Average Error: 14.6 → 0.0
Time: 3.2s
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 r88690 = a;
        double r88691 = r88690 * r88690;
        double r88692 = b;
        double r88693 = r88692 * r88692;
        double r88694 = r88691 - r88693;
        double r88695 = r88694 / r88691;
        double r88696 = fabs(r88695);
        double r88697 = sqrt(r88696);
        return r88697;
}
↓
double f(double a, double b) {
        double r88698 = 1.0;
        double r88699 = a;
        double r88700 = b;
        double r88701 = r88699 + r88700;
        double r88702 = r88699 - r88700;
        double r88703 = r88702 / r88699;
        double r88704 = r88701 * r88703;
        double r88705 = r88699 / r88704;
        double r88706 = r88698 / r88705;
        double r88707 = fabs(r88706);
        double r88708 = sqrt(r88707);
        return r88708;
}

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 14.6

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

  3. Applied difference-of-squares14.6

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

  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 2020036 +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)))))