Average Error: 1.7 → 1.8
Time: 17.3s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.644056109949103 \cdot 10^{+33}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - z \cdot x}{y} + \frac{4}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.644056109949103 \cdot 10^{+33}:\\
\;\;\;\;\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 r707549 = x;
        double r707550 = 4.0;
        double r707551 = r707549 + r707550;
        double r707552 = y;
        double r707553 = r707551 / r707552;
        double r707554 = r707549 / r707552;
        double r707555 = z;
        double r707556 = r707554 * r707555;
        double r707557 = r707553 - r707556;
        double r707558 = fabs(r707557);
        return r707558;
}
↓
double f(double x, double y, double z) {
        double r707559 = x;
        double r707560 = -3.644056109949103e+33;
        bool r707561 = r707559 <= r707560;
        double r707562 = 4.0;
        double r707563 = r707562 + r707559;
        double r707564 = y;
        double r707565 = r707563 / r707564;
        double r707566 = z;
        double r707567 = r707566 / r707564;
        double r707568 = r707559 * r707567;
        double r707569 = r707565 - r707568;
        double r707570 = fabs(r707569);
        double r707571 = r707566 * r707559;
        double r707572 = r707559 - r707571;
        double r707573 = r707572 / r707564;
        double r707574 = r707562 / r707564;
        double r707575 = r707573 + r707574;
        double r707576 = fabs(r707575);
        double r707577 = r707561 ? r707570 : r707576;
        return r707577;
}

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 < -3.644056109949103e+33

    1. Initial program 0.1

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

    3. Applied div-inv0.1

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

    4. Applied associate-*l*0.1

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

    5. Simplified0.1

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

    if -3.644056109949103e+33 < x

    1. Initial program 2.0

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

    2. Taylor expanded around -inf 2.1

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

    3. Simplified2.0

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

    5. Applied associate-*r/2.1

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

    6. Applied sub-div2.1

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

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.644056109949103 \cdot 10^{+33}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x - z \cdot x}{y} + \frac{4}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))