Average Error: 1.6 → 0.3
Time: 13.6s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{4 + x}{y} - \frac{x}{y} \cdot z \le -1.540140242651518318633673548087898092755 \cdot 10^{165} \lor \neg \left(\frac{4 + x}{y} - \frac{x}{y} \cdot z \le 3.227438539306355727994396233811805318431 \cdot 10^{64}\right):\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;\frac{4 + x}{y} - \frac{x}{y} \cdot z \le -1.540140242651518318633673548087898092755 \cdot 10^{165} \lor \neg \left(\frac{4 + x}{y} - \frac{x}{y} \cdot z \le 3.227438539306355727994396233811805318431 \cdot 10^{64}\right):\\
\;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r42634 = x;
        double r42635 = 4.0;
        double r42636 = r42634 + r42635;
        double r42637 = y;
        double r42638 = r42636 / r42637;
        double r42639 = r42634 / r42637;
        double r42640 = z;
        double r42641 = r42639 * r42640;
        double r42642 = r42638 - r42641;
        double r42643 = fabs(r42642);
        return r42643;
}
↓
double f(double x, double y, double z) {
        double r42644 = 4.0;
        double r42645 = x;
        double r42646 = r42644 + r42645;
        double r42647 = y;
        double r42648 = r42646 / r42647;
        double r42649 = r42645 / r42647;
        double r42650 = z;
        double r42651 = r42649 * r42650;
        double r42652 = r42648 - r42651;
        double r42653 = -1.5401402426515183e+165;
        bool r42654 = r42652 <= r42653;
        double r42655 = 3.227438539306356e+64;
        bool r42656 = r42652 <= r42655;
        double r42657 = !r42656;
        bool r42658 = r42654 || r42657;
        double r42659 = fabs(r42652);
        double r42660 = r42650 / r42647;
        double r42661 = r42645 * r42660;
        double r42662 = r42648 - r42661;
        double r42663 = fabs(r42662);
        double r42664 = r42658 ? r42659 : r42663;
        return r42664;
}

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)) < -1.5401402426515183e+165 or 3.227438539306356e+64 < (- (/ (+ x 4.0) y) (* (/ x y) z))

    1. Initial program 0.1

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

    if -1.5401402426515183e+165 < (- (/ (+ x 4.0) y) (* (/ x y) z)) < 3.227438539306356e+64

    1. Initial program 2.6

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    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*0.5

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

    5. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 + x}{y} - \frac{x}{y} \cdot z \le -1.540140242651518318633673548087898092755 \cdot 10^{165} \lor \neg \left(\frac{4 + x}{y} - \frac{x}{y} \cdot z \le 3.227438539306355727994396233811805318431 \cdot 10^{64}\right):\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))