Average Error: 14.8 → 0.0
Time: 21.6s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \sqrt[3]{{\left(\frac{b}{a}\right)}^{6}}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|1 - \sqrt[3]{{\left(\frac{b}{a}\right)}^{6}}\right|}
double f(double a, double b) {
        double r45418 = a;
        double r45419 = r45418 * r45418;
        double r45420 = b;
        double r45421 = r45420 * r45420;
        double r45422 = r45419 - r45421;
        double r45423 = r45422 / r45419;
        double r45424 = fabs(r45423);
        double r45425 = sqrt(r45424);
        return r45425;
}
↓
double f(double a, double b) {
        double r45426 = 1.0;
        double r45427 = b;
        double r45428 = a;
        double r45429 = r45427 / r45428;
        double r45430 = 6.0;
        double r45431 = pow(r45429, r45430);
        double r45432 = cbrt(r45431);
        double r45433 = r45426 - r45432;
        double r45434 = fabs(r45433);
        double r45435 = sqrt(r45434);
        return r45435;
}

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 14.8

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

  2. Simplified14.8

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

  4. Applied add-cbrt-cube26.8

    \[\leadsto \sqrt{\left|1 - \frac{b \cdot b}{a \cdot \color{blue}{\sqrt[3]{\left(a \cdot a\right) \cdot a}}}\right|}\]

  5. Applied add-cbrt-cube26.8

    \[\leadsto \sqrt{\left|1 - \frac{b \cdot b}{\color{blue}{\sqrt[3]{\left(a \cdot a\right) \cdot a}} \cdot \sqrt[3]{\left(a \cdot a\right) \cdot a}}\right|}\]

  6. Applied cbrt-unprod43.3

    \[\leadsto \sqrt{\left|1 - \frac{b \cdot b}{\color{blue}{\sqrt[3]{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)}}}\right|}\]

  7. Applied add-cbrt-cube43.3

    \[\leadsto \sqrt{\left|1 - \frac{b \cdot \color{blue}{\sqrt[3]{\left(b \cdot b\right) \cdot b}}}{\sqrt[3]{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)}}\right|}\]

  8. Applied add-cbrt-cube43.3

    \[\leadsto \sqrt{\left|1 - \frac{\color{blue}{\sqrt[3]{\left(b \cdot b\right) \cdot b}} \cdot \sqrt[3]{\left(b \cdot b\right) \cdot b}}{\sqrt[3]{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)}}\right|}\]

  9. Applied cbrt-unprod43.3

    \[\leadsto \sqrt{\left|1 - \frac{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)}}}{\sqrt[3]{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)}}\right|}\]

  10. Applied cbrt-undiv43.3

    \[\leadsto \sqrt{\left|1 - \color{blue}{\sqrt[3]{\frac{\left(\left(b \cdot b\right) \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot b\right)}{\left(\left(a \cdot a\right) \cdot a\right) \cdot \left(\left(a \cdot a\right) \cdot a\right)}}}\right|}\]

  11. Simplified0.0

    \[\leadsto \sqrt{\left|1 - \sqrt[3]{\color{blue}{{\left(\frac{b}{a}\right)}^{6}}}\right|}\]

  12. Final simplification0.0

    \[\leadsto \sqrt{\left|1 - \sqrt[3]{{\left(\frac{b}{a}\right)}^{6}}\right|}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))