Average Error: 1.9 → 0.4
Time: 2.7s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \le 6.518303178973746577822906909243007013123 \cdot 10^{130}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \end{array}\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\begin{array}{l}
\mathbf{if}\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \le 6.518303178973746577822906909243007013123 \cdot 10^{130}:\\
\;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\

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

\end{array}
double f(double x, double y, double z) {
        double r30479 = x;
        double r30480 = 4.0;
        double r30481 = r30479 + r30480;
        double r30482 = y;
        double r30483 = r30481 / r30482;
        double r30484 = r30479 / r30482;
        double r30485 = z;
        double r30486 = r30484 * r30485;
        double r30487 = r30483 - r30486;
        double r30488 = fabs(r30487);
        return r30488;
}
↓
double f(double x, double y, double z) {
        double r30489 = x;
        double r30490 = 4.0;
        double r30491 = r30489 + r30490;
        double r30492 = y;
        double r30493 = r30491 / r30492;
        double r30494 = r30489 / r30492;
        double r30495 = z;
        double r30496 = r30494 * r30495;
        double r30497 = r30493 - r30496;
        double r30498 = fabs(r30497);
        double r30499 = 6.518303178973747e+130;
        bool r30500 = r30498 <= r30499;
        double r30501 = r30495 / r30492;
        double r30502 = r30489 * r30501;
        double r30503 = r30493 - r30502;
        double r30504 = fabs(r30503);
        double r30505 = r30500 ? r30504 : r30498;
        return r30505;
}

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))) < 6.518303178973747e+130

    1. Initial program 2.9

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

    3. Applied div-inv2.9

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

    4. Applied associate-*l*0.6

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

    5. Simplified0.6

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

    if 6.518303178973747e+130 < (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 simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right| \le 6.518303178973746577822906909243007013123 \cdot 10^{130}:\\ \;\;\;\;\left|\frac{x + 4}{y} - x \cdot \frac{z}{y}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\\ \end{array}\]

Reproduce

herbie shell --seed 2019318 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))