Circumference of an ellipse Challenge
Unlike the circumference of a circle (which is as simple as \$2\pi r\$), the circumference of an ellipse is hard.
Given the semi-major axis \$a\$ and semi-minor axis \$b\$ of an ellipse (see the image below, from Wikipedia), calculate its circumference.
By definition, you can assume \$0 &lt; b \le a\$ for input values. The output value must be within \$10^{-6}\$ relative error from the expected answer for the given test cases.

Standard code-golf rules apply. The shortest code in bytes wins.
Formulas
Relevant information can be found on Wikipedia and MathWorld. \$C\$ is the value of the circumference; \$e\$ and \$h\$ are helper values. The last two are Ramanujan's approximations, the first of which (the crossed-out one) does not meet the error requirements. The second approximation formula (Equation 5) barely does (verification) for up to \$a=5b\$ (which is also the upper limit of the test cases, so you can use it for your answer).
$$
\require{enclose} \\
\begin{align}
e &amp;= \sqrt{1-\frac{b^2}{a^2}} \\
C &amp;= 4aE(e) = 4a\int^{\pi/2}_{0}{\sqrt{1-e^2 \sin^2 \theta} \;d\theta} \tag{1} \\
C &amp;= 2 \pi a \left(1-\sum^{\infty}_{n=1}{\left(\frac{(2n-1)!!}{(2n)!!}\right)^2 \frac{e^{2n}}{2n-1}}\right) \tag{2} \\
h &amp;= \frac{(a-b)^2}{(a+b)^2} \\
C &amp;= \pi (a + b) \left( 1 + \sum^{\infty}_{n=1} { \left( \frac{(2n-1)!!}{2^n n!} \right)^2 \frac{h^n}{(2n-1)^2} } \right) \tag{3} \\
C &amp;= \pi (a + b) \sum^{\infty}_{n=0} { \binom{1/2}{n}^2 h^n } \tag{4} \\
\enclose{horizontalstrike}{C} &amp;\enclose{horizontalstrike}{\approx \pi \left( 3(a+b) - \sqrt{(3a+b)(a+3b)} \right)} \\
C &amp;\approx \pi (a+b) \left( 1+ \frac{3h}{10 + \sqrt{4-3h}} \right) \tag{5}
\end{align}
$$
Test cases
All the values for C (circumference) are calculated using Equation 4 with 1000 terms, and presented with 10 significant figures.
a     b     C
1     1     6.283185307
1.2   1     6.925791195
1.5   1     7.932719795
2     1     9.688448220
3     1     13.36489322
5     1     21.01004454
20    10    96.88448220
123   45    556.6359936
Wolfram Language (Mathematica), 20 bytes
Perimeter[#~Disk~#]&amp;

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