bc(1) GNU Project bc(1)
2006-06-11
NAME
bc - An arbitrary precision calculator language
SYNTAX
bc [ -hlwsqv ] [long-options] [ file ... ]
DESCRIPTION
bc is a language that supports arbitrary precision numbers with
interactive execution of statements. There are some similarities in
the syntax to the C programming language. A standard math library is
available by command line option. If requested, the math library is
defined before processing any files. bc starts by processing code
from all the files listed on the command line in the order listed.
After all files have been processed, bc reads from the standard input.
All code is executed as it is read. (If a file contains a command to
halt the processor, bc will never read from the standard input.)
This version of bc contains several extensions beyond traditional bc
implementations and the POSIX draft standard. Command line options
can cause these extensions to print a warning or to be rejected. This
document describes the language accepted by this processor.
Extensions will be identified as such.
OPTIONS
-h, --help
Print the usage and exit.
-i, --interactive
Force interactive mode.
-l, --mathlib
Define the standard math library.
-w, --warn
Give warnings for extensions to POSIX bc.
-s, --standard
Process exactly the POSIX bc language.
-q, --quiet
Do not print the normal GNU bc welcome.
-v, --version
Print the version number and copyright and quit.
NUMBERS
The most basic element in bc is the number. Numbers are arbitrary
precision numbers. This precision is both in the integer part and the
fractional part. All numbers are represented internally in decimal
and all computation is done in decimal. (This version truncates
results from divide and multiply operations.) There are two
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attributes of numbers, the length and the scale. The length is the
total number of decimal digits used by bc to represent a number and
the scale is the total number of decimal digits after the decimal
point. For example:
.000001 has a length of 6 and scale of 6.
1935.000 has a length of 7 and a scale of 3.
VARIABLES
Numbers are stored in two types of variables, simple variables and
arrays. Both simple variables and array variables are named. Names
begin with a letter followed by any number of letters, digits and
underscores. All letters must be lower case. (Full alpha-numeric
names are an extension. In POSIX bc all names are a single lower case
letter.) The type of variable is clear by the context because all
array variable names will be followed by brackets ([]).
There are four special variables, scale, ibase, obase, and last.
scale defines how some operations use digits after the decimal point.
The default value of scale is 0. ibase and obase define the conversion
base for input and output numbers. The default for both input and
output is base 10. last (an extension) is a variable that has the
value of the last printed number. These will be discussed in further
detail where appropriate. All of these variables may have values
assigned to them as well as used in expressions.
COMMENTS
Comments in bc start with the characters /* and end with the
characters */. Comments may start anywhere and appear as a single
space in the input. (This causes comments to delimit other input
items. For example, a comment can not be found in the middle of a
variable name.) Comments include any newlines (end of line) between
the start and the end of the comment.
To support the use of scripts for bc, a single line comment has been
added as an extension. A single line comment starts at a # character
and continues to the next end of the line. The end of line character
is not part of the comment and is processed normally.
EXPRESSIONS
The numbers are manipulated by expressions and statements. Since the
language was designed to be interactive, statements and expressions
are executed as soon as possible. There is no "main" program.
Instead, code is executed as it is encountered. (Functions, discussed
in detail later, are defined when encountered.)
A simple expression is just a constant. bc converts constants into
internal decimal numbers using the current input base, specified by
the variable ibase. (There is an exception in functions.) The legal
values of ibase are 2 through 36. (Bases greater than 16 are an
extension.) Assigning a value outside this range to ibase will result
in a value of 2 or 36. Input numbers may contain the characters 0-9
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and A-Z. (Note: They must be capitals. Lower case letters are
variable names.) Single digit numbers always have the value of the
digit regardless of the value of ibase. (i.e. A = 10.) For multi-
digit numbers, bc changes all input digits greater or equal to ibase
to the value of ibase-1. This makes the number ZZZ always be the
largest 3 digit number of the input base.
Full expressions are similar to many other high level languages.
Since there is only one kind of number, there are no rules for mixing
types. Instead, there are rules on the scale of expressions. Every
expression has a scale. This is derived from the scale of original
numbers, the operation performed and in many cases, the value of the
variable scale. Legal values of the variable scale are 0 to the
maximum number representable by a C integer.
In the following descriptions of legal expressions, "expr" refers to a
complete expression and "var" refers to a simple or an array variable.
A simple variable is just a
name
and an array variable is specified as
name[expr]
Unless specifically mentioned the scale of the result is the maximum
scale of the expressions involved.
- expr
The result is the negation of the expression.
++ var
The variable is incremented by one and the new value is the
result of the expression.
-- var
The variable is decremented by one and the new value is the
result of the expression.
var ++
The result of the expression is the value of the variable and
then the variable is incremented by one.
var --
The result of the expression is the value of the variable and
then the variable is decremented by one.
expr + expr
The result of the expression is the sum of the two expressions.
expr - expr
The result of the expression is the difference of the two
expressions.
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expr * expr
The result of the expression is the product of the two
expressions.
expr / expr
The result of the expression is the quotient of the two
expressions. The scale of the result is the value of the
variable scale.
expr % expr
The result of the expression is the "remainder" and it is
computed in the following way. To compute a%b, first a/b is
computed to scale digits. That result is used to compute a-
(a/b)*b to the scale of the maximum of scale+scale(b) and
scale(a). If scale is set to zero and both expressions are
integers this expression is the integer remainder function.
expr ^ expr
The result of the expression is the value of the first raised to
the second. The second expression must be an integer. (If the
second expression is not an integer, a warning is generated and
the expression is truncated to get an integer value.) The scale
of the result is scale if the exponent is negative. If the
exponent is positive the scale of the result is the minimum of
the scale of the first expression times the value of the exponent
and the maximum of scale and the scale of the first expression.
(e.g. scale(a^b) = min(scale(a)*b, max( scale, scale(a))).) It
should be noted that expr^0 will always return the value of 1.
( expr )
This alters the standard precedence to force the evaluation of
the expression.
var = expr
The variable is assigned the value of the expression.
var <op>= expr
This is equivalent to "var = var <op> expr" with the exception
that the "var" part is evaluated only once. This can make a
difference if "var" is an array.
Relational expressions are a special kind of expression that always
evaluate to 0 or 1, 0 if the relation is false and 1 if the relation
is true. These may appear in any legal expression. (POSIX bc
requires that relational expressions are used only in if, while, and
for statements and that only one relational test may be done in them.)
The relational operators are
expr1 < expr2
The result is 1 if expr1 is strictly less than expr2.
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expr1 <= expr2
The result is 1 if expr1 is less than or equal to expr2.
expr1 > expr2
The result is 1 if expr1 is strictly greater than expr2.
expr1 >= expr2
The result is 1 if expr1 is greater than or equal to expr2.
expr1 == expr2
The result is 1 if expr1 is equal to expr2.
expr1 != expr2
The result is 1 if expr1 is not equal to expr2.
Boolean operations are also legal. (POSIX bc does NOT have boolean
operations). The result of all boolean operations are 0 and 1 (for
false and true) as in relational expressions. The boolean operators
are:
!expr
The result is 1 if expr is 0.
expr && expr
The result is 1 if both expressions are non-zero.
expr || expr
The result is 1 if either expression is non-zero.
The expression precedence is as follows: (lowest to highest)
|| operator, left associative
&& operator, left associative
! operator, nonassociative
Relational operators, left associative
Assignment operator, right associative
+ and - operators, left associative
*, / and % operators, left associative
^ operator, right associative
unary - operator, nonassociative
++ and -- operators, nonassociative
This precedence was chosen so that POSIX compliant bc programs will
run correctly. This will cause the use of the relational and logical
operators to have some unusual behavior when used with assignment
expressions. Consider the expression:
a = 3 < 5
Most C programmers would assume this would assign the result of "3 <
5" (the value 1) to the variable "a". What this does in bc is assign
the value 3 to the variable "a" and then compare 3 to 5. It is best
to use parenthesis when using relational and logical operators with
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the assignment operators.
There are a few more special expressions that are provided in bc.
These have to do with user defined functions and standard functions.
They all appear as "name(parameters)". See the section on functions
for user defined functions. The standard functions are:
length ( expression )
The value of the length function is the number of significant
digits in the expression.
read ( )
The read function (an extension) will read a number from the
standard input, regardless of where the function occurs.
Beware, this can cause problems with the mixing of data and
program in the standard input. The best use for this function is
in a previously written program that needs input from the user,
but never allows program code to be input from the user. The
value of the read function is the number read from the standard
input using the current value of the variable ibase for the
conversion base.
scale ( expression )
The value of the scale function is the number of digits after the
decimal point in the expression.
sqrt ( expression )
The value of the sqrt function is the square root of the
expression. If the expression is negative, a run time error is
generated.
STATEMENTS
Statements (as in most algebraic languages) provide the sequencing of
expression evaluation. In bc statements are executed "as soon as
possible." Execution happens when a newline in encountered and there
is one or more complete statements. Due to this immediate execution,
newlines are very important in bc. In fact, both a semicolon and a
newline are used as statement separators. An improperly placed
newline will cause a syntax error. Because newlines are statement
separators, it is possible to hide a newline by using the backslash
character. The sequence "\<nl>", where <nl> is the newline appears to
bc as whitespace instead of a newline. A statement list is a series
of statements separated by semicolons and newlines. The following is
a list of bc statements and what they do: (Things enclosed in brackets
([]) are optional parts of the statement.)
expression
This statement does one of two things. If the expression starts
with "<variable> <assignment> ...", it is considered to be an
assignment statement. If the expression is not an assignment
statement, the expression is evaluated and printed to the output.
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After the number is printed, a newline is printed. For example,
"a=1" is an assignment statement and "(a=1)" is an expression
that has an embedded assignment. All numbers that are printed
are printed in the base specified by the variable obase. The
legal values for obase are 2 through BC_BASE_MAX. (See the
section LIMITS.) For bases 2 through 16, the usual method of
writing numbers is used. For bases greater than 16, bc uses a
multi-character digit method of printing the numbers where each
higher base digit is printed as a base 10 number. The multi-
character digits are separated by spaces. Each digit contains
the number of characters required to represent the base ten value
of "obase-1". Since numbers are of arbitrary precision, some
numbers may not be printable on a single output line. These long
numbers will be split across lines using the "\" as the last
character on a line. The maximum number of characters printed
per line is 70. Due to the interactive nature of bc, printing a
number causes the side effect of assigning the printed value to
the special variable last. This allows the user to recover the
last value printed without having to retype the expression that
printed the number. Assigning to last is legal and will
overwrite the last printed value with the assigned value. The
newly assigned value will remain until the next number is printed
or another value is assigned to last. (Some installations may
allow the use of a single period (.) which is not part of a
number as a short hand notation for for last.)
string
The string is printed to the output. Strings start with a double
quote character and contain all characters until the next double
quote character. All characters are take literally, including
any newline. No newline character is printed after the string.
print list
The print statement (an extension) provides another method of
output. The "list" is a list of strings and expressions
separated by commas. Each string or expression is printed in the
order of the list. No terminating newline is printed.
Expressions are evaluated and their value is printed and assigned
to the variable last. Strings in the print statement are printed
to the output and may contain special characters. Special
characters start with the backslash character (\). The special
characters recognized by bc are "a" (alert or bell), "b"
(backspace), "f" (form feed), "n" (newline), "r" (carriage
return), "q" (double quote), "t" (tab), and "\" (backslash). Any
other character following the backslash will be ignored.
{ statement_list }
This is the compound statement. It allows multiple statements to
be grouped together for execution.
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if ( expression ) statement1 [else statement2]
The if statement evaluates the expression and executes statement1
or statement2 depending on the value of the expression. If the
expression is non-zero, statement1 is executed. If statement2 is
present and the value of the expression is 0, then statement2 is
executed. (The else clause is an extension.)
while ( expression ) statement
The while statement will execute the statement while the
expression is non-zero. It evaluates the expression before each
execution of the statement. Termination of the loop is caused
by a zero expression value or the execution of a break statement.
for ( [expression1] ; [expression2] ; [expression3] ) statement
The for statement controls repeated execution of the statement.
Expression1 is evaluated before the loop. Expression2 is
evaluated before each execution of the statement. If it is non-
zero, the statement is evaluated. If it is zero, the loop is
terminated. After each execution of the statement, expression3
is evaluated before the reevaluation of expression2. If
expression1 or expression3 are missing, nothing is evaluated at
the point they would be evaluated. If expression2 is missing, it
is the same as substituting the value 1 for expression2. (The
optional expressions are an extension. POSIX bc requires all
three expressions.) The following is equivalent code for the for
statement:
expression1;
while (expression2) {
statement;
expression3;
}
break
This statement causes a forced exit of the most recent enclosing
while statement or for statement.
continue
The continue statement (an extension) causes the most recent
enclosing for statement to start the next iteration.
halt The halt statement (an extension) is an executed statement that
causes the bc processor to quit only when it is executed. For
example, "if (0 == 1) halt" will not cause bc to terminate
because the halt is not executed.
return
Return the value 0 from a function. (See the section on
functions.)
return ( expression )
Return the value of the expression from a function. (See the
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section on functions.) As an extension, the parenthesis are not
required.
PSEUDO STATEMENTS
These statements are not statements in the traditional sense. They
are not executed statements. Their function is performed at "compile"
time.
limits
Print the local limits enforced by the local version of bc. This
is an extension.
quit When the quit statement is read, the bc processor is terminated,
regardless of where the quit statement is found. For example,
"if (0 == 1) quit" will cause bc to terminate.
warranty
Print a longer warranty notice. This is an extension.
FUNCTIONS
Functions provide a method of defining a computation that can be
executed later. Functions in bc always compute a value and return it
to the caller. Function definitions are "dynamic" in the sense that a
function is undefined until a definition is encountered in the input.
That definition is then used until another definition function for the
same name is encountered. The new definition then replaces the older
definition. A function is defined as follows:
define name ( parameters ) { newline
auto_list statement_list }
A function call is just an expression of the form "name(parameters)".
Parameters are numbers or arrays (an extension). In the function
definition, zero or more parameters are defined by listing their names
separated by commas. All parameters are call by value parameters.
Arrays are specified in the parameter definition by the notation
"name[]". In the function call, actual parameters are full
expressions for number parameters. The same notation is used for
passing arrays as for defining array parameters. The named array is
passed by value to the function. Since function definitions are
dynamic, parameter numbers and types are checked when a function is
called. Any mismatch in number or types of parameters will cause a
runtime error. A runtime error will also occur for the call to an
undefined function.
The auto_list is an optional list of variables that are for "local"
use. The syntax of the auto list (if present) is "auto name, ... ;".
(The semicolon is optional.) Each name is the name of an auto
variable. Arrays may be specified by using the same notation as used
in parameters. These variables have their values pushed onto a stack
at the start of the function. The variables are then initialized to
zero and used throughout the execution of the function. At function
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exit, these variables are popped so that the original value (at the
time of the function call) of these variables are restored. The
parameters are really auto variables that are initialized to a value
provided in the function call. Auto variables are different than
traditional local variables because if function A calls function B, B
may access function A's auto variables by just using the same name,
unless function B has called them auto variables. Due to the fact
that auto variables and parameters are pushed onto a stack, bc
supports recursive functions.
The function body is a list of bc statements. Again, statements are
separated by semicolons or newlines. Return statements cause the
termination of a function and the return of a value. There are two
versions of the return statement. The first form, "return", returns
the value 0 to the calling expression. The second form, "return (
expression )", computes the value of the expression and returns that
value to the calling expression. There is an implied "return (0)" at
the end of every function. This allows a function to terminate and
return 0 without an explicit return statement.
Functions also change the usage of the variable ibase. All constants
in the function body will be converted using the value of ibase at the
time of the function call. Changes of ibase will be ignored during
the execution of the function except for the standard function read,
which will always use the current value of ibase for conversion of
numbers.
Several extensions have been added to functions. First, the format of
the definition has been slightly relaxed. The standard requires the
opening brace be on the same line as the define keyword and all other
parts must be on following lines. This version of bc will allow any
number of newlines before and after the opening brace of the function.
For example, the following definitions are legal.
define d (n) { return (2*n); }
define d (n)
{ return (2*n); }
Functions may be defined as void. A void funtion returns no value and
thus may not be used in any place that needs a value. A void function
does not produce any output when called by itself on an input line.
The key word void is placed between the key word define and the
function name. For example, consider the following session.
define py (y) { print "--->", y, "<---", "\n"; }
define void px (x) { print "--->", x, "<---", "\n"; }
py(1)
--->1<---
0
px(1)
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--->1<---
Since py is not a void function, the call of py(1) prints the desired
output and then prints a second line that is the value of the
function. Since the value of a function that is not given an explicit
return statement is zero, the zero is printed. For px(1), no zero is
printed because the function is a void function.
Also, call by variable for arrays was added. To declare a call by
variable array, the declaration of the array parameter in the function
definition looks like "*name[]". The call to the function remains the
same as call by value arrays.
MATH LIBRARY
If bc is invoked with the -l option, a math library is preloaded and
the default scale is set to 20. The math functions will calculate
their results to the scale set at the time of their call. The math
library defines the following functions:
s (x)
The sine of x, x is in radians.
c (x)
The cosine of x, x is in radians.
a (x)
The arctangent of x, arctangent returns radians.
l (x)
The natural logarithm of x.
e (x)
The exponential function of raising e to the value x.
j (n,x)
The Bessel function of integer order n of x.
EXAMPLES
In /bin/sh, the following will assign the value of "pi" to the shell
variable pi.
pi=$(echo "scale=10; 4*a(1)" | bc -l)
The following is the definition of the exponential function used in
the math library. This function is written in POSIX bc.
scale = 20
/* Uses the fact that e^x = (e^(x/2))^2
When x is small enough, we use the series:
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e^x = 1 + x + x^2/2! + x^3/3! + ...
*/
define e(x) {
auto a, d, e, f, i, m, v, z
/* Check the sign of x. */
if (x<0) {
m = 1
x = -x
}
/* Precondition x. */
z = scale;
scale = 4 + z + .44*x;
while (x > 1) {
f += 1;
x /= 2;
}
/* Initialize the variables. */
v = 1+x
a = x
d = 1
for (i=2; 1; i++) {
e = (a *= x) / (d *= i)
if (e == 0) {
if (f>0) while (f--) v = v*v;
scale = z
if (m) return (1/v);
return (v/1);
}
v += e
}
}
The following is code that uses the extended features of bc to
implement a simple program for calculating checkbook balances. This
program is best kept in a file so that it can be used many times
without having to retype it at every use.
scale=2
print "\nCheck book program!\n"
print " Remember, deposits are negative transactions.\n"
print " Exit by a 0 transaction.\n\n"
print "Initial balance? "; bal = read()
bal /= 1
print "\n"
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while (1) {
"current balance = "; bal
"transaction? "; trans = read()
if (trans == 0) break;
bal -= trans
bal /= 1
}
quit
The following is the definition of the recursive factorial function.
define f (x) {
if (x <= 1) return (1);
return (f(x-1) * x);
}
READLINE AND LIBEDIT OPTIONS
GNU bc can be compiled (via a configure option) to use the GNU
readline input editor library or the BSD libedit library. This allows
the user to do editing of lines before sending them to bc. It also
allows for a history of previous lines typed. When this option is
selected, bc has one more special variable. This special variable,
history is the number of lines of history retained. For readline, a
value of -1 means that an unlimited number of history lines are
retained. Setting the value of history to a positive number restricts
the number of history lines to the number given. The value of 0
disables the history feature. The default value is 100. For more
information, read the user manuals for the GNU readline, history and
BSD libedit libraries. One can not enable both readline and libedit
at the same time.
DIFFERENCES
This version of bc was implemented from the POSIX P1003.2/D11 draft
and contains several differences and extensions relative to the draft
and traditional implementations. It is not implemented in the
traditional way using dc(1). This version is a single process which
parses and runs a byte code translation of the program. There is an
"undocumented" option (-c) that causes the program to output the byte
code to the standard output instead of running it. It was mainly used
for debugging the parser and preparing the math library.
A major source of differences is extensions, where a feature is
extended to add more functionality and additions, where new features
are added. The following is the list of differences and extensions.
LANG environment
This version does not conform to the POSIX standard in the
processing of the LANG environment variable and all environment
variables starting with LC_.
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names
Traditional and POSIX bc have single letter names for functions,
variables and arrays. They have been extended to be multi-
character names that start with a letter and may contain letters,
numbers and the underscore character.
Strings
Strings are not allowed to contain NUL characters. POSIX says
all characters must be included in strings.
last POSIX bc does not have a last variable. Some implementations of
bc use the period (.) in a similar way.
comparisons
POSIX bc allows comparisons only in the if statement, the while
statement, and the second expression of the for statement. Also,
only one relational operation is allowed in each of those
statements.
if statement, else clause
POSIX bc does not have an else clause.
for statement
POSIX bc requires all expressions to be present in the for
statement.
&&, ||, !
POSIX bc does not have the logical operators.
read function
POSIX bc does not have a read function.
print statement
POSIX bc does not have a print statement .
continue statement
POSIX bc does not have a continue statement.
return statement
POSIX bc requires parentheses around the return expression.
array parameters
POSIX bc does not (currently) support array parameters in full.
The POSIX grammar allows for arrays in function definitions, but
does not provide a method to specify an array as an actual
parameter. (This is most likely an oversight in the grammar.)
Traditional implementations of bc have only call by value array
parameters.
function format
POSIX bc requires the opening brace on the same line as the
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define key word and the auto statement on the next line.
=+, =-, =*, =/, =%, =^
POSIX bc does not require these "old style" assignment operators
to be defined. This version may allow these "old style"
assignments. Use the limits statement to see if the installed
version supports them. If it does support the "old style"
assignment operators, the statement "a =- 1" will decrement a by
1 instead of setting a to the value -1.
spaces in numbers
Other implementations of bc allow spaces in numbers. For
example, "x=1 3" would assign the value 13 to the variable x.
The same statement would cause a syntax error in this version of
bc.
errors and execution
This implementation varies from other implementations in terms of
what code will be executed when syntax and other errors are found
in the program. If a syntax error is found in a function
definition, error recovery tries to find the beginning of a
statement and continue to parse the function. Once a syntax
error is found in the function, the function will not be callable
and becomes undefined. Syntax errors in the interactive
execution code will invalidate the current execution block. The
execution block is terminated by an end of line that appears
after a complete sequence of statements. For example,
a = 1
b = 2
has two execution blocks and
{ a = 1
b = 2 }
has one execution block. Any runtime error will terminate the
execution of the current execution block. A runtime warning will not
terminate the current execution block.
Interrupts
During an interactive session, the SIGINT signal (usually
generated by the control-C character from the terminal) will
cause execution of the current execution block to be interrupted.
It will display a "runtime" error indicating which function was
interrupted. After all runtime structures have been cleaned up,
a message will be printed to notify the user that bc is ready for
more input. All previously defined functions remain defined and
the value of all non-auto variables are the value at the point of
interruption. All auto variables and function parameters are
removed during the clean up process. During a non-interactive
session, the SIGINT signal will terminate the entire run of bc.
LIMITS
The following are the limits currently in place for this bc processor.
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bc(1) GNU Project bc(1)
2006-06-11
Some of them may have been changed by an installation. Use the limits
statement to see the actual values.
BC_BASE_MAX
The maximum output base is currently set at 999. The maximum
input base is 16.
BC_DIM_MAX
This is currently an arbitrary limit of 65535 as distributed.
Your installation may be different.
BC_SCALE_MAX
The number of digits after the decimal point is limited to
INT_MAX digits. Also, the number of digits before the decimal
point is limited to INT_MAX digits.
BC_STRING_MAX
The limit on the number of characters in a string is INT_MAX
characters.
exponent
The value of the exponent in the raise operation (^) is limited
to LONG_MAX.
variable names
The current limit on the number of unique names is 32767 for each
of simple variables, arrays and functions.
ENVIRONMENT VARIABLES
The following environment variables are processed by bc:
POSIXLY_CORRECT
This is the same as the -s option.
BC_ENV_ARGS
This is another mechanism to get arguments to bc. The format is
the same as the command line arguments. These arguments are
processed first, so any files listed in the environment arguments
are processed before any command line argument files. This
allows the user to set up "standard" options and files to be
processed at every invocation of bc. The files in the
environment variables would typically contain function
definitions for functions the user wants defined every time bc is
run.
BC_LINE_LENGTH
This should be an integer specifying the number of characters in
an output line for numbers. This includes the backslash and
newline characters for long numbers. As an extension, the value
of zero disables the multi-line feature. Any other value of this
variable that is less than 3 sets the line length to 70.
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bc(1) GNU Project bc(1)
2006-06-11
DIAGNOSTICS
If any file on the command line can not be opened, bc will report that
the file is unavailable and terminate. Also, there are compile and
run time diagnostics that should be self-explanatory.
BUGS
Error recovery is not very good yet.
Email bug reports to bug-bc@gnu.org. Be sure to include the word
``bc'' somewhere in the ``Subject:'' field.
AUTHOR
Philip A. Nelson
philnelson@acm.org
ACKNOWLEDGEMENTS
The author would like to thank Steve Sommars (Steve.Sommars@att.com)
for his extensive help in testing the implementation. Many great
suggestions were given. This is a much better product due to his
involvement.
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