Average Error: 14.8 → 0.0
Time: 26.1s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \frac{b \cdot \frac{b}{a}}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|1 - \frac{b \cdot \frac{b}{a}}{a}\right|}
double f(double a, double b) {
        double r48709 = a;
        double r48710 = r48709 * r48709;
        double r48711 = b;
        double r48712 = r48711 * r48711;
        double r48713 = r48710 - r48712;
        double r48714 = r48713 / r48710;
        double r48715 = fabs(r48714);
        double r48716 = sqrt(r48715);
        return r48716;
}
↓
double f(double a, double b) {
        double r48717 = 1.0;
        double r48718 = b;
        double r48719 = a;
        double r48720 = r48718 / r48719;
        double r48721 = r48718 * r48720;
        double r48722 = r48721 / r48719;
        double r48723 = r48717 - r48722;
        double r48724 = fabs(r48723);
        double r48725 = sqrt(r48724);
        return r48725;
}

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 14.8

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

  2. Simplified14.8

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

  4. Applied associate-/r*0.7

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019303 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))