Average Error: 29.8 → 7.5
Time: 37.5s
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 r7654098 = a;
        double r7654099 = r7654098 * r7654098;
        double r7654100 = b;
        double r7654101 = r7654100 * r7654100;
        double r7654102 = r7654099 - r7654101;
        double r7654103 = r7654102 / r7654099;
        double r7654104 = fabs(r7654103);
        double r7654105 = sqrt(r7654104);
        return r7654105;
}
↓
double f(double a, double b) {
        double r7654106 = 1.0;
        double r7654107 = b;
        double r7654108 = a;
        double r7654109 = r7654107 / r7654108;
        double r7654110 = r7654109 * r7654109;
        double r7654111 = r7654106 - r7654110;
        double r7654112 = fabs(r7654111);
        double r7654113 = sqrt(r7654112);
        return r7654113;
}

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 29.8

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

  2. Simplified7.5

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

  3. Final simplification7.5

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

Reproduce

herbie shell --seed 2019124 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))