diff --git a/jax_scaled_arithmetics/core/__init__.py b/jax_scaled_arithmetics/core/__init__.py
index 22692c4..37890b6 100644
--- a/jax_scaled_arithmetics/core/__init__.py
+++ b/jax_scaled_arithmetics/core/__init__.py
@@ -1,3 +1,3 @@
 # Copyright (c) 2023 Graphcore Ltd. All rights reserved.
-from .datatype import ScaledArray, scaled_array  # noqa: F401
+from .datatype import DTypeLike, ScaledArray, Shape, scaled_array  # noqa: F401
 from .interpreters import autoscale, register_scaled_lax_op, register_scaled_op  # noqa: F401
diff --git a/jax_scaled_arithmetics/core/interpreters.py b/jax_scaled_arithmetics/core/interpreters.py
index 3e32bac..9810535 100644
--- a/jax_scaled_arithmetics/core/interpreters.py
+++ b/jax_scaled_arithmetics/core/interpreters.py
@@ -1,7 +1,7 @@
 # Copyright (c) 2023 Graphcore Ltd. All rights reserved.
 
 from functools import wraps
-from typing import Dict
+from typing import Any, Dict
 
 import jax
 from jax import core
@@ -9,31 +9,48 @@
 
 from ..core import ScaledArray
 
-_scaled_ops_registry = {}
+_scaled_ops_registry: Dict[core.Primitive, Any] = {}
 
 
-def register_scaled_op(lax_func, scaled_func):
-    _scaled_ops_registry[lax_func] = scaled_func
+def register_scaled_op(prim: core.Primitive, scaled_func: Any) -> None:
+    """Register the scaled translation of JAX primitive.
 
+    Raises an error if a scaled translation is already existing for this primitive.
 
-def _get_lax_prim(scaled_func):
+    Args:
+        prim: JAX primitive.
+        scaled_fund: Scaled translation of the primitive. With the same interface.
+    """
+    assert isinstance(prim, core.Primitive)
+    if prim in _scaled_ops_registry:
+        raise KeyError(f"A scaled translation is already registered for the JAX primitive '{prim}'.")
+    _scaled_ops_registry[prim] = scaled_func
+
+
+def _get_lax_prim(scaled_func: Any) -> core.Primitive:
     try:
-        op = getattr(jax.lax, scaled_func.__name__.replace("scaled_", ""))
+        prim_name = scaled_func.__name__.replace("scaled_", "") + "_p"
+        prim = getattr(jax.lax, prim_name)
     except AttributeError:
-        raise AttributeError(f"Could not find corresponding jax.lax primitive for {scaled_func.__name__}")
-    return op
+        raise AttributeError(f"Could not find corresponding 'jax.lax' primitive for '{scaled_func.__name__}'.")
+    # Check as well it is a proper primitive! And not something else also in `jax.lax`
+    if not isinstance(prim, core.Primitive):
+        raise AttributeError(f"The object `{prim}` is not a proper JAX primitive for '{scaled_func.__name__}'.")
+    return prim
 
 
 def register_scaled_lax_op(scaled_func):
     """
-    Registers a scaled function into the scaled_ops_registry by matching
-    the function name with pattern `scaled_{func_name}` to a function in the
+    Registers a scaled function/translation into the scaled_ops_registry by matching
+    the function name with pattern `scaled_{func_name}` to a primitive in the
     `jax.lax` namespace.
 
-    Example: `scaled_mul_p` is matched to `jax.lax.mul_p`
+    Example: `scaled_mul` is matched to `jax.lax.mul_p`
     """
     lax_prim = _get_lax_prim(scaled_func)
     register_scaled_op(lax_prim, scaled_func)
+    # Always return the function in the case of decorator use.
+    return scaled_func
 
 
 def autoscale(fun):
@@ -49,7 +66,7 @@ def wrapped(*args, **kwargs):
     return wrapped
 
 
-def autoscale_jaxpr(jaxpr, consts, *args):
+def autoscale_jaxpr(jaxpr: core.Jaxpr, consts, *args):
     env: Dict[core.Var, ScaledArray] = {}
 
     def read(var):
@@ -67,7 +84,7 @@ def write(var, val):
         invals = safe_map(read, eqn.invars)
         if eqn.primitive not in _scaled_ops_registry:
             raise NotImplementedError(f"{eqn.primitive} does not have an implementation for ScaledArray inputs yet")
-        outvals = _scaled_ops_registry[eqn.primitive](*invals)
+        outvals = _scaled_ops_registry[eqn.primitive](*invals, **eqn.params)
         if not eqn.primitive.multiple_results:
             outvals = [outvals]
         safe_map(write, eqn.outvars, outvals)
diff --git a/jax_scaled_arithmetics/lax/scaled_ops.py b/jax_scaled_arithmetics/lax/scaled_ops.py
index 75e6f21..7611ff9 100644
--- a/jax_scaled_arithmetics/lax/scaled_ops.py
+++ b/jax_scaled_arithmetics/lax/scaled_ops.py
@@ -1,12 +1,39 @@
 # Copyright (c) 2023 Graphcore Ltd. All rights reserved.
+from typing import Optional, Sequence
+
+from jax import lax
 
 from jax_scaled_arithmetics import core
-from jax_scaled_arithmetics.core import ScaledArray
+from jax_scaled_arithmetics.core import DTypeLike, ScaledArray, Shape
 
 
 @core.register_scaled_lax_op
-def scaled_mul_p(A: ScaledArray, B: ScaledArray) -> ScaledArray:
-    return ScaledArray(A.data * B.data, A.scale * B.scale)
+def scaled_broadcast_in_dim(A: ScaledArray, shape: Shape, broadcast_dimensions: Sequence[int]) -> ScaledArray:
+    return ScaledArray(lax.broadcast_in_dim(A.data, shape=shape, broadcast_dimensions=broadcast_dimensions), A.scale)
 
 
-__all__ = ["scaled_mul_p"]
+@core.register_scaled_lax_op
+def scaled_convert_element_type(A: ScaledArray, new_dtype: DTypeLike, weak_type: bool = False) -> ScaledArray:
+    # NOTE: by default, no rescaling done before casting.
+    # Choice of adding an optional rescaling op before is up to the user (and which strategy to use).
+    # NOTE bis: scale not casted as well by default!
+    return ScaledArray(lax.convert_element_type(A.data, new_dtype=new_dtype), A.scale)
+
+
+@core.register_scaled_lax_op
+def scaled_slice(
+    A: ScaledArray, start_indices: Sequence[int], limit_indices: Sequence[int], strides: Optional[Sequence[int]] = None
+) -> ScaledArray:
+    return ScaledArray(
+        lax.slice(A.data, start_indices=start_indices, limit_indices=limit_indices, strides=strides), A.scale
+    )
+
+
+@core.register_scaled_lax_op
+def scaled_transpose(A: ScaledArray, permutation: Sequence[int]) -> ScaledArray:
+    return ScaledArray(lax.transpose(A.data, permutation=permutation), A.scale)
+
+
+@core.register_scaled_lax_op
+def scaled_mul(A: ScaledArray, B: ScaledArray) -> ScaledArray:
+    return ScaledArray(A.data * B.data, A.scale * B.scale)
diff --git a/pyproject.toml b/pyproject.toml
index d90e1e9..728e247 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -29,6 +29,9 @@ Website = "https://github.com/graphcore-research/jax-scaled-arithmetics/#readme"
 [project.optional-dependencies]
 test = ["pytest"]
 
+[tool.setuptools]
+packages = ["jax_scaled_arithmetics"]
+
 [tool.pytest.ini_options]
 minversion = "6.0"
 addopts = ["-ra", "--showlocals", "--strict-config", "-p no:hypothesispytest"]
diff --git a/tests/core/test_interpreter.py b/tests/core/test_interpreter.py
index b906020..affe5ef 100644
--- a/tests/core/test_interpreter.py
+++ b/tests/core/test_interpreter.py
@@ -6,10 +6,14 @@
 import numpy as np
 import numpy.testing as npt
 
-from jax_scaled_arithmetics.core import ScaledArray, autoscale, scaled_array
+from jax_scaled_arithmetics.core import ScaledArray, autoscale, register_scaled_op, scaled_array
 
 
 class AutoScaleInterpreterTests(chex.TestCase):
+    def test__register_scaled_op__error_if_already_registered(self):
+        with self.assertRaises(KeyError):
+            register_scaled_op(jax.lax.mul_p, lambda a, _: a)
+
     @chex.variants(with_jit=True, without_jit=True)
     def test__scaled_identity_function(self):
         def func(x):
@@ -34,7 +38,7 @@ def func(x):
         assert jaxpr.outvars[1].aval.shape == ()
 
     @chex.variants(with_jit=True, without_jit=True)
-    def test__scaled_mul_function(self):
+    def test__scaled_mul__no_attributes(self):
         def func(x, y):
             return x * y
 
@@ -48,3 +52,16 @@ def func(x, y):
         out = asfunc(x, y)
         assert isinstance(out, ScaledArray)
         npt.assert_array_almost_equal(out, expected)
+
+    @chex.variants(with_jit=True, without_jit=True)
+    def test__scaled_convert_element_type__attributes_passing(self):
+        def func(x):
+            return jax.lax.convert_element_type(x, np.float16)
+
+        # Autoscale + (optional) jitting.
+        asfunc = self.variant(autoscale(func))
+        x = scaled_array([-4.0, 2.0], 0.5, dtype=np.float32)
+        out = asfunc(x)
+        assert isinstance(out, ScaledArray)
+        assert out.dtype == np.float16
+        npt.assert_array_almost_equal(out, x)
diff --git a/tests/lax/test_scaled_ops.py b/tests/lax/test_scaled_ops.py
new file mode 100644
index 0000000..cc72fcb
--- /dev/null
+++ b/tests/lax/test_scaled_ops.py
@@ -0,0 +1,51 @@
+# Copyright (c) 2023 Graphcore Ltd. All rights reserved.
+import chex
+import numpy as np
+import numpy.testing as npt
+
+from jax_scaled_arithmetics.core import ScaledArray, scaled_array
+from jax_scaled_arithmetics.lax import (
+    scaled_broadcast_in_dim,
+    scaled_convert_element_type,
+    scaled_mul,
+    scaled_slice,
+    scaled_transpose,
+)
+
+
+class ScaledTranslationPrimitivesTests(chex.TestCase):
+    def test__scaled_broadcast_in_dim__proper_scaling(self):
+        x = scaled_array(np.random.rand(5), 2, dtype=np.float32)
+        z = scaled_broadcast_in_dim(x, shape=(5, 1), broadcast_dimensions=(0,))
+        assert isinstance(z, ScaledArray)
+        npt.assert_array_equal(z.scale, x.scale)
+        npt.assert_array_almost_equal(z.data, x.data.reshape((5, 1)))
+
+    def test__scaled_convert_element_type__proper_scaling(self):
+        x = scaled_array(np.random.rand(5), 2, dtype=np.float32)
+        z = scaled_convert_element_type(x, new_dtype=np.float16)
+        assert isinstance(z, ScaledArray)
+        npt.assert_array_equal(z.scale, x.scale)
+        npt.assert_array_almost_equal(z.data, x.data.astype(z.dtype))
+
+    def test__scaled_transpose__proper_scaling(self):
+        x = scaled_array(np.random.rand(3, 5), 2, dtype=np.float32)
+        z = scaled_transpose(x, (1, 0))
+        assert isinstance(z, ScaledArray)
+        assert z.scale == x.scale
+        npt.assert_array_almost_equal(z.data, x.data.T)
+
+    def test__scaled_slice__proper_scaling(self):
+        x = scaled_array(np.random.rand(5), 2, dtype=np.float32)
+        z = scaled_slice(x, (1,), (4,), (2,))
+        assert isinstance(z, ScaledArray)
+        assert z.scale == x.scale
+        npt.assert_array_almost_equal(z.data, x.data[1:4:2])
+
+    def test__scaled_mul__proper_scaling(self):
+        x = scaled_array([-2.0, 2.0], 3, dtype=np.float32)
+        y = scaled_array([1.5, 1.5], 2, dtype=np.float32)
+        z = scaled_mul(x, y)
+        assert isinstance(z, ScaledArray)
+        assert z.scale == 6
+        npt.assert_array_almost_equal(z, np.asarray(x) * np.asarray(y))