Average Error: 30.8 → 7.8
Time: 25.6s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\left(1 - \frac{b}{a}\right) \cdot \left(\frac{b}{a} + 1\right)\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\left(1 - \frac{b}{a}\right) \cdot \left(\frac{b}{a} + 1\right)\right|}
double f(double a, double b) {
        double r3753156 = a;
        double r3753157 = r3753156 * r3753156;
        double r3753158 = b;
        double r3753159 = r3753158 * r3753158;
        double r3753160 = r3753157 - r3753159;
        double r3753161 = r3753160 / r3753157;
        double r3753162 = fabs(r3753161);
        double r3753163 = sqrt(r3753162);
        return r3753163;
}
↓
double f(double a, double b) {
        double r3753164 = 1.0;
        double r3753165 = b;
        double r3753166 = a;
        double r3753167 = r3753165 / r3753166;
        double r3753168 = r3753164 - r3753167;
        double r3753169 = r3753167 + r3753164;
        double r3753170 = r3753168 * r3753169;
        double r3753171 = fabs(r3753170);
        double r3753172 = sqrt(r3753171);
        return r3753172;
}

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 30.8

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

  2. Simplified7.8

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

  4. Applied *-un-lft-identity7.8

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

  5. Applied difference-of-squares7.8

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

  6. Final simplification7.8

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

Reproduce

herbie shell --seed 2019200 +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)))))