Average Error: 29.4 → 7.6
Time: 19.8s
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 r3721013 = a;
        double r3721014 = r3721013 * r3721013;
        double r3721015 = b;
        double r3721016 = r3721015 * r3721015;
        double r3721017 = r3721014 - r3721016;
        double r3721018 = r3721017 / r3721014;
        double r3721019 = fabs(r3721018);
        double r3721020 = sqrt(r3721019);
        return r3721020;
}
↓
double f(double a, double b) {
        double r3721021 = b;
        double r3721022 = a;
        double r3721023 = r3721021 / r3721022;
        double r3721024 = 1.0;
        double r3721025 = r3721023 + r3721024;
        double r3721026 = r3721024 - r3721023;
        double r3721027 = r3721025 * r3721026;
        double r3721028 = fabs(r3721027);
        double r3721029 = sqrt(r3721028);
        return r3721029;
}

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)))))