Copyright 2001, 2004 Free Software Foundation, Inc.

This file is part of the GNU MP Library.

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it under the terms of either:

  * the GNU Lesser General Public License as published by the Free
    Software Foundation; either version 3 of the License, or (at your
    option) any later version.

or

  * the GNU General Public License as published by the Free Software
    Foundation; either version 2 of the License, or (at your option) any
    later version.

or both in parallel, as here.

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see https://www.gnu.org/licenses/.






                    GMP EXPRESSION EVALUATION
                    -------------------------



THIS CODE IS PRELIMINARY AND MAY BE SUBJECT TO INCOMPATIBLE CHANGES IN
FUTURE VERSIONS OF GMP.



The files in this directory implement a simple scheme of string based
expression parsing and evaluation, supporting mpz, mpq and mpf.

This will be slower than direct GMP library calls, but may be convenient in
various circumstances, such as while prototyping, or for letting a user
enter values in symbolic form.  "2**5723-7" for example is a lot easier to
enter or maintain than the equivalent written out in decimal.



BUILDING

Nothing in this directory is a normal part of libgmp, and nothing is built
or installed, but various Makefile rules are available to compile
everything.

All the functions are available through a little library (there's no shared
library since upward binary compatibility is not guaranteed).

	make libexpr.a

In a program, prototypes are available using

	#include "expr.h"

run-expr.c is a sample program doing evaluations from the command line.

	make run-expr
	./run-expr '1+2*3'

t-expr.c is self-test program, it prints nothing if successful.

	make t-expr
	./t-expr

The expr*.c sources don't depend on gmp-impl.h and can be compiled with just
a standard installed GMP.  This isn't true of t-expr though, since it uses
some of the internal tests/libtests.la.



SIMPLE USAGE

int mpz_expr (mpz_t res, int base, const char *e, ...);
int mpq_expr (mpq_t res, int base, const char *e, ...);
int mpf_expr (mpf_t res, int base, const char *e, ...);

These functions evaluate simple arithmetic expressions.  For example,

	mpz_expr (result, 0, "123+456", NULL);

Numbers are parsed by mpz_expr and mpq_expr the same as mpz_set_str with the
given base.  mpf_expr follows mpf_set_str, but supporting an "0x" prefix for
hex when base==0.

	mpz_expr (result, 0, "0xAAAA * 0x5555", NULL);

White space, as indicated by <ctype.h> isspace(), is ignored except for the
purpose of separating tokens.

Variables can be included in expressions by putting them in the stdarg list
after the string.  "a", "b", "c" etc in the expression string designate
those values.  For example,

        mpq_t  foo, bar;
        ...
	mpq_expr (q, 10, "2/3 + 1/a + b/2", foo, bar, NULL);

Here "a" will be the value from foo and "b" from bar.  Up to 26 variables
can be included this way.  The NULL must be present to indicate the end of
the list.

Variables can also be written "$a", "$b" etc.  This is necessary when using
bases greater than 10 since plain "a", "b" etc will otherwise be interpreted
as numbers.  For example,

        mpf_t  quux;
        mpf_expr (f, 16, "F00F@-6 * $a", quux, NULL);

All the standard C operators are available, with the usual precedences, plus
"**" for exponentiation at the highest precedence (and right associative).

        Operators      Precedence
         **              220
         ~ ! - (unary)   210
         * / %           200
         + -             190
         << >>           180
         <= < >= >       170
         == !=           160
         &               150
         ^               140
         |               130
         &&              120
         ||              110
         ? :             100/101

Currently only mpz_expr has the bitwise ~ % & ^ and | operators.  The
precedence numbers are of interest in the advanced usage described below.

Various functions are available too.  For example,

        mpz_expr (res, 10, "gcd(123,456,789) * abs(a)", var, NULL);

The following is the full set of functions,

        mpz_expr
            abs bin clrbit cmp cmpabs congruent_p divisible_p even_p fib fac
            gcd hamdist invert jacobi kronecker lcm lucnum max min nextprime
            odd_p perfect_power_p perfect_square_p popcount powm
            probab_prime_p root scan0 scan1 setbit sgn sqrt

        mpq_expr
            abs, cmp, den, max, min, num, sgn

        mpf_expr
            abs, ceil, cmp, eq, floor, integer_p, max, min, reldiff, sgn,
            sqrt, trunc

All these are the same as the GMP library functions, except that min and max
don't exist in the library.  Note also that min, max, gcd and lcm take any
number of arguments, not just two.

mpf_expr does all calculations to the precision of the destination variable.


Expression parsing can succeed or fail.  The return value indicates this,
and will be one of the following

	MPEXPR_RESULT_OK
	MPEXPR_RESULT_BAD_VARIABLE
	MPEXPR_RESULT_BAD_TABLE
	MPEXPR_RESULT_PARSE_ERROR
	MPEXPR_RESULT_NOT_UI

BAD_VARIABLE is when a variable is referenced that hasn't been provided.
For example if "c" is used when only two parameters have been passed.
BAD_TABLE is applicable to the advanced usage described below.

PARSE_ERROR is a general syntax error, returned for any mal-formed input
string.

NOT_UI is returned when an attempt is made to use an operand that's bigger
than an "unsigned long" with a function that's restricted to that range.
For example "fib" is mpz_fib_ui and only accepts an "unsigned long".




ADVANCED USAGE

int mpz_expr_a (const struct mpexpr_operator_t *table,
                mpz_ptr res, int base, const char *e, size_t elen,
                mpz_srcptr var[26])
int mpq_expr_a (const struct mpexpr_operator_t *table,
                mpq_ptr res, int base, const char *e, size_t elen,
                mpq_srcptr var[26])
int mpf_expr_a (const struct mpexpr_operator_t *table,
                mpf_ptr res, int base, unsigned long prec,
                const char *e, size_t elen,
                mpf_srcptr var[26])

These functions are an advanced interface to expression parsing.

The string is taken as pointer and length.  This makes it possible to parse
an expression in the middle of somewhere without copying and null
terminating it.

Variables are an array of 26 pointers to the appropriate operands, or NULL
for variables that are not available.  Any combination of variables can be
given, for example just "x" and "y" (var[23] and var[24]) could be set.

Operators and functions are specified with a table.  This makes it possible
to provide additional operators or functions, or to completely change the
syntax.  The standard tables used by the simple functions above are
available as

	const struct mpexpr_operator_t * const mpz_expr_standard_table;
	const struct mpexpr_operator_t * const mpq_expr_standard_table;
	const struct mpexpr_operator_t * const mpf_expr_standard_table;

struct mpexpr_operator_t is the following

	struct mpexpr_operator_t {
	  const char    *name;
	  mpexpr_fun_t  fun;
	  int           type;
	  int           precedence;
	};

        typedef void (*mpexpr_fun_t) (void);

As an example, the standard mpz_expr table entry for multiplication is as
follows.  See the source code for the full set of standard entries.

	{ "*", (mpexpr_fun_t) mpz_mul, MPEXPR_TYPE_BINARY, 200 },

"name" is the string to parse, "fun" is the function to call for it, "type"
indicates what parameters the function takes (among other things), and
"precedence" sets its operator precedence.

A NULL for "name" indicates the end of the table, so for example an mpf
table with nothing but addition could be

        struct mpexpr_operator_t  table[] = {
          { "+", (mpexpr_fun_t) mpf_add, MPEXPR_TYPE_BINARY, 190 },
          { NULL }
        };

A special type MPEXPR_TYPE_NEW_TABLE makes it possible to chain from one
table to another.  For example the following would add a "mod" operator to
the standard mpz table,

        struct mpexpr_operator_t  table[] = {
        { "mod", (mpexpr_fun_t) mpz_fdiv_r, MPEXPR_TYPE_BINARY, 125 },
        { (const char *) mpz_expr_standard_table, NULL, MPEXPR_TYPE_NEW_TABLE }
        };

Notice the low precedence on "mod", so that for instance "45+26 mod 7"
parses as "(45+26)mod7".


Functions are designated by a precedence of 0.  They always occur as
"foo(expr)" and so have no need for a precedence level.  mpq_abs in the
standard mpq table is

	{ "abs", (mpexpr_fun_t) mpq_abs, MPEXPR_TYPE_UNARY },

Functions expecting no arguments as in "foo()" can be given with
MPEXPR_TYPE_0ARY, or actual constants to be parsed as just "foo" are
MPEXPR_TYPE_CONSTANT.  For example if a "void mpf_const_pi(mpf_t f)"
function existed (which it doesn't) it could be,

	{ "pi", (mpexpr_fun_t) mpf_const_pi, MPEXPR_TYPE_CONSTANT },


Parsing of operator names is done by seeking the table entry with the
longest matching name.  So for instance operators "<" and "<=" exist, and
when presented with "x <= y" the parser matches "<=" because it's longer.

Parsing of function names, on the other hand, is done by requiring a whole
alphanumeric word to match.  For example presented with "fib2zz(5)" the
parser will attempt to find a function called "fib2zz".  A function "fib"
wouldn't be used because it doesn't match the whole word.

The flag MPEXPR_TYPE_WHOLEWORD can be ORed into an operator type to override
the default parsing style.  Similarly MPEXPR_TYPE_OPERATOR into a function.


Binary operators are left associative by default, meaning they're evaluated
from left to right, so for example "1+2+3" is treated as "(1+2)+3".
MPEXPR_TYPE_RIGHTASSOC can be ORed into the operator type to work from right
to left as in "1+(2+3)".  This is generally what's wanted for
exponentiation, and for example the standard mpz table has

        { "**", (mpexpr_fun_t) mpz_pow_ui,
          MPEXPR_TYPE_BINARY_UI | MPEXPR_TYPE_RIGHTASSOC, 220 }

Unary operators are postfix by default.  For example a factorial to be used
as "123!" might be

	{ "!", (mpexpr_fun_t) mpz_fac_ui, MPEXPR_TYPE_UNARY_UI, 215 }

MPEXPR_TYPE_PREFIX can be ORed into the type to get a prefix operator.  For
instance negation (unary minus) in the standard mpf table is

	{ "-", (mpexpr_fun_t) mpf_neg,
          MPEXPR_TYPE_UNARY | MPEXPR_TYPE_PREFIX, 210 },


The same operator can exist as a prefix unary and a binary, or as a prefix
and postfix unary, simply by putting two entries in the table.  While
parsing the context determines which style is sought.  But note that the
same operator can't be both a postfix unary and a binary, since the parser
doesn't try to look ahead to decide which ought to be used.

When there's two entries for an operator, both prefix or both postfix (or
binary), then the first in the table will be used.  This makes it possible
to override an entry in a standard table, for example to change the function
it calls, or perhaps its precedence level.  The following would change mpz
division from tdiv to cdiv,

        struct mpexpr_operator_t  table[] = {
          { "/", (mpexpr_fun_t) mpz_cdiv_q, MPEXPR_TYPE_BINARY, 200 },
          { "%", (mpexpr_fun_t) mpz_cdiv_r, MPEXPR_TYPE_BINARY, 200 },
          { (char *) mpz_expr_standard_table, NULL, MPEXPR_TYPE_NEW_TABLE }
        };


The type field indicates what parameters the given function expects.  The
following styles of functions are supported.  mpz_t is shown, but of course
this is mpq_t for mpq_expr_a, mpf_t for mpf_expr_a, etc.

    MPEXPR_TYPE_CONSTANT     void func (mpz_t result);

    MPEXPR_TYPE_0ARY         void func (mpz_t result);
    MPEXPR_TYPE_I_0ARY       int func (void);

    MPEXPR_TYPE_UNARY        void func (mpz_t result, mpz_t op);
    MPEXPR_TYPE_UNARY_UI     void func (mpz_t result, unsigned long op);
    MPEXPR_TYPE_I_UNARY      int func (mpz_t op);
    MPEXPR_TYPE_I_UNARY_UI   int func (unsigned long op);

    MPEXPR_TYPE_BINARY       void func (mpz_t result, mpz_t op1, mpz_t op2);
    MPEXPR_TYPE_BINARY_UI    void func (mpz_t result,
                                        mpz_t op1, unsigned long op2);
    MPEXPR_TYPE_I_BINARY     int func (mpz_t op1, mpz_t op2);
    MPEXPR_TYPE_I_BINARY_UI  int func (mpz_t op1, unsigned long op2);

    MPEXPR_TYPE_TERNARY      void func (mpz_t result,
                                        mpz_t op1, mpz_t op2, mpz_t op3);
    MPEXPR_TYPE_TERNARY_UI   void func (mpz_t result, mpz_t op1, mpz_t op2,
                                        unsigned long op3);
    MPEXPR_TYPE_I_TERNARY    int func (mpz_t op1, mpz_t op2, mpz_t op3);
    MPEXPR_TYPE_I_TERNARY_UI int func (mpz_t op1, mpz_t op2,
                                       unsigned long op3);

Notice the pattern of "UI" for the last parameter as an unsigned long, or
"I" for the result as an "int" return value.

It's important that the declared type for an operator or function matches
the function pointer given.  Any mismatch will have unpredictable results.

For binary functions, a further type attribute is MPEXPR_TYPE_PAIRWISE which
indicates that any number of arguments should be accepted, and evaluated by
applying the given binary function to them pairwise.  This is used by gcd,
lcm, min and max.  For example the standard mpz gcd is

	{ "gcd", (mpexpr_fun_t) mpz_gcd,
	  MPEXPR_TYPE_BINARY | MPEXPR_TYPE_PAIRWISE },

Some special types exist for comparison operators (or functions).
MPEXPR_TYPE_CMP_LT through MPEXPR_TYPE_CMP_GE expect an MPEXPR_TYPE_I_BINARY
function, returning positive, negative or zero like mpz_cmp and similar.
For example the standard mpf "!=" operator is

	{ "!=", (mpexpr_fun_t) mpf_cmp, MPEXPR_TYPE_CMP_NE, 160 },

But there's no obligation to use these types, for instance the standard mpq
table just uses a plain MPEXPR_TYPE_I_BINARY and mpq_equal for "==".

Further special types MPEXPR_TYPE_MIN and MPEXPR_TYPE_MAX exist to implement
the min and max functions, and they take a function like mpf_cmp similarly.
The standard mpf max function is

	{ "max",  (mpexpr_fun_t) mpf_cmp,
          MPEXPR_TYPE_MAX | MPEXPR_TYPE_PAIRWISE },

These can be used as operators too, for instance the following would be the
>? operator which is a feature of GNU C++,

	{ ">?", (mpexpr_fun_t) mpf_cmp, MPEXPR_TYPE_MAX, 175 },

Other special types are used to define "(" ")" parentheses, "," function
argument separator, "!" through "||" logical booleans, ternary "?"  ":", and
the "$" which introduces variables.  See the sources for how they should be
used.


User definable operator tables will have various uses.  For example,

  - a subset of the C operators, to be rid of infrequently used things
  - a more mathematical syntax like "." for multiply, "^" for powering,
    and "!" for factorial
  - a boolean evaluator with "^" for AND, "v" for OR
  - variables introduced with "%" instead of "$"
  - brackets as "[" and "]" instead of "(" and ")"

The only fixed parts of the parsing are the treatment of numbers, whitespace
and the two styles of operator/function name recognition.

As a final example, the following would be a complete mpz table implementing
some operators with a more mathematical syntax.  Notice there's no need to
preserve the standard precedence values, anything can be used so long as
they're in the desired relation to each other.  There's also no need to have
entries in precedence order, but it's convenient to do so to show what comes
where.

        static const struct mpexpr_operator_t  table[] = {
	  { "^",   (mpexpr_fun_t) mpz_pow_ui,
            MPEXPR_TYPE_BINARY_UI | MPEXPR_TYPE_RIGHTASSOC,           9 },

          { "!",   (mpexpr_fun_t) mpz_fac_ui, MPEXPR_TYPE_UNARY_UI,   8 },
          { "-",   (mpexpr_fun_t) mpz_neg,
            MPEXPR_TYPE_UNARY | MPEXPR_TYPE_PREFIX,                   7 },

          { "*",   (mpexpr_fun_t) mpz_mul,    MPEXPR_TYPE_BINARY,     6 },
          { "/",   (mpexpr_fun_t) mpz_fdiv_q, MPEXPR_TYPE_BINARY,     6 },

          { "+",   (mpexpr_fun_t) mpz_add,    MPEXPR_TYPE_BINARY,     5 },
          { "-",   (mpexpr_fun_t) mpz_sub,    MPEXPR_TYPE_BINARY,     5 },

          { "mod", (mpexpr_fun_t) mpz_mod,    MPEXPR_TYPE_BINARY,     6 },

          { ")",   NULL,                      MPEXPR_TYPE_CLOSEPAREN, 4 },
          { "(",   NULL,                      MPEXPR_TYPE_OPENPAREN,  3 },
          { ",",   NULL,                      MPEXPR_TYPE_ARGSEP,     2 },

          { "$",   NULL,                      MPEXPR_TYPE_VARIABLE,   1 },
          { NULL }
        };




INTERNALS

Operator precedence is implemented using a control and data stack, there's
no C recursion.  When an expression like 1+2*3 is read the "+" is held on
the control stack and 1 on the data stack until "*" has been parsed and
applied to 2 and 3.  This happens any time a higher precedence operator
follows a lower one, or when a right-associative operator like "**" is
repeated.

Parentheses are handled by making "(" a special prefix unary with a low
precedence so a whole following expression is read.  The special operator
")" knows to discard the pending "(".  Function arguments are handled
similarly, with the function pretending to be a low precedence prefix unary
operator, and with "," allowed within functions.  The same special ")"
operator recognises a pending function and will invoke it appropriately.

The ternary "? :" operator is also handled using precedences.  ":" is one
level higher than "?", so when a valid a?b:c is parsed the ":" finds a "?"
on the control stack.  It's a parse error for ":" to find anything else.



FUTURE

The ternary "?:" operator evaluates the "false" side of its pair, which is
wasteful, though it ought to be harmless.  It'd be better if it could
evaluate only the "true" side.  Similarly for the logical booleans "&&" and
"||" if they know their result already.

Functions like MPEXPR_TYPE_BINARY could return a status indicating operand
out of range or whatever, to get an error back through mpz_expr etc.  That
would want to be just an option, since plain mpz_add etc have no such
return.

Could have assignments like "a = b*c" modifying the input variables.
Assignment could be an operator attribute, making it expect an lvalue.
There would want to be a standard table without assignments available
though, so user input could be safely parsed.

The closing parenthesis table entry could specify the type of open paren it
expects, so that "(" and ")" could match and "[" and "]" match but not a
mixture of the two.  Currently "[" and "]" can be added, but there's no
error on writing a mixed expression like "2*(3+4]".  Maybe also there could
be a way to say that functions can only be written with one or the other
style of parens.



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