Average Error: 14.6 → 0.0
Time: 4.7s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{a + b}{a} \cdot \frac{a - b}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{a + b}{a} \cdot \frac{a - b}{a}\right|}
double f(double a, double b) {
        double r36626 = a;
        double r36627 = r36626 * r36626;
        double r36628 = b;
        double r36629 = r36628 * r36628;
        double r36630 = r36627 - r36629;
        double r36631 = r36630 / r36627;
        double r36632 = fabs(r36631);
        double r36633 = sqrt(r36632);
        return r36633;
}
↓
double f(double a, double b) {
        double r36634 = a;
        double r36635 = b;
        double r36636 = r36634 + r36635;
        double r36637 = r36636 / r36634;
        double r36638 = r36634 - r36635;
        double r36639 = r36638 / r36634;
        double r36640 = r36637 * r36639;
        double r36641 = fabs(r36640);
        double r36642 = sqrt(r36641);
        return r36642;
}

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 14.6

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

  3. Applied difference-of-squares14.6

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

  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-cbrt-cube0.0

    \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{\left|\frac{a + b}{a} \cdot \frac{a - b}{a}\right|} \cdot \sqrt{\left|\frac{a + b}{a} \cdot \frac{a - b}{a}\right|}\right) \cdot \sqrt{\left|\frac{a + b}{a} \cdot \frac{a - b}{a}\right|}}}\]

  7. Simplified0.0

    \[\leadsto \sqrt[3]{\color{blue}{{\left(\sqrt{\left|\frac{a + b}{a} \cdot \frac{a - b}{a}\right|}\right)}^{3}}}\]
  8. Using strategy rm

  9. Applied add-log-exp0.0

    \[\leadsto \color{blue}{\log \left(e^{\sqrt[3]{{\left(\sqrt{\left|\frac{a + b}{a} \cdot \frac{a - b}{a}\right|}\right)}^{3}}}\right)}\]

  10. Simplified0.0

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

  11. Final simplification0.0

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

Reproduce

herbie shell --seed 2019304 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))