Average Error: 1.6 → 0.4
Time: 10.0s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.5186598216349587 \cdot 10^{+49}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 8.018719735392733 \cdot 10^{-129}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} + \left(\frac{4}{y} - \frac{x}{\frac{y}{z}}\right)\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.5186598216349587 \cdot 10^{+49}:\\
\;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\

\mathbf{elif}\;x \le 8.018719735392733 \cdot 10^{-129}:\\
\;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r1119363 = x;
        double r1119364 = 4.0;
        double r1119365 = r1119363 + r1119364;
        double r1119366 = y;
        double r1119367 = r1119365 / r1119366;
        double r1119368 = r1119363 / r1119366;
        double r1119369 = z;
        double r1119370 = r1119368 * r1119369;
        double r1119371 = r1119367 - r1119370;
        double r1119372 = fabs(r1119371);
        return r1119372;
}
↓
double f(double x, double y, double z) {
        double r1119373 = x;
        double r1119374 = -4.5186598216349587e+49;
        bool r1119375 = r1119373 <= r1119374;
        double r1119376 = 4.0;
        double r1119377 = r1119376 + r1119373;
        double r1119378 = y;
        double r1119379 = r1119377 / r1119378;
        double r1119380 = z;
        double r1119381 = r1119380 / r1119378;
        double r1119382 = r1119373 * r1119381;
        double r1119383 = r1119379 - r1119382;
        double r1119384 = fabs(r1119383);
        double r1119385 = 8.018719735392733e-129;
        bool r1119386 = r1119373 <= r1119385;
        double r1119387 = r1119380 * r1119373;
        double r1119388 = r1119377 - r1119387;
        double r1119389 = r1119388 / r1119378;
        double r1119390 = fabs(r1119389);
        double r1119391 = r1119373 / r1119378;
        double r1119392 = r1119376 / r1119378;
        double r1119393 = r1119378 / r1119380;
        double r1119394 = r1119373 / r1119393;
        double r1119395 = r1119392 - r1119394;
        double r1119396 = r1119391 + r1119395;
        double r1119397 = fabs(r1119396);
        double r1119398 = r1119386 ? r1119390 : r1119397;
        double r1119399 = r1119375 ? r1119384 : r1119398;
        return r1119399;
}

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 < -4.5186598216349587e+49

    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

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

    if -4.5186598216349587e+49 < x < 8.018719735392733e-129

    1. Initial program 2.4

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

    3. Applied associate-*l/0.2

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

    4. Applied sub-div0.2

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

    if 8.018719735392733e-129 < x

    1. Initial program 0.9

      \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
    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*1.0

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

    5. Simplified1.0

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

    6. Taylor expanded around 0 5.8

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

    7. Simplified0.8

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.5186598216349587 \cdot 10^{+49}:\\ \;\;\;\;\left|\frac{4 + x}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{elif}\;x \le 8.018719735392733 \cdot 10^{-129}:\\ \;\;\;\;\left|\frac{\left(4 + x\right) - z \cdot x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x}{y} + \left(\frac{4}{y} - \frac{x}{\frac{y}{z}}\right)\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))