Average Error: 1.8 → 0.2
Time: 2.9s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.761115116279560819897916091491976758654 \cdot 10^{-47}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 2.987849844254570348767285954039240447636 \cdot 10^{-47}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \left(x \cdot z\right) \cdot \frac{1}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right) + 4 \cdot \frac{1}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -5.761115116279560819897916091491976758654 \cdot 10^{-47}:\\
\;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\

\mathbf{elif}\;x \le 2.987849844254570348767285954039240447636 \cdot 10^{-47}:\\
\;\;\;\;\left|\frac{x + 4}{y} - \left(x \cdot z\right) \cdot \frac{1}{y}\right|\\

\mathbf{else}:\\
\;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right) + 4 \cdot \frac{1}{y}\right|\\

\end{array}
double f(double x, double y, double z) {
        double r27435 = x;
        double r27436 = 4.0;
        double r27437 = r27435 + r27436;
        double r27438 = y;
        double r27439 = r27437 / r27438;
        double r27440 = r27435 / r27438;
        double r27441 = z;
        double r27442 = r27440 * r27441;
        double r27443 = r27439 - r27442;
        double r27444 = fabs(r27443);
        return r27444;
}
↓
double f(double x, double y, double z) {
        double r27445 = x;
        double r27446 = -5.761115116279561e-47;
        bool r27447 = r27445 <= r27446;
        double r27448 = 4.0;
        double r27449 = r27445 + r27448;
        double r27450 = y;
        double r27451 = r27449 / r27450;
        double r27452 = z;
        double r27453 = r27452 / r27450;
        double r27454 = r27445 * r27453;
        double r27455 = r27451 - r27454;
        double r27456 = fabs(r27455);
        double r27457 = 2.9878498442545703e-47;
        bool r27458 = r27445 <= r27457;
        double r27459 = r27445 * r27452;
        double r27460 = 1.0;
        double r27461 = r27460 / r27450;
        double r27462 = r27459 * r27461;
        double r27463 = r27451 - r27462;
        double r27464 = fabs(r27463);
        double r27465 = r27445 / r27450;
        double r27466 = r27460 - r27452;
        double r27467 = r27465 * r27466;
        double r27468 = r27448 * r27461;
        double r27469 = r27467 + r27468;
        double r27470 = fabs(r27469);
        double r27471 = r27458 ? r27464 : r27470;
        double r27472 = r27447 ? r27456 : r27471;
        return r27472;
}

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 < -5.761115116279561e-47

    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 -5.761115116279561e-47 < x < 2.9878498442545703e-47

    1. Initial program 3.2

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

    3. Applied div-inv3.2

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

    4. Applied associate-*l*6.6

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

    5. Simplified6.6

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

    7. Applied div-inv6.6

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

    8. Applied associate-*r*0.1

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

    if 2.9878498442545703e-47 < x

    1. Initial program 0.3

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

    2. Taylor expanded around 0 7.2

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

    3. Simplified0.3

      \[\leadsto \left|\color{blue}{\frac{x}{y} \cdot \left(1 - z\right) + 4 \cdot \frac{1}{y}}\right|\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.761115116279560819897916091491976758654 \cdot 10^{-47}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 2.987849844254570348767285954039240447636 \cdot 10^{-47}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \left(x \cdot z\right) \cdot \frac{1}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right) + 4 \cdot \frac{1}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))