Average Error: 1.6 → 1.7
Time: 16.4s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1575874805387.5212:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - z \cdot x}{y} + \frac{4}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -1575874805387.5212:\\
\;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r1198725 = x;
        double r1198726 = 4.0;
        double r1198727 = r1198725 + r1198726;
        double r1198728 = y;
        double r1198729 = r1198727 / r1198728;
        double r1198730 = r1198725 / r1198728;
        double r1198731 = z;
        double r1198732 = r1198730 * r1198731;
        double r1198733 = r1198729 - r1198732;
        double r1198734 = fabs(r1198733);
        return r1198734;
}
↓
double f(double x, double y, double z) {
        double r1198735 = x;
        double r1198736 = -1575874805387.5212;
        bool r1198737 = r1198735 <= r1198736;
        double r1198738 = 4.0;
        double r1198739 = r1198738 + r1198735;
        double r1198740 = y;
        double r1198741 = r1198739 / r1198740;
        double r1198742 = z;
        double r1198743 = r1198742 / r1198740;
        double r1198744 = r1198735 * r1198743;
        double r1198745 = r1198741 - r1198744;
        double r1198746 = fabs(r1198745);
        double r1198747 = r1198742 * r1198735;
        double r1198748 = r1198735 - r1198747;
        double r1198749 = r1198748 / r1198740;
        double r1198750 = r1198738 / r1198740;
        double r1198751 = r1198749 + r1198750;
        double r1198752 = fabs(r1198751);
        double r1198753 = r1198737 ? r1198746 : r1198752;
        return r1198753;
}

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 < -1575874805387.5212

    1. Initial program 0.1

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

    3. Applied div-inv0.2

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

    4. Applied associate-*l*0.2

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

    5. Simplified0.1

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

    if -1575874805387.5212 < x

    1. Initial program 1.9

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

    2. Taylor expanded around 0 2.0

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

    3. Simplified1.9

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

    5. Applied associate-*r/2.0

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

    6. Applied sub-div2.0

      \[\leadsto \left|\frac{4}{y} + \color{blue}{\frac{x - z \cdot x}{y}}\right|\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1575874805387.5212:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - z \cdot x}{y} + \frac{4}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))