mpexpr(TCL)mpexpr(TCL)NAMEmpexpr - Evaluate an expression with multiple precision math
SYNOPSIS
package require Mpexpr
mpexpr arg ?arg arg ...?
mpformat formatString ?arg arg ...?
global mp_precision
DESCRIPTION
Mpexpr is based on Tcl's native expr command, and shares many similari‐
ties with expr. Mpexpr performs all of its calculations using an arbi‐
trary precision math package.
Mpexpr concatenates arg's (adding separator spaces between them), eval‐
uates the result as a Tcl expression, and returns the value. The oper‐
ators permitted in Tcl expressions are a subset of the operators per‐
mitted in C expressions, and they have the same meaning and precedence
as the corresponding C operators. Expressions almost always yield
numeric results (integer or floating-point values). For example, the
expression
mpexpr 8.2 + 6
evaluates to 14.2. Tcl expressions differ from C expressions in the
way that operands are specified. Also, Tcl expressions support non-
numeric operands and string comparisons.
OPERANDS
A Tcl expression consists of a combination of operands, operators, and
parentheses. White space may be used between the operands and opera‐
tors and parentheses; it is ignored by the expression processor. Where
possible, operands are interpreted as integer values. Integer values
may be specified in decimal (the normal case), in octal (if the first
character of the operand is 0), or in hexadecimal (if the first two
characters of the operand are 0x). If an operand does not have one of
the integer formats given above, then it is treated as a floating-point
number if that is possible. Floating-point numbers may be specified in
any of the ways accepted by an ANSI-compliant C compiler (except that
the ``f'', ``F'', ``l'', and ``L'' suffixes will not be permitted in
most installations). For example, all of the following are valid
floating-point numbers: 2.1, 3., 6e4, 7.91e+16. If no numeric inter‐
pretation is possible, then an operand is left as a string (and only a
limited set of operators may be applied to it).
Operands may be specified in any of the following ways:
[1] As an numeric value, either integer or floating-point.
[2] As a Tcl variable, using standard $ notation. The variable's
value will be used as the operand.
[3] As a string enclosed in double-quotes. The expression parser
will perform backslash, variable, and command substitutions on
the information between the quotes, and use the resulting value
as the operand
[4] As a string enclosed in braces. The characters between the open
brace and matching close brace will be used as the operand with‐
out any substitutions.
[5] As a Tcl command enclosed in brackets. The command will be exe‐
cuted and its result will be used as the operand.
[6] As a mathematical function whose arguments have any of the above
forms for operands, such as ``sin($x)''. See below for a list
of defined functions.
Where substitutions occur above (e.g. inside quoted strings), they are
performed by the expression processor. However, an additional layer of
substitution may already have been performed by the command parser
before the expression processor was called. As discussed below, it is
usually best to enclose expressions in braces to prevent the command
parser from performing substitutions on the contents.
For some examples of simple expressions, suppose the variable a has the
value 3 and the variable b has the value 6. Then the command on the
left side of each of the lines below will produce the value on the fol‐
lowing line:
mpexpr 3.1 + $a
6.1
mpexpr 2 + "$a.$b"
5.6
mpexpr 4*[llength "6 2"]
8
mpexpr {{word one} < "word $a"}
0
OPERATORS
The valid operators are listed below, grouped in decreasing order of
precedence:
- + ~ ! Unary minus, unary plus, bit-wise NOT, logical NOT.
None of these operands may be applied to string op‐
erands, and bit-wise NOT may be applied only to
integers.
* / % Multiply, divide, remainder. None of these oper‐
ands may be applied to string operands, and remain‐
der may be applied only to integers. The remainder
will always have the same sign as the divisor and
an absolute value smaller than the divisor.
+ - Add and subtract. Valid for any numeric operands.
<< >> Left and right shift. Valid for integer operands
only. Integers in mpexpr are not limited to a
machine word and do not use two's complement for‐
mat. Therefore shifting will not include a sign
bit.
< > <= >= Boolean less, greater, less than or equal, and
greater than or equal. Each operator produces 1 if
the condition is true, 0 otherwise. These opera‐
tors may be applied to strings as well as numeric
operands, in which case string comparison is used.
== != Boolean equal and not equal. Each operator pro‐
duces a zero/one result. Valid for all operand
types.
& Bit-wise AND. Valid for integer operands only.
^ Bit-wise exclusive OR. Valid for integer operands
only.
| Bit-wise OR. Valid for integer operands only.
&& Logical AND. Produces a 1 result if both operands
are non-zero, 0 otherwise. Valid for numeric oper‐
ands only (integers or floating-point).
|| Logical OR. Produces a 0 result if both operands
are zero, 1 otherwise. Valid for numeric operands
only (integers or floating-point).
x?y:z If-then-else, as in C. If x evaluates to non-zero,
then the result is the value of y. Otherwise the
result is the value of z. The x operand must have
a numeric value.
See the C manual for more details on the results produced by each oper‐
ator. All of the binary operators group left-to-right within the same
precedence level. For example, the command
mpexpr 4*2 < 7
returns 0.
The &&, ||, and ?: operators have ``lazy evaluation'', just as in C,
which means that operands are not evaluated if they are not needed to
determine the outcome. For example, in the command
mpexpr {$v ? [a] : [b]}
only one of [a] or [b] will actually be evaluated, depending on the
value of $v. Note, however, that this is only true if the entire
expression is enclosed in braces; otherwise the Tcl parser will evalu‐
ate both [a] and [b] before invoking the expr command.
MATH FUNCTIONS
Mpexpr supports the following mathematical functions in expressions. x
and y are integer or floating point values; i, j and c are integer val‐
ues;
Math functions compatible with expr:
acos(x) Arc cosine of x.
asin(x) Arc sine of x.
atan(x) Arc tangent of x.
atan2(x,y) Arc tangent of x / y.
ceil(x) Least integral value greater than or equal to x.
cos(x) Cosine of x.
cosh(x) Hyperbolic cosine of x.
exp(x) Exponential function e ** x.
floor(x) Greatest integral value less than or equal to x.
fmod(x,y) Remainder of x divided by y.
hypot(x,y) Euclidean distance of sqrt( x * x + y * y).
log(x) Natural logarithm of x.
log10(x) Base-10 logarithm of x.
pow(x,y) x raised to the y power.
sin(x) Sine of x.
sinh(x) Hyperbolic sine of x.
sqrt(x) Square root of x.
tan(x) Tangent of x.
tanh(x) Hyperbolic tangent of x.
abs(x) Returns the absolute value of x. x may be either inte‐
ger or floating-point, and the result is returned in the
same form.
double(x) If x is a floating value, returns x, otherwise converts
x to floating and returns the converted value.
int(x) If x is an integer value, returns x, otherwise converts
x to integer by truncation and returns the converted
value.
round(x) If x is an integer value, returns x, otherwise converts
x to integer by rounding and returns the converted
value.
Additional mpexpr functions:
root(x,y) The yth root of x.
frem(x,y) Remove all occurance of factory from number x.
minv(x,y) Inverse of x modulo y.
gcd(x,y) Greatest common divisor of x and y.
lcm(x,y) Least common multiple of x and y.
max(x,y) Maximum of x and y.
min(x,y) Minimum of x and y.
pi() Value of pi.
fib(i) Fibonacci number of integer i.
fact(i) Factorial of integer i.
pfact(i) Product of prime numbers up to integer i.
lfactor(i,c) Lowest prime factor of integer i, trying count c times.
iroot(i,j) Integer root j of integer i.
gcdrem(i,j) Relatively prime of greatest common divisior of i
divided by j.
perm(i,j) Permutations of i taking j at a time: i ! / ( i - j ) !.
comb(i,j) Combinations of i taking j at a time: i ! / ( j ! * ( i
- j ) ! ) .
prime(i,c) Return 0 if i is not prime, return 1 if i probably is
prime. Test for primality count c times. The chance of
a non-prime passing this test is less than (1/4)^count.
For example, a count of 100 fails for only 1 in 10^60
numbers.
relprime(i,j) Return 1 if i and j are relatively prime to each other,
0 otherwise.
TYPES, OVERFLOW, AND PRECISION
Computations are performed using arbitrary fixed and floating point
values. Native machine values (int, long, IEEE 754 floating point,
etc. ) and instructions are not used. Conversion among internal rep‐
resentations for integer, floating-point, and string operands is done
automatically as needed. For arithmetic computations, integers are
used until some floating-point number is introduced, after which float‐
ing-point is used. For example,
mpexpr 5 / 4
returns 1, while
mpexpr 5 / 4.0
mpexpr 5 / ( [string length "abcd"] + 0.0 )
both return 1.25. Floating-point values are always returned with a
``.'' or an ``e'' so that they will not look like integer values. For
example,
mpexpr 20.0/5.0
returns ``4.0'', not ``4''.
The global variable mp_precision determines the number of significant
digits that are retained during evaluation. If mp_precision is unset
then 17 digits of precision are used. The maximum value of mp_preci‐
sion is 10000. Note that larger values for mp_precision will require
increasingly longer execution times. Setting mp_precision to an ille‐
gal value will generate an error.
STRING OPERATIONS
String values may be used as operands of the comparison operators,
although the expression evaluator tries to do comparisons as integer or
floating-point when it can. If one of the operands of a comparison is
a string and the other has a numeric value, the numeric operand is con‐
verted back to a string using the C sprintf format specifier %d for
integers and %g for floating-point values. For example, the commands
mpexpr {"0x03" > "2"} mpexpr {"0y" < "0x12"}
both return 1. The first comparison is done using integer comparison,
and the second is done using string comparison after the second operand
is converted to the string ``18''. Because of Tcl's tendency to treat
values as numbers whenever possible, it isn't generally a good idea to
use operators like == when you really want string comparison and the
values of the operands could be arbitrary; it's better in these cases
to use the string compare command instead.
mpformat formats a string in the style of Tcl's native format command.
Mpformat will interpret numeric arguments as arbitrary precision num‐
bers. Mpformat performs limited % substitution on the output string.
The following may be specified:
% [-] [width[.precision]] formatChar
- Specifies left justification; right justification is the
default.
width.precision
Specifies optional width and precision. Default precision is 8.
Width and/or precision may be specified as *, in which the next
argument will be used for the width or precision value.
Format character and result
d Format next argument as integer, truncating after the decimal
point.
f Format next argument in decimal floating point.
e Format next argument in scientific notation.
r, R Format next argument as rational fraction x / y.
N Format next argument as numerator only of rational fraction x /
y.
D Format next argument as denominator only of rational fraction x
/ y.
o Format next argument in octal format, with leading '0'; floating
point argument formatted as octal rational fraction x / y.
x Format next argument in hexadecimal format, with leading '0x';
floating point formatted argument as hexadecimal rational frac‐
tion x / y.
b Format next argument in binary format, with leading '0b'; float‐
ing point argument formatted as binary rational fraction x / y.
s Format next argument as string.
c Format next argument as single character value.
% Format single literal %.
Other characters in format string
\n Format ASCII newline.
\r Format ASCII carriage return.
\t Format ASCII tab.
\f Format ASCII form feed.
\v Format ASCII vertical tab.
\b Format ASCII backspace.
NOTES
Mpexpr is based on Tcl 7.6 'tclExpr.c' and David Bell's 'Calc' program.
This man page is largely borrowed from Tcl 7.6 as well, as is the
mpexpr test suite.
See the files README and INSTALL for additional information.
Tcl 7.6 is Copyright (c) 1987-1994 The Regents of the University of
California and Copyright (c) 1994 Sun Microsystems, Inc.
Calc is Copyright (c) 1994 David I. Bell.
AUTHOR
Tom Poindexter, tpoindex@nyx.net, Talus Technologies, Inc., Highlands
Ranch, CO. http://www.nyx.net/~tpoindex
Version 1.0 released November, 1998.
Copyright 1998 Tom Poindexter. See the file 'LICENSE.TERMS' for addi‐
tional copyright and licensing terms.
Tcl 8 January 1998 mpexpr(TCL)
