Average Error: 30.8 → 7.8
Time: 25.9s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\left(1 - \frac{b}{a}\right) \cdot \left(\frac{b}{a} + 1\right)\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\left(1 - \frac{b}{a}\right) \cdot \left(\frac{b}{a} + 1\right)\right|}
double f(double a, double b) {
        double r3138016 = a;
        double r3138017 = r3138016 * r3138016;
        double r3138018 = b;
        double r3138019 = r3138018 * r3138018;
        double r3138020 = r3138017 - r3138019;
        double r3138021 = r3138020 / r3138017;
        double r3138022 = fabs(r3138021);
        double r3138023 = sqrt(r3138022);
        return r3138023;
}
↓
double f(double a, double b) {
        double r3138024 = 1.0;
        double r3138025 = b;
        double r3138026 = a;
        double r3138027 = r3138025 / r3138026;
        double r3138028 = r3138024 - r3138027;
        double r3138029 = r3138027 + r3138024;
        double r3138030 = r3138028 * r3138029;
        double r3138031 = fabs(r3138030);
        double r3138032 = sqrt(r3138031);
        return r3138032;
}

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 30.8

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

  2. Simplified7.8

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

  4. Applied *-un-lft-identity7.8

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

  5. Applied difference-of-squares7.8

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

  6. Final simplification7.8

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

Reproduce

herbie shell --seed 2019200 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0.0 b a 1.0)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))