Repository: pytorch
Issue: Precision Inconsistent for `half` type compared with NumPy and TensorFlow
Labels: module: numerical-stability,triaged,module: numpy

### 🐛 Describe the bug

When I was using `leaky_relu` in PyTorch with the `half` type, I noticed there was a significant difference between PyTorch's output and NumPy/TensorFlow's output.

To further investigate, I noticed that even if I manually create a `half` tensor with the same value, PyTorch will output a slightly different value compared with NumPy and TensorFlow.

You may see the gist below for details.

```python
import os
os.environ['TF_CPP_MIN_LOG_LEVEL'] = '2'
os.environ['OPENBLAS_NUM_THREADS'] = '1'
os.environ['CUDA_VISIBLE_DEVICES'] = ''
import warnings
warnings.filterwarnings("ignore")
import numpy as np
import torch
import torch.nn.functional as F
import tensorflow as tf

np.random.seed(12)

def pytorch_call(input,negative_slope=0.01):
  return torch.nn.functional.leaky_relu_(input,negative_slope)

def tensorflow_call(input, negative_slope=0.01):
    return tf.nn.leaky_relu(input, alpha=negative_slope)

input_np = np.random.randn(5,7).astype(np.half)
input_torch = torch.tensor(input_np, dtype=torch.half)
input_tf = tf.constant(input_np, dtype='half')
negative_slope = -79.57261578309675

print(f'np input [1, 1]: {input_np[1, 1]}')
print(f'torch input [1, 1]: {input_torch[1, 1]}')
print(f'tf input [1, 1]: {input_tf[1, 1]}')
print()

np_leaky = np.maximum(0, input_np) + negative_slope * np.minimum(0, input_np)
torch_leaky = pytorch_call(input_torch, negative_slope)
tf_leaky = tensorflow_call(input_tf, negative_slope)

print(f'np leaky [1, 1]: {np_leaky[1, 1]}')
print(f'torch leaky [1, 1]: {torch_leaky[1, 1]}')
print(f'tf leaky [1, 1]: {tf_leaky[1, 1]}')
print()

print(f'manually create a half tensor of value "-0.807":')
print(f'np: {np.array([-0.807]).astype(np.half)}')
print(f'torch: {torch.tensor([-0.807], dtype=torch.half)}')
print(f'tf: {tf.constant([-0.807], dtype="half")}')
```

Outputs:
```
np input [1, 1]: -0.80712890625
torch input [1, 1]: -0.80712890625
tf input [1, 1]: -0.80712890625

np leaky [1, 1]: 64.1875
torch leaky [1, 1]: 64.25
tf leaky [1, 1]: 64.1875

manually create a half tensor of value "-0.807":
np: [-0.807]
torch: tensor([-0.8071], dtype=torch.float16)
tf: [-0.807]
```

### Versions

TF version: `2.15.0.post1`

```
Collecting environment information...
PyTorch version: 2.2.1
Is debug build: False
CUDA used to build PyTorch: 12.1
ROCM used to build PyTorch: N/A

OS: AlmaLinux 9.3 (Shamrock Pampas Cat) (x86_64)
GCC version: (GCC) 11.4.1 20230605 (Red Hat 11.4.1-2)
Clang version: 16.0.1 (https://github.com/llvm/llvm-project.git cd89023f797900e4492da58b7bed36f702120011)
CMake version: version 3.23.2
Libc version: glibc-2.34

Python version: 3.11.8 (main, Feb 26 2024, 21:39:34) [GCC 11.2.0] (64-bit runtime)
Python platform: Linux-5.14.0-362.18.1.el9_3.x86_64-x86_64-with-glibc2.34
Is CUDA available: True
CUDA runtime version: 12.1.105
CUDA_MODULE_LOADING set to: LAZY
GPU models and configuration: 
GPU 0: NVIDIA GeForce RTX 2080 Ti
GPU 1: NVIDIA TITAN RTX
GPU 2: NVIDIA GeForce RTX 2080 Ti

Nvidia driver version: 550.54.14
cuDNN version: Could not collect
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
Address sizes:                      46 bits physical, 48 bits virtual
Byte Order:                         Little Endian
CPU(s):                             36
On-line CPU(s) list:                0-35
Vendor ID:                          GenuineIntel
Model name:                         Intel(R) Core(TM) i9-10980XE CPU @ 3.00GHz
CPU family:                         6
Model:                              85
Thread(s) per core:                 2
Core(s) per socket:                 18
Socket(s):                          1
Stepping:                           7
CPU max MHz:                        4800.0000
CPU min MHz:                        1200.0000
BogoMIPS:                           6000.00
Flags:                              fpu vme de pse tsc msr pae mce cx8 apic sep mtrr pge mca cmov pat pse36 clflush dts acpi mmx fxsr sse sse2 ss ht tm pbe syscall nx pdpe1gb rdtscp lm constant_tsc art arch_perfmon pebs bts rep_good nopl xtopology nonstop_tsc cpuid aperfmperf pni pclmulqdq dtes64 monitor ds_cpl vmx est tm2 ssse3 sdbg fma cx16 xtpr pdcm pcid dca sse4_1 sse4_2 x2apic movbe popcnt tsc_deadline_timer aes xsave avx f16c rdrand lahf_lm abm 3dnowprefetch cpuid_fault epb cat_l3 cdp_l3 invpcid_single ssbd mba ibrs ibpb stibp ibrs_enhanced tpr_shadow flexpriority ept vpid ept_ad fsgsbase tsc_adjust bmi1 avx2 smep bmi2 erms invpcid cqm mpx rdt_a avx512f avx512dq rdseed adx smap clflushopt clwb intel_pt avx512cd avx512bw avx512vl xsaveopt xsavec xgetbv1 xsaves cqm_llc cqm_occup_llc cqm_mbm_total cqm_mbm_local dtherm ida arat pln pts hwp hwp_act_window hwp_epp hwp_pkg_req vnmi avx512_vnni md_clear flush_l1d arch_capabilities
Virtualization:                     VT-x
L1d cache:                          576 KiB (18 instances)
L1i cache:                          576 KiB (18 instances)
L2 cache:                           18 MiB (18 instances)
L3 cache:                           24.8 MiB (1 instance)
NUMA node(s):                       1
NUMA node0 CPU(s):                  0-35
Vulnerability Gather data sampling: Mitigation; Microcode
Vulnerability Itlb multihit:        KVM: Mitigation: VMX disabled
Vulnerability L1tf:                 Not affected
Vulnerability Mds:                  Not affected
Vulnerability Meltdown:             Not affected
Vulnerability Mmio stale data:      Mitigation; Clear CPU buffers; SMT vulnerable
Vulnerability Retbleed:             Mitigation; Enhanced IBRS
Vulnerability Spec rstack overflow: Not affected
Vulnerability Spec store bypass:    Mitigation; Speculative Store Bypass disabled via prctl
Vulnerability Spectre v1:           Mitigation; usercopy/swapgs barriers and __user pointer sanitization
Vulnerability Spectre v2:           Mitigation; Enhanced / Automatic IBRS, IBPB conditional, RSB filling, PBRSB-eIBRS SW sequence
Vulnerability Srbds:                Not affected
Vulnerability Tsx async abort:      Mitigation; TSX disabled

Versions of relevant libraries:
[pip3] numpy==1.26.4
[pip3] torch==2.2.1
[pip3] torchaudio==2.2.1
[pip3] torchvision==0.17.1
[pip3] triton==2.2.0
[conda] blas                      1.0                         mkl  
[conda] ffmpeg                    4.3                  hf484d3e_0    pytorch
[conda] libjpeg-turbo             2.0.0                h9bf148f_0    pytorch
[conda] mkl                       2023.1.0         h213fc3f_46344  
[conda] mkl-service               2.4.0           py311h5eee18b_1  
[conda] mkl_fft                   1.3.8           py311h5eee18b_0  
[conda] mkl_random                1.2.4           py311hdb19cb5_0  
[conda] numpy                     1.26.4          py311h08b1b3b_0  
[conda] numpy-base                1.26.4          py311hf175353_0  
[conda] pytorch                   2.2.1           py3.11_cuda12.1_cudnn8.9.2_0    pytorch
[conda] pytorch-cuda              12.1                 ha16c6d3_5    pytorch
[conda] pytorch-mutex             1.0                        cuda    pytorch
[conda] torchaudio                2.2.1               py311_cu121    pytorch
[conda] torchtriton               2.2.0                     py311    pytorch
[conda] torchvision               0.17.1              py311_cu121    pytorch
```

cc @mruberry @rgommers