Average Error: 1.5 → 0.2
Time: 7.0s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \le -1.41012956374722444 \cdot 10^{118} \lor \neg \left(\frac{x + 4}{y} - \frac{x}{y} \cdot z \le 7.6391539351210826 \cdot 10^{58}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \le -1.41012956374722444 \cdot 10^{118} \lor \neg \left(\frac{x + 4}{y} - \frac{x}{y} \cdot z \le 7.6391539351210826 \cdot 10^{58}\right):\\
\;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r137 = x;
        double r138 = 4.0;
        double r139 = r137 + r138;
        double r140 = y;
        double r141 = r139 / r140;
        double r142 = r137 / r140;
        double r143 = z;
        double r144 = r142 * r143;
        double r145 = r141 - r144;
        double r146 = fabs(r145);
        return r146;
}
↓
double f(double x, double y, double z) {
        double r147 = x;
        double r148 = 4.0;
        double r149 = r147 + r148;
        double r150 = y;
        double r151 = r149 / r150;
        double r152 = r147 / r150;
        double r153 = z;
        double r154 = r152 * r153;
        double r155 = r151 - r154;
        double r156 = -1.4101295637472244e+118;
        bool r157 = r155 <= r156;
        double r158 = 7.639153935121083e+58;
        bool r159 = r155 <= r158;
        double r160 = !r159;
        bool r161 = r157 || r160;
        double r162 = fabs(r155);
        double r163 = r153 / r150;
        double r164 = r147 * r163;
        double r165 = r151 - r164;
        double r166 = fabs(r165);
        double r167 = r161 ? r162 : r166;
        return r167;
}

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 4.0) y) (* (/ x y) z)) < -1.4101295637472244e+118 or 7.639153935121083e+58 < (- (/ (+ x 4.0) y) (* (/ x y) z))

    1. Initial program 0.1

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

    if -1.4101295637472244e+118 < (- (/ (+ x 4.0) y) (* (/ x y) z)) < 7.639153935121083e+58

    1. Initial program 2.5

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

    3. Applied div-inv2.6

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

    4. Applied associate-*l*0.3

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

    5. Simplified0.3

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + 4}{y} - \frac{x}{y} \cdot z \le -1.41012956374722444 \cdot 10^{118} \lor \neg \left(\frac{x + 4}{y} - \frac{x}{y} \cdot z \le 7.6391539351210826 \cdot 10^{58}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))