[Doc] Improve type.rst readability

docs/type.rst

@@ -1,8 +1,33 @@
 Type system
 ===========
 
+Taichi supports common numerical data types. Each type is denoted as
+a character indicating its *category* and a number of *precision bits*, e.g., ``i32`` and ``f64``.
+
+The *category* can be one of:
+
+- ``i`` for signed integers, e.g. 233, -666
+- ``u`` for unsigned integers, e.g. 233, 666
+- ``f`` for floating point numbers, e.g. 2.33, 1e-4
+
+The *digital number* can be one of:
+
+- ``8``
+- ``16``
+- ``32``
+- ``64``
+
+It represents how many **bits** are used in storing the data.
+The larger the bit number, the higher the precision is.
+
+For example, the two most commonly used types:
+
+- ``i32`` represents a 32-bit signed integer.
+- ``f32`` represents a 32-bit floating pointer number.
+
 Supported types
 ---------------
+
 Currently, supported basic types in Taichi are
 
 - int8 ``ti.i8``
@@ -46,54 +71,129 @@ Currently, supported basic types in Taichi are
     (OK: supported, EXT: require extension, N/A: not available)
 
 
-Boolean types should be represented using ``ti.i32``.
+.. note::
+
+    Boolean types are represented using ``ti.i32``.
+
 
-Binary operations on different types will give you a promoted type, following the C programming language, e.g.
 
-- ``i32 + f32 = f32``
-- ``f32 + f64 = f64``
-- ``i32 + i64 = i64``
+Type promotion
+--------------
+
+Binary operations on different types will give you a promoted type, following the C programming language convention, e.g.:
+
+- ``i32 + f32 = f32`` (integer + float = float)
+- ``i32 + i64 = i64`` (less-bits + more-bits = more-bits)
+
+Basically it will try to choose the more precise type to contain the result value.
 
 
 .. _default_precisions:
 
 Default precisions
 ------------------
 
-By default, numerical literals have 32-bit precisions.
+By default, all numerical literals have 32-bit precisions.
 For example, ``42`` has type ``ti.i32`` and ``3.14`` has type ``ti.f32``.
+
 Default integer and float-point precisions (``default_ip`` and ``default_fp``) can be specified when initializing Taichi:
 
 .. code-block:: python
 
-    ti.init(..., default_fp=ti.f32)
-    ti.init(..., default_fp=ti.f64)
+    ti.init(default_fp=ti.f32)
+    ti.init(default_fp=ti.f64)
+
+    ti.init(default_ip=ti.i32)
+    ti.init(default_ip=ti.i64)
+
+
+Also note that you may use ``float`` or ``int`` in type definitions as aliases
+for default precisions, e.g.:
+
+.. code-block:: python
+
+    ti.init(default_ip=ti.i64, default_fp=ti.f32)
+
+    x = ti.var(float, 5)
+    y = ti.var(int, 5)
+    # is equivalent to:
+    x = ti.var(ti.f32, 5)
+    y = ti.var(ti.i64, 5)
+
+    def func(a: float) -> int:
+        ...
+
+    # is equivalent to:
+    def func(a: ti.f32) -> ti.i64:
+        ...
 
-    ti.init(..., default_ip=ti.i32)
-    ti.init(..., default_ip=ti.i64)
 
 
 Type casts
 ----------
 
-Use ``ti.cast`` to cast scalar values.
+Implicit casts
+**************
+
+WARNING: The type of a variable is **determinated on it's initialization**.
+
+When a *low-precision* variable is assigned to a *high-precision* variable, it will be
+implicitly promoted to the *wide* type and no warning will be raised:
+
+.. code-block:: python
+
+    a = 1.7
+    a = 1
+    print(a)  # 1.0
+
+When a *high-precision* variable is assigned to a *low-precision* type, it will be
+implicitly down-cast into the *low-precision* type and Taichi will raise a warning:
 
 .. code-block:: python
 
-    a = 1.4
-    b = ti.cast(a, ti.i32)
-    c = ti.cast(b, ti.f32)
+    a = 1
+    a = 1.7
+    print(a)  # 1
+
+Explicit casts
+**************
+
+You may use ``ti.cast`` to explicitly cast scalar values between different types:
+
+.. code-block:: python
 
-    # Equivalently, use ``int()`` and ``float()``
-    # to convert values to default float-point/integer types
-    b = int(a)
-    c = float(b)
+    a = 1.7
+    b = ti.cast(a, ti.i32)  # 1
+    c = ti.cast(b, ti.f32)  # 1.0
 
-    # Element-wise casts in matrices
-    mat = ti.Matrix([[3.0, 0.0], [0.3, 0.1]])
-    mat_int = mat.cast(int)
-    mat_int2 = mat.cast(ti.i32)
+Equivalently, use ``int()`` and ``float()`` to convert values to float-point or
+integer types of default precisions:
+
+.. code-block:: python
+
+    a = 1.7
+    b = int(a)    # 1
+    c = float(a)  # 1.0
+
+Casting vectors and matrices
+****************************
+
+Type casts applied to vectors/matrices are element-wise:
+
+.. code-block:: python
+
+    u = ti.Vector([2.3, 4.7])
+    v = int(u)              # ti.Vector([2, 4])
+    # If you are using ti.i32 as default_ip, this is equivalent to:
+    v = ti.cast(u, ti.i32)  # ti.Vector([2, 4])
+
+Bit casting
+***********
 
 Use ``ti.bit_cast`` to bit-cast a value into another data type. The underlying bits will be preserved in this cast.
 The new type must have the same width as the the old type.
-For example, bit-casting ``i32`` to ``f64`` is not allowed. Use this operation with caution.
+or example, bit-casting ``i32`` to ``f64`` is not allowed. Use this operation with caution.
+
+.. code-block::
+
+    For people from C++, ``ti.bit_cast`` is equivalent to ``reinterpret_cast``.