Average Error: 14.0 → 0.0
Time: 3.5s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{1}{\frac{a}{\left(a + b\right) \cdot \frac{a - b}{a}}}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{1}{\frac{a}{\left(a + b\right) \cdot \frac{a - b}{a}}}\right|}
double f(double a, double b) {
        double r98727 = a;
        double r98728 = r98727 * r98727;
        double r98729 = b;
        double r98730 = r98729 * r98729;
        double r98731 = r98728 - r98730;
        double r98732 = r98731 / r98728;
        double r98733 = fabs(r98732);
        double r98734 = sqrt(r98733);
        return r98734;
}
↓
double f(double a, double b) {
        double r98735 = 1.0;
        double r98736 = a;
        double r98737 = b;
        double r98738 = r98736 + r98737;
        double r98739 = r98736 - r98737;
        double r98740 = r98739 / r98736;
        double r98741 = r98738 * r98740;
        double r98742 = r98736 / r98741;
        double r98743 = r98735 / r98742;
        double r98744 = fabs(r98743);
        double r98745 = sqrt(r98744);
        return r98745;
}

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 14.0

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

  3. Applied difference-of-squares14.0

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

  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 associate-*r/0.0

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

  7. Simplified0.0

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

  9. Applied clear-num0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020033 +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)))))