Average Error: 29.4 → 7.6
Time: 21.0s
Precision: 64
\[0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\left(\frac{b}{a} + 1\right) \cdot \left(1 - \frac{b}{a}\right)\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\left(\frac{b}{a} + 1\right) \cdot \left(1 - \frac{b}{a}\right)\right|}
double f(double a, double b) {
        double r1883042 = a;
        double r1883043 = r1883042 * r1883042;
        double r1883044 = b;
        double r1883045 = r1883044 * r1883044;
        double r1883046 = r1883043 - r1883045;
        double r1883047 = r1883046 / r1883043;
        double r1883048 = fabs(r1883047);
        double r1883049 = sqrt(r1883048);
        return r1883049;
}
↓
double f(double a, double b) {
        double r1883050 = b;
        double r1883051 = a;
        double r1883052 = r1883050 / r1883051;
        double r1883053 = 1.0;
        double r1883054 = r1883052 + r1883053;
        double r1883055 = r1883053 - r1883052;
        double r1883056 = r1883054 * r1883055;
        double r1883057 = fabs(r1883056);
        double r1883058 = sqrt(r1883057);
        return r1883058;
}

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 29.4

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

  2. Simplified7.5

    \[\leadsto \color{blue}{\sqrt{\left|1 - \frac{b}{a} \cdot \frac{b}{a}\right|}}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity7.5

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

  5. Applied difference-of-squares7.6

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

  6. Final simplification7.6

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

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))