Average Error: 1.7 → 1.8
Time: 10.8s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 2.331683792302665078276548190002856814797 \cdot 10^{-52}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{x}{y} + \frac{4}{y}\right) - \frac{x}{\frac{y}{z}}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le 2.331683792302665078276548190002856814797 \cdot 10^{-52}:\\
\;\;\;\;\left|\frac{4 + x}{y} - \frac{z \cdot x}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r19828 = x;
        double r19829 = 4.0;
        double r19830 = r19828 + r19829;
        double r19831 = y;
        double r19832 = r19830 / r19831;
        double r19833 = r19828 / r19831;
        double r19834 = z;
        double r19835 = r19833 * r19834;
        double r19836 = r19832 - r19835;
        double r19837 = fabs(r19836);
        return r19837;
}
↓
double f(double x, double y, double z) {
        double r19838 = x;
        double r19839 = 2.331683792302665e-52;
        bool r19840 = r19838 <= r19839;
        double r19841 = 4.0;
        double r19842 = r19841 + r19838;
        double r19843 = y;
        double r19844 = r19842 / r19843;
        double r19845 = z;
        double r19846 = r19845 * r19838;
        double r19847 = r19846 / r19843;
        double r19848 = r19844 - r19847;
        double r19849 = fabs(r19848);
        double r19850 = r19838 / r19843;
        double r19851 = r19841 / r19843;
        double r19852 = r19850 + r19851;
        double r19853 = r19843 / r19845;
        double r19854 = r19838 / r19853;
        double r19855 = r19852 - r19854;
        double r19856 = fabs(r19855);
        double r19857 = r19840 ? r19849 : r19856;
        return r19857;
}

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 < 2.331683792302665e-52

    1. Initial program 2.1

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

    3. Applied associate-*l/2.3

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

    4. Simplified2.3

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

    if 2.331683792302665e-52 < x

    1. Initial program 0.3

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

    2. Taylor expanded around 0 0.3

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

    3. Simplified0.3

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

    5. Applied *-un-lft-identity0.3

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

    6. Applied associate-*l*0.3

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

    7. Simplified0.3

      \[\leadsto \left|\left(\frac{x}{y} + \frac{4}{y}\right) - 1 \cdot \color{blue}{\frac{x}{\frac{y}{z}}}\right|\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2.331683792302665078276548190002856814797 \cdot 10^{-52}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{x}{y} + \frac{4}{y}\right) - \frac{x}{\frac{y}{z}}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))