Average Error: 14.1 → 0.0
Time: 3.9s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{a + b}{a} \cdot \frac{1}{\frac{a}{a - b}}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{a + b}{a} \cdot \frac{1}{\frac{a}{a - b}}\right|}
double f(double a, double b) {
        double r59478 = a;
        double r59479 = r59478 * r59478;
        double r59480 = b;
        double r59481 = r59480 * r59480;
        double r59482 = r59479 - r59481;
        double r59483 = r59482 / r59479;
        double r59484 = fabs(r59483);
        double r59485 = sqrt(r59484);
        return r59485;
}
↓
double f(double a, double b) {
        double r59486 = a;
        double r59487 = b;
        double r59488 = r59486 + r59487;
        double r59489 = r59488 / r59486;
        double r59490 = 1.0;
        double r59491 = r59486 - r59487;
        double r59492 = r59486 / r59491;
        double r59493 = r59490 / r59492;
        double r59494 = r59489 * r59493;
        double r59495 = fabs(r59494);
        double r59496 = sqrt(r59495);
        return r59496;
}

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 14.1

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

  3. Applied difference-of-squares14.1

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

  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 clear-num0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019353 +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)))))