mpmath version: 1.2.1
OS: Windows 10

When I run the following:

print("Low precision: ", mpmath.hyperu(1, -473.1, 156))
with mpmath.extraprec(1000):
    print("High precision: ", mpmath.hyperu(1, -473.1, 156))

I get:

Low precision:  -1.12058697578989e+44
High precision:  0.001586425835211121087681179361528927983242277888339631178609757381602116352014620984480403121211547027499627345216063020035934706802772039354304125819952326262751451176676820907796632804032104350623713922869560129245748835027779792137220628978091592575872212530734647063184344560862957299239418155652367375142065798074

I believe the correct result is the one that's close to 0.001586. The expected behavior is for the function to either return the value accurate to within 2 ** (-ctx.prec) relative error, or to raise an exception about a failure to converge. The 1e+44 value looks like a catastrophic cancellation error.

I must say though, I'm not so sure if this is easy to fix. Hypergeometric functions are evil.

At least for hyperu(1, x, y) for large x, y of similar magnitude, it empirically seems like the value is close to x / ((x - y) * (1 - x)). But I don't know what asymptotic expansion this comes from, or how fast that converges.