Average Error: 14.6 → 0.0
Time: 7.5s
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 r28250 = a;
        double r28251 = r28250 * r28250;
        double r28252 = b;
        double r28253 = r28252 * r28252;
        double r28254 = r28251 - r28253;
        double r28255 = r28254 / r28251;
        double r28256 = fabs(r28255);
        double r28257 = sqrt(r28256);
        return r28257;
}
↓
double f(double a, double b) {
        double r28258 = 1.0;
        double r28259 = b;
        double r28260 = a;
        double r28261 = r28259 / r28260;
        double r28262 = fabs(r28261);
        double r28263 = 2.0;
        double r28264 = pow(r28262, r28263);
        double r28265 = exp(r28264);
        double r28266 = log(r28265);
        double r28267 = r28258 - r28266;
        double r28268 = fabs(r28267);
        double r28269 = sqrt(r28268);
        return r28269;
}

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 2019305 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))