Average Error: 14.5 → 0.0
Time: 27.2s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \log \left(e^{\frac{b \cdot \frac{b}{a}}{a}}\right)\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|1 - \log \left(e^{\frac{b \cdot \frac{b}{a}}{a}}\right)\right|}
double f(double a, double b) {
        double r50448 = a;
        double r50449 = r50448 * r50448;
        double r50450 = b;
        double r50451 = r50450 * r50450;
        double r50452 = r50449 - r50451;
        double r50453 = r50452 / r50449;
        double r50454 = fabs(r50453);
        double r50455 = sqrt(r50454);
        return r50455;
}
↓
double f(double a, double b) {
        double r50456 = 1.0;
        double r50457 = b;
        double r50458 = a;
        double r50459 = r50457 / r50458;
        double r50460 = r50457 * r50459;
        double r50461 = r50460 / r50458;
        double r50462 = exp(r50461);
        double r50463 = log(r50462);
        double r50464 = r50456 - r50463;
        double r50465 = fabs(r50464);
        double r50466 = sqrt(r50465);
        return r50466;
}

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 14.5

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

  2. Simplified14.5

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

  4. Applied associate-/r*0.6

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

  5. Simplified0.0

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

  7. Applied add-log-exp0.0

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

  8. Final simplification0.0

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

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))