Average Error: 15.0 → 0.0
Time: 3.2s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{a + b}{a} \cdot \frac{1}{\frac{a}{a - b}}\right|}\]
\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 r104722 = a;
        double r104723 = r104722 * r104722;
        double r104724 = b;
        double r104725 = r104724 * r104724;
        double r104726 = r104723 - r104725;
        double r104727 = r104726 / r104723;
        double r104728 = fabs(r104727);
        double r104729 = sqrt(r104728);
        return r104729;
}
↓
double f(double a, double b) {
        double r104730 = a;
        double r104731 = b;
        double r104732 = r104730 + r104731;
        double r104733 = r104732 / r104730;
        double r104734 = 1.0;
        double r104735 = r104730 - r104731;
        double r104736 = r104730 / r104735;
        double r104737 = r104734 / r104736;
        double r104738 = r104733 * r104737;
        double r104739 = fabs(r104738);
        double r104740 = sqrt(r104739);
        return r104740;
}

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 15.0

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

  3. Applied difference-of-squares15.0

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

  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 2020089 +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)))))