Average Error: 14.2 → 0.0
Time: 4.1s
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 r75608 = a;
        double r75609 = r75608 * r75608;
        double r75610 = b;
        double r75611 = r75610 * r75610;
        double r75612 = r75609 - r75611;
        double r75613 = r75612 / r75609;
        double r75614 = fabs(r75613);
        double r75615 = sqrt(r75614);
        return r75615;
}
↓
double f(double a, double b) {
        double r75616 = a;
        double r75617 = b;
        double r75618 = r75616 + r75617;
        double r75619 = r75618 / r75616;
        double r75620 = r75616 - r75617;
        double r75621 = r75620 / r75616;
        double r75622 = r75619 * r75621;
        double r75623 = exp(r75622);
        double r75624 = log(r75623);
        double r75625 = fabs(r75624);
        double r75626 = sqrt(r75625);
        return r75626;
}

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 14.2

    \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
  2. Using strategy rm

  3. Applied difference-of-squares14.2

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

  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 2020020 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))