Average Error: 29.9 → 7.5
Time: 17.1s
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 r12004083 = a;
        double r12004084 = r12004083 * r12004083;
        double r12004085 = b;
        double r12004086 = r12004085 * r12004085;
        double r12004087 = r12004084 - r12004086;
        double r12004088 = r12004087 / r12004084;
        double r12004089 = fabs(r12004088);
        double r12004090 = sqrt(r12004089);
        return r12004090;
}
↓
double f(double a, double b) {
        double r12004091 = 1.0;
        double r12004092 = b;
        double r12004093 = a;
        double r12004094 = r12004092 / r12004093;
        double r12004095 = r12004094 * r12004094;
        double r12004096 = r12004091 - r12004095;
        double r12004097 = fabs(r12004096);
        double r12004098 = sqrt(r12004097);
        return r12004098;
}

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 29.9

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

  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 2019121 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))