Average Error: 14.7 → 0.0
Time: 3.7s
Precision: 64
\[0.0 \le b \le a \le 1\]
\[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
↓
\[\log \left(e^{\sqrt{\left|\frac{\sqrt{a + b}}{1} \cdot \left(\frac{\sqrt{a + b}}{a} \cdot \frac{a - b}{a}\right)\right|}}\right)\]
\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}
↓
\log \left(e^{\sqrt{\left|\frac{\sqrt{a + b}}{1} \cdot \left(\frac{\sqrt{a + b}}{a} \cdot \frac{a - b}{a}\right)\right|}}\right)
double f(double a, double b) {
        double r116575 = a;
        double r116576 = r116575 * r116575;
        double r116577 = b;
        double r116578 = r116577 * r116577;
        double r116579 = r116576 - r116578;
        double r116580 = r116579 / r116576;
        double r116581 = fabs(r116580);
        double r116582 = sqrt(r116581);
        return r116582;
}
↓
double f(double a, double b) {
        double r116583 = a;
        double r116584 = b;
        double r116585 = r116583 + r116584;
        double r116586 = sqrt(r116585);
        double r116587 = 1.0;
        double r116588 = r116586 / r116587;
        double r116589 = r116586 / r116583;
        double r116590 = r116583 - r116584;
        double r116591 = r116590 / r116583;
        double r116592 = r116589 * r116591;
        double r116593 = r116588 * r116592;
        double r116594 = fabs(r116593);
        double r116595 = sqrt(r116594);
        double r116596 = exp(r116595);
        double r116597 = log(r116596);
        return r116597;
}

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 14.7

    \[\sqrt{\left|\frac{a \cdot a - b \cdot b}{a \cdot a}\right|}\]
  2. Using strategy rm

  3. Applied difference-of-squares14.7

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

  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 add-log-exp0.0

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

  8. Applied *-un-lft-identity0.0

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

  9. Applied add-sqr-sqrt0.0

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

  10. Applied times-frac0.0

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

  11. Applied associate-*l*0.0

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

  12. Final simplification0.0

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

Reproduce

herbie shell --seed 2020024 
(FPCore (a b)
  :name "Eccentricity of an ellipse"
  :precision binary64
  :pre (<= 0.0 b a 1)
  (sqrt (fabs (/ (- (* a a) (* b b)) (* a a)))))