Average Error: 14.7 → 0.0
Time: 20.4s
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 r32602 = a;
        double r32603 = r32602 * r32602;
        double r32604 = b;
        double r32605 = r32604 * r32604;
        double r32606 = r32603 - r32605;
        double r32607 = r32606 / r32603;
        double r32608 = fabs(r32607);
        double r32609 = sqrt(r32608);
        return r32609;
}
↓
double f(double a, double b) {
        double r32610 = 1.0;
        double r32611 = b;
        double r32612 = a;
        double r32613 = r32611 / r32612;
        double r32614 = fabs(r32613);
        double r32615 = r32614 * r32614;
        double r32616 = r32610 - r32615;
        double r32617 = fabs(r32616);
        double r32618 = sqrt(r32617);
        return r32618;
}

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 14.7

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

  2. Simplified14.7

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

  4. Applied add-sqr-sqrt14.7

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

  5. Simplified14.7

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

  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 2019325 +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)))))