Revision d4bd1b04-935d-45e7-8e6d-5c983d5211ad - Code Golf Stack Exchange

# [Python 3.8 (pre-release)], 40 bytes
This one works with python integers, giving the correct result even for arbitrarily large integers. Always rounds down.
<!-- language-all: lang-python -->

 lambda x:isqrt(x*x//2)
 from math import*

[Try it online!][TIO-lprj5ssk]

[Python 3.8 (pre-release)]: https://docs.python.org/3.8/
[TIO-lprj5ssk]: https://tio.run/##fZJNbtswEIX3OgWXUiAg809OAZ@kyUJF4kaALSuyCti9vDO03SCbVAuJfEO@efzE@by@HSYu83LZbp4uu2H/62VIpx/j8X1Z29PD6fGRuma7HPZpP6xvadzPh2V9uCyvxz@79Zg26ee2nbq0PSxpSuOU2gb6hH2iPnGfpE8a05A4BIVag3/v26CYiGS0T@0qE0qWwiY5tgLnEMU9DMnFLZOHrSiwuhvlqIfaJydizgRsRSVnLZBrBxHLIlCXuSoa6tdm/3vCUo1KRAFHLJmQMRcpbpiNVKKbMbOyXxdgZLRoJMLChZvuuVmH5ffrDVQl8wnnxidFPqpDvU5q3PpGqDgYAwDX8l3LJfaiYgmQJP49pdgpShoVivQlzEwJHYojmRlHVKgMGCJmHIiUmeJDiF@bxb8JjErBDgJiuCFIEQf2CGXZ0HOIQVSLOsTZ3YJvDi8R9ewAHElLhBBCq/ZxJADz5rmZl3Fa22G3a89ps0nr9Qqd@7Veor/j3N5vWJ/uBLuuu3wA "Python 3.8 (pre-release) – Try It Online"