = -3.644056109949103e+33;
        bool r1138724 = r1138722 <= r1138723;
        double r1138725 = 4.0;
        double r1138726 = y;
        double r1138727 = r1138725 / r1138726;
        double r1138728 = r1138722 / r1138726;
        double r1138729 = r1138727 + r1138728;
        double r1138730 = z;
        double r1138731 = r1138730 / r1138726;
        double r1138732 = r1138731 * r1138722;
        double r1138733 = r1138729 - r1138732;
        double r1138734 = fabs(r1138733);
        double r1138735 = r1138725 + r1138722;
        double r1138736 = r1138722 * r1138730;
        double r1138737 = r1138735 - r1138736;
        double r1138738 = r1138737 / r1138726;
        double r1138739 = fabs(r1138738);
        double r1138740 = r1138724 ? r1138734 : r1138739;
        return r1138740;
}

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 < -3.644056109949103e+33

    1. Initial program 0.1

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

    2. Taylor expanded around 0 0.1

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

    3. Simplified0.1

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

    5. Applied div-inv0.1

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

    6. Applied associate-*l*0.1

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

    7. Simplified0.1

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

    if -3.644056109949103e+33 < x

    1. Initial program 2.0

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

    3. Applied associate-*l/2.1

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

    4. Applied sub-div2.1

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

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.644056109949103 \cdot 10^{+33}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{z}{y} \cdot x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))