Average Error: 29.3 → 7.2
Time: 20.0s
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 r2973026 = a;
        double r2973027 = r2973026 * r2973026;
        double r2973028 = b;
        double r2973029 = r2973028 * r2973028;
        double r2973030 = r2973027 - r2973029;
        double r2973031 = r2973030 / r2973027;
        double r2973032 = fabs(r2973031);
        double r2973033 = sqrt(r2973032);
        return r2973033;
}
↓
double f(double a, double b) {
        double r2973034 = 1.0;
        double r2973035 = b;
        double r2973036 = a;
        double r2973037 = r2973035 / r2973036;
        double r2973038 = r2973037 * r2973037;
        double r2973039 = r2973034 - r2973038;
        double r2973040 = fabs(r2973039);
        double r2973041 = sqrt(r2973040);
        return r2973041;
}

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 29.3

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

  2. Simplified7.2

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

  3. Final simplification7.2

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

Reproduce

herbie shell --seed 2019168 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))