gp2c types and the description systemBy Bill Allombert 
Contents
1 gp2c types
1.1 Introduction
The main feature GP2C adds above GP is the use of types. Types give a
semantic to PARI objects, so that GP2C can generate code that use
specialized (hence faster) PARI functions instead of generic ones. Please
read the section ’Advanced use of GP2C’ in the GP2C manual for how to use
the GP2C types and examples.
Abstractly, a GP2C type is a set of pairs (A,B) where A is a mathematical
object and B its computer representation. Two different types can contain
the same mathematical object, with a different computer representation.
For example the type bool is the set {(true,1L), (false,0L)},
i.e. true and false with true coded by the Clong integer 1 and
false coded by the Clong integer 0; the type negbool is
the set {(true,0L), (false,1L)} which is the same set of mathematical objects,
but now true is coded by the Clong integer 0 and false by the Clong
integer 1.
For each GP2C type, there exists a C type Ct such that for all pairs (A,B)
belonging to the type, the C type of B is Ct. This C type is specified by
the description _typedef.
The GP2C types are preordered. Abstractly, we say that t_{1}≺
t_{2} if and only if there is a map f such that (A,B)↦ (A,f(B))
defines a onetoone mapping from t_{1} to t_{2}.
Practically we restrict the relation ≺ to make it a partial order
such that any two types have an upper bound.
This partial order is defined by the chains in the description
_type_preorder. It can be printed by running gp2c t
.
Figure 1: Example of type preorder 
The process of converting a mathematical object from one type to
another is called casting. The casting methods known to GP2C are
given by the _cast description.
1.2 List of types
In this section, we list the types known to PARI/GP.
The current list is available in the description _typedef.
1.2.1 Basic types

small
 Small integers represented by C long integers.
 int
 Multiprecision integers represented by t_INT GENs.
 real
 Multiprecision floating point real numbers represented by t_REAL GENs.
 mp
 Multiprecision numbers. Union of the types int and real.
 vecsmall
 Vectors of small integers represented by t_VECSMALL GENs.
 vec
 Vectors and matrices of PARI objects, represented by t_VEC,
t_COL or t_MAT GENs.
 var
 Polynomial variables represented by their variable number which is
a C long integer. This is associated to the prototype code ’n’.
 pol
 Polynomials represented by t_POL GENs.
 genstr
 Strings represented by t_STR GENs.
 list
 GP lists represented by t_LIST GENs.
 gen
 Generic PARI objects represented by GENs.
1.2.2 Special types

void
 This type is a set with only one element called void. This is
the return type of functions not returning anything. GP allows to cast it to
0.
 bool
 Boolean values represented as C long integers, where 1 is
true and 0 is false.
 negbool
 Negated boolean values represented as C long integers,
where 0 is true and 1 is false.
 lg
 Vector lengths represented by the lg macro output, i.e.
a C long integer equal to the actual length plus one.
 str
 C strings represented by C const char * strings.
 typ
 GEN types represented by C long integers, as returned by the
typ() macro.
1.2.3 Internal types
The following types are mainly for internal use inside GP2C.

empty
 This type is the empty set. No individual object can be of this
type but a set of objects can. In fact this is a default type for an
unspecified set of objects.
 small_int
 Small integers represented by C int integers.
This is only available for compatibility with PARI function returning
int (instead of long).
 bptr
 Byte pointer. This is used for the primepointer global
variable.
 func_GG
 function with protype GEN f(GEN,GEN). Used by
forvec.
 pari_sp
 This is the stack pointer type pari_sp.
1.2.4 Types for data structure.
These types are mainly defined to allow the use of inline member functions.

nf
 Number field as returned by nfinit.
 bnf
 Big number field as returned by bnfinit.
 bnr
 Ray class field as returned by bnrinit.
 ell
 Elliptic curve as returned by ellinit.
 bell
 Big elliptic curve as returned by ellinit.
 clgp
 Class group as returned by the ’clgp’ member function.
 prid
 Prime ideal as returned by idealprimedec.
 gal
 Galois group as returned by galoisinit.
1.3 CType
A CType is just a level of indirection toward real C types.
Ctypes are defined by the descriptions _decl_base and
_decl_ext. Each type belongs to a Ctype as specified by the
description _typedef.
2 The description system
2.1 Introduction
The description system is a way to describe the PARI application programming
interface in a way understandable by both the GP2C compiler and human
beings. The present document is mostly addressed to this second category.
We start by a simple example:
The description of the GP function sqr is
(int):int sqri($1)
(mp):mp gsqr($1)
(gen):gen gsqr($1)
Each line is called a rule, which in this case consists of three
parts. Let us consider the first one: the parts (int)
, :int
and
sqri($1)
are respectively called the pattern, type,
and action part.
When GP2C compiles sqr(1), it computes the types of the arguments
(here 1 is of type small) and matches them against the patterns
from top to bottom. The “best” rule is used; in case of a tie, the topmost
rule wins. Here, all three rules apply, and the first rule wins. Since the type
of this rule is int, GP2C sets the type of the expression
sqr(1) to int. The action part is sqri($1)
, so GP2C
generates the C code sqri($1) where $1 is replaced by the
code of the argument 1 cast to the pattern type (int). The result
is the C code sqri(gen_1).
Now a more complex example: the description of the GP function exp is
(real):real mpexp($1)
(mp):mp:prec gexp($1, prec)
(gen):gen:prec gexp($1, prec)
When GP2C compiles exp(1), it looks for the "best" rules.
The first rule cannot be used, because there is no way to cast a
small to a real, so it uses the second rule.
This time the result will be
of type mp. The extra part :prec
is called a mode. The
mode ‘prec’ states that the action part will use the special ‘prec’
variable that holds the current real precision. This is obvious from
the action part code, but GP2C do not parse the action part so it
needs this mode. Note that the last rule is also valid and has the
same action part so would generate the exact same code. However, the
type of the expression would be less precise.
The description of the GP function matdet is
(gen, ?0):gen det($1)
(gen, 1):gen det2($1)
(gen, #small):gen $"incorrect flag in matdet"
(gen, small):gen det0($1, $2)
We see several new pattern atoms:

1 matches a literal 1, e.g. matdet(M,1) would match the
second rule.
 ?0 matches an optional literal 0: matdet(M),
matdet(M,0) and matdet(M,) all match the first rule.
 #small matches an unspecified literal small.
Finally, we also see a new action $"…", which causes GP2C
to display the error message and abort.
2.2 Definitions
We now give a formal definition of descriptions.
2.2.1 Description
A description is a lineseparated list of rules.
A rule is a line of the form
(pattern):type:modelist action
Only the pattern part is mandatory, though most rules also include
an action and a type.
A pattern is a commaseparated list of pattern atoms.
The type of a rule is a standard GP2C type.
A modelist is a colonseparated list of modes.
An action is a string (normally a piece of C code) that can
include replacement strings. Replacement strings start by
a $ and are substituted according to the replacement rules.
2.3 Pattern atom
A pattern atom is one of the following, where type is any GP2C type,
n any small integer, "str" any character string and ctype
any Ctype. A pattern atom can match an object.

type. This matches any object of type comparable to type.
 n. This matches a constant small integer value equal to n.
 ?n. This matches an optional small value which defaults to
n.
 ?type. This matches an optional type value with standard default value.
 "str". This matches a constant character string equal to str.
 &type. This matches a reference (the GP &x construction) to
an object of type equal or less than type referencing the same data type.
 nothing. This matches a missing argument.
 #type. This matches a constant value of type type.
 ... This matches any number of arguments matching the previous atom.
This must be the last atom of the pattern. This allows to implement functions
taking an unlimited number of arguments.
 C!ctype. This matches any object of Ctype ctype.
 @type. This matches a variable of type type. This is mainly used
for expressions that evaluate their arguments several times.
 *type. This matches an lvalue of type type. This is used in
constructions that modify their arguments.
2.4 Matching
The best rule is determined as follows:

The result of matching a pattern atom against some GP code is
either ’reject’ or ’match’.
 There are three matching levels: ’partial’, ’normal’ and ’perfect’.
 A pattern matches if all the atoms match.
 A rule matches if its pattern matches.
 The best rule is the matching rule with the higher number of normal
and perfect matches. In case of a tie, the highest number of perfect matches
wins. If there is still a tie, the topmost rule wins.
When matching the pattern atoms type and ?type, the matching
level is determined as follows:

a perfect match occurs when the type of the object is exactly type,
 a normal match when the type is less than type,
 a partial match when the type is bigger than type.
 Rejection happens when the types are uncomparable.
Other pattern atoms always result in a reject or a perfect match.
Modes are used in descriptions to give more information to GP2C about the
action part. They are usually useless to human beings that are smart
enough to understand the action part. The current list of modes is:

prec
 The action uses the prec variable.
 parens
 The action does not have top precedence. GP2C will put it between
parentheses when needed (see $())
 copy
 The action returns data that access memory belonging to other
objects. GP2C will generate calls to gcopy() when needed.
2.6 Lists of replacement strings
The following special sequences can occur in the action part:
 $n. This is replaced by the nth argument of the function.
 $(n). This is replaced by the nth argument of the function
between parenthesis if it has the parens mode.
 $type:n. This is replaced by the nth argument of the
function cast to type type.
 $(type:n). Combination of $(n) and $type:n.
 $%n. This is replaced by the nth argument of the function,
which must be a constant string, with all % characters doubled and no quotes.
This is for use inside format specification.
 $"message". Signals an invalid condition. GP2C will abort by
printing the error message message.
 ${RPN sequence}
The RPN sequence is a space separated list of RPN commands that will be
evaluated by the GP2C internal RPN evaluator.
If the stack is empty at the end of the evaluation, this is replaced by
the empty string, else this is replaced by the integer at the top of the
stack. Some RPN commands generate text, in that case it is pasted just before
the $ sign.
2.7 Lists of RPN commands
The commands are evaluated with respect to a stack of integer values,
initially empty.
The exact list of command supported by a particular GP2C version is the
%accepted_command hash in the script scripts/822_desc.pl.in.

literal integer
 push the integer at the top of the stack.
 :type
 push the type type at the top of the stack.
 add, sub, mul, div, mod
 2ary arithmetic operators
 neg
 1ary arithmetic operator
 and, or, xor
 2ary logical operators
 not
 1ary logical operator
 type
 pop an integer n and push the type of the nth argument.
 value
 pop an integer n and push the value of the nth argument,
provided it is a constant integer.
 code
 pop an integer n and generate the C code for the nth argument.
 cast
 pop an integer n and a type t and generate the C code for the
nth argument cast to type t.
 parens
 this is a flag requesting code and cast to enclose
the C code between parenthesis if the argument has the parens mode.
 str_format, str_raw
 pop an integer n such that the nth argument
is a constant string and display the string without leading and ending ".
Furthermore str_format will display the string in a way suitable for
inclusion in a format string by quoting metacharacters.
The following RPN commands are useful with the ... pattern atom to
implement functions that take an unlimited number of arguments.

nbarg
 push the actual number of arguments of the function.
 format_string, format_args
 pop an integer n such that the nth
argument corresponds to a ... pattern atom and generate a format string
and a list of arguments, see the description of the GP function print.
 code, cast
 If the integer n corresponds to a ... pattern atom,
generate a commaseparated list of C code for the arguments n−1, n, n+1,
…, nbarg, by matching each argument against the n−1 pattern atom.
 stdref
 this is a flag requesting code and type to
prepend a ’&’ before each arguments.
This document was translated from L^{A}T_{E}X by
H^{E}V^{E}A.