Average Error: 14.7 → 0.0
Time: 21.7s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \frac{b \cdot \frac{b}{a}}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|1 - \frac{b \cdot \frac{b}{a}}{a}\right|}
double f(double a, double b) {
        double r29961 = a;
        double r29962 = r29961 * r29961;
        double r29963 = b;
        double r29964 = r29963 * r29963;
        double r29965 = r29962 - r29964;
        double r29966 = r29965 / r29962;
        double r29967 = fabs(r29966);
        double r29968 = sqrt(r29967);
        return r29968;
}
↓
double f(double a, double b) {
        double r29969 = 1.0;
        double r29970 = b;
        double r29971 = a;
        double r29972 = r29970 / r29971;
        double r29973 = r29970 * r29972;
        double r29974 = r29973 / r29971;
        double r29975 = r29969 - r29974;
        double r29976 = fabs(r29975);
        double r29977 = sqrt(r29976);
        return r29977;
}

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 14.7

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

  2. Simplified14.7

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

  4. Applied associate-/r*0.7

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

  5. Simplified0.0

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

  6. Final simplification0.0

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

Reproduce

herbie shell --seed 2019325 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))