Average Error: 14.6 → 0.0
Time: 8.6s
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 r47958 = a;
        double r47959 = r47958 * r47958;
        double r47960 = b;
        double r47961 = r47960 * r47960;
        double r47962 = r47959 - r47961;
        double r47963 = r47962 / r47959;
        double r47964 = fabs(r47963);
        double r47965 = sqrt(r47964);
        return r47965;
}
↓
double f(double a, double b) {
        double r47966 = b;
        double r47967 = a;
        double r47968 = r47966 / r47967;
        double r47969 = -r47968;
        double r47970 = 1.0;
        double r47971 = fma(r47969, r47968, r47970);
        double r47972 = fabs(r47971);
        double r47973 = sqrt(r47972);
        return r47973;
}

Error

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 14.6

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

  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 2019195 +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)))))