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- Copyright 2001, 2004 Free Software Foundation, Inc.
- This file is part of the GNU MP Library.
- The GNU MP Library is free software; you can redistribute it and/or modify
- it under the terms of 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.
- The GNU MP Library is distributed in the hope that it will be useful, but
- WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
- or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
- License for more details.
- You should have received a copy of the GNU Lesser General Public License
- along with the GNU MP Library. If not, see http://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 varargs 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.
- ----------------
- Local variables:
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