Average Error: 1.5 → 0.4
Time: 15.4s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.109110258294614752055774941493075991437 \cdot 10^{97}:\\ \;\;\;\;\left|\frac{1}{\frac{y}{x + 4}} - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 214190368136.352081298828125:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.109110258294614752055774941493075991437 \cdot 10^{97}:\\
\;\;\;\;\left|\frac{1}{\frac{y}{x + 4}} - \frac{x}{y} \cdot z\right|\\

\mathbf{elif}\;x \le 214190368136.352081298828125:\\
\;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r36896 = x;
        double r36897 = 4.0;
        double r36898 = r36896 + r36897;
        double r36899 = y;
        double r36900 = r36898 / r36899;
        double r36901 = r36896 / r36899;
        double r36902 = z;
        double r36903 = r36901 * r36902;
        double r36904 = r36900 - r36903;
        double r36905 = fabs(r36904);
        return r36905;
}
↓
double f(double x, double y, double z) {
        double r36906 = x;
        double r36907 = -1.1091102582946148e+97;
        bool r36908 = r36906 <= r36907;
        double r36909 = 1.0;
        double r36910 = y;
        double r36911 = 4.0;
        double r36912 = r36906 + r36911;
        double r36913 = r36910 / r36912;
        double r36914 = r36909 / r36913;
        double r36915 = r36906 / r36910;
        double r36916 = z;
        double r36917 = r36915 * r36916;
        double r36918 = r36914 - r36917;
        double r36919 = fabs(r36918);
        double r36920 = 214190368136.35208;
        bool r36921 = r36906 <= r36920;
        double r36922 = r36906 * r36916;
        double r36923 = r36912 - r36922;
        double r36924 = r36923 / r36910;
        double r36925 = fabs(r36924);
        double r36926 = r36912 / r36910;
        double r36927 = r36916 / r36910;
        double r36928 = r36906 * r36927;
        double r36929 = r36926 - r36928;
        double r36930 = fabs(r36929);
        double r36931 = r36921 ? r36925 : r36930;
        double r36932 = r36908 ? r36919 : r36931;
        return r36932;
}

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 < -1.1091102582946148e+97

    1. Initial program 0.1

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

    3. Applied clear-num0.3

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

    if -1.1091102582946148e+97 < x < 214190368136.35208

    1. Initial program 2.1

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

    3. Applied associate-*l/0.5

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

    4. Applied sub-div0.5

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

    if 214190368136.35208 < x

    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. Simplified0.1

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.109110258294614752055774941493075991437 \cdot 10^{97}:\\ \;\;\;\;\left|\frac{1}{\frac{y}{x + 4}} - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 214190368136.352081298828125:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))