Discrepancies in Output due to Altered Multiplication Sequence in TensorFlow Operations
@sachinprasadhs I was able to replicate this issue on colab, please find the gist [here](https://colab.research.google.com/gist/sushreebarsa/c25fcb38679e26e98601fd94a22cffce/62475.ipynb). Thank you! Hi, 

This issue is not specific to XLA, mismatch of result happens even without XLA, this is due to the multiple tf.multiply operations happening in the code of below two equations.

```
out1 = tf.multiply(tf.multiply(p0, inputs), tf.multiply(p0, p0))

out2 = tf.multiply(p0, tf.multiply(inputs, tf.multiply(p0, p0)))
```

Here in both cases, the order of multiplication is performed first on the inner most `tf.multiply`, if two `tf.multiply` are in the same level, leftmost operation is performed first and then the value will be ready for the next operation.
During this process, since both takes different code path and different level of multiply operations, there will be precision loss, which is expected and the difference which is observed in not too large.

Below is the simplified code which produces the similar behavior.

```python
import tensorflow as tf
import numpy as np
params = [
tf.random.uniform(shape=[10, 10], dtype=tf.float16),
]
inputs = [tf.constant(-0.7305, dtype=tf.float16)]

p0 = tf.Variable(params[0])

out1 = tf.multiply(tf.multiply(p0, inputs), tf.multiply(p0, p0))

out2 = tf.multiply(p0, tf.multiply(inputs, tf.multiply(p0, p0)))

for i in range(min(len(out1), len(out2))):
    np.testing.assert_allclose(out1[i].numpy(), out2[i].numpy(), rtol=0.0001, atol=0.0001, err_msg=f'at checking {i}th')


AssertionError: 
Not equal to tolerance rtol=0.0001, atol=0.0001
at checking 0th
Mismatched elements: 3 / 10 (30%)
Max absolute difference: 0.0004883
Max relative difference: 0.000791
 x: array([-0.1956 , -0.6167 , -0.00817, -0.4104 , -0.0771 , -0.07294,
       -0.4749 , -0.3445 , -0.501  , -0.1842 ], dtype=float16)
 y: array([-0.1956 , -0.617  , -0.00817, -0.4106 , -0.07715, -0.07294,
       -0.475  , -0.3445 , -0.501  , -0.1842 ], dtype=float16)
```

Gist: https://gist.github.com/sachinprasadhs/f2cbc7d40e91feb6f50aed1e6ca4e29f @sachinprasadhs Hi, Thank you for your response.
I understand that this behavior is caused by a floating-point accuracy problem, which is expected. I have been wondering about **the acceptable tolerance level** for this floating-point accuracy, given that we have uncovered numerous inconsistent behaviors in TensorFlow. We have only selected and reported cases with high differences in values. I would like to know what the acceptable tolerance in TensorFlow is, so that we can internally filter out some false-positive cases. @cantonios , Could you please comment on the above response. Thank you! > @sachinprasadhs Hi, Thank you for your response. I understand that this behavior is caused by a floating-point accuracy problem, which is expected. I have been wondering about **the acceptable tolerance level** for this floating-point accuracy, given that we have uncovered numerous inconsistent behaviors in TensorFlow. We have only selected and reported cases with high differences in values. I would like to know what the acceptable tolerance in TensorFlow is, so that we can internally filter out some false-positive cases.

You're using float16 here, which has very limited precision, so the issue is exasperated.

The "acceptable tolerance level" entirely depends on the sequence of operations you are performing, the dtype, and the magnitudes of your values.  You can estimate it yourself: each floating point value is only accurate to within one unit-of-least-precision (ULP), so you can model each value as `x + delta * x` (error is relative to x here due to the way floating-point values are stored - though this is only an approximation, since the error is really relative to `floor(log2(x))`, but that's harder to reason about).  The more floating-point operations you do, the more these floating-point errors multiply and accumulate.  You can then estimate the expected error for your given set of operations, and you will likely verify that TF (and XLA, and Eigen, pytorch, numpy, etc..) will fall within that.

The only thing TF (and XLA) _does_ sometimes do, is for some operations like convolutions and matrix multiplications, we recognize that types like fp16 and bf16 are prone to this kind of bad error accumulation, particularly when multiplying large matrices, so internally we convert them to higher precision and accumulate in higher precision to mitigate the effect.

The reported error you are seeing seem reasonable for fp16. Are you satisfied with the resolution of your issue?
<a href="https://docs.google.com/forms/d/e/1FAIpQLSfaP12TRhd9xSxjXZjcZFNXPGk4kc1-qMdv3gc6bEP90vY1ew/viewform?entry.85265664=Yes&entry.2137816233=https://github.com/tensorflow/tensorflow/issues/62475">Yes</a>
<a href="https://docs.google.com/forms/d/e/1FAIpQLSfaP12TRhd9xSxjXZjcZFNXPGk4kc1-qMdv3gc6bEP90vY1ew/viewform?entry.85265664=No&entry.2137816233=https://github.com/tensorflow/tensorflow/issues/62475">No</a>

### Issue type

Bug

### Have you reproduced the bug with TensorFlow Nightly?

Yes

### Source

source

### TensorFlow version

2.16.0-dev20231122

### Custom code

Yes

### OS platform and distribution

Ubuntu 22.04.3

### Mobile device

_No response_

### Python version

3.10.12

### Bazel version

_No response_

### GCC/compiler version

_No response_

### CUDA/cuDNN version

_No response_

### GPU model and memory

_No response_

### Current behavior?

I've come across a weird issue in TensorFlow. It seems like rearranging the multiplication operations is somehow giving different results. Check out these two models:
```python
tf.multiply(tf.multiply(self.p0, inp), tf.multiply(self.p0, self.p0))
```

```python
tf.multiply(self.p0, tf.multiply(inp, tf.multiply(self.p0, self.p0)))
```
Mathematically, these two tensor operations should be identical since all I did was switch the order of multiplication. But when I run them in TensorFlow with certain inputs, I'm getting different outcomes, which is really strange

### Standalone code to reproduce the issue

```shell
import tensorflow as tf
import numpy as np
params = [
tf.random.uniform(shape=[10, 10], dtype=tf.float16),
]
class Model1(tf.keras.Model):
    def __init__(self):
        super().__init__()
        # Tensor objects (with comments for shapes)
        self.p0 = tf.Variable(params[0]) # [40, 46] float16

        # Layers or other Keras model objects

    @tf.function(jit_compile=True)
    def __call__(self, inp):
        # Forward pass logic using TensorFlow operations
        # inp: [] : float16
        out = tf.multiply(tf.multiply(self.p0, inp), tf.multiply(self.p0, self.p0))
        # p0 * inp * p0 * p0
        return out


class Model2(tf.keras.Model):
    def __init__(self):
        super().__init__()
        # Tensor objects (with comments for shapes)
        self.p0 = tf.Variable(params[0]) # [40, 46] float16

        # Layers or other Keras model objects

    @tf.function(jit_compile=True)
    def __call__(self, inp):
        # Forward pass logic using TensorFlow operations
        # inp: [] : float16
        out = tf.multiply(self.p0, tf.multiply(inp, tf.multiply(self.p0, self.p0)))
        # p0 * inp * (p0 * p0)
        return out

model1 = Model1()
model2 = Model2()
inputs = [tf.constant(-0.7305, dtype=tf.float16)]
print(f'=========RUNNING WITH {inputs}===========')
print(inputs[0])
with tf.device('cpu'):
    tf.config.run_functions_eagerly(True)
    out1 = model1(*inputs)
    out2 = model2(*inputs)
    print(f'=========eager_output(version:{tf.__version__})================')
    try :
        for i in range(min(len(out1),len(out2))):
            np.testing.assert_allclose(out1[i].numpy(), out2[i].numpy(), rtol=0.0001, atol=0.0001, err_msg=f'at checking {i}th')
        print("XLA_eager does not trigger assertion")
    except AssertionError as e:
        print("XLA_eager triggers assertion")
        print(e)
    tf.config.run_functions_eagerly(False)
    out1 = model1(*inputs)
    out2 = model2(*inputs)
    print(f'=========compiled_output(version:{tf.__version__})================')
    try :
        for i in range(min(len(out1),len(out2))):
            np.testing.assert_allclose(out1[i].numpy(), out2[i].numpy(), rtol=0.0001, atol=0.0001, err_msg=f'at checking {i}th')
        print("XLA_complie does not trigger assertion")
    except AssertionError as e:
        print("XLA_complie triggers assertion")
        print(e)
```


### Relevant log output

```shell
=========RUNNING WITH [<tf.Tensor: shape=(), dtype=float16, numpy=-0.7305>]===========
tf.Tensor(-0.7305, shape=(), dtype=float16)
=========eager_output(version:2.16.0-dev20231122)================
XLA_eager triggers assertion

Not equal to tolerance rtol=0.0001, atol=0.0001
at checking 1th
Mismatched elements: 2 / 10 (20%)
Max absolute difference: 0.0001221
Max relative difference: 0.0008817
 x: array([-2.794e-02, -1.537e-01, -2.107e-02, -1.764e-02, -7.645e-03,
       -2.986e-01, -1.848e-06, -5.126e-04, -1.383e-01, -5.185e-02],
      dtype=float16)
 y: array([-2.794e-02, -1.536e-01, -2.107e-02, -1.764e-02, -7.645e-03,
       -2.986e-01, -1.848e-06, -5.126e-04, -1.384e-01, -5.185e-02],
      dtype=float16)
2023-11-24 13:49:22.632282: I external/local_xla/xla/service/service.cc:144] XLA service 0x9170eb0 initialized for platform CUDA (this does not guarantee that XLA will be used). Devices:
2023-11-24 13:49:22.632327: I external/local_xla/xla/service/service.cc:152]   StreamExecutor device (0): NVIDIA GeForce RTX 2070, Compute Capability 7.5
2023-11-24 13:49:22.632338: I external/local_xla/xla/service/service.cc:152]   StreamExecutor device (1): NVIDIA GeForce RTX 2070, Compute Capability 7.5
2023-11-24 13:49:22.632347: I external/local_xla/xla/service/service.cc:152]   StreamExecutor device (2): NVIDIA GeForce RTX 2070, Compute Capability 7.5
2023-11-24 13:49:22.632357: I external/local_xla/xla/service/service.cc:152]   StreamExecutor device (3): NVIDIA GeForce RTX 2070, Compute Capability 7.5
2023-11-24 13:49:22.700845: I external/local_xla/xla/stream_executor/cuda/cuda_dnn.cc:456] Loaded cuDNN version 8904
WARNING: All log messages before absl::InitializeLog() is called are written to STDERR
I0000 00:00:1700804962.827337 2332797 device_compiler.h:187] Compiled cluster using XLA!  This line is logged at most once for the lifetime of the process.
=========compiled_output(version:2.16.0-dev20231122)================
XLA_complie triggers assertion

Not equal to tolerance rtol=0.0001, atol=0.0001
at checking 1th
Mismatched elements: 2 / 10 (20%)
Max absolute difference: 0.0001221
Max relative difference: 0.0008817
 x: array([-2.794e-02, -1.537e-01, -2.107e-02, -1.764e-02, -7.645e-03,
       -2.986e-01, -1.848e-06, -5.126e-04, -1.383e-01, -5.185e-02],
      dtype=float16)
 y: array([-2.794e-02, -1.536e-01, -2.107e-02, -1.764e-02, -7.645e-03,
       -2.986e-01, -1.848e-06, -5.126e-04, -1.384e-01, -5.185e-02],
      dtype=float16)
```
