Average Error: 14.6 → 0.0
Time: 12.0s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\log \left(e^{\frac{a + b}{a} \cdot \frac{a - b}{a}}\right)\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\log \left(e^{\frac{a + b}{a} \cdot \frac{a - b}{a}}\right)\right|}
double f(double a, double b) {
        double r83448 = a;
        double r83449 = r83448 * r83448;
        double r83450 = b;
        double r83451 = r83450 * r83450;
        double r83452 = r83449 - r83451;
        double r83453 = r83452 / r83449;
        double r83454 = fabs(r83453);
        double r83455 = sqrt(r83454);
        return r83455;
}
↓
double f(double a, double b) {
        double r83456 = a;
        double r83457 = b;
        double r83458 = r83456 + r83457;
        double r83459 = r83458 / r83456;
        double r83460 = r83456 - r83457;
        double r83461 = r83460 / r83456;
        double r83462 = r83459 * r83461;
        double r83463 = exp(r83462);
        double r83464 = log(r83463);
        double r83465 = fabs(r83464);
        double r83466 = sqrt(r83465);
        return r83466;
}

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. Using strategy rm

  3. Applied difference-of-squares14.6

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

  4. Applied times-frac0.0

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

  6. Applied add-log-exp0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019294 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))