Average Error: 1.6 → 0.4
Time: 12.0s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.5186598216349587 \cdot 10^{+49}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 5.426503285374244 \cdot 10^{-130}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.5186598216349587 \cdot 10^{+49}:\\
\;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\

\mathbf{elif}\;x \le 5.426503285374244 \cdot 10^{-130}:\\
\;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{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 r1681731 = x;
        double r1681732 = 4.0;
        double r1681733 = r1681731 + r1681732;
        double r1681734 = y;
        double r1681735 = r1681733 / r1681734;
        double r1681736 = r1681731 / r1681734;
        double r1681737 = z;
        double r1681738 = r1681736 * r1681737;
        double r1681739 = r1681735 - r1681738;
        double r1681740 = fabs(r1681739);
        return r1681740;
}
↓
double f(double x, double y, double z) {
        double r1681741 = x;
        double r1681742 = -4.5186598216349587e+49;
        bool r1681743 = r1681741 <= r1681742;
        double r1681744 = 4.0;
        double r1681745 = r1681744 + r1681741;
        double r1681746 = y;
        double r1681747 = r1681745 / r1681746;
        double r1681748 = z;
        double r1681749 = r1681748 / r1681746;
        double r1681750 = r1681741 * r1681749;
        double r1681751 = r1681747 - r1681750;
        double r1681752 = fabs(r1681751);
        double r1681753 = 5.426503285374244e-130;
        bool r1681754 = r1681741 <= r1681753;
        double r1681755 = r1681748 * r1681741;
        double r1681756 = r1681745 - r1681755;
        double r1681757 = r1681756 / r1681746;
        double r1681758 = fabs(r1681757);
        double r1681759 = r1681754 ? r1681758 : r1681752;
        double r1681760 = r1681743 ? r1681752 : r1681759;
        return r1681760;
}

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 < -4.5186598216349587e+49 or 5.426503285374244e-130 < x

    1. Initial program 0.7

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

    3. Applied div-inv0.7

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

    4. Applied associate-*l*0.8

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

    5. Simplified0.7

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

    if -4.5186598216349587e+49 < x < 5.426503285374244e-130

    1. Initial program 2.4

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

    3. Applied associate-*l/0.2

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

    4. Applied sub-div0.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.5186598216349587 \cdot 10^{+49}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 5.426503285374244 \cdot 10^{-130}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))