Average Error: 14.7 → 0.0
Time: 18.0s
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 r36395 = a;
        double r36396 = r36395 * r36395;
        double r36397 = b;
        double r36398 = r36397 * r36397;
        double r36399 = r36396 - r36398;
        double r36400 = r36399 / r36396;
        double r36401 = fabs(r36400);
        double r36402 = sqrt(r36401);
        return r36402;
}
↓
double f(double a, double b) {
        double r36403 = 1.0;
        double r36404 = b;
        double r36405 = a;
        double r36406 = r36404 / r36405;
        double r36407 = 6.0;
        double r36408 = pow(r36406, r36407);
        double r36409 = cbrt(r36408);
        double r36410 = r36403 - r36409;
        double r36411 = fabs(r36410);
        double r36412 = sqrt(r36411);
        return r36412;
}

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. Simplified14.7

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

  4. Applied add-cbrt-cube26.9

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

  5. Applied add-cbrt-cube26.9

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

  6. Applied cbrt-unprod43.8

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

  7. Applied add-cbrt-cube43.8

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

  8. Applied add-cbrt-cube43.8

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

  9. Applied cbrt-unprod43.8

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

  10. Applied cbrt-undiv43.8

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

  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 2019325 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))