Average Error: 1.7 → 1.7
Time: 3.9s
Precision: 64
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\left|\mathsf{fma}\left(4, \frac{1}{y}, \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\left|\mathsf{fma}\left(4, \frac{1}{y}, \frac{x}{y}\right) - \frac{x}{y} \cdot z\right|
double f(double x, double y, double z) {
        double r29672 = x;
        double r29673 = 4.0;
        double r29674 = r29672 + r29673;
        double r29675 = y;
        double r29676 = r29674 / r29675;
        double r29677 = r29672 / r29675;
        double r29678 = z;
        double r29679 = r29677 * r29678;
        double r29680 = r29676 - r29679;
        double r29681 = fabs(r29680);
        return r29681;
}
↓
double f(double x, double y, double z) {
        double r29682 = 4.0;
        double r29683 = 1.0;
        double r29684 = y;
        double r29685 = r29683 / r29684;
        double r29686 = x;
        double r29687 = r29686 / r29684;
        double r29688 = fma(r29682, r29685, r29687);
        double r29689 = z;
        double r29690 = r29687 * r29689;
        double r29691 = r29688 - r29690;
        double r29692 = fabs(r29691);
        return r29692;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Derivation

  1. Initial program 1.7

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

  2. Taylor expanded around 0 1.7

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

  3. Simplified1.7

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

  4. Final simplification1.7

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

Reproduce

herbie shell --seed 2020036 +o rules:numerics
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4) y) (* (/ x y) z))))