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 r55044 = a;
        double r55045 = r55044 * r55044;
        double r55046 = b;
        double r55047 = r55046 * r55046;
        double r55048 = r55045 - r55047;
        double r55049 = r55048 / r55045;
        double r55050 = fabs(r55049);
        double r55051 = sqrt(r55050);
        return r55051;
}
↓
double f(double a, double b) {
        double r55052 = 1.0;
        double r55053 = b;
        double r55054 = a;
        double r55055 = r55053 / r55054;
        double r55056 = fabs(r55055);
        double r55057 = 2.0;
        double r55058 = pow(r55056, r55057);
        double r55059 = exp(r55058);
        double r55060 = log(r55059);
        double r55061 = r55052 - r55060;
        double r55062 = fabs(r55061);
        double r55063 = sqrt(r55062);
        return r55063;
}

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