.. index:: value
   pair: binary format; value

.. index:: byte
   pair: binary format; byte

:ref:`Bytes <syntax-byte>` encode themselves.

\begin{array}{llcll@{\qquad}l}
\production{byte} & \Bbyte &::=&
  \hex{00} &\Rightarrow& \hex{00} \\ &&|&&
  \dots \\ &&|&
  \hex{FF} &\Rightarrow& \hex{FF} \\
\end{array}

.. index:: integer, unsigned integer, signed integer, uninterpreted integer, LEB128, two's complement
   pair: binary format; integer
   pair: binary format; unsigned integer
   pair: binary format; signed integer
   pair: binary format; uninterpreted integer

All :ref:`integers <syntax-int>` are encoded using the |LEB128|_ variable-length integer encoding, in either unsigned or signed variant.

:ref:`Unsigned integers <syntax-uint>` are encoded in |UnsignedLEB128|_ format. As an additional constraint, the total number of bytes encoding a value of type \uN must not exceed \F{ceil}(N/7) bytes.

\begin{array}{llclll@{\qquad}l}
\production{unsigned integer} & \BuN &::=&
  n{:}\Bbyte &\Rightarrow& n & (\iff n < 2^7 \wedge n < 2^N) \\ &&|&
  n{:}\Bbyte~~m{:}\BuX{(N\B{-7})} &\Rightarrow&
    2^7\cdot m + (n-2^7) & (\iff n \geq 2^7 \wedge N > 7) \\
\end{array}

:ref:`Signed integers <syntax-sint>` are encoded in |SignedLEB128|_ format, which uses a two's complement representation. As an additional constraint, the total number of bytes encoding a value of type \sN must not exceed \F{ceil}(N/7) bytes.

\begin{array}{llclll@{\qquad}l}
\production{signed integer} & \BsN &::=&
  n{:}\Bbyte &\Rightarrow& n & (\iff n < 2^6 \wedge n < 2^{N-1}) \\ &&|&
  n{:}\Bbyte &\Rightarrow& n-2^7 & (\iff 2^6 \leq n < 2^7 \wedge n \geq 2^7-2^{N-1}) \\ &&|&
  n{:}\Bbyte~~m{:}\BsX{(N\B{-7})} &\Rightarrow&
    2^7\cdot m + (n-2^7) & (\iff n \geq 2^7 \wedge N > 7) \\
\end{array}

:ref:`Uninterpreted integers <syntax-int>` are encoded as signed integers.

\begin{array}{llclll@{\qquad\qquad}l}
\production{uninterpreted integer} & \BiN &::=&
  n{:}\BsN &\Rightarrow& i & (\iff n = \signed_{\iN}(i))
\end{array}

.. index:: floating-point number, little endian
   pair: binary format; floating-point number

:ref:`Floating-point <syntax-float>` values are encoded directly by their |IEEE754|_ (Section 3.4) bit pattern in |LittleEndian|_ byte order:

\begin{array}{llclll@{\qquad\qquad}l}
\production{floating-point value} & \BfN &::=&
  b^\ast{:\,}\Bbyte^{N/8} &\Rightarrow& \bytes_{\fN}^{-1}(b^\ast) \\
\end{array}

.. index:: name, byte, Unicode, ! UTF-8
   pair: binary format; name

:ref:`Names <syntax-name>` are encoded as a :ref:`vector <binary-vec>` of bytes containing the |Unicode|_ (Section 3.9) UTF-8 encoding of the name's character sequence.

\begin{array}{llclllll}
\production{name} & \Bname &::=&
  b^\ast{:}\Bvec(\Bbyte) &\Rightarrow& \name
    && (\iff \utf8(\name) = b^\ast) \\
\end{array}

The auxiliary |utf8| function expressing this encoding is defined as follows:

\begin{array}{@{}l@{}}
\begin{array}{@{}lcl@{\qquad}l@{}}
\utf8(c^\ast) &=& (\utf8(c))^\ast \\[1ex]
\utf8(c) &=& b &
  (\begin{array}[t]{@{}c@{~}l@{}}
   \iff & c < \unicode{80} \\
   \wedge & c = b) \\
   \end{array} \\
\utf8(c) &=& b_1~b_2 &
  (\begin{array}[t]{@{}c@{~}l@{}}
   \iff & \unicode{80} \leq c < \unicode{800} \\
   \wedge & c = 2^6(b_1-\hex{C0})+(b_2-\hex{80})) \\
   \end{array} \\
\utf8(c) &=& b_1~b_2~b_3 &
  (\begin{array}[t]{@{}c@{~}l@{}}
   \iff & \unicode{800} \leq c < \unicode{D800} \vee \unicode{E000} \leq c < \unicode{10000} \\
   \wedge & c = 2^{12}(b_1-\hex{E0})+2^6(b_2-\hex{80})+(b_3-\hex{80})) \\
   \end{array} \\
\utf8(c) &=& b_1~b_2~b_3~b_4 &
  (\begin{array}[t]{@{}c@{~}l@{}}
   \iff & \unicode{10000} \leq c < \unicode{110000} \\
   \wedge & c = 2^{18}(b_1-\hex{F0})+2^{12}(b_2-\hex{80})+2^6(b_3-\hex{80})+(b_4-\hex{80})) \\
   \end{array} \\
\end{array} \\
\where b_2, b_3, b_4 < \hex{C0} \\
\end{array}