Average Error: 1.8 → 0.7
Time: 2.7s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.21510854934102217 \cdot 10^{93}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 7.43806599639702096 \cdot 10^{-140}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.21510854934102217 \cdot 10^{93}:\\
\;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\

\mathbf{elif}\;x \le 7.43806599639702096 \cdot 10^{-140}:\\
\;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\

\mathbf{else}:\\
\;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\

\end{array}
double f(double x, double y, double z) {
        double r25425 = x;
        double r25426 = 4.0;
        double r25427 = r25425 + r25426;
        double r25428 = y;
        double r25429 = r25427 / r25428;
        double r25430 = r25425 / r25428;
        double r25431 = z;
        double r25432 = r25430 * r25431;
        double r25433 = r25429 - r25432;
        double r25434 = fabs(r25433);
        return r25434;
}
↓
double f(double x, double y, double z) {
        double r25435 = x;
        double r25436 = -4.215108549341022e+93;
        bool r25437 = r25435 <= r25436;
        double r25438 = 4.0;
        double r25439 = r25435 + r25438;
        double r25440 = y;
        double r25441 = r25439 / r25440;
        double r25442 = r25435 / r25440;
        double r25443 = z;
        double r25444 = r25442 * r25443;
        double r25445 = r25441 - r25444;
        double r25446 = fabs(r25445);
        double r25447 = 7.438065996397021e-140;
        bool r25448 = r25435 <= r25447;
        double r25449 = r25435 * r25443;
        double r25450 = r25439 - r25449;
        double r25451 = r25450 / r25440;
        double r25452 = fabs(r25451);
        double r25453 = r25443 / r25440;
        double r25454 = r25435 * r25453;
        double r25455 = r25441 - r25454;
        double r25456 = fabs(r25455);
        double r25457 = r25448 ? r25452 : r25456;
        double r25458 = r25437 ? r25446 : r25457;
        return r25458;
}

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 3 regimes

  2. if x < -4.215108549341022e+93

    1. Initial program 0.1

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

    if -4.215108549341022e+93 < x < 7.438065996397021e-140

    1. Initial program 2.7

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

    3. Applied associate-*l/0.5

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

    4. Applied sub-div0.5

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

    if 7.438065996397021e-140 < x

    1. Initial program 1.1

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

    3. Applied div-inv1.1

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

    4. Applied associate-*l*1.3

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

    5. Simplified1.2

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.21510854934102217 \cdot 10^{93}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 7.43806599639702096 \cdot 10^{-140}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))