Average Error: 29.4 → 7.6
Time: 20.9s
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 r3216943 = a;
        double r3216944 = r3216943 * r3216943;
        double r3216945 = b;
        double r3216946 = r3216945 * r3216945;
        double r3216947 = r3216944 - r3216946;
        double r3216948 = r3216947 / r3216944;
        double r3216949 = fabs(r3216948);
        double r3216950 = sqrt(r3216949);
        return r3216950;
}
↓
double f(double a, double b) {
        double r3216951 = b;
        double r3216952 = a;
        double r3216953 = r3216951 / r3216952;
        double r3216954 = 1.0;
        double r3216955 = r3216953 + r3216954;
        double r3216956 = r3216954 - r3216953;
        double r3216957 = r3216955 * r3216956;
        double r3216958 = fabs(r3216957);
        double r3216959 = sqrt(r3216958);
        return r3216959;
}

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 29.4

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

  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-squares7.6

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

  6. Final simplification7.6

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))