jax errors: improve leading info in error docs

jax/_src/errors.py

@@ -47,7 +47,8 @@ class JAXIndexError(_JAXErrorMixin, IndexError):
 class ConcretizationTypeError(JAXTypeError):
   """
   This error occurs when a JAX Tracer object is used in a context where a
-  concrete value is required. In some situations, it can be easily fixed by
+  concrete value is required (see :ref:`faq-different-kinds-of-jax-values`
+  for more on what a Tracer is). In some situations, it can be easily fixed by
   marking problematic values as static; in others, it may indicate that your
   program is doing operations that are not directly supported by JAX's JIT
   compilation model.
@@ -283,41 +284,45 @@ def __init__(self, tracer: core.Tracer):
 class TracerArrayConversionError(JAXTypeError):
   """
   This error occurs when a program attempts to convert a JAX Tracer object into
-  a standard NumPy array. It typically occurs in one of a few situations.
+  a standard NumPy array (see :ref:`faq-different-kinds-of-jax-values` for more
+  on what a Tracer is). It typically occurs in one of a few situations.
 
-  Using `numpy` rather than `jax.numpy` functions
-    This error can occur when a JAX Tracer object is passed to a raw numpy
-    function, or a method on a numpy.ndarray object. For example::
+  Using non-JAX functions in JAX transformations
+    This error can occur if you attempt to use a non-JAX library like `numpy`
+    or ``scipy`` inside a JAX transformation (:func:`~jax.jit`, :func:`~jax.grad`,
+    :func:`jax.vmap`, etc.). For example::
 
-      >>> from functools import partial
       >>> from jax import jit
       >>> import numpy as np
-      >>> import jax.numpy as jnp
 
       >>> @jit
       ... def func(x):
       ...   return np.sin(x)
 
-      >>> func(jnp.arange(4))  # doctest: +IGNORE_EXCEPTION_DETAIL
+      >>> func(np.arange(4))  # doctest: +IGNORE_EXCEPTION_DETAIL
       Traceback (most recent call last):
           ...
       TracerArrayConversionError: The numpy.ndarray conversion method
       __array__() was called on the JAX Tracer object
 
-    In this case, check that you are using `jax.numpy` methods rather than
-    `numpy` methods::
+    In this case, you can fix the issue by using :func:`jax.numpy.sin` in place of
+    :func:`numpy.sin`::
 
+      >>> import jax.numpy as jnp
       >>> @jit
       ... def func(x):
       ...   return jnp.sin(x)
 
       >>> func(jnp.arange(4))
       Array([0.        , 0.84147096, 0.9092974 , 0.14112   ], dtype=float32)
 
+    See also `External Callbacks`_ for options for calling back to host-side computations
+    from transformed JAX code.
+
   Indexing a numpy array with a tracer
     If this error arises on a line that involves array indexing, it may be that
-    the array being indexed `x` is a raw numpy.ndarray while the indices `idx`
-    are traced. For example::
+    the array being indexed `x` is a standard numpy.ndarray while the indices `idx`
+    are traced JAX arrays. For example::
 
       >>> x = np.arange(10)
 
@@ -343,6 +348,7 @@ class TracerArrayConversionError(JAXTypeError):
 
     or by declaring the index as a static argument::
 
+      >>> from functools import partial
       >>> @partial(jit, static_argnums=(0,))
       ... def func(i):
       ...   return x[i]
@@ -353,6 +359,8 @@ class TracerArrayConversionError(JAXTypeError):
   To understand more subtleties having to do with tracers vs. regular values,
   and concrete vs. abstract values, you may want to read
   :ref:`faq-different-kinds-of-jax-values`.
+
+  .. _External Callbacks: https://jax.readthedocs.io/en/latest/notebooks/external_callbacks.html
   """
   def __init__(self, tracer: core.Tracer):
     super().__init__(
@@ -364,13 +372,13 @@ def __init__(self, tracer: core.Tracer):
 class TracerIntegerConversionError(JAXTypeError):
   """
   This error can occur when a JAX Tracer object is used in a context where a
-  Python integer is expected. It typically occurs in a few situations.
+  Python integer is expected (see :ref:`faq-different-kinds-of-jax-values` for
+  more on what a Tracer is). It typically occurs in a few situations.
 
   Passing a tracer in place of an integer
-    This error can occur if you attempt to pass a tracer to a function that
-    requires an integer argument; for example::
+    This error can occur if you attempt to pass a traced value to a function
+    that requires a static integer argument; for example::
 
-      >>> from functools import partial
       >>> from jax import jit
       >>> import numpy as np
 
@@ -387,6 +395,7 @@ class TracerIntegerConversionError(JAXTypeError):
     When this happens, the solution is often to mark the problematic argument as
     static::
 
+      >>> from functools import partial
       >>> @partial(jit, static_argnums=1)
       ... def func(x, axis):
       ...   return np.split(x, 2, axis)
@@ -410,7 +419,6 @@ class TracerIntegerConversionError(JAXTypeError):
     quantity.
     For example::
 
-      >>> from functools import partial
       >>> import jax.numpy as jnp
       >>> from jax import jit
 
@@ -437,6 +445,7 @@ class TracerIntegerConversionError(JAXTypeError):
 
     or by declaring the index as a static argument::
 
+      >>> from functools import partial
       >>> @partial(jit, static_argnums=0)
       ... def func(i):
       ...   return L[i]