Average Error: 14.5 → 0.0
Time: 4.5s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{\frac{a - b}{\frac{a}{a + b}}}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{\frac{a - b}{\frac{a}{a + b}}}{a}\right|}
double f(double a, double b) {
        double r81546 = a;
        double r81547 = r81546 * r81546;
        double r81548 = b;
        double r81549 = r81548 * r81548;
        double r81550 = r81547 - r81549;
        double r81551 = r81550 / r81547;
        double r81552 = fabs(r81551);
        double r81553 = sqrt(r81552);
        return r81553;
}
↓
double f(double a, double b) {
        double r81554 = a;
        double r81555 = b;
        double r81556 = r81554 - r81555;
        double r81557 = r81554 + r81555;
        double r81558 = r81554 / r81557;
        double r81559 = r81556 / r81558;
        double r81560 = r81559 / r81554;
        double r81561 = fabs(r81560);
        double r81562 = sqrt(r81561);
        return r81562;
}

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 clear-num0.0

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

  8. Applied associate-*r/0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020025 +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)))))