Average Error: 29.6 → 7.5
Time: 11.6s
Precision: 64
\[0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\left(\frac{b}{a} + 1\right) \cdot \left(1 - \frac{b}{a}\right)\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\left(\frac{b}{a} + 1\right) \cdot \left(1 - \frac{b}{a}\right)\right|}
double f(double a, double b) {
        double r1356909 = a;
        double r1356910 = r1356909 * r1356909;
        double r1356911 = b;
        double r1356912 = r1356911 * r1356911;
        double r1356913 = r1356910 - r1356912;
        double r1356914 = r1356913 / r1356910;
        double r1356915 = fabs(r1356914);
        double r1356916 = sqrt(r1356915);
        return r1356916;
}
↓
double f(double a, double b) {
        double r1356917 = b;
        double r1356918 = a;
        double r1356919 = r1356917 / r1356918;
        double r1356920 = 1.0;
        double r1356921 = r1356919 + r1356920;
        double r1356922 = r1356920 - r1356919;
        double r1356923 = r1356921 * r1356922;
        double r1356924 = fabs(r1356923);
        double r1356925 = sqrt(r1356924);
        return r1356925;
}

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 29.6

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

  2. Simplified7.5

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

  4. Applied *-un-lft-identity7.5

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

  5. Applied difference-of-squares7.5

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

  6. Final simplification7.5

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

Reproduce

herbie shell --seed 2019153 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))