Average Error: 1.6 → 1.6
Time: 15.9s
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 r32457 = x;
        double r32458 = 4.0;
        double r32459 = r32457 + r32458;
        double r32460 = y;
        double r32461 = r32459 / r32460;
        double r32462 = r32457 / r32460;
        double r32463 = z;
        double r32464 = r32462 * r32463;
        double r32465 = r32461 - r32464;
        double r32466 = fabs(r32465);
        return r32466;
}
↓
double f(double x, double y, double z) {
        double r32467 = 4.0;
        double r32468 = y;
        double r32469 = r32467 / r32468;
        double r32470 = x;
        double r32471 = r32470 / r32468;
        double r32472 = r32469 + r32471;
        double r32473 = z;
        double r32474 = r32471 * r32473;
        double r32475 = r32472 - r32474;
        double r32476 = fabs(r32475);
        return r32476;
}

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 1.6

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

  2. Taylor expanded around 0 1.6

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

  3. Simplified1.6

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

  4. Final simplification1.6

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

Reproduce

herbie shell --seed 2019325 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))