Average Error: 14.2 → 0.0
Time: 17.2s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\left(1 + \frac{b}{a}\right) \cdot \frac{a - b}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\left(1 + \frac{b}{a}\right) \cdot \frac{a - b}{a}\right|}
double f(double a, double b) {
        double r62541 = a;
        double r62542 = r62541 * r62541;
        double r62543 = b;
        double r62544 = r62543 * r62543;
        double r62545 = r62542 - r62544;
        double r62546 = r62545 / r62542;
        double r62547 = fabs(r62546);
        double r62548 = sqrt(r62547);
        return r62548;
}
↓
double f(double a, double b) {
        double r62549 = 1.0;
        double r62550 = b;
        double r62551 = a;
        double r62552 = r62550 / r62551;
        double r62553 = r62549 + r62552;
        double r62554 = r62551 - r62550;
        double r62555 = r62554 / r62551;
        double r62556 = r62553 * r62555;
        double r62557 = fabs(r62556);
        double r62558 = sqrt(r62557);
        return r62558;
}

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 14.2

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

  3. Applied difference-of-squares14.2

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

  4. Applied times-frac0.0

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

  5. Taylor expanded around 0 0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))