Power function broken in Version 1.11.4

[KlausC]

The ^ function delivers numerically wrong results for big negative integer exponents.
The second column shows the correct results.

Related to PR #53967, which should be backported to 1.11 and 1.10.

julia> VERSION
v"1.11.4"

julia> E = prevfloat(1.0)
0.9999999999999999

julia> [ E.^(-2 .^(50:63)) Float64.(big(E).^(-2 .^(50:63))) ]
14×2 Matrix{Float64}:
     1.12352         1.13315
     1.23654         1.28403
     1.35914         1.64872
     1.10235e-7      2.71828
   -54.5981          7.38906
 -8942.87           54.5982
    -6.22028e7    2980.96
    -1.18444e15      8.88611e6
    -1.9329e29       7.8963e13
    -2.44925e57      6.23515e27
    -1.91951e113     3.88771e55
    -5.82523e224     1.51143e111
    Inf              2.28441e222
     1.0            Inf

Compare to the fixed version:

julia> VERSION
v"1.12.0-beta1"

julia> E = prevfloat(1.0)
0.9999999999999999

julia> [ E.^(-2 .^(52:63)) Float64.(big(E).^(-2 .^(52:63))) ]
12×2 Matrix{Float64}:
    1.64872         1.64872
    2.71828         2.71828
    7.38906         7.38906
   54.5982         54.5982
 2980.96         2980.96
    8.88611e6       8.88611e6
    7.8963e13       7.8963e13
    6.23515e27      6.23515e27
    3.88771e55      3.88771e55
    1.51143e111     1.51143e111
    2.28441e222     2.28441e222
   Inf             Inf

[giordano]

Related to PR #53967, which should be backported to 1.11 and 1.10.

I believe #53967 was tentatively backported to v1.11, but it broke tests, and the change was reverted.