Average Error: 29.6 → 7.5
Time: 15.5s
Precision: 64
\[0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \frac{\frac{b}{a} \cdot b}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|1 - \frac{\frac{b}{a} \cdot b}{a}\right|}
double f(double a, double b) {
        double r1527816 = a;
        double r1527817 = r1527816 * r1527816;
        double r1527818 = b;
        double r1527819 = r1527818 * r1527818;
        double r1527820 = r1527817 - r1527819;
        double r1527821 = r1527820 / r1527817;
        double r1527822 = fabs(r1527821);
        double r1527823 = sqrt(r1527822);
        return r1527823;
}
↓
double f(double a, double b) {
        double r1527824 = 1.0;
        double r1527825 = b;
        double r1527826 = a;
        double r1527827 = r1527825 / r1527826;
        double r1527828 = r1527827 * r1527825;
        double r1527829 = r1527828 / r1527826;
        double r1527830 = r1527824 - r1527829;
        double r1527831 = fabs(r1527830);
        double r1527832 = sqrt(r1527831);
        return r1527832;
}

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 29.6

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

  2. Simplified7.5

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

  4. Applied associate-*l/7.5

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

  5. Final simplification7.5

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

Reproduce

herbie shell --seed 2019153 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))