diff --git a/test/xpu/skip_list_common.py b/test/xpu/skip_list_common.py
index 7c3aa7f8e..d767e7a91 100644
--- a/test/xpu/skip_list_common.py
+++ b/test/xpu/skip_list_common.py
@@ -3328,4 +3328,5 @@
         "test_sparse_mm_xpu_float64", # - NotImplementedError: Could not run 'aten::addmm' with arguments from the 'SparseXPU' backend. This could be because the operator doesn't exist for this backend, or wa...
         "test_sparse_sum_xpu_float64", # - NotImplementedError: Could not run 'aten::_sparse_sum_backward' with arguments from the 'SparseXPU' backend. This could be because the operator doesn't exist for this...
     ),
+    "test_decomp_xpu.py": None,
 }
diff --git a/test/xpu/test_decomp_xpu.py b/test/xpu/test_decomp_xpu.py
index d659197d9..512aa41ee 100644
--- a/test/xpu/test_decomp_xpu.py
+++ b/test/xpu/test_decomp_xpu.py
@@ -2,7 +2,7 @@
 
 import torch
 from torch.testing._internal.common_device_type import instantiate_device_type_tests
-from torch.testing._internal.common_utils import run_tests
+from torch.testing._internal.common_utils import run_tests, skipIfCrossRef
 
 try:
     from xpu_test_utils import XPUPatchForImport
@@ -11,7 +11,7 @@
 
 with XPUPatchForImport(False):
     import test_decomp
-    from test_decomp import TestDecomp,DecompOneOffTests
+    from test_decomp import TestDecomp,DecompOneOffTests, _getDefaultRtolAndAtol
 
 def _op_assert_ref(test_case, op, test_dtype, i, orig, decomp, ref, args, kwargs):
     assert orig.dtype == decomp.dtype, f"{i} Operation:  {op}"
@@ -49,7 +49,7 @@ def _op_assert_ref(test_case, op, test_dtype, i, orig, decomp, ref, args, kwargs
         (torch.float16, torch.ops.aten.reflection_pad1d_backward.default): 5e-3,
         (torch.bfloat16, torch.ops.aten.reflection_pad1d_backward.default): 5e-3,
         (torch.float16, torch.ops.aten.reflection_pad2d_backward.default): 5e-3,
-        (torch.bfloat16, torch.ops.aten.reflection_pad2d_backward.default): 5e-3,
+        (torch.bfloat16, torch.ops.aten.reflection_pad2d_backward.default): 7e-3, # adjust tolerance for xpu, so hook this func
         (torch.float16, torch.ops.aten.reflection_pad3d_backward.default): 5e-3,
         (torch.bfloat16, torch.ops.aten.reflection_pad3d_backward.default): 5e-2,
         # see https://github.com/pytorch/pytorch/pull/96264
@@ -77,6 +77,106 @@ def _op_assert_ref(test_case, op, test_dtype, i, orig, decomp, ref, args, kwargs
         )
 test_decomp.op_assert_ref=_op_assert_ref
 
+def _op_assert_equal(test_case, op, test_dtype, orig, decomp, args, kwargs):
+    test_case.assertEqual(
+        orig.dtype,
+        decomp.dtype,
+        f"Operation: {op}, orig.dtype: {orig.dtype}, decomp.dtype: {decomp.dtype}, {args}, {kwargs}",
+    )
+    # Before adding an entry to this table, make sure your decomposition is right :)
+    tol_table = {
+        # Due to strange epsilon behaviors, see https://github.com/pytorch/pytorch/issues/73161
+        (torch.float32, torch.ops.aten.native_layer_norm.default): (1e-3, 1e-3),
+        (torch.float32, torch.ops.aten.native_layer_norm_backward.default): (
+            1e-3,
+            1e-3,
+        ),
+        (torch.float64, torch.ops.aten.native_layer_norm.default): (1e-6, 1e-6),
+        # This exceeds default tolerances only on CPU, on CUDA it's fine
+        (torch.float32, torch.ops.aten.grid_sampler_2d.default): (7e-6, 3e-5),
+        # Exceeds tolerances on CUDA, likely due to fma
+        (torch.float32, torch.ops.aten.mv.default): (1e-5, 3e-5),
+        (torch.complex64, torch.ops.aten.mv.default): (5e-5, 5e-5),
+        (torch.float64, torch.ops.aten.upsample_bicubic2d.vec): (1e-5, 5e-4),
+        (torch.float64, torch.ops.aten.upsample_bicubic2d.default): (1e-5, 5e-4),
+        # The decomposition is TOO correct. It computes everything in int64, so sometimes
+        # there's an off-by-one error. See
+        # https://github.com/pytorch/pytorch/issues/81996
+        # https://github.com/pytorch/pytorch/issues/82230
+        (torch.int8, torch.ops.aten.linspace.default): (0, 1),
+        (torch.uint8, torch.ops.aten.linspace.default): (0, 1),
+        (torch.int16, torch.ops.aten.linspace.default): (0, 1),
+        (torch.int32, torch.ops.aten.linspace.default): (0, 1),
+        (torch.int64, torch.ops.aten.linspace.default): (0, 1),
+        (torch.int8, torch.ops.aten.linspace.Tensor_Tensor): (0, 1),
+        (torch.uint8, torch.ops.aten.linspace.Tensor_Tensor): (0, 1),
+        (torch.int16, torch.ops.aten.linspace.Tensor_Tensor): (0, 1),
+        (torch.int32, torch.ops.aten.linspace.Tensor_Tensor): (0, 1),
+        (torch.int64, torch.ops.aten.linspace.Tensor_Tensor): (0, 1),
+        (torch.int8, torch.ops.aten.linspace.Tensor_Scalar): (0, 1),
+        (torch.uint8, torch.ops.aten.linspace.Tensor_Scalar): (0, 1),
+        (torch.int16, torch.ops.aten.linspace.Tensor_Scalar): (0, 1),
+        (torch.int32, torch.ops.aten.linspace.Tensor_Scalar): (0, 1),
+        (torch.int64, torch.ops.aten.linspace.Tensor_Scalar): (0, 1),
+        (torch.int8, torch.ops.aten.linspace.Scalar_Tensor): (0, 1),
+        (torch.uint8, torch.ops.aten.linspace.Scalar_Tensor): (0, 1),
+        (torch.int16, torch.ops.aten.linspace.Scalar_Tensor): (0, 1),
+        (torch.int32, torch.ops.aten.linspace.Scalar_Tensor): (0, 1),
+        (torch.int64, torch.ops.aten.linspace.Scalar_Tensor): (0, 1),
+        (torch.float64,torch.ops.aten._native_batch_norm_legit.default):(3e-7,5e-7), # adjust tolerance for xpu, so hook this func
+    }
+    if (decomp.dtype, op) in tol_table:
+        rtol, atol = tol_table[(decomp.dtype, op)]
+    else:
+        rtol, atol = _getDefaultRtolAndAtol(orig.dtype, decomp.dtype)
+    test_case.assertEqual(
+        orig,
+        decomp,
+        rtol=rtol,
+        atol=atol,
+        msg=f"{op.__name__}\nargs = {args}\nkwargs = {kwargs}",
+    )
+test_decomp.op_assert_equal=_op_assert_equal
+
+@skipIfCrossRef
+def _test_amp_batch_norm_backward(self):
+    device = "xpu"
+    grad_out = torch.randn((1, 2, 16, 16), dtype=torch.float16, device=device)
+    x = torch.randn((1, 2, 16, 16), dtype=torch.float16, device=device)
+    weight = torch.randn((2,), dtype=torch.float32, device=device)
+    rmean = torch.randn((2,), dtype=torch.float32, device=device)
+    rvar = torch.randn((2,), dtype=torch.float32, device=device)
+    mean = torch.randn((0,), dtype=torch.float32, device=device)
+
+    ref = torch.ops.aten.native_batch_norm_backward(
+        grad_out,
+        x,
+        weight,
+        rmean,
+        rvar,
+        mean,
+        mean,
+        False,
+        1e-05,
+        [True, True, True],
+    )
+    res = torch._decomp.decompositions.native_batch_norm_backward(
+        grad_out,
+        x,
+        weight,
+        rmean,
+        rvar,
+        mean,
+        mean,
+        False,
+        1e-05,
+        [True, True, True],
+    )
+    for a, b in zip(ref, res):
+        self.assertEqual(a.stride(), b.stride())
+        self.assertEqual(a.dtype, b.dtype)
+DecompOneOffTests.test_amp_batch_norm_backward=_test_amp_batch_norm_backward
+
 instantiate_device_type_tests(TestDecomp, globals(), only_for="xpu", allow_xpu=True)
 instantiate_device_type_tests(DecompOneOffTests, globals(), only_for="xpu", allow_xpu=True)
 
diff --git a/test/xpu/xpu_test_utils.py b/test/xpu/xpu_test_utils.py
index 6c31415cc..2f851d43e 100644
--- a/test/xpu/xpu_test_utils.py
+++ b/test/xpu/xpu_test_utils.py
@@ -317,6 +317,8 @@
     ("narrow_copy","test_meta_outplace"),
     ("narrow_copy","test_dispatch_meta_outplace"),
     ("narrow_copy","test_dispatch_symbolic_meta_outplace"),
+    ("logspace","test_quick"),
+    ("logspace","test_comprehensive"),
 ]
 
 # some case should adjust tolerance to pass.