Average Error: 1.6 → 1.0
Time: 16.4s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;y \le -7.768829658364721501136974287207896432773 \cdot 10^{128} \lor \neg \left(y \le 5.145815002744266464401609035602130947601 \cdot 10^{-163}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;y \le -7.768829658364721501136974287207896432773 \cdot 10^{128} \lor \neg \left(y \le 5.145815002744266464401609035602130947601 \cdot 10^{-163}\right):\\
\;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r37014 = x;
        double r37015 = 4.0;
        double r37016 = r37014 + r37015;
        double r37017 = y;
        double r37018 = r37016 / r37017;
        double r37019 = r37014 / r37017;
        double r37020 = z;
        double r37021 = r37019 * r37020;
        double r37022 = r37018 - r37021;
        double r37023 = fabs(r37022);
        return r37023;
}
↓
double f(double x, double y, double z) {
        double r37024 = y;
        double r37025 = -7.768829658364722e+128;
        bool r37026 = r37024 <= r37025;
        double r37027 = 5.1458150027442665e-163;
        bool r37028 = r37024 <= r37027;
        double r37029 = !r37028;
        bool r37030 = r37026 || r37029;
        double r37031 = x;
        double r37032 = 4.0;
        double r37033 = r37031 + r37032;
        double r37034 = r37033 / r37024;
        double r37035 = z;
        double r37036 = r37035 / r37024;
        double r37037 = r37031 * r37036;
        double r37038 = r37034 - r37037;
        double r37039 = fabs(r37038);
        double r37040 = r37031 * r37035;
        double r37041 = r37040 / r37024;
        double r37042 = r37034 - r37041;
        double r37043 = fabs(r37042);
        double r37044 = r37030 ? r37039 : r37043;
        return r37044;
}

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 y < -7.768829658364722e+128 or 5.1458150027442665e-163 < y

    1. Initial program 2.5

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

    3. Applied div-inv2.5

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

    4. Applied associate-*l*1.1

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

    5. Simplified1.1

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

    if -7.768829658364722e+128 < y < 5.1458150027442665e-163

    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*7.9

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

    5. Simplified7.9

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

    7. Applied *-un-lft-identity7.9

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

    8. Applied associate-*l*7.9

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

    9. Simplified0.9

      \[\leadsto \left|\frac{x + 4}{y} - 1 \cdot \color{blue}{\frac{x \cdot z}{y}}\right|\]
  3. Recombined 2 regimes into one program.

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -7.768829658364721501136974287207896432773 \cdot 10^{128} \lor \neg \left(y \le 5.145815002744266464401609035602130947601 \cdot 10^{-163}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x \cdot z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))