Average Error: 1.7 → 0.7
Time: 6.9m
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.572611148022768 \cdot 10^{-182}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 8.927747126101457 \cdot 10^{+27}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \left(z \cdot x\right) \cdot \frac{1}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.572611148022768 \cdot 10^{-182}:\\
\;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\

\mathbf{elif}\;x \le 8.927747126101457 \cdot 10^{+27}:\\
\;\;\;\;\left|\frac{4 + x}{y} - \left(z \cdot x\right) \cdot \frac{1}{y}\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 r19197469 = x;
        double r19197470 = 4.0;
        double r19197471 = r19197469 + r19197470;
        double r19197472 = y;
        double r19197473 = r19197471 / r19197472;
        double r19197474 = r19197469 / r19197472;
        double r19197475 = z;
        double r19197476 = r19197474 * r19197475;
        double r19197477 = r19197473 - r19197476;
        double r19197478 = fabs(r19197477);
        return r19197478;
}
↓
double f(double x, double y, double z) {
        double r19197479 = x;
        double r19197480 = -3.572611148022768e-182;
        bool r19197481 = r19197479 <= r19197480;
        double r19197482 = 4.0;
        double r19197483 = r19197482 + r19197479;
        double r19197484 = y;
        double r19197485 = r19197483 / r19197484;
        double r19197486 = z;
        double r19197487 = r19197486 / r19197484;
        double r19197488 = r19197479 * r19197487;
        double r19197489 = r19197485 - r19197488;
        double r19197490 = fabs(r19197489);
        double r19197491 = 8.927747126101457e+27;
        bool r19197492 = r19197479 <= r19197491;
        double r19197493 = r19197486 * r19197479;
        double r19197494 = 1.0;
        double r19197495 = r19197494 / r19197484;
        double r19197496 = r19197493 * r19197495;
        double r19197497 = r19197485 - r19197496;
        double r19197498 = fabs(r19197497);
        double r19197499 = r19197492 ? r19197498 : r19197490;
        double r19197500 = r19197481 ? r19197490 : r19197499;
        return r19197500;
}

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 < -3.572611148022768e-182 or 8.927747126101457e+27 < x

    1. Initial program 1.0

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

    3. Applied div-inv1.0

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

    4. Applied associate-*l*1.3

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

    5. Simplified1.3

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

    if -3.572611148022768e-182 < x < 8.927747126101457e+27

    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*5.7

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

    5. Simplified5.7

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

    7. Applied div-inv5.7

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

    8. Applied associate-*r*0.1

      \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\left(x \cdot z\right) \cdot \frac{1}{y}}\right|\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.572611148022768 \cdot 10^{-182}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 8.927747126101457 \cdot 10^{+27}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \left(z \cdot x\right) \cdot \frac{1}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))