Average Error: 1.8 → 0.5
Time: 15.7s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6.587846277992183793458448564030913674343 \cdot 10^{-111} \lor \neg \left(x \le 1.744861201662154264263076699571251766534 \cdot 10^{-53}\right):\\ \;\;\;\;\left|\left(\frac{x}{y} + \frac{4}{y}\right) - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]
\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{x}{y} + \frac{4}{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 r20679 = x;
        double r20680 = 4.0;
        double r20681 = r20679 + r20680;
        double r20682 = y;
        double r20683 = r20681 / r20682;
        double r20684 = r20679 / r20682;
        double r20685 = z;
        double r20686 = r20684 * r20685;
        double r20687 = r20683 - r20686;
        double r20688 = fabs(r20687);
        return r20688;
}
↓
double f(double x, double y, double z) {
        double r20689 = x;
        double r20690 = -6.587846277992184e-111;
        bool r20691 = r20689 <= r20690;
        double r20692 = 1.7448612016621543e-53;
        bool r20693 = r20689 <= r20692;
        double r20694 = !r20693;
        bool r20695 = r20691 || r20694;
        double r20696 = y;
        double r20697 = r20689 / r20696;
        double r20698 = 4.0;
        double r20699 = r20698 / r20696;
        double r20700 = r20697 + r20699;
        double r20701 = z;
        double r20702 = r20701 / r20696;
        double r20703 = r20689 * r20702;
        double r20704 = r20700 - r20703;
        double r20705 = fabs(r20704);
        double r20706 = r20689 + r20698;
        double r20707 = r20689 * r20701;
        double r20708 = r20706 - r20707;
        double r20709 = r20708 / r20696;
        double r20710 = fabs(r20709);
        double r20711 = r20695 ? r20705 : r20710;
        return r20711;
}

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

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

    5. Applied div-inv0.7

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

    6. Applied associate-*l*0.8

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

    7. Simplified0.8

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

    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

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

Reproduce

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