Average Error: 1.7 → 0.3
Time: 11.6s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6.359236426285955 \cdot 10^{-86}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 2.6596617673403336 \cdot 10^{-59}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - x \cdot \frac{z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -6.359236426285955 \cdot 10^{-86}:\\
\;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\

\mathbf{elif}\;x \le 2.6596617673403336 \cdot 10^{-59}:\\
\;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r892884 = x;
        double r892885 = 4.0;
        double r892886 = r892884 + r892885;
        double r892887 = y;
        double r892888 = r892886 / r892887;
        double r892889 = r892884 / r892887;
        double r892890 = z;
        double r892891 = r892889 * r892890;
        double r892892 = r892888 - r892891;
        double r892893 = fabs(r892892);
        return r892893;
}
↓
double f(double x, double y, double z) {
        double r892894 = x;
        double r892895 = -6.359236426285955e-86;
        bool r892896 = r892894 <= r892895;
        double r892897 = 4.0;
        double r892898 = y;
        double r892899 = r892897 / r892898;
        double r892900 = r892894 / r892898;
        double r892901 = r892899 + r892900;
        double r892902 = z;
        double r892903 = r892900 * r892902;
        double r892904 = r892901 - r892903;
        double r892905 = fabs(r892904);
        double r892906 = 2.6596617673403336e-59;
        bool r892907 = r892894 <= r892906;
        double r892908 = r892897 + r892894;
        double r892909 = r892894 * r892902;
        double r892910 = r892908 - r892909;
        double r892911 = r892910 / r892898;
        double r892912 = fabs(r892911);
        double r892913 = r892902 / r892898;
        double r892914 = r892894 * r892913;
        double r892915 = r892901 - r892914;
        double r892916 = fabs(r892915);
        double r892917 = r892907 ? r892912 : r892916;
        double r892918 = r892896 ? r892905 : r892917;
        return r892918;
}

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 < -6.359236426285955e-86

    1. Initial program 0.7

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

    2. Taylor expanded around 0 0.7

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

    3. Simplified0.7

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

    if -6.359236426285955e-86 < x < 2.6596617673403336e-59

    1. Initial program 3.0

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    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 2.6596617673403336e-59 < 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 div-inv0.3

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

    6. Applied associate-*l*0.4

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

    7. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6.359236426285955 \cdot 10^{-86}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 2.6596617673403336 \cdot 10^{-59}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))