Average Error: 14.5 → 0.0
Time: 3.2s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\log \left(e^{\frac{\left(a + b\right) \cdot \frac{a - b}{a}}{a}}\right)\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\log \left(e^{\frac{\left(a + b\right) \cdot \frac{a - b}{a}}{a}}\right)\right|}
double f(double a, double b) {
        double r77981 = a;
        double r77982 = r77981 * r77981;
        double r77983 = b;
        double r77984 = r77983 * r77983;
        double r77985 = r77982 - r77984;
        double r77986 = r77985 / r77982;
        double r77987 = fabs(r77986);
        double r77988 = sqrt(r77987);
        return r77988;
}
↓
double f(double a, double b) {
        double r77989 = a;
        double r77990 = b;
        double r77991 = r77989 + r77990;
        double r77992 = r77989 - r77990;
        double r77993 = r77992 / r77989;
        double r77994 = r77991 * r77993;
        double r77995 = r77994 / r77989;
        double r77996 = exp(r77995);
        double r77997 = log(r77996);
        double r77998 = fabs(r77997);
        double r77999 = sqrt(r77998);
        return r77999;
}

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 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 add-log-exp0.0

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

  10. Final simplification0.0

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

Reproduce

herbie shell --seed 2020064 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))