Bad news: on Windows, exp2(x) is way worse then pow(2, x). Here I changed the loop of Mark's little driver like so:

    differ = really_bad = 0
    worst = 0.0
    for n in range(100_000):
        x = random.uniform(-1000.0, 999.0) + random.random()
        if exp2(x) != pow(2.0, x):
            differ += 1
            exp2err = exp2_error_ulps(x)
            pow2err = pow2_error_ulps(x)
            assert abs(pow2err) < 0.52
            if abs(exp2err) >= 1.0:
                if abs(exp2err) > abs(worst):
                    worst = exp2err
                really_bad += 1
                if really_bad < 25:
                    print(f"{x.hex():21} {x:22.17f} {exp2err:.5f}, {pow2err:.5f}")
    print(f"{differ=:,}")
    print(f"{really_bad=:,}")
    print(f"worst exp2 ulp error {worst:.5f}")

Then output from one run:

0x1.0946680d45f28p+9   530.55005041041749791 -1.04399, -0.04399
0x1.de4f9662d84f8p+9   956.62177691995657369 -1.00976, -0.00976
-0x1.60f9152be0a09p+4  -22.06081120624188330 1.02472, 0.02472
-0x1.687b056d7a81ap+8 -360.48055156937482479 1.48743, 0.48743
0x1.8e97e9d405622p+9   797.18682337057930454 1.05224, 0.05224
-0x1.2d1e3a03eda7fp+9 -602.23614548782632028 -1.21876, -0.21876
0x1.3af55e79cd45dp+8   314.95847283612766887 -1.10044, -0.10044
0x1.0e7fba610cde6p+9   540.99787533882476964 -1.39782, -0.39782
0x1.9c7d0258e460dp+9   824.97663413192060489 1.19690, 0.19690
0x1.3de5064eb1598p+9   635.78925498637818237 1.75376, -0.24624
-0x1.d5189d23da3d0p+9 -938.19229553371587826 1.07734, 0.07734
0x1.967d0857aa500p+9   812.97681709114112891 1.23630, 0.23630
-0x1.30ee89e018914p+6  -76.23294782781550794 -1.10275, -0.10275
-0x1.e35eb8936dddbp+9 -966.74000780930089149 -1.02686, -0.02686
-0x1.28d40d7693088p+6  -74.20708260795993283 1.00352, 0.00352
-0x1.e965d067d1084p+7 -244.69885563303625986 1.21136, 0.21136
-0x1.b1fbeec1c1ba3p+7 -216.99205594529948371 -1.05536, -0.05536
-0x1.543d715a5824cp+9 -680.48002175620922571 1.24955, 0.24955
0x1.526829d46c034p+9   676.81377654336984051 -1.17826, -0.17826
-0x1.bdaf1d7850c74p+6 -111.42101085656196346 1.08670, 0.08670
-0x1.48218d1605dd0p+9 -656.26211810385029821 1.06103, 0.06103
-0x1.16298bcdb9103p+9 -556.32457896744051595 -1.23732, -0.23732
0x1.39ff24b1a7573p+8   313.99665365539038930 -1.20931, -0.20931
0x1.9cdf1d0101646p+8   412.87153631481157845 -1.23481, -0.23481
differ=38,452
really_bad=7,306
worst exp2 ulp error -1.91748

So they differed in more than a third of the cases; in about a fifth of the differing cases, the exp2 error was at least 1 ulp, and nearly 2 ulp at worst; while in all the differing cases the pow(2, x) error was under 0.52 ulp.