Average Error: 13.7 → 0.0
Time: 4.7s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{a + b}{a} \cdot \frac{a - b}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{a + b}{a} \cdot \frac{a - b}{a}\right|}
double f(double a, double b) {
        double r77676 = a;
        double r77677 = r77676 * r77676;
        double r77678 = b;
        double r77679 = r77678 * r77678;
        double r77680 = r77677 - r77679;
        double r77681 = r77680 / r77677;
        double r77682 = fabs(r77681);
        double r77683 = sqrt(r77682);
        return r77683;
}
↓
double f(double a, double b) {
        double r77684 = a;
        double r77685 = b;
        double r77686 = r77684 + r77685;
        double r77687 = r77686 / r77684;
        double r77688 = r77684 - r77685;
        double r77689 = r77688 / r77684;
        double r77690 = r77687 * r77689;
        double r77691 = fabs(r77690);
        double r77692 = sqrt(r77691);
        return r77692;
}

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 13.7

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

  3. Applied difference-of-squares13.7

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

  4. Applied times-frac0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2020060 +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)))))