Average Error: 1.5 → 0.4
Time: 15.0s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.109110258294614752055774941493075991437 \cdot 10^{97}:\\ \;\;\;\;\left|\frac{1}{\frac{y}{x + 4}} - 1 \cdot \frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \le 4.00953015364486885541650984422639355434 \cdot 10^{48}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1}{y} \cdot \left(x + 4\right) - \frac{x}{y} \cdot z\right|\\ \end{array}\]
\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}} - 1 \cdot \frac{z}{\frac{y}{x}}\right|\\

\mathbf{elif}\;x \le 4.00953015364486885541650984422639355434 \cdot 10^{48}:\\
\;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r29874 = x;
        double r29875 = 4.0;
        double r29876 = r29874 + r29875;
        double r29877 = y;
        double r29878 = r29876 / r29877;
        double r29879 = r29874 / r29877;
        double r29880 = z;
        double r29881 = r29879 * r29880;
        double r29882 = r29878 - r29881;
        double r29883 = fabs(r29882);
        return r29883;
}
↓
double f(double x, double y, double z) {
        double r29884 = x;
        double r29885 = -1.1091102582946148e+97;
        bool r29886 = r29884 <= r29885;
        double r29887 = 1.0;
        double r29888 = y;
        double r29889 = 4.0;
        double r29890 = r29884 + r29889;
        double r29891 = r29888 / r29890;
        double r29892 = r29887 / r29891;
        double r29893 = z;
        double r29894 = r29888 / r29884;
        double r29895 = r29893 / r29894;
        double r29896 = r29887 * r29895;
        double r29897 = r29892 - r29896;
        double r29898 = fabs(r29897);
        double r29899 = 4.009530153644869e+48;
        bool r29900 = r29884 <= r29899;
        double r29901 = r29884 * r29893;
        double r29902 = r29890 - r29901;
        double r29903 = r29902 / r29888;
        double r29904 = fabs(r29903);
        double r29905 = r29887 / r29888;
        double r29906 = r29905 * r29890;
        double r29907 = r29884 / r29888;
        double r29908 = r29907 * r29893;
        double r29909 = r29906 - r29908;
        double r29910 = fabs(r29909);
        double r29911 = r29900 ? r29904 : r29910;
        double r29912 = r29886 ? r29898 : r29911;
        return r29912;
}

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

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

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

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

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

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

    7. Applied times-frac0.3

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

    8. Applied associate-*l*0.3

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

    9. Simplified0.3

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

    if -1.1091102582946148e+97 < x < 4.009530153644869e+48

    1. Initial program 2.0

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    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 4.009530153644869e+48 < x

    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

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

    5. Applied div-inv0.3

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

    6. Applied add-cube-cbrt0.3

      \[\leadsto \left|\frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{y \cdot \frac{1}{x + 4}} - \frac{x}{y} \cdot z\right|\]

    7. Applied times-frac0.3

      \[\leadsto \left|\color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{y} \cdot \frac{\sqrt[3]{1}}{\frac{1}{x + 4}}} - \frac{x}{y} \cdot z\right|\]

    8. Simplified0.3

      \[\leadsto \left|\color{blue}{\frac{1}{y}} \cdot \frac{\sqrt[3]{1}}{\frac{1}{x + 4}} - \frac{x}{y} \cdot z\right|\]

    9. Simplified0.3

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.109110258294614752055774941493075991437 \cdot 10^{97}:\\ \;\;\;\;\left|\frac{1}{\frac{y}{x + 4}} - 1 \cdot \frac{z}{\frac{y}{x}}\right|\\ \mathbf{elif}\;x \le 4.00953015364486885541650984422639355434 \cdot 10^{48}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{1}{y} \cdot \left(x + 4\right) - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Reproduce

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