preserves/_includes/cheatsheet-binary.md

For a value <span class="postcard-grammar binarysyntax">*V*</span>, we write <span
class="postcard-grammar binarysyntax">«*V*»</span> for the binary encoding of <span
class="postcard-grammar binarysyntax">*V*</span>.

{:.postcard-grammar.binarysyntax}
«`#f`» | = | `80`
«`#t`» | = | `81`

{:.postcard-grammar.binarysyntax}
«`@`*W* *V*» | = | `85` «*W*» «*V*»
«`#!`*V*» | = | `86` «*V*»

{:.postcard-grammar.binarysyntax}
«*V*» | = | `87``04` **binary32**(*V*) | if *V* ∈ Float
«*V*» | = | `87``08` **binary64**(*V*) | if *V* ∈ Double

{:.postcard-grammar.binarysyntax}
«*V*» | = | `B0` **varint**(&#124;**intbytes**(*V*)&#124;) **intbytes**(*V*) | if *V* ∈ SignedInteger
«*V*» | = | `B1` **varint**(&#124;**utf8**(*V*)&#124;) **utf8**(*V*)         | if *V* ∈ String
«*V*» | = | `B2` **varint**(&#124;*V*&#124;) *V*                             | if *V* ∈ ByteString
«*V*» | = | `B3` **varint**(&#124;**utf8**(*V*)&#124;) **utf8**(*V*)         | if *V* ∈ Symbol

{:.postcard-grammar.binarysyntax}
«`<`*L* *F*<sub>1</sub> ... *F*<sub>m</sub>`>`» | = | `B4` «*L*» «*F*<sub>1</sub>» ... «*F*<sub>m</sub>» `84`
«`[`*X*<sub>1</sub> ... *X*<sub>m</sub>`]`» | = | `B5` «*X*<sub>1</sub>» ... «*X*<sub>m</sub>» `84`
«`#{`*E*<sub>1</sub> ... *E*<sub>m</sub>`}`» | = | `B6` «*E*<sub>1</sub>» ... «*E*<sub>m</sub>» `84`
«`{`*K*<sub>1</sub>`:`*V*<sub>1</sub> ... *K*<sub>m</sub>`:`*V*<sub>m</sub>`}`» | = | `B7` «*K*<sub>1</sub>» «*V*<sub>1</sub>» ... «*K*<sub>m</sub>» «*V*<sub>m</sub>» `84`

{:.postcard-grammar.binarysyntax}
**varint**(*n*) | = | <span class="outputish">*n*</span> | if *n* &lt; 128
| | <span class="outputish">(*n* &amp; 127) &#124; 128</span> **varint**(*n* &gt;&gt; 7) | if *n* ≥ 128

{:.postcard-grammar.binarysyntax}
**intbytes**(*n*) | = | <span class="roman">(the empty sequence)</span> | if *n* = 0
| | **signedBigEndian**(*n*) | otherwise
**signedBigEndian**(*n*) | = | <span class="outputish">*n* &amp; 255</span> | if -128 ≤ *n* ≤ 127
| | **signedBigEndian**(*n* &gt;&gt; 8) <span class="outputish">*n* &amp; 255</span> | otherwise

The functions <span class="postcard-grammar binarysyntax">**binary32**(*F*)</span> and <span
class="postcard-grammar binarysyntax">**binary64**(*D*)</span> yield big-endian 4- and 8-byte
IEEE 754 binary representations of <span class="postcard-grammar binarysyntax">*F*</span> and
<span class="postcard-grammar binarysyntax">*D*</span>, respectively.

<!--
Together, <span class="postcard-grammar binarysyntax">**div**</span> and <span
class="postcard-grammar binarysyntax">**mod**</span> give [Euclidean
division](https://en.wikipedia.org/wiki/Euclidean_division); that is, if
<span class="postcard-grammar binarysyntax">*n* **div** *d* = *q*</span> and
<span class="postcard-grammar binarysyntax">*n* **mod** *d* = *r*</span>, then
<span class="postcard-grammar binarysyntax">*n* = *dq* + *r*</span> and
<span class="postcard-grammar binarysyntax">0 ≤ *r* &lt; |d|</span>.
-->

<!--
For a value `V`, we write `«V»` for the binary encoding of `V`.

                          «#f» = [0x80]
                          «#t» = [0x81]

                        «@W V» = [0x85] ++ «W» ++ «V»
                         «#!V» = [0x86] ++ «V»

      «V» if V ∈ Float         = [0x87, 0x04] ++ binary32(V)
      «V» if V ∈ Double        = [0x87, 0x08] ++ binary64(V)

      «V» if V ∈ SignedInteger = [0xB0] ++ varint(|intbytes(V)|) ++ intbytes(V)
      «V» if V ∈ String        = [0xB1] ++ varint(|utf8(V)|) ++ utf8(V)
      «V» if V ∈ ByteString    = [0xB2] ++ varint(|V|) ++ V
      «V» if V ∈ Symbol        = [0xB3] ++ varint(|utf8(V)|) ++ utf8(V)

               «<L F_1...F_m>» = [0xB4] ++ «L» ++ «F_1» ++...++ «F_m» ++ [0x84]
                 «[X_1...X_m]» = [0xB5] ++ «X_1» ++...++ «X_m» ++ [0x84]
                «#{E_1...E_m}» = [0xB6] ++ «E_1» ++...++ «E_m» ++ [0x84]
         «{K_1:V_1...K_m:V_m}» = [0xB7] ++ «K_1» ++ «V_1» ++...++ «K_m» ++ «V_m» ++ [0x84]

                     varint(v) = [v] if v < 128
                                 [(v & 0x7F) + 128] ++ varint(v >> 7) if v ≥ 128

The functions `binary32(F)` and `binary64(D)` yield big-endian 4- and
8-byte IEEE 754 binary representations of `F` and `D`, respectively.

The function `intbytes(x)` is a big-endian two's-complement signed binary representation of
`x`, taking exactly as many whole bytes as needed to unambiguously identify the value and its
sign. In particular, `intbytes(0)` is the empty byte sequence.
-->