Average Error: 14.2 → 0.0
Time: 3.8s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\sqrt[3]{{\left(\frac{a + b}{a} \cdot \frac{a - b}{a}\right)}^{3}}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\sqrt[3]{{\left(\frac{a + b}{a} \cdot \frac{a - b}{a}\right)}^{3}}\right|}
double f(double a, double b) {
        double r63386 = a;
        double r63387 = r63386 * r63386;
        double r63388 = b;
        double r63389 = r63388 * r63388;
        double r63390 = r63387 - r63389;
        double r63391 = r63390 / r63387;
        double r63392 = fabs(r63391);
        double r63393 = sqrt(r63392);
        return r63393;
}
↓
double f(double a, double b) {
        double r63394 = a;
        double r63395 = b;
        double r63396 = r63394 + r63395;
        double r63397 = r63396 / r63394;
        double r63398 = r63394 - r63395;
        double r63399 = r63398 / r63394;
        double r63400 = r63397 * r63399;
        double r63401 = 3.0;
        double r63402 = pow(r63400, r63401);
        double r63403 = cbrt(r63402);
        double r63404 = fabs(r63403);
        double r63405 = sqrt(r63404);
        return r63405;
}

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 14.2

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

  3. Applied difference-of-squares14.2

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

  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-cube27.3

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

  7. Applied add-cbrt-cube26.8

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

  8. Applied cbrt-undiv26.8

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

  9. Applied add-cbrt-cube27.3

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

  10. Applied add-cbrt-cube26.8

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

  11. Applied cbrt-undiv26.8

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

  12. Applied cbrt-unprod26.8

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

  13. Simplified0.0

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

  14. Final simplification0.0

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

Reproduce

herbie shell --seed 2020057 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))