Average Error: 14.6 → 0.0
Time: 9.2s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \left|\frac{b}{a}\right| \cdot \left|\frac{b}{a}\right|\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|1 - \left|\frac{b}{a}\right| \cdot \left|\frac{b}{a}\right|\right|}
double f(double a, double b) {
        double r50563 = a;
        double r50564 = r50563 * r50563;
        double r50565 = b;
        double r50566 = r50565 * r50565;
        double r50567 = r50564 - r50566;
        double r50568 = r50567 / r50564;
        double r50569 = fabs(r50568);
        double r50570 = sqrt(r50569);
        return r50570;
}
↓
double f(double a, double b) {
        double r50571 = 1.0;
        double r50572 = b;
        double r50573 = a;
        double r50574 = r50572 / r50573;
        double r50575 = fabs(r50574);
        double r50576 = r50575 * r50575;
        double r50577 = r50571 - r50576;
        double r50578 = fabs(r50577);
        double r50579 = sqrt(r50578);
        return r50579;
}

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 14.6

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

  2. Simplified14.6

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

  4. Applied add-sqr-sqrt14.6

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

  5. Simplified14.5

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

  6. Simplified0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2019235 +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)))))