diff --git a/fuzz_tests/test_generate.py b/fuzz_tests/test_generate.py
index 2a52a78..e94e1f1 100644
--- a/fuzz_tests/test_generate.py
+++ b/fuzz_tests/test_generate.py
@@ -1,7 +1,7 @@
 from hypothesis import given
 
+from tensora.compile import BroadcastTargetIndexError, TensorMethod
 from tensora.desugar import DiagonalAccessError, NoKernelFoundError
-from tensora.function import BroadcastTargetIndexError, TensorMethod
 
 from .strategies import problem_and_tensors
 
diff --git a/noxfile.py b/noxfile.py
index 87c31ee..077a33d 100644
--- a/noxfile.py
+++ b/noxfile.py
@@ -1,7 +1,7 @@
 from nox import options, parametrize
 from nox_poetry import Session, session
 
-options.sessions = ["test", "test_taco", "test_numpy", "coverage", "lint"]
+options.sessions = ["test", "test_taco", "test_cffi", "test_numpy", "coverage", "lint"]
 
 
 @session(python=["3.10", "3.11", "3.12"])
@@ -18,6 +18,13 @@ def test_taco(s: Session):
     s.run("python", "-m", "pytest", "--cov", "tensora", "tests/taco")
 
 
+@session(python=["3.10", "3.11", "3.12"])
+def test_cffi(s: Session):
+    s.install(".[cffi]", "pytest", "pytest-cov")
+    s.env["COVERAGE_FILE"] = f".coverage.cffi.{s.python}"
+    s.run("python", "-m", "pytest", "--cov", "tensora", "tests_cffi")
+
+
 @session(python=["3.10", "3.11", "3.12"])
 def test_numpy(s: Session):
     s.install(".[numpy]", "pytest", "pytest-cov")
diff --git a/poetry.lock b/poetry.lock
index fd6a385..94a29d3 100644
--- a/poetry.lock
+++ b/poetry.lock
@@ -1,14 +1,14 @@
-# This file is automatically @generated by Poetry 1.7.1 and should not be changed by hand.
+# This file is automatically @generated by Poetry 1.8.3 and should not be changed by hand.
 
 [[package]]
 name = "argcomplete"
-version = "3.2.2"
+version = "3.5.0"
 description = "Bash tab completion for argparse"
 optional = false
 python-versions = ">=3.8"
 files = [
-    {file = "argcomplete-3.2.2-py3-none-any.whl", hash = "sha256:e44f4e7985883ab3e73a103ef0acd27299dbfe2dfed00142c35d4ddd3005901d"},
-    {file = "argcomplete-3.2.2.tar.gz", hash = "sha256:f3e49e8ea59b4026ee29548e24488af46e30c9de57d48638e24f54a1ea1000a2"},
+    {file = "argcomplete-3.5.0-py3-none-any.whl", hash = "sha256:d4bcf3ff544f51e16e54228a7ac7f486ed70ebf2ecfe49a63a91171c76bf029b"},
+    {file = "argcomplete-3.5.0.tar.gz", hash = "sha256:4349400469dccfb7950bb60334a680c58d88699bff6159df61251878dc6bf74b"},
 ]
 
 [package.extras]
@@ -16,32 +16,32 @@ test = ["coverage", "mypy", "pexpect", "ruff", "wheel"]
 
 [[package]]
 name = "attrs"
-version = "23.2.0"
+version = "24.2.0"
 description = "Classes Without Boilerplate"
 optional = false
 python-versions = ">=3.7"
 files = [
-    {file = "attrs-23.2.0-py3-none-any.whl", hash = "sha256:99b87a485a5820b23b879f04c2305b44b951b502fd64be915879d77a7e8fc6f1"},
-    {file = "attrs-23.2.0.tar.gz", hash = "sha256:935dc3b529c262f6cf76e50877d35a4bd3c1de194fd41f47a2b7ae8f19971f30"},
+    {file = "attrs-24.2.0-py3-none-any.whl", hash = "sha256:81921eb96de3191c8258c199618104dd27ac608d9366f5e35d011eae1867ede2"},
+    {file = "attrs-24.2.0.tar.gz", hash = "sha256:5cfb1b9148b5b086569baec03f20d7b6bf3bcacc9a42bebf87ffaaca362f6346"},
 ]
 
 [package.extras]
-cov = ["attrs[tests]", "coverage[toml] (>=5.3)"]
-dev = ["attrs[tests]", "pre-commit"]
-docs = ["furo", "myst-parser", "sphinx", "sphinx-notfound-page", "sphinxcontrib-towncrier", "towncrier", "zope-interface"]
-tests = ["attrs[tests-no-zope]", "zope-interface"]
-tests-mypy = ["mypy (>=1.6)", "pytest-mypy-plugins"]
-tests-no-zope = ["attrs[tests-mypy]", "cloudpickle", "hypothesis", "pympler", "pytest (>=4.3.0)", "pytest-xdist[psutil]"]
+benchmark = ["cloudpickle", "hypothesis", "mypy (>=1.11.1)", "pympler", "pytest (>=4.3.0)", "pytest-codspeed", "pytest-mypy-plugins", "pytest-xdist[psutil]"]
+cov = ["cloudpickle", "coverage[toml] (>=5.3)", "hypothesis", "mypy (>=1.11.1)", "pympler", "pytest (>=4.3.0)", "pytest-mypy-plugins", "pytest-xdist[psutil]"]
+dev = ["cloudpickle", "hypothesis", "mypy (>=1.11.1)", "pre-commit", "pympler", "pytest (>=4.3.0)", "pytest-mypy-plugins", "pytest-xdist[psutil]"]
+docs = ["cogapp", "furo", "myst-parser", "sphinx", "sphinx-notfound-page", "sphinxcontrib-towncrier", "towncrier (<24.7)"]
+tests = ["cloudpickle", "hypothesis", "mypy (>=1.11.1)", "pympler", "pytest (>=4.3.0)", "pytest-mypy-plugins", "pytest-xdist[psutil]"]
+tests-mypy = ["mypy (>=1.11.1)", "pytest-mypy-plugins"]
 
 [[package]]
 name = "babel"
-version = "2.14.0"
+version = "2.16.0"
 description = "Internationalization utilities"
 optional = false
-python-versions = ">=3.7"
+python-versions = ">=3.8"
 files = [
-    {file = "Babel-2.14.0-py3-none-any.whl", hash = "sha256:efb1a25b7118e67ce3a259bed20545c29cb68be8ad2c784c83689981b7a57287"},
-    {file = "Babel-2.14.0.tar.gz", hash = "sha256:6919867db036398ba21eb5c7a0f6b28ab8cbc3ae7a73a44ebe34ae74a4e7d363"},
+    {file = "babel-2.16.0-py3-none-any.whl", hash = "sha256:368b5b98b37c06b7daf6696391c3240c938b37767d4584413e8438c5c435fa8b"},
+    {file = "babel-2.16.0.tar.gz", hash = "sha256:d1f3554ca26605fe173f3de0c65f750f5a42f924499bf134de6423582298e316"},
 ]
 
 [package.extras]
@@ -49,33 +49,33 @@ dev = ["freezegun (>=1.0,<2.0)", "pytest (>=6.0)", "pytest-cov"]
 
 [[package]]
 name = "black"
-version = "24.2.0"
+version = "24.8.0"
 description = "The uncompromising code formatter."
 optional = false
 python-versions = ">=3.8"
 files = [
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+    {file = "black-24.8.0-cp39-cp39-win_amd64.whl", hash = "sha256:73bbf84ed136e45d451a260c6b73ed674652f90a2b3211d6a35e78054563a9bb"},
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+    {file = "black-24.8.0.tar.gz", hash = "sha256:2500945420b6784c38b9ee885af039f5e7471ef284ab03fa35ecdde4688cd83f"},
 ]
 
 [package.dependencies]
@@ -95,85 +95,100 @@ uvloop = ["uvloop (>=0.15.2)"]
 
 [[package]]
 name = "blinker"
-version = "1.7.0"
+version = "1.8.2"
 description = "Fast, simple object-to-object and broadcast signaling"
 optional = false
 python-versions = ">=3.8"
 files = [
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-    {file = "blinker-1.7.0.tar.gz", hash = "sha256:e6820ff6fa4e4d1d8e2747c2283749c3f547e4fee112b98555cdcdae32996182"},
+    {file = "blinker-1.8.2-py3-none-any.whl", hash = "sha256:1779309f71bf239144b9399d06ae925637cf6634cf6bd131104184531bf67c01"},
+    {file = "blinker-1.8.2.tar.gz", hash = "sha256:8f77b09d3bf7c795e969e9486f39c2c5e9c39d4ee07424be2bc594ece9642d83"},
 ]
 
 [[package]]
 name = "certifi"
-version = "2024.2.2"
+version = "2024.7.4"
 description = "Python package for providing Mozilla's CA Bundle."
 optional = false
 python-versions = ">=3.6"
 files = [
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 ]
 
 [[package]]
 name = "cffi"
-version = "1.16.0"
+version = "1.17.0"
 description = "Foreign Function Interface for Python calling C code."
 optional = false
 python-versions = ">=3.8"
 files = [
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 [package.dependencies]
@@ -322,63 +337,83 @@ development = ["black", "flake8", "mypy", "pytest", "types-colorama"]
 
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 ]
 
 [package.dependencies]
@@ -389,13 +424,13 @@ toml = ["tomli"]
 
 [[package]]
 name = "dash"
-version = "2.16.0"
+version = "2.17.1"
 description = "A Python framework for building reactive web-apps. Developed by Plotly."
 optional = false
 python-versions = ">=3.8"
 files = [
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 ]
 
 [package.dependencies]
@@ -414,11 +449,11 @@ Werkzeug = "<3.1"
 
 [package.extras]
 celery = ["celery[redis] (>=5.1.2)", "redis (>=3.5.3)"]
-ci = ["black (==22.3.0)", "dash-dangerously-set-inner-html", "dash-flow-example (==0.0.5)", "flake8 (==7.0.0)", "flaky (==3.7.0)", "flask-talisman (==1.0.0)", "jupyterlab (<4.0.0)", "mimesis (<=11.1.0)", "mock (==4.0.3)", "numpy (<=1.26.3)", "openpyxl", "orjson (==3.9.12)", "pandas (>=1.4.0)", "pyarrow", "pylint (==3.0.3)", "pytest-mock", "pytest-rerunfailures", "pytest-sugar (==0.9.6)", "pyzmq (==25.1.2)", "xlrd (>=2.0.1)"]
+ci = ["black (==22.3.0)", "dash-dangerously-set-inner-html", "dash-flow-example (==0.0.5)", "flake8 (==7.0.0)", "flaky (==3.8.1)", "flask-talisman (==1.0.0)", "jupyterlab (<4.0.0)", "mimesis (<=11.1.0)", "mock (==4.0.3)", "numpy (<=1.26.3)", "openpyxl", "orjson (==3.10.3)", "pandas (>=1.4.0)", "pyarrow", "pylint (==3.0.3)", "pytest-mock", "pytest-rerunfailures", "pytest-sugar (==0.9.6)", "pyzmq (==25.1.2)", "xlrd (>=2.0.1)"]
 compress = ["flask-compress"]
 dev = ["PyYAML (>=5.4.1)", "coloredlogs (>=15.0.1)", "fire (>=0.4.0)"]
 diskcache = ["diskcache (>=5.2.1)", "multiprocess (>=0.70.12)", "psutil (>=5.8.0)"]
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 [[package]]
 name = "dash-core-components"
@@ -466,13 +501,13 @@ files = [
 
 [[package]]
 name = "exceptiongroup"
-version = "1.2.0"
+version = "1.2.2"
 description = "Backport of PEP 654 (exception groups)"
 optional = false
 python-versions = ">=3.7"
 files = [
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 ]
 
 [package.extras]
@@ -480,29 +515,29 @@ test = ["pytest (>=6)"]
 
 [[package]]
 name = "filelock"
-version = "3.13.1"
+version = "3.15.4"
 description = "A platform independent file lock."
 optional = false
 python-versions = ">=3.8"
 files = [
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 ]
 
 [package.extras]
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-testing = ["covdefaults (>=2.3)", "coverage (>=7.3.2)", "diff-cover (>=8)", "pytest (>=7.4.3)", "pytest-cov (>=4.1)", "pytest-mock (>=3.12)", "pytest-timeout (>=2.2)"]
+docs = ["furo (>=2023.9.10)", "sphinx (>=7.2.6)", "sphinx-autodoc-typehints (>=1.25.2)"]
+testing = ["covdefaults (>=2.3)", "coverage (>=7.3.2)", "diff-cover (>=8.0.1)", "pytest (>=7.4.3)", "pytest-asyncio (>=0.21)", "pytest-cov (>=4.1)", "pytest-mock (>=3.12)", "pytest-timeout (>=2.2)", "virtualenv (>=20.26.2)"]
 typing = ["typing-extensions (>=4.8)"]
 
 [[package]]
 name = "flask"
-version = "3.0.2"
+version = "3.0.3"
 description = "A simple framework for building complex web applications."
 optional = false
 python-versions = ">=3.8"
 files = [
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+    {file = "flask-3.0.3-py3-none-any.whl", hash = "sha256:34e815dfaa43340d1d15a5c3a02b8476004037eb4840b34910c6e21679d288f3"},
+    {file = "flask-3.0.3.tar.gz", hash = "sha256:ceb27b0af3823ea2737928a4d99d125a06175b8512c445cbd9a9ce200ef76842"},
 ]
 
 [package.dependencies]
@@ -560,13 +595,13 @@ pytrace = ["flask-cors (>=3.0.10)", "pycrunch-trace (>=0.1.6)"]
 
 [[package]]
 name = "hypothesis"
-version = "6.98.15"
+version = "6.111.2"
 description = "A library for property-based testing"
 optional = false
 python-versions = ">=3.8"
 files = [
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-    {file = "hypothesis-6.98.15.tar.gz", hash = "sha256:1e31210951511b24ce8b3b6e04d791c466385a30ac3af571bf2223954b025d77"},
+    {file = "hypothesis-6.111.2-py3-none-any.whl", hash = "sha256:055e8228958e22178d6077e455fd86a72044d02dac130dbf9c8b31e161b9809c"},
+    {file = "hypothesis-6.111.2.tar.gz", hash = "sha256:0496ad28c7240ee9ba89fcc7fb1dc74e89f3e40fbcbbb5f73c0091558dec8e6e"},
 ]
 
 [package.dependencies]
@@ -578,9 +613,10 @@ rich = {version = ">=9.0.0", optional = true, markers = "extra == \"cli\""}
 sortedcontainers = ">=2.1.0,<3.0.0"
 
 [package.extras]
-all = ["backports.zoneinfo (>=0.2.1)", "black (>=19.10b0)", "click (>=7.0)", "django (>=3.2)", "dpcontracts (>=0.4)", "lark (>=0.10.1)", "libcst (>=0.3.16)", "numpy (>=1.17.3)", "pandas (>=1.1)", "pytest (>=4.6)", "python-dateutil (>=1.4)", "pytz (>=2014.1)", "redis (>=3.0.0)", "rich (>=9.0.0)", "tzdata (>=2024.1)"]
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 cli = ["black (>=19.10b0)", "click (>=7.0)", "rich (>=9.0.0)"]
 codemods = ["libcst (>=0.3.16)"]
+crosshair = ["crosshair-tool (>=0.0.70)", "hypothesis-crosshair (>=0.0.13)"]
 dateutil = ["python-dateutil (>=1.4)"]
 django = ["django (>=3.2)"]
 dpcontracts = ["dpcontracts (>=0.4)"]
@@ -595,33 +631,33 @@ zoneinfo = ["backports.zoneinfo (>=0.2.1)", "tzdata (>=2024.1)"]
 
 [[package]]
 name = "idna"
-version = "3.6"
+version = "3.8"
 description = "Internationalized Domain Names in Applications (IDNA)"
 optional = false
-python-versions = ">=3.5"
+python-versions = ">=3.6"
 files = [
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 ]
 
 [[package]]
 name = "importlib-metadata"
-version = "7.0.1"
+version = "8.4.0"
 description = "Read metadata from Python packages"
 optional = false
 python-versions = ">=3.8"
 files = [
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 [package.dependencies]
 zipp = ">=0.5"
 
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+doc = ["furo", "jaraco.packaging (>=9.3)", "jaraco.tidelift (>=1.4)", "rst.linker (>=1.9)", "sphinx (>=3.5)", "sphinx-lint"]
 perf = ["ipython"]
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+test = ["flufl.flake8", "importlib-resources (>=1.3)", "jaraco.test (>=5.4)", "packaging", "pyfakefs", "pytest (>=6,!=8.1.*)", "pytest-checkdocs (>=2.4)", "pytest-cov", "pytest-enabler (>=2.2)", "pytest-mypy", "pytest-perf (>=0.9.2)", "pytest-ruff (>=0.2.1)"]
 
 [[package]]
 name = "iniconfig"
@@ -636,24 +672,24 @@ files = [
 
 [[package]]
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-version = "2.1.2"
+version = "2.2.0"
 description = "Safely pass data to untrusted environments and back."
 optional = false
-python-versions = ">=3.7"
+python-versions = ">=3.8"
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+version = "3.1.4"
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 python-versions = ">=3.7"
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 ]
 
 [package.dependencies]
@@ -664,55 +700,83 @@ i18n = ["Babel (>=2.7)"]
 
 [[package]]
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+version = "1.4.0"
 description = "A concrete syntax tree with AST-like properties for Python 3.0 through 3.12 programs."
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 python-versions = ">=3.9"
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 name = "markdown"
-version = "3.5.2"
+version = "3.7"
 description = "Python implementation of John Gruber's Markdown."
 optional = false
 python-versions = ">=3.8"
 files = [
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 ]
 
 [package.extras]
@@ -836,13 +900,13 @@ files = [
 
 [[package]]
 name = "mkdocs"
-version = "1.5.3"
+version = "1.6.0"
 description = "Project documentation with Markdown."
 optional = false
-python-versions = ">=3.7"
+python-versions = ">=3.8"
 files = [
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 [package.dependencies]
@@ -850,29 +914,45 @@ click = ">=7.0"
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 ghp-import = ">=1.0"
 jinja2 = ">=2.11.1"
-markdown = ">=3.2.1"
+markdown = ">=3.3.6"
 markupsafe = ">=2.0.1"
 mergedeep = ">=1.3.4"
+mkdocs-get-deps = ">=0.2.0"
 packaging = ">=20.5"
 pathspec = ">=0.11.1"
-platformdirs = ">=2.2.0"
 pyyaml = ">=5.1"
 pyyaml-env-tag = ">=0.1"
 watchdog = ">=2.0"
 
 [package.extras]
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+min-versions = ["babel (==2.9.0)", "click (==7.0)", "colorama (==0.4)", "ghp-import (==1.0)", "importlib-metadata (==4.4)", "jinja2 (==2.11.1)", "markdown (==3.3.6)", "markupsafe (==2.0.1)", "mergedeep (==1.3.4)", "mkdocs-get-deps (==0.2.0)", "packaging (==20.5)", "pathspec (==0.11.1)", "pyyaml (==5.1)", "pyyaml-env-tag (==0.1)", "watchdog (==2.0)"]
+
+[[package]]
+name = "mkdocs-get-deps"
+version = "0.2.0"
+description = "MkDocs extension that lists all dependencies according to a mkdocs.yml file"
+optional = false
+python-versions = ">=3.8"
+files = [
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+
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+platformdirs = ">=2.2.0"
+pyyaml = ">=5.1"
 
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 name = "mkdocs-material"
-version = "9.5.12"
+version = "9.5.33"
 description = "Documentation that simply works"
 optional = false
 python-versions = ">=3.8"
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@@ -880,7 +960,7 @@ babel = ">=2.10,<3.0"
 colorama = ">=0.4,<1.0"
 jinja2 = ">=3.0,<4.0"
 markdown = ">=3.2,<4.0"
-mkdocs = ">=1.5.3,<1.6.0"
+mkdocs = ">=1.6,<2.0"
 mkdocs-material-extensions = ">=1.3,<2.0"
 paginate = ">=0.5,<1.0"
 pygments = ">=2.16,<3.0"
@@ -928,23 +1008,25 @@ files = [
 
 [[package]]
 name = "nox"
-version = "2023.4.22"
+version = "2024.4.15"
 description = "Flexible test automation."
 optional = false
 python-versions = ">=3.7"
 files = [
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 [package.dependencies]
 argcomplete = ">=1.9.4,<4.0"
 colorlog = ">=2.6.1,<7.0.0"
 packaging = ">=20.9"
-virtualenv = ">=14"
+tomli = {version = ">=1", markers = "python_version < \"3.11\""}
+virtualenv = ">=20.14.1"
 
 [package.extras]
-tox-to-nox = ["jinja2", "tox (<4)"]
+tox-to-nox = ["jinja2", "tox"]
+uv = ["uv (>=0.1.6)"]
 
 [[package]]
 name = "nox-poetry"
@@ -1009,68 +1091,73 @@ files = [
 
 [[package]]
 name = "packaging"
-version = "23.2"
+version = "24.1"
 description = "Core utilities for Python packages"
 optional = false
-python-versions = ">=3.7"
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 ]
 
 [[package]]
 name = "paginate"
-version = "0.5.6"
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 description = "Divides large result sets into pages for easier browsing"
 optional = false
 python-versions = "*"
 files = [
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 ]
 
+[package.extras]
+dev = ["pytest", "tox"]
+lint = ["black"]
+
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+version = "2.2.2"
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 python-versions = ">=3.9"
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 ]
 
 [package.dependencies]
 numpy = [
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-    {version = ">=1.23.2,<2", markers = "python_version == \"3.11\""},
-    {version = ">=1.26.0,<2", markers = "python_version >= \"3.12\""},
+    {version = ">=1.22.4", markers = "python_version < \"3.11\""},
+    {version = ">=1.23.2", markers = "python_version == \"3.11\""},
+    {version = ">=1.26.0", markers = "python_version >= \"3.12\""},
 ]
 python-dateutil = ">=2.8.2"
 pytz = ">=2020.1"
@@ -1128,28 +1215,29 @@ files = [
 
 [[package]]
 name = "platformdirs"
-version = "4.2.0"
-description = "A small Python package for determining appropriate platform-specific dirs, e.g. a \"user data dir\"."
+version = "4.2.2"
+description = "A small Python package for determining appropriate platform-specific dirs, e.g. a `user data dir`."
 optional = false
 python-versions = ">=3.8"
 files = [
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 [package.extras]
 docs = ["furo (>=2023.9.10)", "proselint (>=0.13)", "sphinx (>=7.2.6)", "sphinx-autodoc-typehints (>=1.25.2)"]
 test = ["appdirs (==1.4.4)", "covdefaults (>=2.3)", "pytest (>=7.4.3)", "pytest-cov (>=4.1)", "pytest-mock (>=3.12)"]
+type = ["mypy (>=1.8)"]
 
 [[package]]
 name = "plotly"
-version = "5.19.0"
+version = "5.23.0"
 description = "An open-source, interactive data visualization library for Python"
 optional = false
 python-versions = ">=3.8"
 files = [
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 ]
 
 [package.dependencies]
@@ -1158,13 +1246,13 @@ tenacity = ">=6.2.0"
 
 [[package]]
 name = "pluggy"
-version = "1.4.0"
+version = "1.5.0"
 description = "plugin and hook calling mechanisms for python"
 optional = false
 python-versions = ">=3.8"
 files = [
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 ]
 
 [package.extras]
@@ -1173,27 +1261,28 @@ testing = ["pytest", "pytest-benchmark"]
 
 [[package]]
 name = "psutil"
-version = "5.9.8"
+version = "6.0.0"
 description = "Cross-platform lib for process and system monitoring in Python."
 optional = false
-python-versions = ">=2.7, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*, !=3.4.*, !=3.5.*"
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+    {file = "psutil-6.0.0-cp37-abi3-win_amd64.whl", hash = "sha256:33ea5e1c975250a720b3a6609c490db40dae5d83a4eb315170c4fe0d8b1f34b3"},
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+    {file = "psutil-6.0.0.tar.gz", hash = "sha256:8faae4f310b6d969fa26ca0545338b21f73c6b15db7c4a8d934a5482faa818f2"},
 ]
 
 [package.extras]
@@ -1201,43 +1290,42 @@ test = ["enum34", "ipaddress", "mock", "pywin32", "wmi"]
 
 [[package]]
 name = "pycparser"
-version = "2.21"
+version = "2.22"
 description = "C parser in Python"
 optional = false
-python-versions = ">=2.7, !=3.0.*, !=3.1.*, !=3.2.*, !=3.3.*"
+python-versions = ">=3.8"
 files = [
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-    {file = "pycparser-2.21.tar.gz", hash = "sha256:e644fdec12f7872f86c58ff790da456218b10f863970249516d60a5eaca77206"},
+    {file = "pycparser-2.22-py3-none-any.whl", hash = "sha256:c3702b6d3dd8c7abc1afa565d7e63d53a1d0bd86cdc24edd75470f4de499cfcc"},
+    {file = "pycparser-2.22.tar.gz", hash = "sha256:491c8be9c040f5390f5bf44a5b07752bd07f56edf992381b05c701439eec10f6"},
 ]
 
 [[package]]
 name = "pygments"
-version = "2.17.2"
+version = "2.18.0"
 description = "Pygments is a syntax highlighting package written in Python."
 optional = false
-python-versions = ">=3.7"
+python-versions = ">=3.8"
 files = [
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+    {file = "pygments-2.18.0.tar.gz", hash = "sha256:786ff802f32e91311bff3889f6e9a86e81505fe99f2735bb6d60ae0c5004f199"},
 ]
 
 [package.extras]
-plugins = ["importlib-metadata"]
 windows-terminal = ["colorama (>=0.4.6)"]
 
 [[package]]
 name = "pymdown-extensions"
-version = "10.7"
+version = "10.9"
 description = "Extension pack for Python Markdown."
 optional = false
 python-versions = ">=3.8"
 files = [
-    {file = "pymdown_extensions-10.7-py3-none-any.whl", hash = "sha256:6ca215bc57bc12bf32b414887a68b810637d039124ed9b2e5bd3325cbb2c050c"},
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+    {file = "pymdown_extensions-10.9-py3-none-any.whl", hash = "sha256:d323f7e90d83c86113ee78f3fe62fc9dee5f56b54d912660703ea1816fed5626"},
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 ]
 
 [package.dependencies]
-markdown = ">=3.5"
+markdown = ">=3.6"
 pyyaml = "*"
 
 [package.extras]
@@ -1245,13 +1333,13 @@ extra = ["pygments (>=2.12)"]
 
 [[package]]
 name = "pytest"
-version = "8.0.2"
+version = "8.3.2"
 description = "pytest: simple powerful testing with Python"
 optional = false
 python-versions = ">=3.8"
 files = [
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-    {file = "pytest-8.0.2.tar.gz", hash = "sha256:d4051d623a2e0b7e51960ba963193b09ce6daeb9759a451844a21e4ddedfc1bd"},
+    {file = "pytest-8.3.2-py3-none-any.whl", hash = "sha256:4ba08f9ae7dcf84ded419494d229b48d0903ea6407b030eaec46df5e6a73bba5"},
+    {file = "pytest-8.3.2.tar.gz", hash = "sha256:c132345d12ce551242c87269de812483f5bcc87cdbb4722e48487ba194f9fdce"},
 ]
 
 [package.dependencies]
@@ -1259,21 +1347,21 @@ colorama = {version = "*", markers = "sys_platform == \"win32\""}
 exceptiongroup = {version = ">=1.0.0rc8", markers = "python_version < \"3.11\""}
 iniconfig = "*"
 packaging = "*"
-pluggy = ">=1.3.0,<2.0"
-tomli = {version = ">=1.0.0", markers = "python_version < \"3.11\""}
+pluggy = ">=1.5,<2"
+tomli = {version = ">=1", markers = "python_version < \"3.11\""}
 
 [package.extras]
-testing = ["argcomplete", "attrs (>=19.2.0)", "hypothesis (>=3.56)", "mock", "nose", "pygments (>=2.7.2)", "requests", "setuptools", "xmlschema"]
+dev = ["argcomplete", "attrs (>=19.2)", "hypothesis (>=3.56)", "mock", "pygments (>=2.7.2)", "requests", "setuptools", "xmlschema"]
 
 [[package]]
 name = "pytest-cov"
-version = "4.1.0"
+version = "5.0.0"
 description = "Pytest plugin for measuring coverage."
 optional = false
-python-versions = ">=3.7"
+python-versions = ">=3.8"
 files = [
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 ]
 
 [package.dependencies]
@@ -1281,7 +1369,7 @@ coverage = {version = ">=5.2.1", extras = ["toml"]}
 pytest = ">=4.6"
 
 [package.extras]
-testing = ["fields", "hunter", "process-tests", "pytest-xdist", "six", "virtualenv"]
+testing = ["fields", "hunter", "process-tests", "pytest-xdist", "virtualenv"]
 
 [[package]]
 name = "python-dateutil"
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 ]
 
 [[package]]
@@ -1373,115 +1474,101 @@ pyyaml = "*"
 
 [[package]]
 name = "regex"
-version = "2023.12.25"
+version = "2024.7.24"
 description = "Alternative regular expression module, to replace re."
 optional = false
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 ]
 
 [package.dependencies]
-numpy = ">=1.22.4,<1.29.0"
+numpy = ">=1.23.5,<2.3"
 
 [package.extras]
-dev = ["click", "cython-lint (>=0.12.2)", "doit (>=0.36.0)", "mypy", "pycodestyle", "pydevtool", "rich-click", "ruff", "types-psutil", "typing_extensions"]
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+dev = ["cython-lint (>=0.12.2)", "doit (>=0.36.0)", "mypy (==1.10.0)", "pycodestyle", "pydevtool", "rich-click", "ruff (>=0.0.292)", "types-psutil", "typing_extensions"]
+doc = ["jupyterlite-pyodide-kernel", "jupyterlite-sphinx (>=0.13.1)", "jupytext", "matplotlib (>=3.5)", "myst-nb", "numpydoc", "pooch", "pydata-sphinx-theme (>=0.15.2)", "sphinx (>=5.0.0,<=7.3.7)", "sphinx-design (>=0.4.0)"]
+test = ["Cython", "array-api-strict (>=2.0)", "asv", "gmpy2", "hypothesis (>=6.30)", "meson", "mpmath", "ninja", "pooch", "pytest", "pytest-cov", "pytest-timeout", "pytest-xdist", "scikit-umfpack", "threadpoolctl"]
 
 [[package]]
 name = "setuptools"
-version = "69.1.1"
+version = "69.5.1"
 description = "Easily download, build, install, upgrade, and uninstall Python packages"
 optional = false
 python-versions = ">=3.8"
 files = [
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 ]
 
 [package.extras]
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+docs = ["furo", "jaraco.packaging (>=9.3)", "jaraco.tidelift (>=1.4)", "pygments-github-lexers (==0.0.5)", "rst.linker (>=1.9)", "sphinx (>=3.5)", "sphinx-favicon", "sphinx-inline-tabs", "sphinx-lint", "sphinx-notfound-page (>=1,<2)", "sphinx-reredirects", "sphinxcontrib-towncrier"]
+testing = ["build[virtualenv]", "filelock (>=3.4.0)", "importlib-metadata", "ini2toml[lite] (>=0.9)", "jaraco.develop (>=7.21)", "jaraco.envs (>=2.2)", "jaraco.path (>=3.2.0)", "mypy (==1.9)", "packaging (>=23.2)", "pip (>=19.1)", "pytest (>=6,!=8.1.1)", "pytest-checkdocs (>=2.4)", "pytest-cov", "pytest-enabler (>=2.2)", "pytest-home (>=0.5)", "pytest-mypy", "pytest-perf", "pytest-ruff (>=0.2.1)", "pytest-timeout", "pytest-xdist (>=3)", "tomli", "tomli-w (>=1.0.0)", "virtualenv (>=13.0.0)", "wheel"]
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 [[package]]
@@ -1651,17 +1746,18 @@ files = [
 
 [[package]]
 name = "tenacity"
-version = "8.2.3"
+version = "9.0.0"
 description = "Retry code until it succeeds"
 optional = false
-python-versions = ">=3.7"
+python-versions = ">=3.8"
 files = [
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 ]
 
 [package.extras]
-doc = ["reno", "sphinx", "tornado (>=4.5)"]
+doc = ["reno", "sphinx"]
+test = ["pytest", "tornado (>=4.5)", "typeguard"]
 
 [[package]]
 name = "tensora-taco"
@@ -1689,24 +1785,24 @@ files = [
 
 [[package]]
 name = "tomlkit"
-version = "0.12.4"
+version = "0.13.2"
 description = "Style preserving TOML library"
 optional = false
-python-versions = ">=3.7"
+python-versions = ">=3.8"
 files = [
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 [[package]]
 name = "typer"
-version = "0.9.0"
+version = "0.9.4"
 description = "Typer, build great CLIs. Easy to code. Based on Python type hints."
 optional = false
 python-versions = ">=3.6"
 files = [
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 [package.dependencies]
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 [[package]]
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 python-versions = ">=3.8"
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-python-versions = "*"
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-
 [[package]]
 name = "tzdata"
 version = "2024.1"
@@ -1758,13 +1839,13 @@ files = [
 
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 name = "urllib3"
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+version = "2.2.2"
 description = "HTTP library with thread-safe connection pooling, file post, and more."
 optional = false
 python-versions = ">=3.8"
 files = [
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 [package.extras]
@@ -1775,13 +1856,13 @@ zstd = ["zstandard (>=0.18.0)"]
 
 [[package]]
 name = "virtualenv"
-version = "20.25.1"
+version = "20.26.3"
 description = "Virtual Python Environment builder"
 optional = false
 python-versions = ">=3.7"
 files = [
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 [package.dependencies]
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 platformdirs = ">=3.9.1,<5"
 
 [package.extras]
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 [[package]]
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 description = "Filesystem events monitoring"
 optional = false
 python-versions = ">=3.8"
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+    {file = "watchdog-4.0.2-py3-none-manylinux2014_x86_64.whl", hash = "sha256:c0b14488bd336c5b1845cee83d3e631a1f8b4e9c5091ec539406e4a324f882d8"},
+    {file = "watchdog-4.0.2-py3-none-win32.whl", hash = "sha256:0d8a7e523ef03757a5aa29f591437d64d0d894635f8a50f370fe37f913ce4e19"},
+    {file = "watchdog-4.0.2-py3-none-win_amd64.whl", hash = "sha256:c344453ef3bf875a535b0488e3ad28e341adbd5a9ffb0f7d62cefacc8824ef2b"},
+    {file = "watchdog-4.0.2-py3-none-win_ia64.whl", hash = "sha256:baececaa8edff42cd16558a639a9b0ddf425f93d892e8392a56bf904f5eff22c"},
+    {file = "watchdog-4.0.2.tar.gz", hash = "sha256:b4dfbb6c49221be4535623ea4474a4d6ee0a9cef4a80b20c28db4d858b64e270"},
 ]
 
 [package.extras]
@@ -1836,13 +1923,13 @@ watchmedo = ["PyYAML (>=3.10)"]
 
 [[package]]
 name = "werkzeug"
-version = "3.0.1"
+version = "3.0.4"
 description = "The comprehensive WSGI web application library."
 optional = false
 python-versions = ">=3.8"
 files = [
-    {file = "werkzeug-3.0.1-py3-none-any.whl", hash = "sha256:90a285dc0e42ad56b34e696398b8122ee4c681833fb35b8334a095d82c56da10"},
-    {file = "werkzeug-3.0.1.tar.gz", hash = "sha256:507e811ecea72b18a404947aded4b3390e1db8f826b494d76550ef45bb3b1dcc"},
+    {file = "werkzeug-3.0.4-py3-none-any.whl", hash = "sha256:02c9eb92b7d6c06f31a782811505d2157837cea66aaede3e217c7c27c039476c"},
+    {file = "werkzeug-3.0.4.tar.gz", hash = "sha256:34f2371506b250df4d4f84bfe7b0921e4762525762bbd936614909fe25cd7306"},
 ]
 
 [package.dependencies]
@@ -1853,25 +1940,26 @@ watchdog = ["watchdog (>=2.3)"]
 
 [[package]]
 name = "zipp"
-version = "3.17.0"
+version = "3.20.0"
 description = "Backport of pathlib-compatible object wrapper for zip files"
 optional = false
 python-versions = ">=3.8"
 files = [
-    {file = "zipp-3.17.0-py3-none-any.whl", hash = "sha256:0e923e726174922dce09c53c59ad483ff7bbb8e572e00c7f7c46b88556409f31"},
-    {file = "zipp-3.17.0.tar.gz", hash = "sha256:84e64a1c28cf7e91ed2078bb8cc8c259cb19b76942096c8d7b84947690cabaf0"},
+    {file = "zipp-3.20.0-py3-none-any.whl", hash = "sha256:58da6168be89f0be59beb194da1250516fdaa062ccebd30127ac65d30045e10d"},
+    {file = "zipp-3.20.0.tar.gz", hash = "sha256:0145e43d89664cfe1a2e533adc75adafed82fe2da404b4bbb6b026c0157bdb31"},
 ]
 
 [package.extras]
-docs = ["furo", "jaraco.packaging (>=9.3)", "jaraco.tidelift (>=1.4)", "rst.linker (>=1.9)", "sphinx (<7.2.5)", "sphinx (>=3.5)", "sphinx-lint"]
-testing = ["big-O", "jaraco.functools", "jaraco.itertools", "more-itertools", "pytest (>=6)", "pytest-black (>=0.3.7)", "pytest-checkdocs (>=2.4)", "pytest-cov", "pytest-enabler (>=2.2)", "pytest-ignore-flaky", "pytest-mypy (>=0.9.1)", "pytest-ruff"]
+doc = ["furo", "jaraco.packaging (>=9.3)", "jaraco.tidelift (>=1.4)", "rst.linker (>=1.9)", "sphinx (>=3.5)", "sphinx-lint"]
+test = ["big-O", "importlib-resources", "jaraco.functools", "jaraco.itertools", "jaraco.test", "more-itertools", "pytest (>=6,!=8.1.*)", "pytest-checkdocs (>=2.4)", "pytest-cov", "pytest-enabler (>=2.2)", "pytest-ignore-flaky", "pytest-mypy", "pytest-ruff (>=0.2.1)"]
 
 [extras]
+cffi = ["setuptools"]
 numpy = ["numpy"]
 scipy = ["scipy"]
-taco = ["tensora-taco"]
+taco = ["setuptools", "tensora-taco"]
 
 [metadata]
 lock-version = "2.0"
 python-versions = "^3.10"
-content-hash = "1acc24ea01fbef60d73ecd521062d989185736563a0b046603b872b6078e2ee0"
+content-hash = "9b51176f0473d286943bd5d3a0ff6f42d555c8d02b9d550f63ea26084afa2374"
diff --git a/pyproject.toml b/pyproject.toml
index 2365f04..8635292 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -21,9 +21,10 @@ classifiers = [
 python = "^3.10"
 returns = "^0.20|^0.21|^0.22"
 cffi = "^1.16"
-# setuptools is required by cffi on Python 3.12 and up
+# setuptools is required for cffi compilation on Python 3.12 and up
 # https://cffi.readthedocs.io/en/latest/whatsnew.html#v1-16-0rc1
-setuptools = { version = "^69", python = ">=3.12" }
+setuptools = { version = "^69", python = ">=3.12", optional = true }
+llvmlite = "^0.42"
 parsita = "^2.0"
 typer = "^0.9"
 tensora-taco = { version = "^0.1.2", optional = true }
@@ -51,7 +52,8 @@ mkdocs-material = "^9"
 tensora = "tensora.cli:app"
 
 [tool.poetry.extras]
-taco = ["tensora-taco"]
+taco = ["tensora-taco", "setuptools"]
+cffi = ["setuptools"]
 numpy = ["numpy"]
 scipy = ["scipy"]
 
@@ -98,7 +100,7 @@ extend-immutable-calls = ["hypothesis.strategies.integers"]
 
 [tool.ruff.lint.per-file-ignores]
 # Allow glob imports and uppercase identifiers in tests
-"tests/*" = ["F403", "F405", "N802", "N806"]
+"tests*/*" = ["F403", "F405", "N802", "N806"]
 
 # F401: Allow unused imports in __init__.py
 "__init__.py" = ["F401"]
diff --git a/src/tensora/__init__.py b/src/tensora/__init__.py
index 9b3a383..b04ba8f 100644
--- a/src/tensora/__init__.py
+++ b/src/tensora/__init__.py
@@ -1,4 +1,4 @@
+from .compile import evaluate, evaluate_taco, evaluate_tensora, tensor_method
 from .format import Format, Mode
-from .function import evaluate, evaluate_taco, evaluate_tensora, tensor_method
 from .generate import TensorCompiler
 from .tensor import Tensor
diff --git a/src/tensora/cli.py b/src/tensora/cli.py
index a475052..3cc3e3a 100644
--- a/src/tensora/cli.py
+++ b/src/tensora/cli.py
@@ -9,7 +9,7 @@
 
 from .expression import parse_assignment
 from .format import parse_named_format
-from .generate import TensorCompiler, generate_c_code
+from .generate import Language, TensorCompiler, generate_code
 from .kernel_type import KernelType
 from .problem import make_problem
 
@@ -52,6 +52,14 @@ def tensora(
             help="The tensor algebra compiler to use to generate the kernel.",
         ),
     ] = TensorCompiler.tensora,
+    language: Annotated[
+        Language,
+        typer.Option(
+            "--language",
+            "-l",
+            help="The language in which to generate the kernel.",
+        ),
+    ] = Language.c,
     output_path: Annotated[
         Optional[Path],
         typer.Option(
@@ -107,7 +115,7 @@ def tensora(
             raise NotImplementedError()
 
     # Generate code
-    match generate_c_code(problem, kernel_types, tensor_compiler):
+    match generate_code(problem, kernel_types, tensor_compiler, language):
         case Failure(error):
             typer.echo(str(error), err=True)
             raise typer.Exit(1)
diff --git a/src/tensora/codegen/__init__.py b/src/tensora/codegen/__init__.py
index 8b546ef..f732e66 100644
--- a/src/tensora/codegen/__init__.py
+++ b/src/tensora/codegen/__init__.py
@@ -1 +1,2 @@
 from ._ir_to_c import ir_to_c, ir_to_c_function_definition, ir_to_c_statement
+from ._ir_to_llvm import ir_to_llvm
diff --git a/src/tensora/codegen/_ir_to_c.py b/src/tensora/codegen/_ir_to_c.py
index 69edeb6..0a399b3 100644
--- a/src/tensora/codegen/_ir_to_c.py
+++ b/src/tensora/codegen/_ir_to_c.py
@@ -160,13 +160,13 @@ def ir_to_c_boolean_to_integer(self: BooleanToInteger) -> str:
 
 @ir_to_c_expression.register(ArrayAllocate)
 def ir_to_c_array_allocate(self: ArrayAllocate) -> str:
-    return f"malloc(sizeof({type_to_c(self.type)}) * {parens(self.n_elements, (Add, Subtract))})"
+    return f"malloc(sizeof({type_to_c(self.element_type)}) * {parens(self.n_elements, (Add, Subtract))})"
 
 
 @ir_to_c_expression.register(ArrayReallocate)
 def ir_to_c_array_reallocate(self: ArrayReallocate) -> str:
     old = ir_to_c_expression(self.old)
-    return f"realloc({old}, sizeof({type_to_c(self.type)}) * {parens(self.n_elements, (Add, Subtract))})"
+    return f"realloc({old}, sizeof({type_to_c(self.element_type)}) * {parens(self.n_elements, (Add, Subtract))})"
 
 
 def ir_to_c_declaration(self: Declaration) -> str:
diff --git a/src/tensora/codegen/_ir_to_llvm.py b/src/tensora/codegen/_ir_to_llvm.py
new file mode 100644
index 0000000..3980d88
--- /dev/null
+++ b/src/tensora/codegen/_ir_to_llvm.py
@@ -0,0 +1,537 @@
+__all__ = ["ir_to_llvm"]
+
+from functools import singledispatch
+
+import llvmlite.ir as llvm
+
+from ..compile import target
+from ..ir.ast import (
+    Add,
+    And,
+    ArrayAllocate,
+    ArrayIndex,
+    ArrayReallocate,
+    Assignable,
+    Assignment,
+    AttributeAccess,
+    Block,
+    BooleanLiteral,
+    BooleanToInteger,
+    Branch,
+    Declaration,
+    DeclarationAssignment,
+    Equal,
+    Expression,
+    FloatLiteral,
+    FunctionDefinition,
+    GreaterThan,
+    GreaterThanOrEqual,
+    IntegerLiteral,
+    LessThan,
+    LessThanOrEqual,
+    Loop,
+    Max,
+    Min,
+    Module,
+    Multiply,
+    NotEqual,
+    Or,
+    Return,
+    Statement,
+    Subtract,
+    Variable,
+)
+from ._type_to_llvm import (
+    attribute_indexes,
+    llvm_boolean_type,
+    llvm_float_type,
+    llvm_integer_type,
+    llvm_size_type,
+    type_to_llvm,
+)
+
+# Create a target machine from the target so that we can get the word sizes
+target_machine = target.create_target_machine()
+
+
+@singledispatch
+def get_element_pointer(
+    self: Assignable, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> llvm.Value:
+    raise NotImplementedError(f"get_element_pointer not implemented for {type(self)}: {self}")
+
+
+@get_element_pointer.register(Variable)
+def get_element_pointer_variable(
+    self: Variable, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> llvm.Value:
+    return locals[self.name]
+
+
+@get_element_pointer.register(AttributeAccess)
+def get_element_pointer_attribute_access(
+    self: AttributeAccess, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> llvm.Value:
+    stem = ir_to_llvm_expression(self.target, builder, locals)
+    return builder.gep(
+        stem,
+        [
+            llvm.Constant(llvm_size_type, 0),
+            # All structs are indexed with i32 in LLVM
+            llvm.Constant(llvm.IntType(32), attribute_indexes[self.attribute]),
+        ],
+    )
+
+
+@get_element_pointer.register(ArrayIndex)
+def get_element_pointer_array_index(
+    self: ArrayIndex, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> llvm.Value:
+    target = ir_to_llvm_expression(self.target, builder, locals)
+    index = ir_to_llvm_expression(self.index, builder, locals)
+    return builder.gep(target, [index])
+
+
+@singledispatch
+def ir_to_llvm_expression(
+    self: Expression, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> llvm.Value:
+    raise NotImplementedError(f"ir_to_llvm_expression not implemented for {type(self)}: {self}")
+
+
+@ir_to_llvm_expression.register(Variable)
+def ir_to_llvm_variable(
+    self: Variable, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> llvm.Value:
+    return builder.load(locals[self.name])
+
+
+@ir_to_llvm_expression.register(AttributeAccess)
+def ir_to_llvm_attribute_access(
+    self: AttributeAccess, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> llvm.Value:
+    stem = ir_to_llvm_expression(self.target, builder, locals)
+    return builder.load(
+        builder.gep(
+            stem,
+            [
+                llvm.Constant(llvm_size_type, 0),
+                # All structs are indexed with i32 in LLVM
+                llvm.Constant(llvm.IntType(32), attribute_indexes[self.attribute]),
+            ],
+        )
+    )
+
+
+@ir_to_llvm_expression.register(ArrayIndex)
+def ir_to_llvm_array_index(
+    self: ArrayIndex, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> llvm.Value:
+    target = ir_to_llvm_expression(self.target, builder, locals)
+    index = ir_to_llvm_expression(self.index, builder, locals)
+    return builder.load(builder.gep(target, [index]))
+
+
+@ir_to_llvm_expression.register(IntegerLiteral)
+def ir_to_llvm_integer_literal(
+    self: IntegerLiteral, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> llvm.Value:
+    return llvm.Constant(llvm_integer_type, self.value)
+
+
+@ir_to_llvm_expression.register(FloatLiteral)
+def ir_to_llvm_float_literal(
+    self: FloatLiteral, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> llvm.Value:
+    return llvm.Constant(llvm_float_type, self.value)
+
+
+@ir_to_llvm_expression.register(BooleanLiteral)
+def ir_to_llvm_boolean_literal(
+    self: BooleanLiteral, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> llvm.Value:
+    return llvm.Constant(llvm_boolean_type, self.value)
+
+
+@ir_to_llvm_expression.register(Add)
+def ir_to_llvm_add(
+    self: Add, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> llvm.Value:
+    left = ir_to_llvm_expression(self.left, builder, locals)
+    right = ir_to_llvm_expression(self.right, builder, locals)
+    match left.type, right.type:
+        case (llvm.IntType(), llvm.IntType()):
+            return builder.add(left, right)
+        case (llvm.IntType(), llvm.DoubleType()):
+            left = builder.sitofp(left, llvm_float_type)
+            return builder.fadd(left, right)
+        case (llvm.DoubleType(), llvm.IntType()):
+            right = builder.sitofp(right, llvm_float_type)
+            return builder.fadd(left, right)
+        case (llvm.DoubleType(), llvm.DoubleType()):
+            return builder.fadd(left, right)
+        case (llvm.PointerType(), llvm.IntType()):
+            return builder.gep(left, [right])
+        case _:
+            raise TypeError(f"Cannot add {left.type} and {right.type}")
+
+
+@ir_to_llvm_expression.register(Subtract)
+def ir_to_llvm_subtract(
+    self: Subtract, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> llvm.Value:
+    left = ir_to_llvm_expression(self.left, builder, locals)
+    right = ir_to_llvm_expression(self.right, builder, locals)
+    match left.type, right.type:
+        case (llvm.IntType(), llvm.IntType()):
+            return builder.sub(left, right)
+        case (llvm.IntType(), llvm.DoubleType()):
+            left = builder.sitofp(left, llvm_float_type)
+            return builder.fsub(left, right)
+        case (llvm.DoubleType(), llvm.IntType()):
+            right = builder.sitofp(right, llvm_float_type)
+            return builder.fsub(left, right)
+        case (llvm.DoubleType(), llvm.DoubleType()):
+            return builder.fsub(left, right)
+        case _:
+            raise TypeError(f"Cannot add {left.type} and {right.type}")
+
+
+@ir_to_llvm_expression.register(Multiply)
+def ir_to_llvm_multiply(
+    self: Multiply, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> llvm.Value:
+    left = ir_to_llvm_expression(self.left, builder, locals)
+    right = ir_to_llvm_expression(self.right, builder, locals)
+    match left.type, right.type:
+        case (llvm.IntType(), llvm.IntType()):
+            return builder.mul(left, right)
+        case (llvm.IntType(), llvm.DoubleType()):
+            left = builder.sitofp(left, llvm_float_type)
+            return builder.fmul(left, right)
+        case (llvm.DoubleType(), llvm.IntType()):
+            right = builder.sitofp(right, llvm_float_type)
+            return builder.fmul(left, right)
+        case (llvm.DoubleType(), llvm.DoubleType()):
+            return builder.fmul(left, right)
+        case _:
+            raise TypeError(f"Cannot add {left.type} and {right.type}")
+
+
+@ir_to_llvm_expression.register(Equal)
+def ir_to_llvm_equal(
+    self: Equal, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> llvm.Value:
+    return builder.icmp_signed(
+        "==",
+        ir_to_llvm_expression(self.left, builder, locals),
+        ir_to_llvm_expression(self.right, builder, locals),
+    )
+
+
+@ir_to_llvm_expression.register(NotEqual)
+def ir_to_llvm_not_equal(
+    self: NotEqual, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> llvm.Value:
+    return builder.icmp_signed(
+        "!=",
+        ir_to_llvm_expression(self.left, builder, locals),
+        ir_to_llvm_expression(self.right, builder, locals),
+    )
+
+
+@ir_to_llvm_expression.register(GreaterThan)
+def ir_to_llvm_greater_than(
+    self: GreaterThan, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> llvm.Value:
+    return builder.icmp_signed(
+        ">",
+        ir_to_llvm_expression(self.left, builder, locals),
+        ir_to_llvm_expression(self.right, builder, locals),
+    )
+
+
+@ir_to_llvm_expression.register(LessThan)
+def ir_to_llvm_less_than(
+    self: LessThan, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> llvm.Value:
+    return builder.icmp_signed(
+        "<",
+        ir_to_llvm_expression(self.left, builder, locals),
+        ir_to_llvm_expression(self.right, builder, locals),
+    )
+
+
+@ir_to_llvm_expression.register(GreaterThanOrEqual)
+def ir_to_llvm_greater_than_or_equal(
+    self: GreaterThanOrEqual, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> llvm.Value:
+    return builder.icmp_signed(
+        ">=",
+        ir_to_llvm_expression(self.left, builder, locals),
+        ir_to_llvm_expression(self.right, builder, locals),
+    )
+
+
+@ir_to_llvm_expression.register(LessThanOrEqual)
+def ir_to_llvm_less_than_or_equal(
+    self: LessThanOrEqual, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> llvm.Value:
+    return builder.icmp_signed(
+        "<=",
+        ir_to_llvm_expression(self.left, builder, locals),
+        ir_to_llvm_expression(self.right, builder, locals),
+    )
+
+
+@ir_to_llvm_expression.register(And)
+def ir_to_llvm_and(
+    self: And, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> llvm.Value:
+    left_block = builder.block
+    right_block = builder.append_basic_block()
+    end_block = builder.append_basic_block()
+
+    left = ir_to_llvm_expression(self.left, builder, locals)
+
+    builder.cbranch(left, right_block, end_block)
+
+    builder.position_at_end(right_block)
+    right = ir_to_llvm_expression(self.right, builder, locals)
+    builder.branch(end_block)
+
+    builder.position_at_end(end_block)
+    phi = builder.phi(llvm_boolean_type)
+    phi.add_incoming(llvm.Constant(llvm_boolean_type, 0), left_block)
+    phi.add_incoming(right, right_block)
+
+    return phi
+
+
+@ir_to_llvm_expression.register(Or)
+def ir_to_llvm_or(self: Or, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]) -> llvm.Value:
+    left_block = builder.block
+    right_block = builder.append_basic_block()
+    end_block = builder.append_basic_block()
+
+    left = ir_to_llvm_expression(self.left, builder, locals)
+
+    builder.cbranch(left, end_block, right_block)
+
+    builder.position_at_end(right_block)
+    right = ir_to_llvm_expression(self.right, builder, locals)
+    builder.branch(end_block)
+
+    builder.position_at_end(end_block)
+    phi = builder.phi(llvm_boolean_type)
+    phi.add_incoming(llvm.Constant(llvm_boolean_type, 1), left_block)
+    phi.add_incoming(right, right_block)
+
+    return phi
+
+
+@ir_to_llvm_expression.register(Max)
+def ir_to_llvm_max(
+    self: Max, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> llvm.Value:
+    left = ir_to_llvm_expression(self.left, builder, locals)
+    right = ir_to_llvm_expression(self.right, builder, locals)
+    condition = builder.icmp_signed(">", left, right)
+    return builder.select(condition, left, right)
+
+
+@ir_to_llvm_expression.register(Min)
+def ir_to_llvm_min(
+    self: Min, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> llvm.Value:
+    left = ir_to_llvm_expression(self.left, builder, locals)
+    right = ir_to_llvm_expression(self.right, builder, locals)
+    condition = builder.icmp_signed("<", left, right)
+    return builder.select(condition, left, right)
+
+
+@ir_to_llvm_expression.register(BooleanToInteger)
+def ir_to_llvm_boolean_to_integer(
+    self: BooleanToInteger, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> llvm.Value:
+    expression = ir_to_llvm_expression(self.expression, builder, locals)
+    return builder.zext(expression, llvm_integer_type)
+
+
+@ir_to_llvm_expression.register(ArrayAllocate)
+def ir_to_llvm_array_allocate(
+    self: ArrayAllocate, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> llvm.Value:
+    element_size = llvm.Constant(
+        llvm_integer_type, type_to_llvm(self.element_type).get_abi_size(target_machine.target_data)
+    )
+    n_elements = ir_to_llvm_expression(self.n_elements, builder, locals)
+    memory_size = builder.zext(builder.mul(element_size, n_elements), llvm_size_type)
+    memory_pointer = builder.call(locals["malloc"], [memory_size])
+    array_pointer = builder.bitcast(memory_pointer, type_to_llvm(self.element_type).as_pointer())
+    return array_pointer
+
+
+@ir_to_llvm_expression.register(ArrayReallocate)
+def ir_to_llvm_array_reallocate(
+    self: ArrayReallocate, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> llvm.Value:
+    old_array_pointer = get_element_pointer(self.old, builder, locals)
+    old_memory_pointer = builder.bitcast(old_array_pointer, llvm.IntType(8).as_pointer())
+    element_size = llvm.Constant(
+        llvm_integer_type, type_to_llvm(self.element_type).get_abi_size(target_machine.target_data)
+    )
+    n_elements = ir_to_llvm_expression(self.n_elements, builder, locals)
+    memory_size = builder.zext(builder.mul(element_size, n_elements), llvm_size_type)
+    memory_pointer = builder.call(locals["realloc"], [old_memory_pointer, memory_size])
+    array_pointer = builder.bitcast(memory_pointer, type_to_llvm(self.element_type).as_pointer())
+    return array_pointer
+
+
+def ir_to_llvm_declaration(
+    self: Declaration, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> llvm.Value:
+    variable = builder.alloca(type_to_llvm(self.type), name=self.name.name)
+    locals[self.name.name] = variable
+    return variable
+
+
+@singledispatch
+def ir_to_llvm_statement(
+    self: Statement, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> None:
+    raise NotImplementedError(f"ir_to_llvm_statement not implemented for {type(self)}: {self}")
+
+
+@ir_to_llvm_statement.register(Expression)
+def convert_expression_to_statement(
+    self: Expression, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> None:
+    # An expression can also be a statement; run it here and throw away the result
+    _ = ir_to_llvm_expression(self)
+
+
+@ir_to_llvm_statement.register(Declaration)
+def convert_declaration_to_statement(
+    self: Declaration, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> None:
+    # A declaration can also be a statement; declare the variable and throw away the result
+    _ = ir_to_llvm_declaration(self, builder, locals)
+
+
+@ir_to_llvm_statement.register(Assignment)
+def ir_to_llvm_assignment(
+    self: Assignment, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> None:
+    value = ir_to_llvm_expression(self.value, builder, locals)
+    target = get_element_pointer(self.target, builder, locals)
+    match value.type, target.type.pointee:
+        case (llvm.IntType(), llvm.DoubleType()):
+            value = builder.sitofp(value, llvm_float_type)
+        case _:
+            pass
+    builder.store(value, target)
+
+
+@ir_to_llvm_statement.register(DeclarationAssignment)
+def ir_to_llvm_declaration_assignment(
+    self: DeclarationAssignment, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> None:
+    value = ir_to_llvm_expression(self.value, builder, locals)
+    target = ir_to_llvm_declaration(self.target, builder, locals)
+    match value.type, target.type:
+        case (llvm.IntType(), llvm.DoubleType()):
+            value = builder.sitofp(value, llvm_float_type)
+        case _:
+            pass
+    builder.store(value, target)
+
+
+@ir_to_llvm_statement.register(Block)
+def ir_to_llvm_block(self: Block, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]) -> None:
+    if self.comment is not None:
+        # Add comment if it there is one
+        builder.comment(self.comment)
+
+    for statement in self.statements:
+        ir_to_llvm_statement(statement, builder, locals)
+
+
+@ir_to_llvm_statement.register(Branch)
+def ir_to_llvm_branch(
+    self: Branch, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> None:
+    condition = ir_to_llvm_expression(self.condition, builder, locals)
+    with builder.if_else(condition) as (then, otherwise):
+        with then:
+            ir_to_llvm_statement(self.if_true, builder, locals)
+        with otherwise:
+            ir_to_llvm_statement(self.if_false, builder, locals)
+
+
+@ir_to_llvm_statement.register(Loop)
+def ir_to_llvm_loop(self: Loop, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]) -> None:
+    condition_block = builder.append_basic_block()
+    body_block = builder.append_basic_block()
+    end_block = builder.append_basic_block()
+
+    builder.branch(condition_block)
+
+    builder.position_at_end(condition_block)
+    condition = ir_to_llvm_expression(self.condition, builder, locals)
+    builder.cbranch(condition, body_block, end_block)
+
+    builder.position_at_end(body_block)
+    ir_to_llvm_statement(self.body, builder, locals)
+    builder.branch(condition_block)
+
+    builder.position_at_end(end_block)
+
+
+@ir_to_llvm_statement.register(Return)
+def ir_to_llvm_return(
+    self: Return, builder: llvm.IRBuilder, locals: dict[str, llvm.Value]
+) -> None:
+    builder.ret(ir_to_llvm_expression(self.value, builder, locals))
+
+
+def ir_to_llvm_function_definition(
+    self: FunctionDefinition, module: llvm.Module, functions: dict[str, llvm.Function]
+) -> None:
+    return_type = type_to_llvm(self.return_type)
+    argument_types = [type_to_llvm(parameter.type) for parameter in self.parameters]
+    function_type = llvm.FunctionType(return_type, argument_types)
+    function = llvm.Function(module, function_type, name=self.name.name)
+
+    body = llvm.IRBuilder(function.append_basic_block())
+    locals = {}
+
+    # Assign each function parameter to a local variable
+    for parameter, llvm_parameter in zip(self.parameters, function.args):
+        variable = body.alloca(type_to_llvm(parameter.type), name=parameter.name.name)
+        body.store(llvm_parameter, variable)
+        locals[parameter.name.name] = variable
+
+    ir_to_llvm_statement(self.body, body, {**locals, **functions})
+
+
+def ir_to_llvm(self: Module) -> llvm.Module:
+    module = llvm.Module()
+
+    # Declare malloc
+    malloc_signature = llvm.FunctionType(llvm.IntType(8).as_pointer(), [llvm_size_type])
+    malloc = llvm.Function(module, malloc_signature, name="malloc")
+
+    # Declare realloc
+    realloc_signature = llvm.FunctionType(
+        llvm.IntType(8).as_pointer(), [llvm.IntType(8).as_pointer(), llvm_size_type]
+    )
+    realloc = llvm.Function(module, realloc_signature, name="realloc")
+
+    # This is not compatible with defined functions ever being called
+    functions = {"malloc": malloc, "realloc": realloc}
+
+    for function in self.definitions:
+        ir_to_llvm_function_definition(function, module, functions)
+
+    return module
diff --git a/src/tensora/codegen/_type_to_llvm.py b/src/tensora/codegen/_type_to_llvm.py
new file mode 100644
index 0000000..a9423b3
--- /dev/null
+++ b/src/tensora/codegen/_type_to_llvm.py
@@ -0,0 +1,85 @@
+__all__ = [
+    "llvm_integer_type",
+    "llvm_float_type",
+    "llvm_boolean_type",
+    "llvm_mode_type",
+    "llvm_size_type",
+    "type_to_llvm",
+    "attribute_indexes",
+]
+
+from functools import singledispatch
+
+import llvmlite.ir as llvm
+
+from ..ir.types import Array, FixedArray, Float, Integer, Mode, Pointer, Tensor, Type
+
+llvm_integer_type = llvm.IntType(32)
+llvm_float_type = llvm.DoubleType()
+llvm_boolean_type = llvm.IntType(1)
+llvm_mode_type = llvm.IntType(8)
+llvm_size_type = llvm.IntType(64)
+
+
+@singledispatch
+def type_to_llvm(self: Type) -> llvm.Type:
+    raise NotImplementedError(f"type_to_llvm not implemented for {type(self)}: {self}")
+
+
+@type_to_llvm.register(Integer)
+def type_to_llvm_integer(self: Integer) -> llvm.Type:
+    return llvm_integer_type
+
+
+@type_to_llvm.register(Float)
+def type_to_llvm_float(self: Float) -> llvm.Type:
+    return llvm_float_type
+
+
+tensor_attribute_indexes = {
+    "dimensions": 1,
+    "indices": 5,
+    "vals": 6,
+}
+
+
+@type_to_llvm.register(Tensor)
+def type_to_llvm_tensor(self: Tensor) -> llvm.Type:
+    return llvm.LiteralStructType(
+        [
+            llvm_integer_type,  # order
+            llvm.PointerType(llvm_integer_type),  # dimensions
+            llvm_integer_type,  # csize
+            llvm.PointerType(llvm_integer_type),  # mode_ordering
+            llvm_mode_type,  # mode_types
+            llvm.PointerType(llvm.PointerType(llvm.PointerType(llvm_integer_type))),  # indices
+            llvm.PointerType(llvm_float_type),  # vals
+            llvm_integer_type,  # vals_size
+        ]
+    )
+
+
+@type_to_llvm.register(Mode)
+def type_to_llvm_mode(self: Mode) -> llvm.Type:
+    return llvm_mode_type
+
+
+@type_to_llvm.register(Pointer)
+def type_to_llvm_pointer(self: Pointer) -> llvm.Type:
+    return llvm.PointerType(type_to_llvm(self.target))
+
+
+@type_to_llvm.register(Array)
+def type_to_llvm_array(self: Array) -> llvm.Type:
+    return llvm.PointerType(type_to_llvm(self.element))
+
+
+@type_to_llvm.register(FixedArray)
+def type_to_llvm_fixed_array(self: FixedArray) -> llvm.Type:
+    return llvm.ArrayType(type_to_llvm(self.element), self.n)
+
+
+# LLVM index for all attributes.
+# If we ever reuse an attribute name, and it goes to a different index, the whole IR will have to
+# be redesigned.
+attribute_indexes = tensor_attribute_indexes
diff --git a/src/tensora/compile/__init__.py b/src/tensora/compile/__init__.py
new file mode 100644
index 0000000..ae0428e
--- /dev/null
+++ b/src/tensora/compile/__init__.py
@@ -0,0 +1,16 @@
+from ._cffi_ownership import (
+    allocate_taco_structure,
+    taco_structure_to_cffi,
+    take_ownership_of_arrays,
+    take_ownership_of_tensor,
+    take_ownership_of_tensor_members,
+    tensor_cdefs,
+)
+from ._initialize_llvm import target
+from ._porcelain import evaluate, evaluate_cffi, evaluate_taco, evaluate_tensora, tensor_method
+from ._tensor_method import (
+    BackendCompiler,
+    BroadcastTargetIndexError,
+    TensorMethod,
+    UnsupportedBackendCompilerError,
+)
diff --git a/src/tensora/compile.py b/src/tensora/compile/_cffi_ownership.py
similarity index 81%
rename from src/tensora/compile.py
rename to src/tensora/compile/_cffi_ownership.py
index 6547edb..f1bc7ee 100644
--- a/src/tensora/compile.py
+++ b/src/tensora/compile/_cffi_ownership.py
@@ -1,29 +1,16 @@
 __all__ = [
-    "generate_library",
     "allocate_taco_structure",
     "taco_structure_to_cffi",
     "take_ownership_of_arrays",
-    "tensor_cdefs",
     "take_ownership_of_tensor_members",
+    "take_ownership_of_tensor",
+    "taco_type_header",
+    "tensor_cdefs",
 ]
 
-import re
-import tempfile
-import threading
-from pathlib import Path
-from typing import Any
 from weakref import WeakKeyDictionary
 
 from cffi import FFI
-from returns.result import Failure, Success
-
-from .generate import TensorCompiler, generate_c_code
-from .kernel_type import KernelType
-from .problem import Problem
-
-lock = threading.Lock()
-
-taco_binary = Path(__file__).parent.joinpath("taco/bin/taco")
 
 global_weakkeydict = WeakKeyDictionary()
 
@@ -58,14 +45,6 @@
 #   coordinate of each value
 # vals_size: Deprecated https://github.com/tensor-compiler/taco/issues/208#issuecomment-476314322
 
-taco_define_header = """
-    #ifndef TACO_C_HEADERS
-    #define TACO_C_HEADERS
-    #define TACO_MIN(_a,_b) ((_a) < (_b) ? (_a) : (_b))
-    #define TACO_MAX(_a,_b) ((_a) > (_b) ? (_a) : (_b))
-    #endif
-"""
-
 taco_type_header = """
     typedef enum { taco_mode_dense, taco_mode_sparse } taco_mode_t;
 
@@ -76,7 +55,7 @@
       int32_t*     mode_ordering; // mode storage ordering
       taco_mode_t* mode_types;    // mode storage types
       int32_t***   indices;       // tensor index data (per mode)
-      double*     vals;          // tensor values
+      double*      vals;          // tensor values
       int32_t      vals_size;     // values array size
     } taco_tensor_t;
 
@@ -96,71 +75,10 @@
 tensor_lib = tensor_cdefs.dlopen(None)
 
 
-def generate_library(
-    problem: Problem, compiler: TensorCompiler = TensorCompiler.tensora
-) -> tuple[list[str], Any]:
-    """Generate source, compile it, and load it.
-
-    Given a problem:
-    (1) invoke the tensor algebra compiler to generate C code for evaluate
-    (2) parse the signature in the source to determine the order of arguments
-    (3) compile the source with cffi
-    (4) return the list of parameter names and the compiled library
-
-    Args:
-        problem: A valid tensor algebra expression and associated tensor formats
-        compiler: The tensor algebra compiler to use to generate the C code
-
-    Returns:
-        A tuple where the first element is the list of variable names in the order they appear in the function
-        signature, and the second element is the compiled FFILibrary which has a single method `evaluate` which expects
-        cffi pointers to taco_tensor_t instances in order specified by the list of variable names.
-    """
-    match generate_c_code(problem, [KernelType.evaluate], compiler):
-        case Failure(error):
-            raise error
-        case Success(source):
-            pass
-        case _:
-            raise NotImplementedError()
-
-    # Determine signature
-    # 1) Find function by name and capture its parameter list
-    # 2) Find each parameter by `*` and capture its name
-    signature_match = re.search(r"int(32_t)? evaluate\(([^)]*)\)", source)
-    signature = signature_match.group(0)
-    parameter_list_matches = re.finditer(r"\*([ ]*restrict[ ]*)?([^,]*)", signature_match.group(2))
-    parameter_names = [match.group(2) for match in parameter_list_matches]
-
-    # Use cffi to compile the kernels
-    ffibuilder = FFI()
-    ffibuilder.include(tensor_cdefs)
-    ffibuilder.cdef(signature + ";")
-    ffibuilder.set_source(
-        "taco_kernel",
-        taco_define_header + taco_type_header + source,
-        extra_compile_args=["-Wno-unused-variable", "-Wno-unknown-pragmas"],
-    )
-
-    with tempfile.TemporaryDirectory() as temp_dir:
-        # Lock because FFI.compile is not thread safe: https://foss.heptapod.net/pypy/cffi/-/issues/490
-        with lock:
-            # Create shared object in temporary directory
-            lib_path = ffibuilder.compile(tmpdir=temp_dir)
-
-        # Load the shared object
-        lib = ffibuilder.dlopen(lib_path)
-
-    # Return the parameter names because we need to know the order in which to send the arguments. It appears that this
-    # order is always the order in which the name first appears in the expression left-to-right, but it is not clear
-    # that this is guaranteed.
-    # Return the entire library rather than just the function because it appears that the memory containing the compiled
-    # code is freed as soon as the library goes out of scope: https://stackoverflow.com/q/55323592/1485877
-    return parameter_names, lib
-
-
 def allocate_taco_structure(
-    mode_types: tuple[int, ...], dimensions: tuple[int, ...], mode_ordering: tuple[int, ...]
+    mode_types: tuple[int, ...],
+    dimensions: tuple[int, ...],
+    mode_ordering: tuple[int, ...],
 ):
     """Allocate all parts of a taco tensor except growable arrays.
 
diff --git a/src/tensora/compile/_compile_cffi.py b/src/tensora/compile/_compile_cffi.py
new file mode 100644
index 0000000..7cc468a
--- /dev/null
+++ b/src/tensora/compile/_compile_cffi.py
@@ -0,0 +1,59 @@
+__all__ = ["compile_evaluate"]
+
+import re
+import tempfile
+import threading
+from typing import Any
+
+from cffi import FFI
+
+from ._cffi_ownership import taco_type_header, tensor_cdefs
+
+lock = threading.Lock()
+
+taco_define_header = """
+    #ifndef TACO_C_HEADERS
+    #define TACO_C_HEADERS
+    #define TACO_MIN(_a,_b) ((_a) < (_b) ? (_a) : (_b))
+    #define TACO_MAX(_a,_b) ((_a) > (_b) ? (_a) : (_b))
+    #endif
+"""
+
+
+def compile_evaluate(source: str) -> Any:
+    """Compile evaluate kernel in C code using CFFI.
+
+    Args:
+        source: C code containing one evaluate function
+
+    Returns:
+        The compiled FFILibrary which has a single method `evaluate` which
+        expects cffi pointers to taco_tensor_t instances.
+    """
+    # Extract signature
+    # This needs to be provided alone to cdef
+    signature_match = re.search(r"int(32_t)? evaluate\(([^)]*)\)", source)
+    signature = signature_match.group(0)
+
+    # Use cffi to compile the kernels
+    ffibuilder = FFI()
+    ffibuilder.include(tensor_cdefs)
+    ffibuilder.cdef(signature + ";")
+    ffibuilder.set_source(
+        "taco_kernel",
+        taco_define_header + taco_type_header + source,
+        extra_compile_args=["-Wno-unused-variable", "-Wno-unknown-pragmas"],
+    )
+
+    with tempfile.TemporaryDirectory() as temp_dir:
+        # Lock because FFI.compile is not thread safe: https://foss.heptapod.net/pypy/cffi/-/issues/490
+        with lock:
+            # Create shared object in temporary directory
+            lib_path = ffibuilder.compile(tmpdir=temp_dir)
+
+        # Load the shared object
+        lib = ffibuilder.dlopen(lib_path)
+
+    # Return the entire library rather than just the function because it appears that the memory containing the compiled
+    # code is freed as soon as the library goes out of scope: https://stackoverflow.com/q/55323592/1485877
+    return lib
diff --git a/src/tensora/compile/_compile_llvm.py b/src/tensora/compile/_compile_llvm.py
new file mode 100644
index 0000000..a58e461
--- /dev/null
+++ b/src/tensora/compile/_compile_llvm.py
@@ -0,0 +1,31 @@
+__all__ = ["compile_module"]
+
+import llvmlite.binding as llvm
+
+from ..codegen import ir_to_llvm
+from ..ir.ast import Module
+from ._initialize_llvm import target
+
+
+def compile_module(module: Module) -> llvm.ExecutionEngine:
+    llvm_ir = ir_to_llvm(module)
+
+    # Compile the module
+    llvm_module = llvm.parse_assembly(str(llvm_ir))
+    llvm_module.verify()
+
+    # Create target machine
+    # We have to recreate this for every module because create_mcjit_compiler
+    # takes ownership of target_machine and frees it when the engine goes out of scope
+    target_machine = target.create_target_machine()
+
+    # Create execution engine
+    backing_mod = llvm.parse_assembly("")
+    engine = llvm.create_mcjit_compiler(backing_mod, target_machine)
+
+    # Add the module to the engine and make sure it is ready for execution
+    engine.add_module(llvm_module)
+    engine.finalize_object()
+    engine.run_static_constructors()
+
+    return engine
diff --git a/src/tensora/compile/_initialize_llvm.py b/src/tensora/compile/_initialize_llvm.py
new file mode 100644
index 0000000..3424299
--- /dev/null
+++ b/src/tensora/compile/_initialize_llvm.py
@@ -0,0 +1,12 @@
+__all__ = ["target"]
+
+import llvmlite.binding as llvm
+
+# Initialize the LLVM
+# https://llvmlite.readthedocs.io/en/latest/user-guide/binding/examples.html
+llvm.initialize()
+llvm.initialize_native_target()
+llvm.initialize_native_asmprinter()
+
+# Create the target representing the current host
+target = llvm.Target.from_default_triple()
diff --git a/src/tensora/compile/_porcelain.py b/src/tensora/compile/_porcelain.py
new file mode 100644
index 0000000..1844a46
--- /dev/null
+++ b/src/tensora/compile/_porcelain.py
@@ -0,0 +1,88 @@
+__all__ = ["tensor_method", "evaluate", "evaluate_taco", "evaluate_cffi", "evaluate_tensora"]
+
+from functools import lru_cache
+
+from returns.functions import raise_exception
+
+from ..expression import parse_assignment
+from ..format import parse_format
+from ..generate import TensorCompiler
+from ..problem import Problem, make_problem
+from ..tensor import Tensor
+from ._tensor_method import BackendCompiler, TensorMethod
+
+
+@lru_cache()
+def cachable_tensor_method(
+    problem: Problem, compiler: TensorCompiler, backend: BackendCompiler
+) -> TensorMethod:
+    return TensorMethod(problem, compiler, backend)
+
+
+def tensor_method(
+    assignment: str,
+    formats: dict[str, str],
+    compiler: TensorCompiler = TensorCompiler.tensora,
+    backend: BackendCompiler | None = None,
+) -> TensorMethod:
+    parsed_assignment = parse_assignment(assignment).alt(raise_exception).unwrap()
+    parsed_formats = {
+        name: parse_format(format).alt(raise_exception).unwrap()
+        for name, format in formats.items()
+    }
+
+    problem = make_problem(parsed_assignment, parsed_formats).alt(raise_exception).unwrap()
+
+    if backend is None:
+        if compiler == TensorCompiler.tensora:
+            backend = BackendCompiler.llvm
+        else:
+            backend = BackendCompiler.cffi
+
+    return cachable_tensor_method(problem, compiler, backend)
+
+
+def evaluate_taco(assignment: str, output_format: str, **inputs: Tensor) -> Tensor:
+    parsed_assignment = parse_assignment(assignment).alt(raise_exception).unwrap()
+    input_formats = {name: tensor.format for name, tensor in inputs.items()}
+    parsed_output_format = parse_format(output_format).alt(raise_exception).unwrap()
+
+    formats = {parsed_assignment.target.name: parsed_output_format} | input_formats
+
+    problem = make_problem(parsed_assignment, formats).alt(raise_exception).unwrap()
+
+    function = cachable_tensor_method(problem, TensorCompiler.taco, BackendCompiler.cffi)
+
+    return function(**inputs)
+
+
+def evaluate_cffi(assignment: str, output_format: str, **inputs: Tensor) -> Tensor:
+    parsed_assignment = parse_assignment(assignment).alt(raise_exception).unwrap()
+    input_formats = {name: tensor.format for name, tensor in inputs.items()}
+    parsed_output_format = parse_format(output_format).alt(raise_exception).unwrap()
+
+    formats = {parsed_assignment.target.name: parsed_output_format} | input_formats
+
+    problem = make_problem(parsed_assignment, formats).alt(raise_exception).unwrap()
+
+    function = cachable_tensor_method(problem, TensorCompiler.tensora, BackendCompiler.cffi)
+
+    return function(**inputs)
+
+
+def evaluate_tensora(assignment: str, output_format: str, **inputs: Tensor) -> Tensor:
+    parsed_assignment = parse_assignment(assignment).alt(raise_exception).unwrap()
+    input_formats = {name: tensor.format for name, tensor in inputs.items()}
+    parsed_output_format = parse_format(output_format).alt(raise_exception).unwrap()
+
+    formats = {parsed_assignment.target.name: parsed_output_format} | input_formats
+
+    problem = make_problem(parsed_assignment, formats).alt(raise_exception).unwrap()
+
+    function = cachable_tensor_method(problem, TensorCompiler.tensora, BackendCompiler.llvm)
+
+    return function(**inputs)
+
+
+def evaluate(assignment: str, output_format: str, **inputs: Tensor) -> Tensor:
+    return evaluate_tensora(assignment, output_format, **inputs)
diff --git a/src/tensora/function.py b/src/tensora/compile/_tensor_method.py
similarity index 65%
rename from src/tensora/function.py
rename to src/tensora/compile/_tensor_method.py
index de16153..b31af79 100644
--- a/src/tensora/function.py
+++ b/src/tensora/compile/_tensor_method.py
@@ -1,25 +1,22 @@
 __all__ = [
-    "tensor_method",
-    "evaluate",
-    "evaluate_taco",
-    "evaluate_tensora",
     "TensorMethod",
+    "BackendCompiler",
     "BroadcastTargetIndexError",
+    "UnsupportedBackendCompilerError",
 ]
 
 from dataclasses import dataclass
-from functools import lru_cache
+from enum import Enum
 from inspect import Parameter, Signature
 
-from returns.functions import raise_exception
+from returns.result import Failure, Success
 
-from .compile import allocate_taco_structure, generate_library, take_ownership_of_arrays
-from .expression import parse_assignment
-from .expression.ast import Assignment
-from .format import parse_format
-from .generate import TensorCompiler
-from .problem import Problem, make_problem
-from .tensor import Tensor
+from ..expression.ast import Assignment
+from ..generate import Language, TensorCompiler, generate_code, generate_module_tensora
+from ..kernel_type import KernelType
+from ..problem import Problem
+from ..tensor import Tensor
+from ._cffi_ownership import allocate_taco_structure, take_ownership_of_arrays, tensor_cdefs
 
 
 @dataclass(frozen=True, slots=True)
@@ -37,10 +34,29 @@ def __str__(self):
         )
 
 
+class BackendCompiler(Enum):
+    llvm = "llvm"
+    cffi = "cffi"
+
+
+@dataclass(frozen=True, slots=True)
+class UnsupportedBackendCompilerError(Exception):
+    backend_compiler: BackendCompiler
+    tensor_compiler: TensorCompiler
+
+    def __str__(self):
+        return f"Backend compiler {self.backend_compiler} is not supported for tensor compiler {self.tensor_compiler}"
+
+
 class TensorMethod:
     """A function taking specific tensor arguments."""
 
-    def __init__(self, problem: Problem, compiler: TensorCompiler = TensorCompiler.tensora):
+    def __init__(
+        self,
+        problem: Problem,
+        compiler: TensorCompiler = TensorCompiler.tensora,
+        backend: BackendCompiler = BackendCompiler.llvm,
+    ):
         # Reject broadcasting to outputs because there is no way to specify output dimensions that
         # do not have a corresponding input dimension
         input_indexes = set(problem.assignment.expression.index_participants().keys())
@@ -64,8 +80,35 @@ def __init__(self, problem: Problem, compiler: TensorCompiler = TensorCompiler.t
             ]
         )
 
-        # Compile taco function
-        self._parameter_order, self._cffi_lib = generate_library(problem, compiler)
+        match backend:
+            case BackendCompiler.llvm:
+                match compiler:
+                    case TensorCompiler.tensora:
+                        match generate_module_tensora(problem, [KernelType.evaluate]):
+                            case Failure(error):
+                                raise error
+                            case Success(tensora_module):
+                                from ._compile_llvm import compile_module
+
+                                self._lib = compile_module(tensora_module)
+
+                                # Convert ctypes function to cffi function
+                                function_type = (
+                                    f"int32_t (*)({', '.join(['void *'] * len(problem.formats))})"
+                                )
+                                function_pointer = self._lib.get_function_address("evaluate")
+                                self._evaluate = tensor_cdefs.cast(function_type, function_pointer)
+                    case TensorCompiler.taco:
+                        raise UnsupportedBackendCompilerError(backend, compiler)
+            case BackendCompiler.cffi:
+                match generate_code(problem, [KernelType.evaluate], compiler, Language.c):
+                    case Failure(error):
+                        raise error
+                    case Success(c_code):
+                        from ._compile_cffi import compile_evaluate
+
+                        self._lib = compile_evaluate(c_code)
+                        self._evaluate = self._lib.evaluate
 
     def __call__(self, *args, **kwargs):
         # Handle arguments like normal Python function
@@ -137,9 +180,9 @@ def __call__(self, *args, **kwargs):
 
         all_arguments = {self._output_name: output, **bound_arguments}
 
-        cffi_args = [all_arguments[name].cffi_tensor for name in self._parameter_order]
+        cffi_args = [all_arguments[name].cffi_tensor for name in self._problem.formats.keys()]
 
-        return_value = self._cffi_lib.evaluate(*cffi_args)
+        return_value = self._evaluate(*cffi_args)
 
         take_ownership_of_arrays(cffi_output)
 
@@ -147,56 +190,3 @@ def __call__(self, *args, **kwargs):
             raise RuntimeError(f"Kernel evaluation failed with error code {return_value}")
 
         return output
-
-
-@lru_cache()
-def cachable_tensor_method(problem: Problem, compiler: TensorCompiler) -> TensorMethod:
-    return TensorMethod(problem, compiler)
-
-
-def tensor_method(
-    assignment: str,
-    formats: dict[str, str],
-    compiler: TensorCompiler = TensorCompiler.tensora,
-) -> TensorMethod:
-    parsed_assignment = parse_assignment(assignment).alt(raise_exception).unwrap()
-    parsed_formats = {
-        name: parse_format(format).alt(raise_exception).unwrap()
-        for name, format in formats.items()
-    }
-
-    problem = make_problem(parsed_assignment, parsed_formats).alt(raise_exception).unwrap()
-
-    return cachable_tensor_method(problem, compiler)
-
-
-def evaluate_taco(assignment: str, output_format: str, **inputs: Tensor) -> Tensor:
-    parsed_assignment = parse_assignment(assignment).alt(raise_exception).unwrap()
-    input_formats = {name: tensor.format for name, tensor in inputs.items()}
-    parsed_output_format = parse_format(output_format).alt(raise_exception).unwrap()
-
-    formats = {parsed_assignment.target.name: parsed_output_format} | input_formats
-
-    problem = make_problem(parsed_assignment, formats).alt(raise_exception).unwrap()
-
-    function = cachable_tensor_method(problem, TensorCompiler.taco)
-
-    return function(**inputs)
-
-
-def evaluate_tensora(assignment: str, output_format: str, **inputs: Tensor) -> Tensor:
-    parsed_assignment = parse_assignment(assignment).alt(raise_exception).unwrap()
-    input_formats = {name: tensor.format for name, tensor in inputs.items()}
-    parsed_output_format = parse_format(output_format).alt(raise_exception).unwrap()
-
-    formats = {parsed_assignment.target.name: parsed_output_format} | input_formats
-
-    problem = make_problem(parsed_assignment, formats).alt(raise_exception).unwrap()
-
-    function = cachable_tensor_method(problem, TensorCompiler.tensora)
-
-    return function(**inputs)
-
-
-def evaluate(assignment: str, output_format: str, **inputs: Tensor) -> Tensor:
-    return evaluate_tensora(assignment, output_format, **inputs)
diff --git a/src/tensora/generate/__init__.py b/src/tensora/generate/__init__.py
index 09f1183..9567d6f 100644
--- a/src/tensora/generate/__init__.py
+++ b/src/tensora/generate/__init__.py
@@ -1,2 +1,3 @@
-from ._base import TensorCompiler, generate_c_code
+from ._base import Language, TensorCompiler, generate_code
 from ._taco import TacoError
+from ._tensora import generate_module_tensora
diff --git a/src/tensora/generate/_base.py b/src/tensora/generate/_base.py
index 2e7dfbf..69ba94a 100644
--- a/src/tensora/generate/_base.py
+++ b/src/tensora/generate/_base.py
@@ -1,13 +1,15 @@
-__all__ = ["TensorCompiler", "generate_c_code"]
+__all__ = ["TensorCompiler", "generate_code"]
 
+from dataclasses import dataclass
 from enum import Enum
 
-from returns.result import Result
+from returns.result import Failure, Result, Success
 
+from ..codegen import ir_to_c, ir_to_llvm
 from ..kernel_type import KernelType
 from ..problem import Problem
 from ._taco import generate_c_code_taco
-from ._tensora import generate_c_code_tensora
+from ._tensora import generate_module_tensora
 
 
 class TensorCompiler(str, Enum):
@@ -19,11 +21,46 @@ def __str__(self) -> str:
         return self.name
 
 
-def generate_c_code(
-    problem: Problem, kernel_types: list[KernelType], tensor_compiler: TensorCompiler
+class Language(str, Enum):
+    # Python 3.10 does not support StrEnum, so do it manually
+    c = "c"
+    llvm = "llvm"
+
+    def __str__(self) -> str:
+        return self.name
+
+
+@dataclass(frozen=True, slots=True)
+class UnsupportedLanguageError(Exception):
+    tensor_compiler: TensorCompiler
+    language: Language
+
+    def __str__(self):
+        return f"For tensor compiler {self.tensor_compiler}, language {self.language} is not supported."
+
+
+def generate_code(
+    problem: Problem,
+    kernel_types: list[KernelType],
+    tensor_compiler: TensorCompiler,
+    language: Language,
 ) -> Result[str, Exception]:
     match tensor_compiler:
         case TensorCompiler.tensora:
-            return generate_c_code_tensora(problem, kernel_types)
+            match generate_module_tensora(problem, kernel_types):
+                case Failure(error):
+                    return Failure(error)
+                case Success(module):
+                    match language:
+                        case Language.c:
+                            return Success(ir_to_c(module))
+                        case Language.llvm:
+                            return Success(str(ir_to_llvm(module)))
+                case _:
+                    raise NotImplementedError()
         case TensorCompiler.taco:
-            return generate_c_code_taco(problem, kernel_types)
+            match language:
+                case Language.c:
+                    return generate_c_code_taco(problem, kernel_types)
+                case Language.llvm:
+                    return Failure(UnsupportedLanguageError(tensor_compiler, language))
diff --git a/src/tensora/generate/_taco.py b/src/tensora/generate/_taco.py
index b549414..7427869 100644
--- a/src/tensora/generate/_taco.py
+++ b/src/tensora/generate/_taco.py
@@ -1,5 +1,6 @@
 __all__ = ["generate_c_code_taco", "TacoError"]
 
+import re
 from dataclasses import dataclass
 
 from returns.result import Failure, Result, Success
@@ -51,6 +52,16 @@ def generate_c_code_taco(
 
     match taco_cli(taco_arguments):
         case Success(code):
-            return Success(code)
+            declaration_regex = r"^int (evaluate|compute|assemble)\([^(]\)"
+
+            argument_string = ", ".join(
+                [f"taco_tensora_t *{name}" for name in problem.formats.keys()]
+            )
+
+            reordered_code = re.sub(
+                declaration_regex, rf"int \1({argument_string})", code, flags=re.MULTILINE
+            )
+
+            return Success(reordered_code)
         case Failure(message):
             return Failure(TacoError(message))
diff --git a/src/tensora/generate/_tensora.py b/src/tensora/generate/_tensora.py
index c789234..828bf34 100644
--- a/src/tensora/generate/_tensora.py
+++ b/src/tensora/generate/_tensora.py
@@ -1,8 +1,7 @@
-__all__ = ["generate_c_code_tensora"]
+__all__ = ["generate_module_tensora"]
 
 from returns.result import Failure, Result, Success
 
-from ..codegen import ir_to_c
 from ..desugar import (
     DiagonalAccessError,
     NoKernelFoundError,
@@ -18,9 +17,9 @@
 from ..problem import Problem
 
 
-def generate_c_code_tensora(
+def generate_module_tensora(
     problem: Problem, kernel_types: list[KernelType]
-) -> Result[str, DiagonalAccessError | NoKernelFoundError]:
+) -> Result[Module, DiagonalAccessError | NoKernelFoundError]:
     formats = problem.formats
 
     desugar = desugar_assignment(problem.assignment)
@@ -40,4 +39,4 @@ def generate_c_code_tensora(
     functions = [generate_ir(definition, graph, kernel_type) for kernel_type in kernel_types]
     module = Module(functions)
 
-    return Success(ir_to_c(peephole(module)))
+    return Success(peephole(module))
diff --git a/src/tensora/ir/ast.py b/src/tensora/ir/ast.py
index 5026397..d4890e9 100644
--- a/src/tensora/ir/ast.py
+++ b/src/tensora/ir/ast.py
@@ -44,10 +44,12 @@
 from .types import Type
 
 
-def to_expression(expression: Expression | int | str) -> Expression:
+def to_expression(expression: Expression | float | int | str) -> Expression:
     match expression:
         case Expression():
             return expression
+        case float():
+            return FloatLiteral(expression)
         case int():
             return IntegerLiteral(expression)
         case str():
@@ -84,11 +86,11 @@ def idx(self, index: Expression | int | str) -> Assignable:
         index = to_expression(index)
         return ArrayIndex(self, index)
 
-    def assign(self, value: Expression | int | str) -> Statement:
+    def assign(self, value: Expression | float | int | str) -> Statement:
         value = to_expression(value)
         return Assignment(self, value)
 
-    def increment(self, amount: Expression | int | str = 1) -> Statement:
+    def increment(self, amount: Expression | float | int | str = 1) -> Statement:
         amount = to_expression(amount)
         return self.assign(self.plus(amount))
 
@@ -103,12 +105,16 @@ def declare(self, type: Type) -> Declaration:
 
 @dataclass(frozen=True, slots=True)
 class AttributeAccess(Assignable):
+    # For languages that care, this represents attribute access to an object on
+    # the heap. There is no way to use an object on the stack in this IR.
     target: Assignable
     attribute: str
 
 
 @dataclass(frozen=True, slots=True)
 class ArrayIndex(Assignable):
+    # For languages that care, this represents attribute access to an array on
+    # the heap. There is no way to use an array on the stack in this IR.
     target: Assignable
     index: Expression
 
@@ -243,14 +249,14 @@ class BooleanToInteger(Expression):
 
 @dataclass(frozen=True, slots=True)
 class ArrayAllocate(Expression):
-    type: Type
+    element_type: Type
     n_elements: Expression
 
 
 @dataclass(frozen=True, slots=True)
 class ArrayReallocate(Expression):
     old: Assignable
-    type: Type
+    element_type: Type
     n_elements: Expression
 
 
diff --git a/src/tensora/tensor.py b/src/tensora/tensor.py
index 0360c34..f226370 100644
--- a/src/tensora/tensor.py
+++ b/src/tensora/tensor.py
@@ -423,7 +423,7 @@ def recurse(node, i_level):
 def evaluate_binary_operator(
     left: Union[Tensor, Real], right: Union[Tensor, Real], operator: str
 ) -> Tensor:
-    from .function import evaluate_tensora
+    from .compile import evaluate_tensora
 
     def indexes_string(tensor):
         return ",".join(f"i{i}" for i in range(tensor.order))
@@ -499,7 +499,7 @@ def indexes_string(tensor):
 
 
 def evaluate_matrix_multiplication_operator(left: Tensor, right: Tensor):
-    from .function import evaluate_tensora
+    from .compile import evaluate_tensora
 
     if isinstance(left, Tensor) and isinstance(right, Tensor):
         if left.order == 1 and right.order == 1:
diff --git a/tests/test_cli.py b/tests/test_cli.py
index 3e082fd..7b443a3 100644
--- a/tests/test_cli.py
+++ b/tests/test_cli.py
@@ -36,6 +36,13 @@ def test_multiple_kernels():
     assert "assemble" in result.stdout
 
 
+def test_llvm_language():
+    result = runner.invoke(app, ["y(i) = A(i,j) * x(j)", "-l", "llvm"])
+
+    assert result.exit_code == 0
+    assert "getelementptr" in result.stdout
+
+
 def test_write_to_file():
     with NamedTemporaryFile(suffix=".c") as f:
         result = runner.invoke(app, ["y(i) = A(i,j) * x(j)", "-o", f.name])
diff --git a/tests/test_compile.py b/tests/test_compile.py
deleted file mode 100644
index 0712ab5..0000000
--- a/tests/test_compile.py
+++ /dev/null
@@ -1,115 +0,0 @@
-import tempfile
-
-from cffi import FFI
-
-from tensora import Tensor
-from tensora.compile import (
-    lock,
-    taco_define_header,
-    taco_type_header,
-    take_ownership_of_tensor,
-    take_ownership_of_tensor_members,
-    tensor_cdefs,
-)
-
-source = """
-taco_tensor_t create_tensor() {
-    int32_t order = 2;
-
-    int32_t* dimensions = malloc(sizeof(int32_t) * order);
-    dimensions[0] = 3;
-    dimensions[1] = 4;
-
-    int32_t* mode_ordering = malloc(sizeof(int32_t) * order);
-    mode_ordering[0] = 0;
-    mode_ordering[1] = 1;
-
-    taco_mode_t* mode_types = malloc(sizeof(taco_mode_t) * order);
-    mode_types[0] = taco_mode_dense;
-    mode_types[1] = taco_mode_sparse;
-
-    int32_t* pos = malloc(sizeof(*pos) * 4);
-    pos[0] = 0;
-    pos[1] = 3;
-    pos[2] = 3;
-    pos[3] = 5;
-    int32_t* crd = malloc(sizeof(*crd) * 5);
-    crd[0] = 0;
-    crd[1] = 2;
-    crd[2] = 3;
-    crd[3] = 0;
-    crd[4] = 3;
-
-    int32_t** indices1 = malloc(sizeof(*indices1) * 2);
-    indices1[0] = pos;
-    indices1[1] = crd;
-
-    int32_t*** indices = malloc(sizeof(*indices) * order);
-    indices[0] = NULL;
-    indices[1] = indices1;
-
-    double* vals = malloc(sizeof(*vals) * 5);
-    vals[0] = 6;
-    vals[1] = 9;
-    vals[2] = 8;
-    vals[3] = 5;
-    vals[4] = 7;
-
-    taco_tensor_t tensor = {
-        .order = order,
-        .dimensions = dimensions,
-        .csize = 8,
-        .mode_ordering = mode_ordering,
-        .mode_types = mode_types,
-        .indices = (uint8_t***)indices,
-        .vals = (uint8_t*)vals,
-        .vals_size = 5
-    };
-    return tensor;
-}
-
-taco_tensor_t* create_pointer_to_tensor() {
-    taco_tensor_t* tensor = malloc(sizeof(taco_tensor_t));
-    *tensor = create_tensor();
-    return tensor;
-}
-"""
-
-ffi = FFI()
-ffi.include(tensor_cdefs)
-ffi.cdef(
-    """
-taco_tensor_t create_tensor();
-taco_tensor_t* create_pointer_to_tensor();
-"""
-)
-ffi.set_source(
-    "taco_kernel",
-    taco_define_header + taco_type_header + source,
-    extra_compile_args=["-Wno-unused-variable", "-Wno-unknown-pragmas"],
-)
-
-expected_tensor = Tensor.from_lol([[6, 0, 9, 8], [0, 0, 0, 0], [5, 0, 0, 7]], format="ds")
-
-with tempfile.TemporaryDirectory() as temp_dir:
-    # Lock because FFI.compile is not thread safe: https://foss.heptapod.net/pypy/cffi/-/issues/490
-    with lock:
-        # Create shared object in temporary directory
-        lib_path = ffi.compile(tmpdir=temp_dir)
-
-    # Load the shared object
-    lib = ffi.dlopen(lib_path)
-
-
-def test_take_ownership_of_tensor_on_returned_struct():
-    cffi_tensor = lib.create_tensor()
-    take_ownership_of_tensor_members(cffi_tensor)
-    tensor = Tensor(cffi_tensor)
-    assert tensor == expected_tensor
-
-
-def test_take_ownership_of_tensor_on_returned_pointer_to_struct():
-    cffi_tensor = lib.create_pointer_to_tensor()
-    take_ownership_of_tensor(cffi_tensor)
-    tensor = Tensor(cffi_tensor)
-    assert tensor == expected_tensor
diff --git a/tests/test_tensor_method.py b/tests/test_tensor_method.py
index c2ec498..6eadc30 100644
--- a/tests/test_tensor_method.py
+++ b/tests/test_tensor_method.py
@@ -1,8 +1,8 @@
 import pytest
 
 from tensora import Tensor, tensor_method
+from tensora.compile import BroadcastTargetIndexError
 from tensora.desugar import DiagonalAccessError, NoKernelFoundError
-from tensora.function import BroadcastTargetIndexError
 
 
 def test_tensor_method():
diff --git a/tests_cffi/__init__.py b/tests_cffi/__init__.py
new file mode 100644
index 0000000..6ea0a06
--- /dev/null
+++ b/tests_cffi/__init__.py
@@ -0,0 +1,3 @@
+# This folder cannot be nested until the import needs of the base packages and
+# the cffi extra are more differentiated. Right now, the cffi extra needs only
+# the setuptools package and only in Python 3.12.
diff --git a/tests_cffi/test_combinatorically.py b/tests_cffi/test_combinatorically.py
new file mode 100644
index 0000000..60c21be
--- /dev/null
+++ b/tests_cffi/test_combinatorically.py
@@ -0,0 +1,154 @@
+import pytest
+
+from tensora import Tensor
+from tensora.compile import evaluate_cffi as evaluate
+
+
+def assert_same_as_dense(expression, format_out, **tensor_pairs):
+    tensors_in_format = {
+        name: Tensor.from_lol(data, format=format) for name, (data, format) in tensor_pairs.items()
+    }
+    tensors_as_dense = {name: Tensor.from_lol(data) for name, (data, _) in tensor_pairs.items()}
+
+    dense_format = "d" * (format_out.count("d") + format_out.count("s"))
+    actual = evaluate(expression, format_out, **tensors_in_format)
+    expected = evaluate(expression, dense_format, **tensors_as_dense)
+    assert actual == expected
+
+
+@pytest.mark.parametrize("dense", [[3, 2, 4], [0, 0, 0]])
+@pytest.mark.parametrize("format_in", ["s", "d"])
+@pytest.mark.parametrize("format_out", ["s", "d"])
+def test_copy_1(dense, format_in, format_out):
+    a = Tensor.from_lol(dense, format=format_in)
+    actual = evaluate("b(i) = a(i)", format_out, a=a)
+    assert actual == a
+
+
+@pytest.mark.parametrize("dense", [[[0, 2, 4], [0, -1, 0]], [[0, 0, 0], [0, 0, 0]]])
+@pytest.mark.parametrize("format_in", ["ss", "dd", "sd", "ds", "d1d0"])
+@pytest.mark.parametrize("format_out", ["ss", "dd", "sd", "ds", "d1d0"])
+def test_copy_2(dense, format_in, format_out):
+    a = Tensor.from_lol(dense, format=format_in)
+    actual = evaluate("b(i,j) = a(i,j)", format_out, a=a)
+    assert actual == a
+
+
+@pytest.mark.parametrize("dense", [[[0, 2, 4], [0, -1, 0]], [[0, 0, 0], [0, 0, 0]]])
+@pytest.mark.parametrize("format_in", ["s1s0", "d1d0", "s1d0", "d1s0", "dd"])
+@pytest.mark.parametrize("format_out", ["s1s0", "d1d0", "s1d0", "d1s0", "dd"])
+def test_copy_2_backwards(dense, format_in, format_out):
+    a = Tensor.from_lol(dense, format=format_in)
+    actual = evaluate("b(i,j) = a(i,j)", format_out, a=a)
+    assert actual == a
+
+
+@pytest.mark.parametrize("expression", [0, 1])
+def test_constant_scalar(expression):
+    actual = evaluate(f"a() = {expression}", "")
+    assert actual == Tensor.from_lol(expression)
+
+
+@pytest.mark.parametrize("dense1", [[0, 2, 4, 0], [0, 0, 0, 0]])
+@pytest.mark.parametrize("dense2", [[-1, 3.5, 0, 0], [0, 0, 0, 0]])
+@pytest.mark.parametrize("format1", ["s", "d"])
+@pytest.mark.parametrize("format2", ["s", "d"])
+def test_vector_dot(dense1, dense2, format1, format2):
+    assert_same_as_dense(
+        "out() = in1(i) * in2(i)", "", in1=(dense1, format1), in2=(dense2, format2)
+    )
+
+
+@pytest.mark.parametrize("dense1", [[0, 2, 4, 0], [0, 0, 0, 0]])
+@pytest.mark.parametrize("dense2", [[-1, 3.5, 0, 0], [0, 0, 0, 0]])
+@pytest.mark.parametrize("format1", ["s", "d"])
+@pytest.mark.parametrize("format2", ["s", "d"])
+@pytest.mark.parametrize("format_out", ["s", "d"])
+@pytest.mark.parametrize("operator", ["+", "-", "*"])
+def test_vector_binary(operator, dense1, dense2, format1, format2, format_out):
+    assert_same_as_dense(
+        f"out(i) = in1(i) {operator} in2(i)",
+        format_out,
+        in1=(dense1, format1),
+        in2=(dense2, format2),
+    )
+
+
+@pytest.mark.parametrize("dense1", [[[0, 2, 4], [0, -1, 0]], [[0, 0, 0], [0, 0, 0]]])
+@pytest.mark.parametrize("dense2", [[[-1, 3.5], [0, 0], [4, 0]], [[0, 0], [0, 0], [0, 0]]])
+@pytest.mark.parametrize("format1", ["ss", "dd", "sd", "ds", "d1d0"])
+@pytest.mark.parametrize("format2", ["ss", "dd", "sd", "ds", "d1d0"])
+@pytest.mark.parametrize("format_out", ["dd", "d1d0"])
+def test_matrix_dot(dense1, dense2, format1, format2, format_out):
+    assert_same_as_dense(
+        "out(i,k) = in1(i,j) * in2(j,k)", format_out, in1=(dense1, format1), in2=(dense2, format2)
+    )
+
+
+@pytest.mark.parametrize("dense1", [[[0, 2, 4], [0, -1, 0]], [[0, 0, 0], [0, 0, 0]]])
+@pytest.mark.parametrize("dense2", [[-1, 3.5, 0], [0, 0, 0]])
+@pytest.mark.parametrize("format1", ["ss", "dd", "sd", "ds", "s1s0", "d1d0", "s1d0", "d1s0"])
+@pytest.mark.parametrize("format2", ["s", "d"])
+@pytest.mark.parametrize("format_out", ["d"])
+def test_matrix_vector_product(dense1, dense2, format1, format2, format_out):
+    assert_same_as_dense(
+        "out(i) = in1(i,j) * in2(j)", format_out, in1=(dense1, format1), in2=(dense2, format2)
+    )
+
+
+@pytest.mark.parametrize("dense1", [[[0, 2, 4], [0, -1, 0]], [[0, 0, 0], [0, 0, 0]]])
+@pytest.mark.parametrize("dense2", [[[-1, 3.5], [0, 0], [4, 0]], [[0, 0], [0, 0], [0, 0]]])
+@pytest.mark.parametrize("dense3", [[[-3, 0], [7, 0]], [[0, 0], [0, 0]]])
+@pytest.mark.parametrize("format1", ["dd", "ds"])
+@pytest.mark.parametrize("format2", ["dd"])
+@pytest.mark.parametrize("format3", ["dd", "ds"])
+@pytest.mark.parametrize("format_out", ["dd"])
+def test_matrix_multiply_add(dense1, dense2, dense3, format1, format2, format3, format_out):
+    assert_same_as_dense(
+        "out(i,k) = in1(i,j) * in2(j,k) + in3(i,k)",
+        format_out,
+        in1=(dense1, format1),
+        in2=(dense2, format2),
+        in3=(dense3, format3),
+    )
+
+
+@pytest.mark.parametrize(
+    "dense_b",
+    [
+        [
+            [[0, 2, 4, 0], [0, -1, 0, 3], [1, -1, 0, 0]],
+            [[-2, 4, 0, 0], [0, 0, 0, 3], [1, 1, 0, 0]],
+        ],
+        [
+            [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]],
+            [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0]],
+        ],
+    ],
+)
+@pytest.mark.parametrize(
+    "dense_d",
+    [
+        [[-1, 3.5, 1, 2, 0], [0, 2, 6, 3, 0], [4, 0, 0, 1, -1], [0, 0, 3, 6, 9]],
+        [[0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0]],
+    ],
+)
+@pytest.mark.parametrize(
+    "dense_c",
+    [
+        [[0, 0, 1, 2, 7], [7, 0, 5, 2, 0], [-1, 0, 0, 2, 1]],
+        [[0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0]],
+    ],
+)
+@pytest.mark.parametrize("format_b", ["ddd", "dss", "sss", "ssd", "d1d0s2", "s0d2d1"])
+@pytest.mark.parametrize("format_d", ["dd", "ds", "ss"])
+@pytest.mark.parametrize("format_c", ["dd", "ds", "ss"])
+@pytest.mark.parametrize("format_out", ["dd", "d1d0"])
+def test_mttkrp(dense_b, dense_d, dense_c, format_b, format_d, format_c, format_out):
+    assert_same_as_dense(
+        "A(i,j) = B(i,k,l) * D(l,j) * C(k,j)",
+        format_out,
+        B=(dense_b, format_b),
+        D=(dense_d, format_d),
+        C=(dense_c, format_c),
+    )
diff --git a/tests_cffi/test_evaluate.py b/tests_cffi/test_evaluate.py
new file mode 100644
index 0000000..5633f7e
--- /dev/null
+++ b/tests_cffi/test_evaluate.py
@@ -0,0 +1,45 @@
+from tensora import Tensor
+from tensora.compile import evaluate_cffi as evaluate
+
+
+def test_csr_matrix_vector_product():
+    A = Tensor.from_aos([(1, 0), (0, 1), (1, 2)], [2.0, -2.0, 4.0], dimensions=(2, 3), format="ds")
+
+    x = Tensor.from_aos([(0,), (1,), (2,)], [3.0, 2.5, 2.0], dimensions=(3,), format="d")
+
+    expected = Tensor.from_aos([(0,), (1,)], [-5.0, 14.0], dimensions=(2,), format="d")
+
+    actual = evaluate("y(i) = A(i,j) * x(j)", "d", A=A, x=x)
+
+    assert actual == expected
+
+
+def test_csc_matrix_vector_product():
+    A = Tensor.from_aos(
+        [(1, 0), (0, 1), (1, 2)], [2.0, -2.0, 4.0], dimensions=(2, 3), format="d1s0"
+    )
+
+    x = Tensor.from_aos([(0,), (1,), (2,)], [3.0, 2.5, 2.0], dimensions=(3,), format="d")
+
+    expected = Tensor.from_aos([(0,), (1,)], [-5.0, 14.0], dimensions=(2,), format="d")
+
+    actual = evaluate("y(i) = A(i,j) * x(j)", "d", A=A, x=x)
+
+    assert actual == expected
+
+
+def test_csr_matrix_plus_csr_matrix():
+    A = Tensor.from_aos([(1, 0), (0, 1), (1, 2)], [2.0, -2.0, 4.0], dimensions=(2, 3), format="ds")
+
+    B = Tensor.from_aos([(1, 1), (1, 2), (0, 2)], [-3.0, 4.0, 3.5], dimensions=(2, 3), format="ds")
+
+    expected = Tensor.from_aos(
+        [(1, 0), (0, 1), (1, 2), (1, 1), (0, 2)],
+        [2.0, -2.0, 8.0, -3.0, 3.5],
+        dimensions=(2, 3),
+        format="ds",
+    )
+
+    actual = evaluate("C(i,j) = A(i,j) + B(i,j)", "ds", A=A, B=B)
+
+    assert actual == expected