Average Error: 14.5 → 0.0
Time: 9.5s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{\frac{a - b}{\frac{a}{a + b}}}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{\frac{a - b}{\frac{a}{a + b}}}{a}\right|}
double f(double a, double b) {
        double r283 = a;
        double r284 = r283 * r283;
        double r285 = b;
        double r286 = r285 * r285;
        double r287 = r284 - r286;
        double r288 = r287 / r284;
        double r289 = fabs(r288);
        double r290 = sqrt(r289);
        return r290;
}
↓
double f(double a, double b) {
        double r291 = a;
        double r292 = b;
        double r293 = r291 - r292;
        double r294 = r291 + r292;
        double r295 = r291 / r294;
        double r296 = r293 / r295;
        double r297 = r296 / r291;
        double r298 = fabs(r297);
        double r299 = sqrt(r298);
        return r299;
}

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 14.5

    \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
  2. Using strategy rm

  3. Applied difference-of-squares14.5

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

  4. Applied times-frac0.0

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

  6. Applied clear-num0.0

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

  8. Applied associate-*r/0.0

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

  9. Simplified0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020025 +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)))))