Average Error: 1.5 → 0.3
Time: 2.6s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.434071340375765 \cdot 10^{73} \lor \neg \left(x \le 2.74765514668075791 \cdot 10^{-60}\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right) + 4 \cdot \frac{1}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -5.434071340375765 \cdot 10^{73} \lor \neg \left(x \le 2.74765514668075791 \cdot 10^{-60}\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 r24030 = x;
        double r24031 = 4.0;
        double r24032 = r24030 + r24031;
        double r24033 = y;
        double r24034 = r24032 / r24033;
        double r24035 = r24030 / r24033;
        double r24036 = z;
        double r24037 = r24035 * r24036;
        double r24038 = r24034 - r24037;
        double r24039 = fabs(r24038);
        return r24039;
}
↓
double f(double x, double y, double z) {
        double r24040 = x;
        double r24041 = -5.434071340375765e+73;
        bool r24042 = r24040 <= r24041;
        double r24043 = 2.747655146680758e-60;
        bool r24044 = r24040 <= r24043;
        double r24045 = !r24044;
        bool r24046 = r24042 || r24045;
        double r24047 = y;
        double r24048 = r24040 / r24047;
        double r24049 = 1.0;
        double r24050 = z;
        double r24051 = r24049 - r24050;
        double r24052 = r24048 * r24051;
        double r24053 = 4.0;
        double r24054 = r24049 / r24047;
        double r24055 = r24053 * r24054;
        double r24056 = r24052 + r24055;
        double r24057 = fabs(r24056);
        double r24058 = r24040 + r24053;
        double r24059 = r24040 * r24050;
        double r24060 = r24058 - r24059;
        double r24061 = r24060 / r24047;
        double r24062 = fabs(r24061);
        double r24063 = r24046 ? r24057 : r24062;
        return r24063;
}

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 < -5.434071340375765e+73 or 2.747655146680758e-60 < x

    1. Initial program 0.3

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

    2. Taylor expanded around 0 8.1

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

    3. Simplified0.3

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

    if -5.434071340375765e+73 < x < 2.747655146680758e-60

    1. Initial program 2.4

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

    3. Applied associate-*l/0.3

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

    4. Applied sub-div0.3

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.434071340375765 \cdot 10^{73} \lor \neg \left(x \le 2.74765514668075791 \cdot 10^{-60}\right):\\ \;\;\;\;\left|\frac{x}{y} \cdot \left(1 - z\right) + 4 \cdot \frac{1}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020033 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))