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How libcurl connects

When libcurl is about to do an Internet transfer, it first resolves the host name to get a number of IP addresses for the host. A hostname needs to have at least one address for libcurl to be able to connect to it.
A hostname can have both IPv4 addresses and IPv6 addresses and they can have a set of both.
If the host only returned addresses of a single IP family, libcurl iterates over each address and tries to connect. If the connect attempt fails for an IP, libcurl continues to try the next entry until the entire list is exhausted.
An application can limit which IP versions libcurl uses by setting CURLOPT_IPRESOLVE.

Happy Eyeballs

When it has received both IPv4 and IPv6 addresses for a host, libcurl first tries to connect to an IPv6 address and after a short delay it tries connecting to the first IPv4 address - at the same time and in parallel. Once one of the attempts succeeds, the others are discarded. This method of attempting to connect using both families at the same time is called Happy Eyeballs and is the widely accepted best practice for Internet clients.
An application can set the delay with which the second family connect attempt starts in the Happy Eyeball procedure by using CURLOPT_HAPPY_EYEBALLS_TIMEOUT_MS.

Timeout and halving

The connection phase has a maximum allowed time (set with CURLOPT_CONNECTTIMEOUT_MS), which defaults to 300 seconds. The entire connect procedure is deemed failed if no connect has succeeded within that time.
When libcurl has multiple addresses left to try to connect to, and there is more than 600 millisecond left, it will at most allow half the remaining time for this attempt. This is to avoid a single sink-hole address make libcurl spend its entire timeout on that bad entry.
For example: if there are 1000 milliseconds left of the timeout and there are two IP addresses left to try to connect to, libcurl then only allows 500 milliseconds on the next attempt.
If there instead only are 600 milliseconds left of the timeout and there are two IP addresses left to try to connect to, libcurl allows the entire remaining timeout period on the next attempt, in order to not make it too short to succeed. The timeout halving approach is only done as long as there is more than 600 milliseconds remaining.

HTTP/3

For applications that ask libcurl to use HTTP/3, it adds another layer of Happy Eyeballs. HTTP/3 works over QUIC and QUIC is a different transport protocol than TCP and a mechanism that sometimes is blocked or otherwise does not work as well as TCP. In an effort to smooth out the problems this brings, libcurl performs QUIC connects in parallel with regular TCP connects in addition to the different IP version connects described above.
When libcurl get both IPv4 and IPv6 addresses for a host, and it wants to do HTTP/3 with the host, it proceeds like this:
  1. 1.
    Start an IPv6 QUIC connect attempt, iterate over the IPv6 addresses
  2. 2.
    After a short delay, start an IPv4 QUIC connect attempt, iterate over the IPv4 addresses
  3. 3.
    After a short delay, start an IPv6 TCP connect attempt, iterate over the IPv6 addresses
  4. 4.
    After a short delay, start an IPv4 TCP connect attempt, iterate over the IPv4 addresses
Once a connect attempt is successful, all the other ones are immediately discarded.
The HTTP/3 happy eyeballing is done when libcurl is asked to use CURL_HTTP_VERSION_3 but not if set to CURL_HTTP_VERSION_3ONLY.