Average Error: 1.7 → 0.3
Time: 17.9s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.872419455404728917343841935601901663261 \cdot 10^{86}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 1.895662615647486210566329068575767552188 \cdot 10^{-36}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.872419455404728917343841935601901663261 \cdot 10^{86}:\\
\;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\

\mathbf{elif}\;x \le 1.895662615647486210566329068575767552188 \cdot 10^{-36}:\\
\;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r1951462 = x;
        double r1951463 = 4.0;
        double r1951464 = r1951462 + r1951463;
        double r1951465 = y;
        double r1951466 = r1951464 / r1951465;
        double r1951467 = r1951462 / r1951465;
        double r1951468 = z;
        double r1951469 = r1951467 * r1951468;
        double r1951470 = r1951466 - r1951469;
        double r1951471 = fabs(r1951470);
        return r1951471;
}
↓
double f(double x, double y, double z) {
        double r1951472 = x;
        double r1951473 = -1.872419455404729e+86;
        bool r1951474 = r1951472 <= r1951473;
        double r1951475 = 4.0;
        double r1951476 = y;
        double r1951477 = r1951475 / r1951476;
        double r1951478 = r1951472 / r1951476;
        double r1951479 = r1951477 + r1951478;
        double r1951480 = z;
        double r1951481 = r1951478 * r1951480;
        double r1951482 = r1951479 - r1951481;
        double r1951483 = fabs(r1951482);
        double r1951484 = 1.8956626156474862e-36;
        bool r1951485 = r1951472 <= r1951484;
        double r1951486 = r1951475 + r1951472;
        double r1951487 = r1951472 * r1951480;
        double r1951488 = r1951486 - r1951487;
        double r1951489 = r1951488 / r1951476;
        double r1951490 = fabs(r1951489);
        double r1951491 = r1951485 ? r1951490 : r1951483;
        double r1951492 = r1951474 ? r1951483 : r1951491;
        return r1951492;
}

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 < -1.872419455404729e+86 or 1.8956626156474862e-36 < x

    1. Initial program 0.2

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

    2. Taylor expanded around 0 0.2

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

    3. Simplified0.2

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

    if -1.872419455404729e+86 < x < 1.8956626156474862e-36

    1. Initial program 2.6

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

    3. Applied associate-*l/0.4

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

    4. Applied sub-div0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.872419455404728917343841935601901663261 \cdot 10^{86}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \mathbf{elif}\;x \le 1.895662615647486210566329068575767552188 \cdot 10^{-36}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - x \cdot z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))