BUG: `^(::Float64, ::Union{Int, Float64})` incorrect for very large negative exponents

[KlausC]

The floating point power function fails for a border case of integer argument:

julia> 2.0 ^ typemin(Int) # should be 0.0
0.5

julia> (-1.0) ^ typemin(Int) # should be 1.0
-1.0

The bug seems to be here: Base.Math.pow_body(::Float64, ::Integer)

[KlausC]

Further investigation shows, that the powers of big integers are unreliable for ~ n < -2^53:

The first column uses Int, the second Float64 exponent type. Third column the correct value.

julia> [45:62 prevfloat(1.0).^(-2 .^ (45:62)) prevfloat(1.0).^float(-2 .^ (45:62)) exp.(log(big(prevfloat(1.0))) .* (-2 .^ (45:62)))]
18×4 Matrix{BigFloat}:
 45.0      1.00391           1.00391          1.00391
 46.0      1.00781           1.00781          1.00784
 47.0      1.01562           1.01562          1.01575
 48.0      1.03123           1.03123          1.03174
 49.0      1.06233           1.06233          1.06449
 50.0      1.12352           1.12352          1.13315
 51.0      1.23654           1.23654          1.28403
 52.0      1.35914           1.35914          1.64872
 53.0      1.10235e-07       1.10235e-07      2.71828
 54.0    -54.5981          -54.5981           7.38906
 55.0  -8942.87          -8942.87            54.5982
 56.0     -6.22028e+07      -6.22028e+07   2980.96
 57.0     -1.18444e+15      -1.18444e+15      8.88611e+06
 58.0     -1.9329e+29       -1.9329e+29       7.8963e+13
 59.0     -2.44925e+57      -2.44925e+57      6.23515e+27
 60.0     -1.91951e+113     -1.91951e+113     3.88771e+55
 61.0     -5.82523e+224     -5.82523e+224     1.51143e+111
 62.0     Inf               Inf               2.28441e+222

[mbauman]

The issue here is that inv(x) returns a rounded Float64 of the inverse — with 52 bits precision. In fact, we start losing 1ulp accuracy at -2^27 and it just goes downhill from there. Seems we need a better strategy here.

julia/base/math.jl

Line 1322 in e07c0f1

rx = inv(x)

[oscardssmith]

@mbauman that gets compensated for on

julia/base/math.jl

Line 1324 in e07c0f1

isfinite(x) && (xnlo = -fma(x, rx, -1.) * rx)

I think something must be going wrong in the error compensation though. Investigating now.

[mbauman]

Ah, I see. We still are missing compensation somewhere here:

julia> ulps(b, x) = abs(b^x - Float64(big(b)^x))/eps(b^x)
ulps (generic function with 3 methods)

julia> [ulps(prevfloat(1.0), -2^e) for e in 26:54]
29-element Vector{Float64}:
      0.0
      1.0
      2.0
      8.0
     32.0
    128.0
    512.0
   2048.0
   8192.0
  32768.0
 131073.0
 524302.0
      2.097258e6
      8.389461e6
      3.3561258e7
      1.34272348e8
      5.37307984e8
      2.15098174e9
      8.617942835e9
      3.4584177964e10
      1.3924045658e11
      5.6426422088e11
      2.316653480914e12
      9.762831672521e12
      4.3352610026219e13
      2.13851005933614e14
      1.304153939836051e15
      2.0538755963422595e23
      8.723923242651639e15

[KlausC]

When I fixed the first issue (typemin(Int)), I found the second (inaccuracy) disappeared. Will drop PR soon.