Average Error: 29.5 → 7.4
Time: 1.1m
Precision: 64
\[0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}
double f(double a, double b) {
        double r9773072 = a;
        double r9773073 = r9773072 * r9773072;
        double r9773074 = b;
        double r9773075 = r9773074 * r9773074;
        double r9773076 = r9773073 - r9773075;
        double r9773077 = r9773076 / r9773073;
        double r9773078 = fabs(r9773077);
        double r9773079 = sqrt(r9773078);
        return r9773079;
}
↓
double f(double a, double b) {
        double r9773080 = 1.0;
        double r9773081 = b;
        double r9773082 = a;
        double r9773083 = r9773081 / r9773082;
        double r9773084 = r9773083 * r9773083;
        double r9773085 = r9773080 - r9773084;
        double r9773086 = fabs(r9773085);
        double r9773087 = sqrt(r9773086);
        return r9773087;
}

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 29.5

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

  2. Simplified7.4

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

  3. Final simplification7.4

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

Reproduce

herbie shell --seed 2019125 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))