Average Error: 29.1 → 7.7
Time: 32.5s
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 r3119310 = a;
        double r3119311 = r3119310 * r3119310;
        double r3119312 = b;
        double r3119313 = r3119312 * r3119312;
        double r3119314 = r3119311 - r3119313;
        double r3119315 = r3119314 / r3119311;
        double r3119316 = fabs(r3119315);
        double r3119317 = sqrt(r3119316);
        return r3119317;
}
↓
double f(double a, double b) {
        double r3119318 = 1.0;
        double r3119319 = b;
        double r3119320 = a;
        double r3119321 = r3119319 / r3119320;
        double r3119322 = r3119321 * r3119321;
        double r3119323 = r3119318 - r3119322;
        double r3119324 = fabs(r3119323);
        double r3119325 = sqrt(r3119324);
        return r3119325;
}

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 29.1

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

  2. Simplified7.7

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

  3. Final simplification7.7

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

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))