Average Error: 14.3 → 0.0
Time: 9.6s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\mathsf{fma}\left(-\frac{b}{a}, \frac{b}{a}, 1\right)\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\mathsf{fma}\left(-\frac{b}{a}, \frac{b}{a}, 1\right)\right|}
double f(double a, double b) {
        double r92604 = a;
        double r92605 = r92604 * r92604;
        double r92606 = b;
        double r92607 = r92606 * r92606;
        double r92608 = r92605 - r92607;
        double r92609 = r92608 / r92605;
        double r92610 = fabs(r92609);
        double r92611 = sqrt(r92610);
        return r92611;
}
↓
double f(double a, double b) {
        double r92612 = b;
        double r92613 = a;
        double r92614 = r92612 / r92613;
        double r92615 = -r92614;
        double r92616 = 1.0;
        double r92617 = fma(r92615, r92614, r92616);
        double r92618 = fabs(r92617);
        double r92619 = sqrt(r92618);
        return r92619;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.3

    \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\sqrt{\left|\mathsf{fma}\left(-\frac{b}{a}, \frac{b}{a}, 1\right)\right|}}\]

  3. Final simplification0.0

    \[\leadsto \sqrt{\left|\mathsf{fma}\left(-\frac{b}{a}, \frac{b}{a}, 1\right)\right|}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0.0 b a 1.0)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))