Average Error: 1.6 → 0.1
Time: 10.3s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.4899595848850141 \cdot 10^{46} \lor \neg \left(x \le 2.0322004459479788 \cdot 10^{-8}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.4899595848850141 \cdot 10^{46} \lor \neg \left(x \le 2.0322004459479788 \cdot 10^{-8}\right):\\
\;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r36787 = x;
        double r36788 = 4.0;
        double r36789 = r36787 + r36788;
        double r36790 = y;
        double r36791 = r36789 / r36790;
        double r36792 = r36787 / r36790;
        double r36793 = z;
        double r36794 = r36792 * r36793;
        double r36795 = r36791 - r36794;
        double r36796 = fabs(r36795);
        return r36796;
}
↓
double f(double x, double y, double z) {
        double r36797 = x;
        double r36798 = -1.489959584885014e+46;
        bool r36799 = r36797 <= r36798;
        double r36800 = 2.0322004459479788e-08;
        bool r36801 = r36797 <= r36800;
        double r36802 = !r36801;
        bool r36803 = r36799 || r36802;
        double r36804 = 4.0;
        double r36805 = r36797 + r36804;
        double r36806 = y;
        double r36807 = r36805 / r36806;
        double r36808 = z;
        double r36809 = r36808 / r36806;
        double r36810 = r36797 * r36809;
        double r36811 = r36807 - r36810;
        double r36812 = fabs(r36811);
        double r36813 = r36797 * r36808;
        double r36814 = r36805 - r36813;
        double r36815 = r36814 / r36806;
        double r36816 = fabs(r36815);
        double r36817 = r36803 ? r36812 : r36816;
        return r36817;
}

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 < -1.489959584885014e+46 or 2.0322004459479788e-08 < x

    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 -1.489959584885014e+46 < x < 2.0322004459479788e-08

    1. Initial program 2.4

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

    3. Applied associate-*l/0.2

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

    4. Applied sub-div0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.4899595848850141 \cdot 10^{46} \lor \neg \left(x \le 2.0322004459479788 \cdot 10^{-8}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))