Average Error: 1.6 → 0.9
Time: 2.4s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.957413419224405531608231597965624479539 \cdot 10^{110} \lor \neg \left(x \le 1.549094226672070469905729483396986365053 \cdot 10^{-189}\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right) + 4 \cdot \frac{1}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -5.957413419224405531608231597965624479539 \cdot 10^{110} \lor \neg \left(x \le 1.549094226672070469905729483396986365053 \cdot 10^{-189}\right):\\
\;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right) + 4 \cdot \frac{1}{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 r23987 = x;
        double r23988 = 4.0;
        double r23989 = r23987 + r23988;
        double r23990 = y;
        double r23991 = r23989 / r23990;
        double r23992 = r23987 / r23990;
        double r23993 = z;
        double r23994 = r23992 * r23993;
        double r23995 = r23991 - r23994;
        double r23996 = fabs(r23995);
        return r23996;
}
↓
double f(double x, double y, double z) {
        double r23997 = x;
        double r23998 = -5.9574134192244055e+110;
        bool r23999 = r23997 <= r23998;
        double r24000 = 1.5490942266720705e-189;
        bool r24001 = r23997 <= r24000;
        double r24002 = !r24001;
        bool r24003 = r23999 || r24002;
        double r24004 = y;
        double r24005 = r23997 / r24004;
        double r24006 = 1.0;
        double r24007 = z;
        double r24008 = r24006 - r24007;
        double r24009 = r24005 * r24008;
        double r24010 = 4.0;
        double r24011 = r24006 / r24004;
        double r24012 = r24010 * r24011;
        double r24013 = r24009 + r24012;
        double r24014 = fabs(r24013);
        double r24015 = r23997 + r24010;
        double r24016 = r23997 * r24007;
        double r24017 = r24015 - r24016;
        double r24018 = r24017 / r24004;
        double r24019 = fabs(r24018);
        double r24020 = r24003 ? r24014 : r24019;
        return r24020;
}

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 < -5.9574134192244055e+110 or 1.5490942266720705e-189 < x

    1. Initial program 1.2

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

    2. Taylor expanded around 0 6.0

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

    3. Simplified1.2

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

    if -5.9574134192244055e+110 < x < 1.5490942266720705e-189

    1. Initial program 2.0

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

    3. Applied associate-*l/0.6

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

    4. Applied sub-div0.6

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.957413419224405531608231597965624479539 \cdot 10^{110} \lor \neg \left(x \le 1.549094226672070469905729483396986365053 \cdot 10^{-189}\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right) + 4 \cdot \frac{1}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))