Average Error: 1.4 → 0.1
Time: 8.7s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right| \le 871440318472255963136:\\ \;\;\;\;\left|\frac{\left(4 - x \cdot z\right) + x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right| \le 871440318472255963136:\\
\;\;\;\;\left|\frac{\left(4 - x \cdot z\right) + x}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r16971 = x;
        double r16972 = 4.0;
        double r16973 = r16971 + r16972;
        double r16974 = y;
        double r16975 = r16973 / r16974;
        double r16976 = r16971 / r16974;
        double r16977 = z;
        double r16978 = r16976 * r16977;
        double r16979 = r16975 - r16978;
        double r16980 = fabs(r16979);
        return r16980;
}
↓
double f(double x, double y, double z) {
        double r16981 = 4.0;
        double r16982 = x;
        double r16983 = r16981 + r16982;
        double r16984 = y;
        double r16985 = r16983 / r16984;
        double r16986 = r16982 / r16984;
        double r16987 = z;
        double r16988 = r16986 * r16987;
        double r16989 = r16985 - r16988;
        double r16990 = fabs(r16989);
        double r16991 = 8.71440318472256e+20;
        bool r16992 = r16990 <= r16991;
        double r16993 = r16982 * r16987;
        double r16994 = r16981 - r16993;
        double r16995 = r16994 + r16982;
        double r16996 = r16995 / r16984;
        double r16997 = fabs(r16996);
        double r16998 = r16992 ? r16997 : r16990;
        return r16998;
}

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 (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))) < 8.71440318472256e+20

    1. Initial program 3.2

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

    3. Applied associate-*l/0.1

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

    4. Applied sub-div0.1

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

    5. Simplified0.1

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

    if 8.71440318472256e+20 < (fabs (- (/ (+ x 4.0) y) (* (/ x y) z)))

    1. Initial program 0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right| \le 871440318472255963136:\\ \;\;\;\;\left|\frac{\left(4 - x \cdot z\right) + x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))