Average Error: 1.8 → 0.5
Time: 15.7s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6.587846277992183793458448564030913674343 \cdot 10^{-111} \lor \neg \left(x \le 1.744861201662154264263076699571251766534 \cdot 10^{-53}\right):\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -6.587846277992183793458448564030913674343 \cdot 10^{-111} \lor \neg \left(x \le 1.744861201662154264263076699571251766534 \cdot 10^{-53}\right):\\
\;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - 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 r27861 = x;
        double r27862 = 4.0;
        double r27863 = r27861 + r27862;
        double r27864 = y;
        double r27865 = r27863 / r27864;
        double r27866 = r27861 / r27864;
        double r27867 = z;
        double r27868 = r27866 * r27867;
        double r27869 = r27865 - r27868;
        double r27870 = fabs(r27869);
        return r27870;
}
↓
double f(double x, double y, double z) {
        double r27871 = x;
        double r27872 = -6.587846277992184e-111;
        bool r27873 = r27871 <= r27872;
        double r27874 = 1.7448612016621543e-53;
        bool r27875 = r27871 <= r27874;
        double r27876 = !r27875;
        bool r27877 = r27873 || r27876;
        double r27878 = 4.0;
        double r27879 = y;
        double r27880 = r27878 / r27879;
        double r27881 = r27871 / r27879;
        double r27882 = r27880 + r27881;
        double r27883 = z;
        double r27884 = r27883 / r27879;
        double r27885 = r27871 * r27884;
        double r27886 = r27882 - r27885;
        double r27887 = fabs(r27886);
        double r27888 = r27871 + r27878;
        double r27889 = r27871 * r27883;
        double r27890 = r27888 - r27889;
        double r27891 = r27890 / r27879;
        double r27892 = fabs(r27891);
        double r27893 = r27877 ? r27887 : r27892;
        return r27893;
}

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 < -6.587846277992184e-111 or 1.7448612016621543e-53 < x

    1. Initial program 0.6

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

    2. Taylor expanded around 0 0.6

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

    3. Simplified0.6

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

    5. Applied div-inv0.7

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

    6. Applied associate-*l*0.8

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

    7. Simplified0.8

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

    if -6.587846277992184e-111 < x < 1.7448612016621543e-53

    1. Initial program 3.2

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    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 simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6.587846277992183793458448564030913674343 \cdot 10^{-111} \lor \neg \left(x \le 1.744861201662154264263076699571251766534 \cdot 10^{-53}\right):\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))