Average Error: 14.6 → 0.0
Time: 6.9s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \log \left(e^{{\left(\left|\frac{b}{a}\right|\right)}^{2}}\right)\right|}\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\sqrt{\left|1 - \log \left(e^{{\left(\left|\frac{b}{a}\right|\right)}^{2}}\right)\right|}
double f(double a, double b) {
        double r60905 = a;
        double r60906 = r60905 * r60905;
        double r60907 = b;
        double r60908 = r60907 * r60907;
        double r60909 = r60906 - r60908;
        double r60910 = r60909 / r60906;
        double r60911 = fabs(r60910);
        double r60912 = sqrt(r60911);
        return r60912;
}
↓
double f(double a, double b) {
        double r60913 = 1.0;
        double r60914 = b;
        double r60915 = a;
        double r60916 = r60914 / r60915;
        double r60917 = fabs(r60916);
        double r60918 = 2.0;
        double r60919 = pow(r60917, r60918);
        double r60920 = exp(r60919);
        double r60921 = log(r60920);
        double r60922 = r60913 - r60921;
        double r60923 = fabs(r60922);
        double r60924 = sqrt(r60923);
        return r60924;
}

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. Simplified14.6

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

  4. Applied add-sqr-sqrt14.6

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

  5. Simplified14.6

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

  6. Simplified0.0

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

  8. Applied add-log-exp0.0

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

  9. Simplified0.0

    \[\leadsto \sqrt{\left|1 - \log \color{blue}{\left(e^{{\left(\left|\frac{b}{a}\right|\right)}^{2}}\right)}\right|}\]

  10. Final simplification0.0

    \[\leadsto \sqrt{\left|1 - \log \left(e^{{\left(\left|\frac{b}{a}\right|\right)}^{2}}\right)\right|}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0.0 b a 1.0)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))