Average Error: 14.6 → 0.0
Time: 11.2s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}
double f(double a, double b) {
        double r56926 = a;
        double r56927 = r56926 * r56926;
        double r56928 = b;
        double r56929 = r56928 * r56928;
        double r56930 = r56927 - r56929;
        double r56931 = r56930 / r56927;
        double r56932 = fabs(r56931);
        double r56933 = sqrt(r56932);
        return r56933;
}
↓
double f(double a, double b) {
        double r56934 = 1.0;
        double r56935 = b;
        double r56936 = a;
        double r56937 = r56935 / r56936;
        double r56938 = r56937 * r56937;
        double r56939 = r56934 - r56938;
        double r56940 = fabs(r56939);
        double r56941 = sqrt(r56940);
        return r56941;
}

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 14.6

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

  2. Simplified14.6

    \[\leadsto \color{blue}{\sqrt{\left|1 - \frac{b \cdot b}{a \cdot a}\right|}}\]
  3. Using strategy rm

  4. Applied times-frac0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019235 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))