Average Error: 1.6 → 0.2
Time: 11.5s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -393523888002957312000 \lor \neg \left(x \le 4.65838789560541676126034813512897433168 \cdot 10^{-85}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -393523888002957312000 \lor \neg \left(x \le 4.65838789560541676126034813512897433168 \cdot 10^{-85}\right):\\
\;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r25974 = x;
        double r25975 = 4.0;
        double r25976 = r25974 + r25975;
        double r25977 = y;
        double r25978 = r25976 / r25977;
        double r25979 = r25974 / r25977;
        double r25980 = z;
        double r25981 = r25979 * r25980;
        double r25982 = r25978 - r25981;
        double r25983 = fabs(r25982);
        return r25983;
}
↓
double f(double x, double y, double z) {
        double r25984 = x;
        double r25985 = -3.935238880029573e+20;
        bool r25986 = r25984 <= r25985;
        double r25987 = 4.658387895605417e-85;
        bool r25988 = r25984 <= r25987;
        double r25989 = !r25988;
        bool r25990 = r25986 || r25989;
        double r25991 = 4.0;
        double r25992 = r25984 + r25991;
        double r25993 = y;
        double r25994 = r25992 / r25993;
        double r25995 = z;
        double r25996 = r25995 / r25993;
        double r25997 = r25984 * r25996;
        double r25998 = r25994 - r25997;
        double r25999 = fabs(r25998);
        double r26000 = r25984 * r25995;
        double r26001 = r25992 - r26000;
        double r26002 = r26001 / r25993;
        double r26003 = fabs(r26002);
        double r26004 = r25990 ? r25999 : r26003;
        return r26004;
}

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 x < -3.935238880029573e+20 or 4.658387895605417e-85 < x

    1. Initial program 0.4

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

    3. Applied div-inv0.4

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

    4. Applied associate-*l*0.4

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

    5. Simplified0.4

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

    if -3.935238880029573e+20 < x < 4.658387895605417e-85

    1. Initial program 2.6

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

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -393523888002957312000 \lor \neg \left(x \le 4.65838789560541676126034813512897433168 \cdot 10^{-85}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))