Average Error: 14.5 → 0.0
Time: 4.8s
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 r67952 = a;
        double r67953 = r67952 * r67952;
        double r67954 = b;
        double r67955 = r67954 * r67954;
        double r67956 = r67953 - r67955;
        double r67957 = r67956 / r67953;
        double r67958 = fabs(r67957);
        double r67959 = sqrt(r67958);
        return r67959;
}
↓
double f(double a, double b) {
        double r67960 = a;
        double r67961 = b;
        double r67962 = r67960 - r67961;
        double r67963 = r67960 + r67961;
        double r67964 = r67960 / r67963;
        double r67965 = r67962 / r67964;
        double r67966 = r67965 / r67960;
        double r67967 = fabs(r67966);
        double r67968 = sqrt(r67967);
        return r67968;
}

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 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))