Average Error: 1.5 → 1.5
Time: 10.2s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\left|\left(\frac{4}{y} + \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|
double f(double x, double y, double z) {
        double r33347 = x;
        double r33348 = 4.0;
        double r33349 = r33347 + r33348;
        double r33350 = y;
        double r33351 = r33349 / r33350;
        double r33352 = r33347 / r33350;
        double r33353 = z;
        double r33354 = r33352 * r33353;
        double r33355 = r33351 - r33354;
        double r33356 = fabs(r33355);
        return r33356;
}
↓
double f(double x, double y, double z) {
        double r33357 = 4.0;
        double r33358 = y;
        double r33359 = r33357 / r33358;
        double r33360 = x;
        double r33361 = r33360 / r33358;
        double r33362 = r33359 + r33361;
        double r33363 = z;
        double r33364 = r33361 * r33363;
        double r33365 = r33362 - r33364;
        double r33366 = fabs(r33365);
        return r33366;
}

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. Initial program 1.5

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

  2. Taylor expanded around 0 1.5

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

  3. Simplified1.5

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

  4. Final simplification1.5

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

Reproduce

herbie shell --seed 2020045 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))