Average Error: 30.4 → 8.1
Time: 24.0s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}\]
\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 r4370959 = a;
        double r4370960 = r4370959 * r4370959;
        double r4370961 = b;
        double r4370962 = r4370961 * r4370961;
        double r4370963 = r4370960 - r4370962;
        double r4370964 = r4370963 / r4370960;
        double r4370965 = fabs(r4370964);
        double r4370966 = sqrt(r4370965);
        return r4370966;
}
↓
double f(double a, double b) {
        double r4370967 = 1.0;
        double r4370968 = b;
        double r4370969 = a;
        double r4370970 = r4370968 / r4370969;
        double r4370971 = r4370970 * r4370970;
        double r4370972 = r4370967 - r4370971;
        double r4370973 = fabs(r4370972);
        double r4370974 = sqrt(r4370973);
        return r4370974;
}

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 30.4

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

  2. Simplified8.1

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

  3. Final simplification8.1

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0.0 b a 1.0)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))