Average Error: 14.2 → 0.0
Time: 20.2s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{a - b \cdot \frac{b}{a}}{a}\right|}\]
\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 r56042 = a;
        double r56043 = r56042 * r56042;
        double r56044 = b;
        double r56045 = r56044 * r56044;
        double r56046 = r56043 - r56045;
        double r56047 = r56046 / r56043;
        double r56048 = fabs(r56047);
        double r56049 = sqrt(r56048);
        return r56049;
}
↓
double f(double a, double b) {
        double r56050 = a;
        double r56051 = b;
        double r56052 = r56051 / r56050;
        double r56053 = r56051 * r56052;
        double r56054 = r56050 - r56053;
        double r56055 = r56054 / r56050;
        double r56056 = fabs(r56055);
        double r56057 = sqrt(r56056);
        return r56057;
}

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)))))