This system was solved in May 2020, using solveSystem in Macaulay2 v1.15 with an Intel(R) Core(TM) i5-5250U CPU at 1.60GHz.
There were 48 solutions found in 1.708 seconds (with a Bezout bound of 256).
Reference: "Computing all solutions to polynomial systems using homotopy continuation" by A. Morgan and A. Sommese (pages 115-138).
See also: http://homepages.math.uic.edu/~jan/Demo/ipp.html.
i1 : ipp(RR_53)
2 2 2 2 2 2 2 2
o1 = {x + x - 1, x + x - 1, x + x - 1, x + x - 1, - .249151x x +
1 2 3 4 5 6 7 8 1 3
------------------------------------------------------------------------
.279423x x + 1.60914x x + 1.4348x x - .800528x x + .400264x x +
2 3 1 4 2 4 6 7 5 8
------------------------------------------------------------------------
.0740524x - .08305x - .38616x - .755266x + .504202x - 1.09163x +
1 2 3 4 5 6
------------------------------------------------------------------------
.400264x + .0492073, .125016x x - .119228x x - .686607x x -
8 1 3 2 3 1 4
------------------------------------------------------------------------
.71994x x - .432419x x - .864839x x - .0371573x + .0354369x +
2 4 5 7 6 8 1 2
------------------------------------------------------------------------
.0853835x - .039252x - .432419x + .013873, - .63555x x - .666404x x
3 5 7 1 3 2 3
------------------------------------------------------------------------
- .11572x x + .110362x x + .290702x x - .629388x x + 1.25878x x +
1 4 2 4 5 7 6 7 5 8
------------------------------------------------------------------------
.581404x x + .195947x - 1.22803x - .0790342x + .0263879x -
6 8 1 2 4 5
------------------------------------------------------------------------
.0571314x - 1.16281x + 1.25878x + 2.16257, 1.48948x x + 1.32811x x
6 7 8 1 3 2 3
------------------------------------------------------------------------
+ .230623x x - .258645x x + 1.16517x x + .53817x x - .269085x x +
1 4 2 4 5 7 6 7 5 8
------------------------------------------------------------------------
.582586x x - .20817x + 2.68683x - .699103x + .357444x + 1.24991x +
6 8 1 2 3 4 5
------------------------------------------------------------------------
1.46774x + 1.16517x + 1.10763x - .696868}
6 7 8
o1 : List
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The object ipp is a method function.