Average Error: 1.4 → 0.4
Time: 19.9s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.1618774464985056 \cdot 10^{+63}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 3.5037736689937804 \cdot 10^{-122}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.1618774464985056 \cdot 10^{+63}:\\
\;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\

\mathbf{elif}\;x \le 3.5037736689937804 \cdot 10^{-122}:\\
\;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r1232639 = x;
        double r1232640 = 4.0;
        double r1232641 = r1232639 + r1232640;
        double r1232642 = y;
        double r1232643 = r1232641 / r1232642;
        double r1232644 = r1232639 / r1232642;
        double r1232645 = z;
        double r1232646 = r1232644 * r1232645;
        double r1232647 = r1232643 - r1232646;
        double r1232648 = fabs(r1232647);
        return r1232648;
}
↓
double f(double x, double y, double z) {
        double r1232649 = x;
        double r1232650 = -2.1618774464985056e+63;
        bool r1232651 = r1232649 <= r1232650;
        double r1232652 = 4.0;
        double r1232653 = y;
        double r1232654 = r1232652 / r1232653;
        double r1232655 = r1232649 / r1232653;
        double r1232656 = r1232654 + r1232655;
        double r1232657 = z;
        double r1232658 = r1232655 * r1232657;
        double r1232659 = r1232656 - r1232658;
        double r1232660 = fabs(r1232659);
        double r1232661 = 3.5037736689937804e-122;
        bool r1232662 = r1232649 <= r1232661;
        double r1232663 = r1232652 + r1232649;
        double r1232664 = r1232649 * r1232657;
        double r1232665 = r1232663 - r1232664;
        double r1232666 = r1232665 / r1232653;
        double r1232667 = fabs(r1232666);
        double r1232668 = r1232662 ? r1232667 : r1232660;
        double r1232669 = r1232651 ? r1232660 : r1232668;
        return r1232669;
}

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 < -2.1618774464985056e+63 or 3.5037736689937804e-122 < x

    1. Initial program 0.4

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

    2. Taylor expanded around 0 0.5

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

    3. Simplified0.5

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

    if -2.1618774464985056e+63 < x < 3.5037736689937804e-122

    1. Initial program 2.3

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    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-div0.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.1618774464985056 \cdot 10^{+63}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 3.5037736689937804 \cdot 10^{-122}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))