Average Error: 1.7 → 1.2
Time: 7.9s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.775517148444541890019330037710355464013 \cdot 10^{50} \lor \neg \left(x \le 2.295099393231360663003372969170030877664 \cdot 10^{-243}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.775517148444541890019330037710355464013 \cdot 10^{50} \lor \neg \left(x \le 2.295099393231360663003372969170030877664 \cdot 10^{-243}\right):\\
\;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{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 r18117 = x;
        double r18118 = 4.0;
        double r18119 = r18117 + r18118;
        double r18120 = y;
        double r18121 = r18119 / r18120;
        double r18122 = r18117 / r18120;
        double r18123 = z;
        double r18124 = r18122 * r18123;
        double r18125 = r18121 - r18124;
        double r18126 = fabs(r18125);
        return r18126;
}
↓
double f(double x, double y, double z) {
        double r18127 = x;
        double r18128 = -2.775517148444542e+50;
        bool r18129 = r18127 <= r18128;
        double r18130 = 2.2950993932313607e-243;
        bool r18131 = r18127 <= r18130;
        double r18132 = !r18131;
        bool r18133 = r18129 || r18132;
        double r18134 = 4.0;
        double r18135 = r18127 + r18134;
        double r18136 = y;
        double r18137 = r18135 / r18136;
        double r18138 = z;
        double r18139 = r18138 / r18136;
        double r18140 = r18127 * r18139;
        double r18141 = r18137 - r18140;
        double r18142 = fabs(r18141);
        double r18143 = r18127 * r18138;
        double r18144 = r18135 - r18143;
        double r18145 = r18144 / r18136;
        double r18146 = fabs(r18145);
        double r18147 = r18133 ? r18142 : r18146;
        return r18147;
}

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 < -2.775517148444542e+50 or 2.2950993932313607e-243 < x

    1. Initial program 1.2

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

    3. Applied div-inv1.2

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

    4. Applied associate-*l*2.0

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

    5. Simplified2.0

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

    if -2.775517148444542e+50 < x < 2.2950993932313607e-243

    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 simplification1.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.775517148444541890019330037710355464013 \cdot 10^{50} \lor \neg \left(x \le 2.295099393231360663003372969170030877664 \cdot 10^{-243}\right):\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))