Average Error: 1.4 → 0.4
Time: 14.8s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.1618774464985056 \cdot 10^{+63}:\\ \;\;\;\;\left|\frac{4}{y} - \left(\frac{x}{y} \cdot z - \frac{x}{y}\right)\right|\\ \mathbf{elif}\;x \le 3.5037736689937804 \cdot 10^{-122}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4}{y} - \left(\frac{x}{y} \cdot z - \frac{x}{y}\right)\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.1618774464985056 \cdot 10^{+63}:\\
\;\;\;\;\left|\frac{4}{y} - \left(\frac{x}{y} \cdot z - \frac{x}{y}\right)\right|\\

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

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

\end{array}
double f(double x, double y, double z) {
        double r875776 = x;
        double r875777 = 4.0;
        double r875778 = r875776 + r875777;
        double r875779 = y;
        double r875780 = r875778 / r875779;
        double r875781 = r875776 / r875779;
        double r875782 = z;
        double r875783 = r875781 * r875782;
        double r875784 = r875780 - r875783;
        double r875785 = fabs(r875784);
        return r875785;
}
↓
double f(double x, double y, double z) {
        double r875786 = x;
        double r875787 = -2.1618774464985056e+63;
        bool r875788 = r875786 <= r875787;
        double r875789 = 4.0;
        double r875790 = y;
        double r875791 = r875789 / r875790;
        double r875792 = r875786 / r875790;
        double r875793 = z;
        double r875794 = r875792 * r875793;
        double r875795 = r875794 - r875792;
        double r875796 = r875791 - r875795;
        double r875797 = fabs(r875796);
        double r875798 = 3.5037736689937804e-122;
        bool r875799 = r875786 <= r875798;
        double r875800 = r875789 + r875786;
        double r875801 = r875786 * r875793;
        double r875802 = r875800 - r875801;
        double r875803 = r875802 / r875790;
        double r875804 = fabs(r875803);
        double r875805 = r875799 ? r875804 : r875797;
        double r875806 = r875788 ? r875797 : r875805;
        return r875806;
}

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 6.1

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

    3. Simplified0.5

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

    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

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

Reproduce

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