Average Error: 14.7 → 0.0
Time: 3.4s
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 r62701 = a;
        double r62702 = r62701 * r62701;
        double r62703 = b;
        double r62704 = r62703 * r62703;
        double r62705 = r62702 - r62704;
        double r62706 = r62705 / r62702;
        double r62707 = fabs(r62706);
        double r62708 = sqrt(r62707);
        return r62708;
}
↓
double f(double a, double b) {
        double r62709 = a;
        double r62710 = b;
        double r62711 = r62709 + r62710;
        double r62712 = r62709 - r62710;
        double r62713 = r62712 / r62709;
        double r62714 = r62711 * r62713;
        double r62715 = r62714 / r62709;
        double r62716 = fabs(r62715);
        double r62717 = sqrt(r62716);
        return r62717;
}

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. Using strategy rm

  3. Applied difference-of-squares14.7

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

  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 2020021 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))