Average Error: 1.7 → 0.6
Time: 12.8s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.05718793893972162 \cdot 10^{142} \lor \neg \left(x \le 8.6806295500210938 \cdot 10^{-56}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{x}{y}, 1 - z, \frac{4}{y}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.05718793893972162 \cdot 10^{142} \lor \neg \left(x \le 8.6806295500210938 \cdot 10^{-56}\right):\\
\;\;\;\;\left|\mathsf{fma}\left(\frac{x}{y}, 1 - z, \frac{4}{y}\right)\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 r18975 = x;
        double r18976 = 4.0;
        double r18977 = r18975 + r18976;
        double r18978 = y;
        double r18979 = r18977 / r18978;
        double r18980 = r18975 / r18978;
        double r18981 = z;
        double r18982 = r18980 * r18981;
        double r18983 = r18979 - r18982;
        double r18984 = fabs(r18983);
        return r18984;
}
↓
double f(double x, double y, double z) {
        double r18985 = x;
        double r18986 = -2.0571879389397216e+142;
        bool r18987 = r18985 <= r18986;
        double r18988 = 8.680629550021094e-56;
        bool r18989 = r18985 <= r18988;
        double r18990 = !r18989;
        bool r18991 = r18987 || r18990;
        double r18992 = y;
        double r18993 = r18985 / r18992;
        double r18994 = 1.0;
        double r18995 = z;
        double r18996 = r18994 - r18995;
        double r18997 = 4.0;
        double r18998 = r18997 / r18992;
        double r18999 = fma(r18993, r18996, r18998);
        double r19000 = fabs(r18999);
        double r19001 = r18985 + r18997;
        double r19002 = r18985 * r18995;
        double r19003 = r19001 - r19002;
        double r19004 = r19003 / r18992;
        double r19005 = fabs(r19004);
        double r19006 = r18991 ? r19000 : r19005;
        return r19006;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Split input into 2 regimes

  2. if x < -2.0571879389397216e+142 or 8.680629550021094e-56 < x

    1. Initial program 0.3

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

    2. Taylor expanded around 0 9.2

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

    3. Simplified0.3

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

    if -2.0571879389397216e+142 < x < 8.680629550021094e-56

    1. Initial program 2.5

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

    3. Applied associate-*l/0.8

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

    4. Applied sub-div0.7

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.05718793893972162 \cdot 10^{142} \lor \neg \left(x \le 8.6806295500210938 \cdot 10^{-56}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{x}{y}, 1 - z, \frac{4}{y}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019199 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))