Average Error: 14.5 → 0.0
Time: 4.4s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{a + b}{\frac{a}{\frac{a - b}{a}}}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{a + b}{\frac{a}{\frac{a - b}{a}}}\right|}
double f(double a, double b) {
        double r68631 = a;
        double r68632 = r68631 * r68631;
        double r68633 = b;
        double r68634 = r68633 * r68633;
        double r68635 = r68632 - r68634;
        double r68636 = r68635 / r68632;
        double r68637 = fabs(r68636);
        double r68638 = sqrt(r68637);
        return r68638;
}
↓
double f(double a, double b) {
        double r68639 = a;
        double r68640 = b;
        double r68641 = r68639 + r68640;
        double r68642 = r68639 - r68640;
        double r68643 = r68642 / r68639;
        double r68644 = r68639 / r68643;
        double r68645 = r68641 / r68644;
        double r68646 = fabs(r68645);
        double r68647 = sqrt(r68646);
        return r68647;
}

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. Using strategy rm

  3. Applied difference-of-squares14.5

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

  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 associate-*r/0.0

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

  7. Simplified0.0

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

  9. Applied associate-/l*0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2019354 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))