Average Error: 1.5 → 0.2
Time: 6.2s
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 r159 = x;
        double r160 = 4.0;
        double r161 = r159 + r160;
        double r162 = y;
        double r163 = r161 / r162;
        double r164 = r159 / r162;
        double r165 = z;
        double r166 = r164 * r165;
        double r167 = r163 - r166;
        double r168 = fabs(r167);
        return r168;
}
↓
double f(double x, double y, double z) {
        double r169 = x;
        double r170 = 4.0;
        double r171 = r169 + r170;
        double r172 = y;
        double r173 = r171 / r172;
        double r174 = r169 / r172;
        double r175 = z;
        double r176 = r174 * r175;
        double r177 = r173 - r176;
        double r178 = -1.4101295637472244e+118;
        bool r179 = r177 <= r178;
        double r180 = 7.639153935121083e+58;
        bool r181 = r177 <= r180;
        double r182 = !r181;
        bool r183 = r179 || r182;
        double r184 = fabs(r177);
        double r185 = r175 / r172;
        double r186 = r169 * r185;
        double r187 = r173 - r186;
        double r188 = fabs(r187);
        double r189 = r183 ? r184 : r188;
        return r189;
}

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