Average Error: 14.7 → 0.0
Time: 3.8s
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 r67335 = a;
        double r67336 = r67335 * r67335;
        double r67337 = b;
        double r67338 = r67337 * r67337;
        double r67339 = r67336 - r67338;
        double r67340 = r67339 / r67336;
        double r67341 = fabs(r67340);
        double r67342 = sqrt(r67341);
        return r67342;
}
↓
double f(double a, double b) {
        double r67343 = a;
        double r67344 = b;
        double r67345 = r67343 + r67344;
        double r67346 = r67345 / r67343;
        double r67347 = 1.0;
        double r67348 = r67343 - r67344;
        double r67349 = r67343 / r67348;
        double r67350 = r67347 / r67349;
        double r67351 = r67346 * r67350;
        double r67352 = fabs(r67351);
        double r67353 = sqrt(r67352);
        return r67353;
}

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. Using strategy rm

  3. Applied difference-of-squares14.7

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

  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 2020056 +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)))))