# Statistics

Concrete analyzes all compiled circuits and calculates some statistics. These statistics can be used to find bottlenecks and compare circuits. Statistics are calculated in terms of basic operations. There are 6 basic operations in Concrete:

- **clear addition:** x + y where x is encrypted and y is clear
- **encrypted addition:** x + y where both x and y are encrypted
- **clear multiplication:** x * y where x is encrypted and y is clear
- **encrypted negation:** -x where x is encrypted
- **key switch:** building block for table lookups
- **packing key switch:** building block for table lookups
- **programmable bootstrapping:** building block for table lookups

You can print all statistics using `show_statistics` configuration option:

```python
from concrete import fhe

@fhe.compiler({"x": "encrypted"})
def f(x):
    return (x**2) + (2*x) + 4

inputset = range(2**2)
circuit = f.compile(inputset, show_statistics=True)
```

This code will print:

```
Statistics
--------------------------------------------------------------------------------
size_of_secret_keys: 22648
size_of_bootstrap_keys: 51274176
size_of_keyswitch_keys: 64092720
size_of_inputs: 16392
size_of_outputs: 16392
p_error: 9.627450598589458e-06
global_p_error: 9.627450598589458e-06
complexity: 99198195.0
programmable_bootstrap_count: 1
programmable_bootstrap_count_per_parameter: {
    BootstrapKeyParam(polynomial_size=2048, glwe_dimension=1, input_lwe_dimension=781, level=1, base_log=23, variance=9.940977002694397e-32): 1
}
key_switch_count: 1
key_switch_count_per_parameter: {
    KeyswitchKeyParam(level=5, base_log=3, variance=1.939836732335308e-11): 1
}
packing_key_switch_count: 0
clear_addition_count: 1
clear_addition_count_per_parameter: {
    LweSecretKeyParam(dimension=2048): 1
}
encrypted_addition_count: 1
encrypted_addition_count_per_parameter: {
    LweSecretKeyParam(dimension=2048): 1
}
clear_multiplication_count: 1
clear_multiplication_count_per_parameter: {
    LweSecretKeyParam(dimension=2048): 1
}
encrypted_negation_count: 0
--------------------------------------------------------------------------------
```

{% hint style="info" %}
Each of these properties can be directly accessed on the circuit (e.g., `circuit.programmable_bootstrap_count`).
{% endhint %}

## Tags

Circuit analysis also considers [tags](../tutorial/tagging.md)!

Imagine you have a neural network with 10 layers, each of them tagged. You can easily see the amount of additions and multiplications required for matrix multiplications per layer:

```
Statistics
--------------------------------------------------------------------------------
clear_multiplication_count_per_tag: {
    /model/model: 53342
    /model/model.0/Gemm: 14720
    /model/model.0/Gemm.matmul: 14720
    /model/model.2/Gemm: 11730
    /model/model.2/Gemm.matmul: 11730
    /model/model.4/Gemm: 9078
    /model/model.4/Gemm.matmul: 9078
    /model/model.6/Gemm: 6764
    /model/model.6/Gemm.matmul: 6764
    /model/model.8/Gemm: 4788
    /model/model.8/Gemm.matmul: 4788
    /model/model.10/Gemm: 3150
    /model/model.10/Gemm.matmul: 3150
    /model/model.12/Gemm: 1850
    /model/model.12/Gemm.matmul: 1850
    /model/model.14/Gemm: 888
    /model/model.14/Gemm.matmul: 888
    /model/model.16/Gemm: 264
    /model/model.16/Gemm.matmul: 264
    /model/model.18/Gemm: 110
    /model/model.18/Gemm.matmul: 110
}
encrypted_addition_count_per_tag: {
    /model/model: 53342
    /model/model.0/Gemm: 14720
    /model/model.0/Gemm.matmul: 14720
    /model/model.2/Gemm: 11730
    /model/model.2/Gemm.matmul: 11730
    /model/model.4/Gemm: 9078
    /model/model.4/Gemm.matmul: 9078
    /model/model.6/Gemm: 6764
    /model/model.6/Gemm.matmul: 6764
    /model/model.8/Gemm: 4788
    /model/model.8/Gemm.matmul: 4788
    /model/model.10/Gemm: 3150
    /model/model.10/Gemm.matmul: 3150
    /model/model.12/Gemm: 1850
    /model/model.12/Gemm.matmul: 1850
    /model/model.14/Gemm: 888
    /model/model.14/Gemm.matmul: 888
    /model/model.16/Gemm: 264
    /model/model.16/Gemm.matmul: 264
    /model/model.18/Gemm: 110
    /model/model.18/Gemm.matmul: 110
}
--------------------------------------------------------------------------------
```