Average Error: 14.2 → 0.0
Time: 19.5s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{a - b \cdot \frac{b}{a}}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{a - b \cdot \frac{b}{a}}{a}\right|}
double f(double a, double b) {
        double r76054 = a;
        double r76055 = r76054 * r76054;
        double r76056 = b;
        double r76057 = r76056 * r76056;
        double r76058 = r76055 - r76057;
        double r76059 = r76058 / r76055;
        double r76060 = fabs(r76059);
        double r76061 = sqrt(r76060);
        return r76061;
}
↓
double f(double a, double b) {
        double r76062 = a;
        double r76063 = b;
        double r76064 = r76063 / r76062;
        double r76065 = r76063 * r76064;
        double r76066 = r76062 - r76065;
        double r76067 = r76066 / r76062;
        double r76068 = fabs(r76067);
        double r76069 = sqrt(r76068);
        return r76069;
}

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 associate-/r*14.4

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

  4. Simplified0.0

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

  5. Final simplification0.0

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

Reproduce

herbie shell --seed 2019323 +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)))))