Average Error: 14.7 → 0.0
Time: 9.5s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{1}{\frac{a}{a - b \cdot \frac{b}{a}}}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{1}{\frac{a}{a - b \cdot \frac{b}{a}}}\right|}
double f(double a, double b) {
        double r61932 = a;
        double r61933 = r61932 * r61932;
        double r61934 = b;
        double r61935 = r61934 * r61934;
        double r61936 = r61933 - r61935;
        double r61937 = r61936 / r61933;
        double r61938 = fabs(r61937);
        double r61939 = sqrt(r61938);
        return r61939;
}
↓
double f(double a, double b) {
        double r61940 = 1.0;
        double r61941 = a;
        double r61942 = b;
        double r61943 = r61942 / r61941;
        double r61944 = r61942 * r61943;
        double r61945 = r61941 - r61944;
        double r61946 = r61941 / r61945;
        double r61947 = r61940 / r61946;
        double r61948 = fabs(r61947);
        double r61949 = sqrt(r61948);
        return r61949;
}

Error

Bits error versus a

Bits error versus b

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.7

    \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
  2. Using strategy rm

  3. Applied clear-num14.7

    \[\leadsto \sqrt{\left|\color{blue}{\frac{1}{\frac{a \cdot a}{a \cdot a - b \cdot b}}}\right|}\]

  4. Simplified0.0

    \[\leadsto \sqrt{\left|\frac{1}{\color{blue}{\frac{a}{a - b \cdot \frac{b}{a}}}}\right|}\]

  5. Final simplification0.0

    \[\leadsto \sqrt{\left|\frac{1}{\frac{a}{a - b \cdot \frac{b}{a}}}\right|}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))