Average Error: 1.5 → 0.2
Time: 2.5s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.071982188238494 \cdot 10^{-20}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 6.107865634613209 \cdot 10^{-56}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right) + 4 \cdot \frac{1}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.071982188238494 \cdot 10^{-20}:\\
\;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\

\mathbf{elif}\;x \le 6.107865634613209 \cdot 10^{-56}:\\
\;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\

\mathbf{else}:\\
\;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right) + 4 \cdot \frac{1}{y}\right|\\

\end{array}
double f(double x, double y, double z) {
        double r22901 = x;
        double r22902 = 4.0;
        double r22903 = r22901 + r22902;
        double r22904 = y;
        double r22905 = r22903 / r22904;
        double r22906 = r22901 / r22904;
        double r22907 = z;
        double r22908 = r22906 * r22907;
        double r22909 = r22905 - r22908;
        double r22910 = fabs(r22909);
        return r22910;
}
↓
double f(double x, double y, double z) {
        double r22911 = x;
        double r22912 = -1.0719821882384945e-20;
        bool r22913 = r22911 <= r22912;
        double r22914 = 4.0;
        double r22915 = r22911 + r22914;
        double r22916 = y;
        double r22917 = r22915 / r22916;
        double r22918 = z;
        double r22919 = r22918 / r22916;
        double r22920 = r22911 * r22919;
        double r22921 = r22917 - r22920;
        double r22922 = fabs(r22921);
        double r22923 = 6.107865634613209e-56;
        bool r22924 = r22911 <= r22923;
        double r22925 = r22911 * r22918;
        double r22926 = r22915 - r22925;
        double r22927 = r22926 / r22916;
        double r22928 = fabs(r22927);
        double r22929 = r22911 / r22916;
        double r22930 = 1.0;
        double r22931 = r22930 - r22918;
        double r22932 = r22929 * r22931;
        double r22933 = r22930 / r22916;
        double r22934 = r22914 * r22933;
        double r22935 = r22932 + r22934;
        double r22936 = fabs(r22935);
        double r22937 = r22924 ? r22928 : r22936;
        double r22938 = r22913 ? r22922 : r22937;
        return r22938;
}

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 3 regimes

  2. if x < -1.0719821882384945e-20

    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. Simplified0.2

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

    if -1.0719821882384945e-20 < x < 6.107865634613209e-56

    1. Initial program 2.7

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

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

    if 6.107865634613209e-56 < x

    1. Initial program 0.3

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

    2. Taylor expanded around 0 7.2

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

    3. Simplified0.4

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.071982188238494 \cdot 10^{-20}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 6.107865634613209 \cdot 10^{-56}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right) + 4 \cdot \frac{1}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))