Average Error: 1.5 → 0.4
Time: 15.3s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.109110258294614752055774941493075991437 \cdot 10^{97}:\\ \;\;\;\;\left|\frac{1}{\frac{y}{x + 4}} - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 214190368136.352081298828125:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.109110258294614752055774941493075991437 \cdot 10^{97}:\\
\;\;\;\;\left|\frac{1}{\frac{y}{x + 4}} - \frac{x}{y} \cdot z\right|\\

\mathbf{elif}\;x \le 214190368136.352081298828125:\\
\;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\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 r45574 = x;
        double r45575 = 4.0;
        double r45576 = r45574 + r45575;
        double r45577 = y;
        double r45578 = r45576 / r45577;
        double r45579 = r45574 / r45577;
        double r45580 = z;
        double r45581 = r45579 * r45580;
        double r45582 = r45578 - r45581;
        double r45583 = fabs(r45582);
        return r45583;
}
↓
double f(double x, double y, double z) {
        double r45584 = x;
        double r45585 = -1.1091102582946148e+97;
        bool r45586 = r45584 <= r45585;
        double r45587 = 1.0;
        double r45588 = y;
        double r45589 = 4.0;
        double r45590 = r45584 + r45589;
        double r45591 = r45588 / r45590;
        double r45592 = r45587 / r45591;
        double r45593 = r45584 / r45588;
        double r45594 = z;
        double r45595 = r45593 * r45594;
        double r45596 = r45592 - r45595;
        double r45597 = fabs(r45596);
        double r45598 = 214190368136.35208;
        bool r45599 = r45584 <= r45598;
        double r45600 = r45584 * r45594;
        double r45601 = r45590 - r45600;
        double r45602 = r45601 / r45588;
        double r45603 = fabs(r45602);
        double r45604 = r45590 / r45588;
        double r45605 = r45594 / r45588;
        double r45606 = r45584 * r45605;
        double r45607 = r45604 - r45606;
        double r45608 = fabs(r45607);
        double r45609 = r45599 ? r45603 : r45608;
        double r45610 = r45586 ? r45597 : r45609;
        return r45610;
}

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 3 regimes

  2. if x < -1.1091102582946148e+97

    1. Initial program 0.1

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

    3. Applied clear-num0.3

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

    if -1.1091102582946148e+97 < x < 214190368136.35208

    1. Initial program 2.1

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

    3. Applied associate-*l/0.5

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

    4. Applied sub-div0.5

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

    if 214190368136.35208 < x

    1. Initial program 0.1

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

    3. Applied div-inv0.2

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

    4. Applied associate-*l*0.2

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

    5. Simplified0.1

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.109110258294614752055774941493075991437 \cdot 10^{97}:\\ \;\;\;\;\left|\frac{1}{\frac{y}{x + 4}} - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 214190368136.352081298828125:\\ \;\;\;\;\left|\frac{\left(x + 4\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))