Average Error: 1.4 → 0.1
Time: 8.4s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right| \le 871440318472255963136:\\ \;\;\;\;\left|\frac{\left(4 - x \cdot z\right) + x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right| \le 871440318472255963136:\\
\;\;\;\;\left|\frac{\left(4 - x \cdot z\right) + x}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r18495 = x;
        double r18496 = 4.0;
        double r18497 = r18495 + r18496;
        double r18498 = y;
        double r18499 = r18497 / r18498;
        double r18500 = r18495 / r18498;
        double r18501 = z;
        double r18502 = r18500 * r18501;
        double r18503 = r18499 - r18502;
        double r18504 = fabs(r18503);
        return r18504;
}
↓
double f(double x, double y, double z) {
        double r18505 = 4.0;
        double r18506 = x;
        double r18507 = r18505 + r18506;
        double r18508 = y;
        double r18509 = r18507 / r18508;
        double r18510 = r18506 / r18508;
        double r18511 = z;
        double r18512 = r18510 * r18511;
        double r18513 = r18509 - r18512;
        double r18514 = fabs(r18513);
        double r18515 = 8.71440318472256e+20;
        bool r18516 = r18514 <= r18515;
        double r18517 = r18506 * r18511;
        double r18518 = r18505 - r18517;
        double r18519 = r18518 + r18506;
        double r18520 = r18519 / r18508;
        double r18521 = fabs(r18520);
        double r18522 = r18516 ? r18521 : r18514;
        return r18522;
}

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 (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))) < 8.71440318472256e+20

    1. Initial program 3.2

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

    3. Applied associate-*l/0.1

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

    4. Applied sub-div0.1

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

    5. Simplified0.1

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

    if 8.71440318472256e+20 < (fabs (- (/ (+ x 4.0) y) (* (/ x y) z)))

    1. Initial program 0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right| \le 871440318472255963136:\\ \;\;\;\;\left|\frac{\left(4 - x \cdot z\right) + x}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{4 + x}{y} - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x y z)
  :name "fabs fraction 1"
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))