Average Error: 1.6 → 0.2
Time: 3.8s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6.8540150644513744 \cdot 10^{56} \lor \neg \left(x \le 3.6071842534283706 \cdot 10^{-82}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -6.8540150644513744 \cdot 10^{56} \lor \neg \left(x \le 3.6071842534283706 \cdot 10^{-82}\right):\\
\;\;\;\;\left|\frac{x + 4}{y} - 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 r34592 = x;
        double r34593 = 4.0;
        double r34594 = r34592 + r34593;
        double r34595 = y;
        double r34596 = r34594 / r34595;
        double r34597 = r34592 / r34595;
        double r34598 = z;
        double r34599 = r34597 * r34598;
        double r34600 = r34596 - r34599;
        double r34601 = fabs(r34600);
        return r34601;
}
↓
double f(double x, double y, double z) {
        double r34602 = x;
        double r34603 = -6.854015064451374e+56;
        bool r34604 = r34602 <= r34603;
        double r34605 = 3.6071842534283706e-82;
        bool r34606 = r34602 <= r34605;
        double r34607 = !r34606;
        bool r34608 = r34604 || r34607;
        double r34609 = 4.0;
        double r34610 = r34602 + r34609;
        double r34611 = y;
        double r34612 = r34610 / r34611;
        double r34613 = z;
        double r34614 = r34613 / r34611;
        double r34615 = r34602 * r34614;
        double r34616 = r34612 - r34615;
        double r34617 = fabs(r34616);
        double r34618 = r34602 * r34613;
        double r34619 = r34610 - r34618;
        double r34620 = r34619 / r34611;
        double r34621 = fabs(r34620);
        double r34622 = r34608 ? r34617 : r34621;
        return r34622;
}

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 < -6.854015064451374e+56 or 3.6071842534283706e-82 < x

    1. Initial program 0.4

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

    3. Applied div-inv0.4

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

    4. Applied associate-*l*0.3

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

    5. Simplified0.3

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

    if -6.854015064451374e+56 < x < 3.6071842534283706e-82

    1. Initial program 2.5

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    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 simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6.8540150644513744 \cdot 10^{56} \lor \neg \left(x \le 3.6071842534283706 \cdot 10^{-82}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))