Average Error: 1.6 → 1.6
Time: 14.6s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\left|\mathsf{fma}\left(\frac{x}{y}, 1 - z, \frac{4}{y}\right)\right|\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\left|\mathsf{fma}\left(\frac{x}{y}, 1 - z, \frac{4}{y}\right)\right|
double f(double x, double y, double z) {
        double r33774 = x;
        double r33775 = 4.0;
        double r33776 = r33774 + r33775;
        double r33777 = y;
        double r33778 = r33776 / r33777;
        double r33779 = r33774 / r33777;
        double r33780 = z;
        double r33781 = r33779 * r33780;
        double r33782 = r33778 - r33781;
        double r33783 = fabs(r33782);
        return r33783;
}
↓
double f(double x, double y, double z) {
        double r33784 = x;
        double r33785 = y;
        double r33786 = r33784 / r33785;
        double r33787 = 1.0;
        double r33788 = z;
        double r33789 = r33787 - r33788;
        double r33790 = 4.0;
        double r33791 = r33790 / r33785;
        double r33792 = fma(r33786, r33789, r33791);
        double r33793 = fabs(r33792);
        return r33793;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 1.6

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

  2. Taylor expanded around 0 3.4

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

  3. Simplified1.6

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

  4. Final simplification1.6

    \[\leadsto \left|\mathsf{fma}\left(\frac{x}{y}, 1 - z, \frac{4}{y}\right)\right|\]

Reproduce

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