[PATCH] Define midpoint and lerp functions for C++20 (P0811R3)

[Joseph Myers]

On Tue, 5 Mar 2019, Jonathan Wakely wrote:

> The midpoint and lerp functions for floating point types come straight
> from the P0811R3 proposal, with no attempt at optimization.

I don't know whether P0811R3 states different requirements from the public 
P0811R2, but the implementation of midpoint using isnormal does *not* 
satisfy "at most one inexact operation occurs" and is *not* correctly 
rounded, contrary to the claims made in P0811R2.

Consider e.g. midpoint(DBL_MIN + DBL_TRUE_MIN, DBL_MIN + DBL_TRUE_MIN).  
The value DBL_MIN + DBL_TRUE_MIN is normal, but dividing it by 2 is 
inexact (and so that midpoint implementation would produce DBL_MIN as 
result, so failing to satisfy midpoint(x, x) == x).

Replacing isnormal(x) by something like isgreaterequal(fabs(x), MIN*2) 
would avoid those inexact divisions, but there would still be spurious 
overflows in non-default rounding modes for e.g. midpoint(DBL_MAX, 
DBL_TRUE_MIN) in FE_UPWARD mode, so failing "No overflow occurs" if that's 
meant to apply in all rounding modes.

-- 
Joseph S. Myers
joseph@codesourcery.com