From b55cf2958a305184e073e76a8d32d10225188475 Mon Sep 17 00:00:00 2001
From: Umut <umutsahin@protonmail.com>
Date: Wed, 8 Dec 2021 11:05:57 +0300
Subject: [PATCH] docs: write some missing sections and update some outdated
 ones

---
 .../artifacts/auto/1.initial.graph.png        | Bin 7755 -> 7054 bytes
 .../artifacts/auto/2.final.graph.png          | Bin 0 -> 7054 bytes
 .../artifacts/manual/1.initial.graph.png      | Bin 23985 -> 33573 bytes
 .../manual/2.after-float-fuse-0.graph.png     | Bin 0 -> 17280 bytes
 .../artifacts/manual/2.final.graph.png        | Bin 23985 -> 0 bytes
 .../artifacts/manual/3.final.graph.png        | Bin 0 -> 17280 bytes
 docs/dev/explanation/COMPILATION.md           |  25 +-
 docs/user/benchmarks.md                       |  21 +-
 docs/user/howto/COMPILING_AND_EXECUTING.md    |   1 -
 .../user/howto/DEBUG_SUPPORT_SUBMIT_ISSUES.md |  41 +--
 docs/user/howto/PRINTING_AND_DRAWING.md       |   1 +
 docs/user/howto/REDUCE_NEEDED_PRECISION.md    |  52 +++-
 docs/user/index.rst                           |   1 -
 docs/user/tutorial/ARITHMETIC_OPERATIONS.md   | 245 ------------------
 docs/user/tutorial/COMPILATION_ARTIFACTS.md   | 183 ++++++++-----
 docs/user/tutorial/INDEXING.md                | 104 +++++++-
 docs/user/tutorial/TABLE_LOOKUP.md            |  68 ++++-
 17 files changed, 365 insertions(+), 377 deletions(-)
 create mode 100644 docs/_static/tutorials/artifacts/auto/2.final.graph.png
 create mode 100644 docs/_static/tutorials/artifacts/manual/2.after-float-fuse-0.graph.png
 delete mode 100644 docs/_static/tutorials/artifacts/manual/2.final.graph.png
 create mode 100644 docs/_static/tutorials/artifacts/manual/3.final.graph.png
 delete mode 100644 docs/user/tutorial/ARITHMETIC_OPERATIONS.md

diff --git a/docs/_static/tutorials/artifacts/auto/1.initial.graph.png b/docs/_static/tutorials/artifacts/auto/1.initial.graph.png
index e88663dffc60707a81a63d987e12b32008f08ca2..50495286b6ec941da79074e70988da1ac030105f 100644
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literal 0
HcmV?d00001

diff --git a/docs/dev/explanation/COMPILATION.md b/docs/dev/explanation/COMPILATION.md
index dfa9ca971..be0eee789 100644
--- a/docs/dev/explanation/COMPILATION.md
+++ b/docs/dev/explanation/COMPILATION.md
@@ -1,9 +1,3 @@
-
-```{warning}
-FIXME(umut): update with the new API
-```
-
-
 # Compilation Pipeline In Depth
 
 ## What is **concretefhe**?
@@ -23,15 +17,12 @@ import concrete.numpy as hnp
 def f(x, y):
     return (2 * x) + y
 
-# Define the inputs of homomorphized function
-x = hnp.EncryptedScalar(hnp.UnsignedInteger(2))
-y = hnp.EncryptedScalar(hnp.UnsignedInteger(1))
+# Create a NumPy FHE Compiler
+compiler = hnp.NPFHECompiler(f, {"x": "encrypted", "y": "encrypted"})
 
-# Compile the function to its homomorphic equivalent
-circuit = hnp.compile_numpy_function(
-    f, {"x": x, "y": y},
-    [(0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1), (3, 0), (3, 1)],
-)
+# Compile an FHE Circuit using an inputset
+compiler.eval_on_inputset([(0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1), (3, 0), (3, 1)])
+circuit = compiler.get_compiled_fhe_circuit()
 
 # Make homomorphic inference
 circuit.run(1, 0)
@@ -223,7 +214,7 @@ def f(x):
 ### Parameters
 
 ```
-x = EncryptedScalar(UnsignedInteger(2))
+x = "encrypted"
 ```
 
 #### Corresponding operation graph
@@ -272,8 +263,8 @@ def f(x, y):
 ### Parameters
 
 ```
-x = EncryptedScalar(UnsignedInteger(3))
-y = EncryptedScalar(UnsignedInteger(1))
+x = "encrypted"
+y = "encrypted"
 ```
 
 #### Corresponding operation graph
diff --git a/docs/user/benchmarks.md b/docs/user/benchmarks.md
index 8b6db9bd1..387e0335f 100644
--- a/docs/user/benchmarks.md
+++ b/docs/user/benchmarks.md
@@ -1,22 +1,21 @@
 # Benchmarks
 
-```{warning}
-FIXME(Umut): update a bit
-```
-
-In order to track our progress over time, we have set a [public benchmark](https://progress.zama.ai) containing:
-- a list of functions that we want to compile
+To track our progress over time, we have created a [progress tracker](https://progress.zama.ai) that have
+- list of targets that we want to compile
 - status on the compilation of these functions
-- compilation time
-- evaluation time on different hardware's
+- compilation and evaluation times on different hardware
 - accuracy of the functions for which it makes sense
 - loss of the functions for which it makes sense
 
-Remark that we are not limited to these, and we'll certainly add more information later, as key generation time, encryption and decryption time, and more evaluation time once the explicit inference API is available.
+Note that we are not limited to these, and we'll certainly add more information (e.g., key generation time, encryption time, inference time, decryption time, etc.) once the explicit inference API is available.
 
-The benchmark can be used by competitive frameworks or technologies, in order to compare fairly with the **Concrete Framework**. Notably, one can see:
+```{warning}
+FIXME(all): update the sentence above when the encrypt, decrypt, run_inference, keygen API's are available
+```
+
+Our public benchmarks can be used by competing frameworks or technologies for comparison with **Concrete Framework**. Notably, one can see:
 - if the same functions can be compiled
 - what are discrepancies in the exactness of the evaluations
 - how do evaluation times compare
 
-If one wants to see more functions in the benchmark or if there is more information you would like the benchmark to track, don't hesitate to drop an email to <hello@zama.ai>.
\ No newline at end of file
+If you want to see more functions in the progress tracker or if there is another metric you would like to track, don't hesitate to drop an email to <hello@zama.ai>.
diff --git a/docs/user/howto/COMPILING_AND_EXECUTING.md b/docs/user/howto/COMPILING_AND_EXECUTING.md
index f0482fe30..afdcec90b 100644
--- a/docs/user/howto/COMPILING_AND_EXECUTING.md
+++ b/docs/user/howto/COMPILING_AND_EXECUTING.md
@@ -85,7 +85,6 @@ FIXME(benoit): explain the API to encrypt, run_inference, decrypt, keygen etc wh
 
 ## Further reading
 
-- [Arithmetic Operations Tutorial](../tutorial/ARITHMETIC_OPERATIONS.md)
 - [Working With Floating Points Tutorial](../tutorial/WORKING_WITH_FLOATING_POINTS.md)
 - [Table Lookup Tutorial](../tutorial/TABLE_LOOKUP.md)
 - [Compiling a torch model](../tutorial/COMPILING_TORCH_MODEL.md)
diff --git a/docs/user/howto/DEBUG_SUPPORT_SUBMIT_ISSUES.md b/docs/user/howto/DEBUG_SUPPORT_SUBMIT_ISSUES.md
index b23cf6700..c462e8b25 100644
--- a/docs/user/howto/DEBUG_SUPPORT_SUBMIT_ISSUES.md
+++ b/docs/user/howto/DEBUG_SUPPORT_SUBMIT_ISSUES.md
@@ -43,31 +43,34 @@ Hopefully, it is just a misunderstanding or a small mistake on your side, that o
 
 When things are more complicated, or if you want to have a look by yourself, you may want to have a look to the compilation reports, which are called artifacts. This is as simple as described in [here](../tutorial/COMPILATION_ARTIFACTS.md)
 
-This function will create a directory, containing notably:
-```{warning}
-FIXME(Umut): check it is still accurate
-```
-- bounds.txt: a file describing the expected ranges of data in the different steps of the computation
-- cryptographic_parameters.txt: a file describing the different keys
-- ir_nodes.txt: a file describing the different nodes in the intermediate representation (IR)
-- optimizations_applied.txt: a file describing the different optimizations which were applied
-- target_nodes.txt: a file describing the different nodes in the VM graph
+The artifact system will create a directory, containing:
+- **environment.txt:** information about your system
+- **requirements.txt:** information about your python dependencies
+- **function.txt:** source code of the function you are compiling
+- **parameters.txt:** parameters you specified for compilation
+- **1.initial.graph.txt:** textual representation of the initial computation graph right after tracing
+- **1.initial.graph.png:** visual representation of the initial computation graph right after tracing
+- ...
+- **X.description.graph.txt:** textual representation of the Xth computation graph after topological transforms
+- **X.description.graph.png:** visual representation of the Xth computation graph after topological transforms
+- ...
+- **N.final.graph.txt:** textual representation of the final computation graph right before MLIR conversion
+- **N.final.graph.png:** visual representation of the final computation graph right before MLIR conversion
+- **bounds.txt:** ranges of data in the different steps of the computation for the final graph that is being compiled
+- **mlir.txt**: resulting MLIR code that is sent to the compiler (if compilation succeeded)
+- **traceback.txt**: information about the error you encountered (if compilation failed)
+
 
 Attaching the artifact with your issue or Slack message may help people to have a look at the core of the problem.
 The more precise your bug, the more likely we can reproduce and fix
 
-```{warning}
-FIXME(Umut): is it still needed or do we already have some of those information in artifacts?
-```
-
-In order to simplify our work and let us reproduce your bug easily, any information is useful. Notably, in addition to the python script, some information like:
-- the OS version
-- the python version
-- the python packages you use
-- the reproducibility rate you see on your side
+To simplify our work and let us reproduce your bug easily, we need all the information we can get. So, in addition to your python script, the following information would be very useful.
+- compilation artifacts
+- reproducibility rate you see on your side
 - any insight you might have on the bug
 - any workaround you have been able to find
-may be useful to us. Don't remember, **Concrete** is a project where we are open to contribution, more information at Contributing (TODO: add a link).
+
+Remember, **Concrete Framework** is a project where we are open to contributions, more information at [Contributing](../../dev/howto/CONTRIBUTING.md).
 
 ## Submitting an issue
 
diff --git a/docs/user/howto/PRINTING_AND_DRAWING.md b/docs/user/howto/PRINTING_AND_DRAWING.md
index 8bb477350..4c06ac6f6 100644
--- a/docs/user/howto/PRINTING_AND_DRAWING.md
+++ b/docs/user/howto/PRINTING_AND_DRAWING.md
@@ -1,5 +1,6 @@
 ```{warning}
 FIXME(all): should we update to the new API or have it for both?
+FIXME(all): we should add an example drawing and printing with fusing (so subgraphs)
 ```
 
 # Printing and Drawing
diff --git a/docs/user/howto/REDUCE_NEEDED_PRECISION.md b/docs/user/howto/REDUCE_NEEDED_PRECISION.md
index eb4f413e6..a1d7b1c04 100644
--- a/docs/user/howto/REDUCE_NEEDED_PRECISION.md
+++ b/docs/user/howto/REDUCE_NEEDED_PRECISION.md
@@ -1,8 +1,54 @@
-```{warning}
-FIXME(Umut): put somewhere an example of an error dump if the user asks for too much precision and explain
+# Having a Function Which Requires Less Precision
+
+With our current technology, we cannot represent integers with more than 7 bits.
+We are actively working on supporting larger integers, so it should get better in the future.
+
+## What happens when you have larger values?
+
+You get a compilation error. Here is an example:
+
+<!--python-test:skip-->
+```python
+import concrete.numpy as hnp
+
+def f(x):
+    return 42 * x
+
+compiler = hnp.NPFHECompiler(f, {"x": "encrypted"})
+compiler.eval_on_inputset(range(2 ** 3))
+compiler.get_compiled_fhe_circuit()
 ```
 
-# Having a Function Which Requires Less Precision
+results in
+
+```
+Traceback (most recent call last):
+  File "/home/default/Documents/Projects/Zama/hdk/dist/demo.py", line 9, in <module>
+    circuit = compiler.get_compiled_fhe_circuit()
+  File "/home/default/Documents/Projects/Zama/hdk/concrete/numpy/np_fhe_compiler.py", line 274, in get_compiled_fhe_circuit
+    return compile_op_graph_to_fhe_circuit(
+  File "/home/default/Documents/Projects/Zama/hdk/concrete/numpy/compile.py", line 676, in compile_op_graph_to_fhe_circuit
+    result = run_compilation_function_with_error_management(
+  File "/home/default/Documents/Projects/Zama/hdk/concrete/numpy/compile.py", line 141, in run_compilation_function_with_error_management
+    return compilation_function()
+  File "/home/default/Documents/Projects/Zama/hdk/concrete/numpy/compile.py", line 674, in compilation_function
+    return _compile_op_graph_to_fhe_circuit_internal(op_graph, show_mlir, compilation_artifacts)
+  File "/home/default/Documents/Projects/Zama/hdk/concrete/numpy/compile.py", line 626, in _compile_op_graph_to_fhe_circuit_internal
+    prepare_op_graph_for_mlir(op_graph)
+  File "/home/default/Documents/Projects/Zama/hdk/concrete/numpy/compile.py", line 603, in prepare_op_graph_for_mlir
+    update_bit_width_for_mlir(op_graph)
+  File "/home/default/Documents/Projects/Zama/hdk/concrete/common/mlir/utils.py", line 204, in update_bit_width_for_mlir
+    raise RuntimeError(
+RuntimeError: max_bit_width of some nodes is too high for the current version of the compiler (maximum must be 7) which is not compatible with:
+
+%0 = x                  # EncryptedScalar<uint3>
+%1 = 42                 # ClearScalar<uint6>
+%2 = mul(%0, %1)        # EncryptedScalar<uint9>
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ 9 bits is not supported for the time being
+return %2
+```
+
+when you try to run.
 
 ## Why can some computation work with less precision?
 
diff --git a/docs/user/index.rst b/docs/user/index.rst
index d47d4646f..ac8009b1a 100644
--- a/docs/user/index.rst
+++ b/docs/user/index.rst
@@ -15,7 +15,6 @@ Getting Started
    :caption: Tutorial
 
    tutorial/COMPILING_TORCH_MODEL.md
-   tutorial/ARITHMETIC_OPERATIONS.md
    tutorial/TABLE_LOOKUP.md
    tutorial/WORKING_WITH_FLOATING_POINTS.md
    tutorial/NUMPY_SUPPORT.md
diff --git a/docs/user/tutorial/ARITHMETIC_OPERATIONS.md b/docs/user/tutorial/ARITHMETIC_OPERATIONS.md
deleted file mode 100644
index 4264b224d..000000000
--- a/docs/user/tutorial/ARITHMETIC_OPERATIONS.md
+++ /dev/null
@@ -1,245 +0,0 @@
-```{warning}
-FIXME(Umut): update a bit, with the new API
-FIXME(Umut/Arthur): update a bit to explain things with the tensors and new operations. At the same time, I think we can exhaustively give examples of every supported functions, since we start to have a lot, so maybe, we would just explain a bit?
-FIXME(all): actually, I am not even sure we should keep this .md, it can't be exhaustive enough, and looks pretty trivial. What do you think
-
-```
-
-# Arithmetic Operations
-
-In this tutorial, we are going to go over all arithmetic operations available in **Concrete**. Please read [Compiling and Executing](../howto/COMPILING_AND_EXECUTING.md) before reading further to see how you can compile the functions below.
-
-## Addition
-
-### Static ClearScalar and EncryptedScalar
-
-<!--python-test:skip-->
-```python
-def f(x):
-    return x + 42
-```
-
-or
-
-<!--python-test:skip-->
-```python
-def f(x):
-    return 42 + x
-```
-
-where
-
-- `x = EncryptedScalar(UnsignedInteger(bits))`
-
-results in
-
-<!--python-test:skip-->
-```python
-circuit.run(3) == 45
-circuit.run(0) == 42
-```
-
-### Dynamic ClearScalar and EncryptedScalar
-
-<!--python-test:skip-->
-```python
-def f(x, y):
-    return x + y
-```
-
-or
-
-<!--python-test:skip-->
-```python
-def f(x, y):
-    return y + x
-```
-
-results in
-
-<!--python-test:skip-->
-```python
-circuit.run(6, 4) == 10
-circuit.run(1, 1) == 2
-```
-
-where
-
-- `x = EncryptedScalar(UnsignedInteger(bits))`
-- `y = ClearScalar(UnsignedInteger(bits))`
-
-### EncryptedScalar and EncryptedScalar
-
-<!--python-test:skip-->
-```python
-def f(x, y):
-    return x + y
-```
-
-where
-
-- `x = EncryptedScalar(UnsignedInteger(bits))`
-- `y = EncryptedScalar(UnsignedInteger(bits))`
-
-results in
-
-<!--python-test:skip-->
-```python
-circuit.run(7, 7) == 14
-circuit.run(3, 4) == 7
-```
-
-## Subtraction
-
-### Static ClearScalar and EncryptedScalar 
-
-<!--python-test:skip-->
-```python
-def f(x):
-    return 3 - x
-```
-
-where
-
-- `x = EncryptedScalar(UnsignedInteger(bits))`
-
-results in
-
-<!--python-test:skip-->
-```python
-circuit.run(2) == 1
-circuit.run(3) == 0
-```
-
-### Dynamic ClearScalar and EncryptedScalar
-
-<!--python-test:skip-->
-```python
-def f(x, y):
-    return y - x
-```
-
-where
-
-- `x = EncryptedScalar(UnsignedInteger(bits))`
-- `y = ClearScalar(UnsignedInteger(bits))`
-
-results in
-
-<!--python-test:skip-->
-```python
-circuit.run(2, 4) == 2
-circuit.run(1, 7) == 6
-```
-
-## Multiplication
-
-### Static ClearScalar and EncryptedScalar
-
-<!--python-test:skip-->
-```python
-def f(x):
-    return x * 2
-```
-
-or
-
-<!--python-test:skip-->
-```python
-def f(x):
-    return 2 * x
-```
-
-where
-
-- `x = EncryptedScalar(UnsignedInteger(bits))`
-
-results in
-
-<!--python-test:skip-->
-```python
-circuit.run(2) == 4
-circuit.run(5) == 10
-```
-
-### Dynamic ClearScalar and EncryptedScalar
-
-<!--python-test:skip-->
-```python
-def f(x, y):
-    return x * y
-```
-
-or
-
-<!--python-test:skip-->
-```python
-def f(x, y):
-    return y * x
-```
-
-where
-
-- `x = EncryptedScalar(UnsignedInteger(bits))`
-- `y = ClearScalar(UnsignedInteger(bits))`
-
-results in
-
-<!--python-test:skip-->
-```python
-circuit.run(2, 3) == 6
-circuit.run(1, 7) == 7
-```
-
-## Dot Product
-
-### Dynamic ClearTensor and EncryptedTensor
-
-<!--python-test:skip-->
-```python
-def f(x, y):
-    return np.dot(x, y)
-```
-
-or
-
-<!--python-test:skip-->
-```python
-def f(x, y):
-    return np.dot(y, x)
-```
-
-where
-
-- `x = EncryptedTensor(UnsignedInteger(bits), shape=(2,))`
-- `y = ClearTensor(UnsignedInteger(bits), shape=(2,))`
-
-results in
-
-<!--python-test:skip-->
-```python
-circuit.run([1, 1], [2, 3]) == 5
-circuit.run([2, 3], [2, 3]) == 13
-```
-
-## Combining all together
-
-<!--python-test:skip-->
-```python
-def f(x, y, z):
-    return 100 - (2 * (np.dot(x, y) + z))
-```
-
-where
-
-- `x = EncryptedTensor(UnsignedInteger(bits), shape=(2,))`
-- `y = ClearTensor(UnsignedInteger(bits), shape=(2,))`
-- `z = EncryptedScalar(UnsignedInteger(bits))`
-
-results in
-
-<!--python-test:skip-->
-```python
-circuit.run([1, 2], [4, 3], 10) == 60
-circuit.run([2, 3], [3, 2], 5) == 66
-```
diff --git a/docs/user/tutorial/COMPILATION_ARTIFACTS.md b/docs/user/tutorial/COMPILATION_ARTIFACTS.md
index 4ba939dd4..a42ef60da 100644
--- a/docs/user/tutorial/COMPILATION_ARTIFACTS.md
+++ b/docs/user/tutorial/COMPILATION_ARTIFACTS.md
@@ -1,7 +1,3 @@
-```{warning}
-FIXME(Umut): check it is still valid. I guess yes, but some files may have gone or renamed.
-```
-
 # Compilation Artifacts
 
 In this tutorial, we are going to go over the artifact system, which is designed to inspect/debug the compilation process easily. 
@@ -16,7 +12,7 @@ def f(x):
     return np.sin(x)
 ```
 
-This function fails (for now) to compile because `Concrete` doesn't support floating point outputs. When you try to compile it (you might want to check [this](../howto/COMPILING_AND_EXECUTING.md) to see how you can do that), an exception will be raised and the artifacts will be exported automatically.
+This function fails to compile because **Concrete Framework** doesn't support floating point outputs. When you try to compile it (you might want to check [this](../howto/COMPILING_AND_EXECUTING.md) to see how you can do that), an exception will be raised and the artifacts will be exported automatically.
 
 ### environment.txt
 
@@ -60,34 +56,58 @@ x :: EncryptedScalar<Integer<unsigned, 3 bits>>
 
 ### 1.initial.graph.txt
 
-This file contains information about the initial computation graph of the function you are trying to compile.
+This file contains textual representation of the initial computation graph right after tracing.
 
 ```
-%0 = x                                   # EncryptedScalar<Integer<unsigned, 3 bits>>
-%1 = np.sin(0)                           # EncryptedScalar<Float<64 bits>>
-return(%1)
+%0 = x              # EncryptedScalar<uint3>
+%1 = sin(%0)        # EncryptedScalar<float64>
+return %1
 ```
 
 ### 1.initial.graph.png
 
-This file contains the visualization of the initial computation graph of the function you are trying to compile.
+This file contains the visual representation of the initial computation graph right after tracing.
 
 ![](../../_static/tutorials/artifacts/auto/1.initial.graph.png)
 
+### 2.final.graph.txt
+
+This file contains textual representation of the final computation graph right before MLIR conversion.
+
+```
+%0 = x              # EncryptedScalar<uint3>
+%1 = sin(%0)        # EncryptedScalar<float64>
+return %1
+```
+
+### 2.final.graph.png
+
+This file contains the visual representation of the final computation graph right before MLIR conversion.
+
+![](../../_static/tutorials/artifacts/auto/2.final.graph.png)
+
 ### traceback.txt
 
 This file contains information about the error you got.
 
 ```
 Traceback (most recent call last):
-  File "/src/concrete/numpy/compile.py", line 301, in compile_numpy_function
+  File "/home/default/Documents/Projects/Zama/hdk/concrete/numpy/compile.py", line 141, in run_compilation_function_with_error_management
+    return compilation_function()
+  File "/home/default/Documents/Projects/Zama/hdk/concrete/numpy/compile.py", line 769, in compilation_function
     return _compile_numpy_function_internal(
-  File "/src/concrete/numpy/compile.py", line 234, in _compile_numpy_function_internal
-    op_graph = _compile_numpy_function_into_op_graph_internal(
-  File "/src/concrete/numpy/compile.py", line 103, in _compile_numpy_function_into_op_graph_internal
-    raise ValueError(
-ValueError: <lambda> cannot be compiled as it has nodes with either float inputs or outputs.
-Offending nodes : <concrete.common.representation.intermediate.GenericFunction object at 0x7f6689fd37f0>
+  File "/home/default/Documents/Projects/Zama/hdk/concrete/numpy/compile.py", line 722, in _compile_numpy_function_internal
+    fhe_circuit = _compile_op_graph_to_fhe_circuit_internal(
+  File "/home/default/Documents/Projects/Zama/hdk/concrete/numpy/compile.py", line 626, in _compile_op_graph_to_fhe_circuit_internal
+    prepare_op_graph_for_mlir(op_graph)
+  File "/home/default/Documents/Projects/Zama/hdk/concrete/numpy/compile.py", line 597, in prepare_op_graph_for_mlir
+    raise RuntimeError(
+RuntimeError: function you are trying to compile isn't supported for MLIR lowering
+
+%0 = x              # EncryptedScalar<uint3>
+%1 = sin(%0)        # EncryptedScalar<float64>
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ only integer outputs are supported
+return %1
 ```
 
 ## Manual export
@@ -96,74 +116,114 @@ Manual exports are mostly used for visualization. Nonetheless, they can be very
 
 ```python
 import concrete.numpy as hnp
+import numpy as np
 import pathlib
 
+def f(x):
+    return 127 - (50 * (np.sin(x) + 1)).astype(np.uint32)
+
 artifacts = hnp.CompilationArtifacts(pathlib.Path("/tmp/custom/export/path"))
-hnp.compile_numpy_function(
-    lambda x: 100 - (3 * (x + 2)),
-    {"x": hnp.EncryptedScalar(hnp.UnsignedInteger(3))},
-    inputset=range(2 ** 3),
-    compilation_artifacts=artifacts,
-)
+
+compiler = hnp.NPFHECompiler(f, {"x": "encrypted"}, compilation_artifacts=artifacts)
+compiler.eval_on_inputset(range(2 ** 3))
+
 artifacts.export()
 ```
 
-Since this example compiles, we can see some new artifacts. 
-
 ### 1.initial.graph.txt
 
-This file contains information about the initial computation graph of the function you are trying to compile.
+This file contains textual representation of the initial computation graph right after tracing.
 
 ```
-%0 = Constant(100)                       # ClearScalar<Integer<unsigned, 7 bits>>
-%1 = Constant(3)                         # ClearScalar<Integer<unsigned, 2 bits>>
-%2 = x                                   # EncryptedScalar<Integer<unsigned, 3 bits>>
-%3 = Constant(2)                         # ClearScalar<Integer<unsigned, 2 bits>>
-%4 = Add(2, 3)                           # EncryptedScalar<Integer<unsigned, 3 bits>>
-%5 = Mul(4, 1)                           # EncryptedScalar<Integer<unsigned, 3 bits>>
-%6 = Sub(0, 5)                           # EncryptedScalar<Integer<unsigned, 7 bits>>
-return(%6)
+%0 = 127                             # ClearScalar<uint7>
+%1 = 50                              # ClearScalar<uint6>
+%2 = 1                               # ClearScalar<uint1>
+%3 = x                               # EncryptedScalar<uint3>
+%4 = sin(%3)                         # EncryptedScalar<float64>
+%5 = add(%4, %2)                     # EncryptedScalar<float64>
+%6 = mul(%5, %1)                     # EncryptedScalar<float64>
+%7 = astype(%6, dtype=uint32)        # EncryptedScalar<uint32>
+%8 = sub(%0, %7)                     # EncryptedScalar<uint32>
+return %8
 ```
 
 ### 1.initial.graph.png
 
-This file contains the visualization of the initial computation graph of the function you are trying to compile.
+This file contains the visual representation of the initial computation graph right after tracing.
 
 ![](../../_static/tutorials/artifacts/manual/1.initial.graph.png)
 
-### 2.final.graph.txt
+### 2.after-float-fuse-0.graph.txt
 
-This file contains information about the final computation graph of the function you are trying to compile.
+This file contains textual representation of the intermediate computation graph after fusing.
 
 ```
-%0 = Constant(100)                       # ClearScalar<Integer<unsigned, 7 bits>>
-%1 = Constant(3)                         # ClearScalar<Integer<unsigned, 2 bits>>
-%2 = x                                   # EncryptedScalar<Integer<unsigned, 3 bits>>
-%3 = Constant(2)                         # ClearScalar<Integer<unsigned, 2 bits>>
-%4 = Add(2, 3)                           # EncryptedScalar<Integer<unsigned, 4 bits>>
-%5 = Mul(4, 1)                           # EncryptedScalar<Integer<unsigned, 5 bits>>
-%6 = Sub(0, 5)                           # EncryptedScalar<Integer<unsigned, 7 bits>>
-return(%6)
+%0 = 127                 # ClearScalar<uint7>
+%1 = x                   # EncryptedScalar<uint3>
+%2 = subgraph(%1)        # EncryptedScalar<uint32>
+%3 = sub(%0, %2)         # EncryptedScalar<uint32>
+return %3
+
+Subgraphs:
+
+    %2 = subgraph(%1):
+
+        %0 = 50                              # ClearScalar<uint6>
+        %1 = 1                               # ClearScalar<uint1>
+        %2 = float_subgraph_input            # EncryptedScalar<uint3>
+        %3 = sin(%2)                         # EncryptedScalar<float64>
+        %4 = add(%3, %1)                     # EncryptedScalar<float64>
+        %5 = mul(%4, %0)                     # EncryptedScalar<float64>
+        %6 = astype(%5, dtype=uint32)        # EncryptedScalar<uint32>
+        return %6
 ```
 
-### 2.final.graph.png
+### 2.after-float-fuse-0.graph.png
 
-This file contains the visualization of the final computation graph of the function you are trying to compile.
+This file contains the visual representation of the intermediate computation graph after fusing.
 
-![](../../_static/tutorials/artifacts/manual/2.final.graph.png)
+![](../../_static/tutorials/artifacts/manual/2.after-float-fuse-0.graph.png)
+
+### 3.final.graph.txt
+
+This file contains textual representation of the final computation graph right before MLIR conversion.
+
+```
+%0 = 127                 # ClearScalar<uint7>
+%1 = x                   # EncryptedScalar<uint3>
+%2 = subgraph(%1)        # EncryptedScalar<uint7>
+%3 = sub(%0, %2)         # EncryptedScalar<uint7>
+return %3
+
+Subgraphs:
+
+    %2 = subgraph(%1):
+
+        %0 = 50                              # ClearScalar<uint6>
+        %1 = 1                               # ClearScalar<uint1>
+        %2 = float_subgraph_input            # EncryptedScalar<uint3>
+        %3 = sin(%2)                         # EncryptedScalar<float64>
+        %4 = add(%3, %1)                     # EncryptedScalar<float64>
+        %5 = mul(%4, %0)                     # EncryptedScalar<float64>
+        %6 = astype(%5, dtype=uint32)        # EncryptedScalar<uint7>
+        return %6
+```
+
+### 3.final.graph.png
+
+This file contains the visual representation of the final computation graph right before MLIR conversion.
+
+![](../../_static/tutorials/artifacts/manual/3.final.graph.png)
 
 ### bounds.txt
 
 This file contains information about the bounds of the final computation graph of the function you are trying to compile using the input set you provide.
 
 ```
-%0 :: [100, 100]
-%1 :: [3, 3]
-%2 :: [0, 7]
-%3 :: [2, 2]
-%4 :: [2, 9]
-%5 :: [6, 27]
-%6 :: [73, 94]
+%0 :: [127, 127]
+%1 :: [0, 7]
+%2 :: [2, 95]
+%3 :: [32, 125]
 ```
 
 You can learn what bounds are [here](../../dev/explanation/TERMINOLOGY_AND_STRUCTURE.md).
@@ -175,15 +235,14 @@ This file contains information about the MLIR of the function you are trying to
 ```
 module  {
   func @main(%arg0: !HLFHE.eint<7>) -> !HLFHE.eint<7> {
-    %c100_i8 = constant 100 : i8
-    %c3_i8 = constant 3 : i8
-    %c2_i8 = constant 2 : i8
-    %0 = "HLFHE.add_eint_int"(%arg0, %c2_i8) : (!HLFHE.eint<7>, i8) -> !HLFHE.eint<7>
-    %1 = "HLFHE.mul_eint_int"(%0, %c3_i8) : (!HLFHE.eint<7>, i8) -> !HLFHE.eint<7>
-    %2 = "HLFHE.sub_int_eint"(%c100_i8, %1) : (i8, !HLFHE.eint<7>) -> !HLFHE.eint<7>
-    return %2 : !HLFHE.eint<7>
+    %c127_i8 = arith.constant 127 : i8
+    %cst = arith.constant dense<"..."> : tensor<128xi64>
+    %0 = "HLFHE.apply_lookup_table"(%arg0, %cst) : (!HLFHE.eint<7>, tensor<128xi64>) -> !HLFHE.eint<7>
+    %1 = "HLFHE.sub_int_eint"(%c127_i8, %0) : (i8, !HLFHE.eint<7>) -> !HLFHE.eint<7>
+    return %1 : !HLFHE.eint<7>
   }
 }
+
 ```
 
 You can learn more about MLIR [here](../../dev/explanation/MLIR.md).
diff --git a/docs/user/tutorial/INDEXING.md b/docs/user/tutorial/INDEXING.md
index 4a9e077bf..9dc2695d2 100644
--- a/docs/user/tutorial/INDEXING.md
+++ b/docs/user/tutorial/INDEXING.md
@@ -1,9 +1,103 @@
-```{warning}
-FIXME(Umut): to be done
-```
-
 # Indexing
 
-# Slicing
+## Constant Indexing
 
+Constant indexing refers to the index being static (i.e., known during compilation).
 
+Here are some examples of constant indexing:
+
+### Extracting a single element
+
+```python
+import concrete.numpy as hnp
+import numpy as np
+
+def f(x):
+    return x[1]
+
+inputset = [np.random.randint(0, 2 ** 3, size=(3,), dtype=np.uint8) for _ in range(10)]
+
+compiler = hnp.NPFHECompiler(f, {"x": "encrypted"})
+compiler.eval_on_inputset(inputset)
+circuit = compiler.get_compiled_fhe_circuit()
+
+test_input = np.array([4, 2, 6], dtype=np.uint8)
+expected_output = 2
+
+assert np.array_equal(circuit.run(test_input), expected_output)
+```
+
+You can use negative indexing.
+
+```python
+import concrete.numpy as hnp
+import numpy as np
+
+def f(x):
+    return x[-1]
+
+inputset = [np.random.randint(0, 2 ** 3, size=(3,), dtype=np.uint8) for _ in range(10)]
+
+compiler = hnp.NPFHECompiler(f, {"x": "encrypted"})
+compiler.eval_on_inputset(inputset)
+circuit = compiler.get_compiled_fhe_circuit()
+
+test_input = np.array([4, 2, 6], dtype=np.uint8)
+expected_output = 6
+
+assert np.array_equal(circuit.run(test_input), expected_output)
+```
+
+You can use multidimensional indexing as well.
+
+```python
+import concrete.numpy as hnp
+import numpy as np
+
+def f(x):
+    return x[-1, 1]
+
+inputset = [np.random.randint(0, 2 ** 3, size=(3, 2), dtype=np.uint8) for _ in range(10)]
+
+compiler = hnp.NPFHECompiler(f, {"x": "encrypted"})
+compiler.eval_on_inputset(inputset)
+circuit = compiler.get_compiled_fhe_circuit()
+
+test_input = np.array([[4, 2], [1, 5], [7, 6]], dtype=np.uint8)
+expected_output = 6
+
+assert np.array_equal(circuit.run(test_input), expected_output)
+```
+
+### Extracting a slice
+
+```python
+import concrete.numpy as hnp
+import numpy as np
+
+def f(x):
+    return x[1:4]
+
+inputset = [np.random.randint(0, 2 ** 3, size=(5,), dtype=np.uint8) for _ in range(10)]
+
+compiler = hnp.NPFHECompiler(f, {"x": "encrypted"})
+compiler.eval_on_inputset(inputset)
+circuit = compiler.get_compiled_fhe_circuit()
+
+test_input = np.array([4, 2, 6, 1, 7], dtype=np.uint8)
+expected_output = np.array([2, 6, 1], dtype=np.uint8)
+
+assert np.array_equal(circuit.run(test_input), expected_output)
+```
+
+You can use multidimensional slicing as well.
+
+#### Note
+
+There are certain limitations of slicing due to MLIR. So if you stumple into `RuntimeError: Compilation failed: Failed to lower to LLVM dialect`, know that we are aware of it, and we are trying to make such cases compilable.
+
+## Dynamic Indexing
+
+Dynamic indexing refers to the index being dynamic (i.e., can change during runtime).
+Such indexing is especially useful for things like decision trees.
+Unfortunately, we don't support dynamic indexing for the time being.
diff --git a/docs/user/tutorial/TABLE_LOOKUP.md b/docs/user/tutorial/TABLE_LOOKUP.md
index 30074e43a..4ccaa4312 100644
--- a/docs/user/tutorial/TABLE_LOOKUP.md
+++ b/docs/user/tutorial/TABLE_LOOKUP.md
@@ -1,20 +1,15 @@
-```{warning}
-FIXME(Umut): update a bit, with the new API
-FIXME(Umut): add the multiTLU case
-```
-
 # Table Lookup
 
-In this tutorial, we are going to go over the ways to perform table lookups in **Concrete**. Please read [Compiling and Executing](../howto/COMPILING_AND_EXECUTING.md) before reading further to see how you can compile the functions below.
+In this tutorial, we are going to go over the ways to perform direct table lookups in **Concrete Framework**. Please read [Compiling and Executing](../howto/COMPILING_AND_EXECUTING.md) before reading further to see how you can compile the functions below.
 
 ## Direct table lookup
 
-**Concrete** provides a special class to allow direct table lookups. Here is how to import and use it:
+**Concrete Framework** provides a special class to allow direct table lookups. Here is how to use it:
 
 ```python
-from concrete.common.extensions.table import LookupTable
+import concrete.numpy as hnp
 
-table = LookupTable([2, 1, 3, 0])
+table = hnp.LookupTable([2, 1, 3, 0])
 
 def f(x):
     return table[x]
@@ -22,7 +17,7 @@ def f(x):
 
 where
 
-- `x = EncryptedScalar(UnsignedInteger(2))`
+- `x = "encrypted"` scalar
 
 results in
 
@@ -34,9 +29,56 @@ circuit.run(2) == 3
 circuit.run(3) == 0
 ```
 
+Moreover, direct lookup tables can be used with tensors where the same table lookup is applied to each value in the tensor, so
+
+- `x = "encrypted"` tensor of shape `(2, 3)`
+
+results in
+
+<!--python-test:skip-->
+```python
+input = np.array([[0, 1, 3], [2, 3, 1]], dtype=np.uint8)
+circuit.run(input) == [[2, 1, 0], [3, 0, 1]]
+```
+
+## Direct Multi Table Lookup
+
+Sometimes you may want to apply a different lookup table to each value in a tensor. That's where direct multi lookup table becomes handy. Here is how to use it:
+
+<!--python-test:skip-->
+```python
+import concrete.numpy as hnp
+
+squared = hnp.LookupTable([i ** 2 for i in range(4)])
+cubed = hnp.LookupTable([i ** 3 for i in range(4)])
+
+table = hnp.MultiLookupTable([
+    [squared, cubed],
+    [squared, cubed],
+    [squared, cubed],
+])
+
+def f(x):
+    return table[x]
+```
+
+where
+
+- `x = "encrypted"` tensor of shape `(3, 2)`
+
+results in
+
+<!--python-test:skip-->
+```python
+input = np.array([[2, 3], [1, 2], [3, 0]], dtype=np.uint8)
+circuit.run(input) == [[4, 27], [1, 8], [9, 0]]
+```
+
+Basically, we applied `squared` table to the first column and `cubed` to the second one.
+
 ## Fused table lookup
 
-Direct tables are tedious to prepare by hand. When possible, **Concrete** fuses the floating point operations into a single table lookup automatically. There are some limitations on fusing operations, which you can learn more about on the next tutorial, [Working With Floating Points](./WORKING_WITH_FLOATING_POINTS.md).
+Direct tables are tedious to prepare by hand. When possible, **Concrete Framework** fuses the floating point operations into table lookups automatically. There are some limitations on fusing operations, which you can learn more about on the next tutorial, [Working With Floating Points](./WORKING_WITH_FLOATING_POINTS.md).
 
 Here is an example function that results in fused table lookup:
 
@@ -48,7 +90,7 @@ def f(x):
 
 where
 
-- `x = EncryptedScalar(UnsignedInteger(3))`
+- `x = "encrypted"` scalar
 
 results in
 
@@ -76,7 +118,7 @@ Internally, it uses the following lookup table
 
 <!--python-test:skip-->
 ```python
-table = LookupTable([50, 92, 95, 57, 12, 2, 36, 82])
+table = hnp.LookupTable([50, 92, 95, 57, 12, 2, 36, 82])
 ```
 
 which is calculated by: