Repository: pytorch
Issue: Unexpected results with torch.nn.functional.layer_norm
Labels: module: numerical-stability,triaged,module: norms and normalization

### 🐛 Describe the bug

Consider the following inputs:
```
a = torch.tensor([-3.], dtype=torch.float64)
weight = torch.tensor([1e10], dtype=torch.float64)
bias = torch.tensor([-1], dtype=torch.float64)
normalaxis = (1,)
eps = 0.1
torch.nn.functional.layer_norm(a, normalaxis, weight= weight, bias=bias, eps=eps)
```

This returns -1. as expected. In fact, when the weight is between 1e0 to 1e10, the return value is -1. However, when the weight is further increased from 1e10, the return value changes (first decreases and then increases). This makes no sense to me since the mean in every case is -3 and therefore, the weight will always get multiplied by 0 according to the layer_norm update equation which means the answer should always be -1 (equal to bias). What am I missing here?

https://discuss.pytorch.org/t/unexpected-results-with-torch-nn-functional-layer-norm/176064?u=hello-fri-end

### Versions

```
--2023-03-31 10:35:36--  https://raw.githubusercontent.com/pytorch/pytorch/master/torch/utils/collect_env.py
Resolving raw.githubusercontent.com (raw.githubusercontent.com)... 185.199.108.133, 185.199.109.133, 185.199.110.133, ...
Connecting to raw.githubusercontent.com (raw.githubusercontent.com)|185.199.108.133|:443... connected.
HTTP request sent, awaiting response... 200 OK
Length: 21501 (21K) [text/plain]
Saving to: ‘collect_env.py’

collect_env.py      100%[===================>]  21.00K  --.-KB/s    in 0.001s  

2023-03-31 10:35:36 (22.6 MB/s) - ‘collect_env.py’ saved [21501/21501]

Collecting environment information...
PyTorch version: 1.13.1+cu116
Is debug build: False
CUDA used to build PyTorch: 11.6
ROCM used to build PyTorch: N/A

OS: Ubuntu 20.04.5 LTS (x86_64)
GCC version: (Ubuntu 9.4.0-1ubuntu1~20.04.1) 9.4.0
Clang version: 10.0.0-4ubuntu1 
CMake version: version 3.25.2
Libc version: glibc-2.31

Python version: 3.9.16 (main, Dec  7 2022, 01:11:51)  [GCC 9.4.0] (64-bit runtime)
Python platform: Linux-5.10.147+-x86_64-with-glibc2.31
Is CUDA available: False
CUDA runtime version: 11.8.89
CUDA_MODULE_LOADING set to: N/A
GPU models and configuration: Could not collect
Nvidia driver version: Could not collect
cuDNN version: Probably one of the following:
/usr/lib/x86_64-linux-gnu/libcudnn.so.8.7.0
/usr/lib/x86_64-linux-gnu/libcudnn_adv_infer.so.8.7.0
/usr/lib/x86_64-linux-gnu/libcudnn_adv_train.so.8.7.0
/usr/lib/x86_64-linux-gnu/libcudnn_cnn_infer.so.8.7.0
/usr/lib/x86_64-linux-gnu/libcudnn_cnn_train.so.8.7.0
/usr/lib/x86_64-linux-gnu/libcudnn_ops_infer.so.8.7.0
/usr/lib/x86_64-linux-gnu/libcudnn_ops_train.so.8.7.0
HIP runtime version: N/A
MIOpen runtime version: N/A
Is XNNPACK available: True

CPU:
Architecture:                    x86_64
CPU op-mode(s):                  32-bit, 64-bit
Byte Order:                      Little Endian
Address sizes:                   46 bits physical, 48 bits virtual
CPU(s):                          2
On-line CPU(s) list:             0,1
Thread(s) per core:              2
Core(s) per socket:              1
Socket(s):                       1
NUMA node(s):                    1
Vendor ID:                       GenuineIntel
CPU family:                      6
Model:                           79
Model name:                      Intel(R) Xeon(R) CPU @ 2.20GHz
Stepping:                        0
CPU MHz:                         2200.164
BogoMIPS:                        4400.32
Hypervisor vendor:               KVM
Virtualization type:             full
L1d cache:                       32 KiB
L1i cache:                       32 KiB
L2 cache:                        256 KiB
L3 cache:                        55 MiB
NUMA node0 CPU(s):               0,1
Vulnerability Itlb multihit:     Not affected
Vulnerability L1tf:              Mitigation; PTE Inversion
Vulnerability Mds:               Vulnerable; SMT Host state unknown
Vulnerability Meltdown:          Vulnerable
Vulnerability Mmio stale data:   Vulnerable
Vulnerability Retbleed:          Vulnerable
Vulnerability Spec store bypass: Vulnerable
Vulnerability Spectre v1:        Vulnerable: __user pointer sanitization and usercopy barriers only; no swapgs barriers
Vulnerability Spectre v2:        Vulnerable, IBPB: disabled, STIBP: disabled, PBRSB-eIBRS: Not affected
Vulnerability Srbds:             Not affected
Vulnerability Tsx async abort:   Vulnerable
Flags:                           fpu vme de pse tsc msr pae mce cx8 apic sep mtrr pge mca cmov pat pse36 clflush mmx fxsr sse sse2 ss ht syscall nx pdpe1gb rdtscp lm constant_tsc rep_good nopl xtopology nonstop_tsc cpuid tsc_known_freq pni pclmulqdq ssse3 fma cx16 pcid sse4_1 sse4_2 x2apic movbe popcnt aes xsave avx f16c rdrand hypervisor lahf_lm abm 3dnowprefetch invpcid_single ssbd ibrs ibpb stibp fsgsbase tsc_adjust bmi1 hle avx2 smep bmi2 erms invpcid rtm rdseed adx smap xsaveopt arat md_clear arch_capabilities

Versions of relevant libraries:
[pip3] numpy==1.22.4
[pip3] torch==1.13.1+cu116
[pip3] torchaudio==0.13.1+cu116
[pip3] torchsummary==1.5.1
[pip3] torchtext==0.14.1
[pip3] torchvision==0.14.1+cu116
[conda] Could not collect
```