Average Error: 14.8 → 0.0
Time: 26.9s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \frac{b}{a \cdot \frac{a}{b}}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|1 - \frac{b}{a \cdot \frac{a}{b}}\right|}
double f(double a, double b) {
        double r28154 = a;
        double r28155 = r28154 * r28154;
        double r28156 = b;
        double r28157 = r28156 * r28156;
        double r28158 = r28155 - r28157;
        double r28159 = r28158 / r28155;
        double r28160 = fabs(r28159);
        double r28161 = sqrt(r28160);
        return r28161;
}
↓
double f(double a, double b) {
        double r28162 = 1.0;
        double r28163 = b;
        double r28164 = a;
        double r28165 = r28164 / r28163;
        double r28166 = r28164 * r28165;
        double r28167 = r28163 / r28166;
        double r28168 = r28162 - r28167;
        double r28169 = fabs(r28168);
        double r28170 = sqrt(r28169);
        return r28170;
}

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 14.8

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

  2. Simplified14.8

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

  4. Applied associate-/l*14.7

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))