Average Error: 14.3 → 0.0
Time: 2.3s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{\left(a + b\right) \cdot \frac{a - b}{a}}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{\left(a + b\right) \cdot \frac{a - b}{a}}{a}\right|}
double f(double a, double b) {
        double r49799 = a;
        double r49800 = r49799 * r49799;
        double r49801 = b;
        double r49802 = r49801 * r49801;
        double r49803 = r49800 - r49802;
        double r49804 = r49803 / r49800;
        double r49805 = fabs(r49804);
        double r49806 = sqrt(r49805);
        return r49806;
}
↓
double f(double a, double b) {
        double r49807 = a;
        double r49808 = b;
        double r49809 = r49807 + r49808;
        double r49810 = r49807 - r49808;
        double r49811 = r49810 / r49807;
        double r49812 = r49809 * r49811;
        double r49813 = r49812 / r49807;
        double r49814 = fabs(r49813);
        double r49815 = sqrt(r49814);
        return r49815;
}

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 14.3

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

  3. Applied difference-of-squares14.3

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

  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. Final simplification0.0

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

Reproduce

herbie shell --seed 2020018 +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)))))