Average Error: 13.9 → 0.0
Time: 12.8s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\sqrt{\left|1 - \log \left(e^{{\left(\left|\frac{b}{a}\right|\right)}^{2}}\right)\right|}\]
\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 r58524 = a;
        double r58525 = r58524 * r58524;
        double r58526 = b;
        double r58527 = r58526 * r58526;
        double r58528 = r58525 - r58527;
        double r58529 = r58528 / r58525;
        double r58530 = fabs(r58529);
        double r58531 = sqrt(r58530);
        return r58531;
}
↓
double f(double a, double b) {
        double r58532 = 1.0;
        double r58533 = b;
        double r58534 = a;
        double r58535 = r58533 / r58534;
        double r58536 = fabs(r58535);
        double r58537 = 2.0;
        double r58538 = pow(r58536, r58537);
        double r58539 = exp(r58538);
        double r58540 = log(r58539);
        double r58541 = r58532 - r58540;
        double r58542 = fabs(r58541);
        double r58543 = sqrt(r58542);
        return r58543;
}

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 13.9

    \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]

  2. Simplified13.9

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

  4. Applied add-sqr-sqrt13.9

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

  5. Simplified13.9

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

  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 2019199 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :pre (<= 0.0 b a 1.0)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))