Average Error: 14.8 → 0.0
Time: 3.1s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|\frac{\left(a - b\right) + \frac{a - b}{a} \cdot b}{a}\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|\frac{\left(a - b\right) + \frac{a - b}{a} \cdot b}{a}\right|}
double f(double a, double b) {
        double r76928 = a;
        double r76929 = r76928 * r76928;
        double r76930 = b;
        double r76931 = r76930 * r76930;
        double r76932 = r76929 - r76931;
        double r76933 = r76932 / r76929;
        double r76934 = fabs(r76933);
        double r76935 = sqrt(r76934);
        return r76935;
}
↓
double f(double a, double b) {
        double r76936 = a;
        double r76937 = b;
        double r76938 = r76936 - r76937;
        double r76939 = r76938 / r76936;
        double r76940 = r76939 * r76937;
        double r76941 = r76938 + r76940;
        double r76942 = r76941 / r76936;
        double r76943 = fabs(r76942);
        double r76944 = sqrt(r76943);
        return r76944;
}

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. Using strategy rm

  3. Applied difference-of-squares14.8

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

  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. Using strategy rm

  9. Applied *-commutative0.0

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

  11. Applied distribute-lft-in0.0

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

  12. Simplified0.0

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

  13. Final simplification0.0

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

Reproduce

herbie shell --seed 2019362 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))