Average Error: 1.7 → 0.2
Time: 4.0s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \le -3692424100045327958016 \lor \neg \left(\frac{x + 4}{y} - \frac{x}{y} \cdot z \le 44132419733092834724901161765291029799370000\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \le -3692424100045327958016 \lor \neg \left(\frac{x + 4}{y} - \frac{x}{y} \cdot z \le 44132419733092834724901161765291029799370000\right):\\
\;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\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 r30395 = x;
        double r30396 = 4.0;
        double r30397 = r30395 + r30396;
        double r30398 = y;
        double r30399 = r30397 / r30398;
        double r30400 = r30395 / r30398;
        double r30401 = z;
        double r30402 = r30400 * r30401;
        double r30403 = r30399 - r30402;
        double r30404 = fabs(r30403);
        return r30404;
}
↓
double f(double x, double y, double z) {
        double r30405 = x;
        double r30406 = 4.0;
        double r30407 = r30405 + r30406;
        double r30408 = y;
        double r30409 = r30407 / r30408;
        double r30410 = r30405 / r30408;
        double r30411 = z;
        double r30412 = r30410 * r30411;
        double r30413 = r30409 - r30412;
        double r30414 = -3.692424100045328e+21;
        bool r30415 = r30413 <= r30414;
        double r30416 = 4.4132419733092835e+43;
        bool r30417 = r30413 <= r30416;
        double r30418 = !r30417;
        bool r30419 = r30415 || r30418;
        double r30420 = fabs(r30413);
        double r30421 = r30405 * r30411;
        double r30422 = r30407 - r30421;
        double r30423 = r30422 / r30408;
        double r30424 = fabs(r30423);
        double r30425 = r30419 ? r30420 : r30424;
        return r30425;
}

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 4.0) y) (* (/ x y) z)) < -3.692424100045328e+21 or 4.4132419733092835e+43 < (- (/ (+ x 4.0) y) (* (/ x y) z))

    1. Initial program 0.1

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

    if -3.692424100045328e+21 < (- (/ (+ x 4.0) y) (* (/ x y) z)) < 4.4132419733092835e+43

    1. Initial program 3.6

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

    3. Applied associate-*l/0.3

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \le -3692424100045327958016 \lor \neg \left(\frac{x + 4}{y} - \frac{x}{y} \cdot z \le 44132419733092834724901161765291029799370000\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))