Average Error: 14.0 → 0.0
Time: 4.3s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\log \left(e^{\frac{a + b}{a} \cdot \frac{a - b}{a}}\right)\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\log \left(e^{\frac{a + b}{a} \cdot \frac{a - b}{a}}\right)\right|}
double f(double a, double b) {
        double r82763 = a;
        double r82764 = r82763 * r82763;
        double r82765 = b;
        double r82766 = r82765 * r82765;
        double r82767 = r82764 - r82766;
        double r82768 = r82767 / r82764;
        double r82769 = fabs(r82768);
        double r82770 = sqrt(r82769);
        return r82770;
}
↓
double f(double a, double b) {
        double r82771 = a;
        double r82772 = b;
        double r82773 = r82771 + r82772;
        double r82774 = r82773 / r82771;
        double r82775 = r82771 - r82772;
        double r82776 = r82775 / r82771;
        double r82777 = r82774 * r82776;
        double r82778 = exp(r82777);
        double r82779 = log(r82778);
        double r82780 = fabs(r82779);
        double r82781 = sqrt(r82780);
        return r82781;
}

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 14.0

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

  3. Applied difference-of-squares14.0

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

  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 add-log-exp0.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2020033 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))