bc - an arbitrary precision calculator language


bc [-lws] [file ...]


bc(1) 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(1) begins by processing code from all the files listed on the command line in the order listed. After all files have been processed, bc(1) reads from the standard input. All code is executed as it is read. (If a file contains a command to halt the processor, bc(1) will never read from the standard input.)

This version of bc(1) contains several extensions beyond traditional bc(1) 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.


Define the standard math library.
Give warnings for extensions to POSIX bc(1).
Process exactly the POSIX bc(1) language.


The most basic element in bc(1) 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 attributes of numbers: length and scale. The length is the total number of significant decimal digits in 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.


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 alphanumeric names are an extension. In POSIX bc(1) 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.

Defines how some operations use digits after the decimal point. The default value of scale is 0.
Defines the conversion base for input numbers. The default is base 10.
Defines the conversion base for output numbers. The default is base 10. If you attempt to convert hex to decimal using these commands, it will not work:
This is because you have set the value of obase to the same value as ibase; the input value assigned to obase is treated as if it were base 16 (the value of ibase).
(An extension) is a variable that has the value of the last printed number. In some versions of bc(1), this is variable has the name . (dot).

All of these variables may have values assigned to them. They may also be used in expressions. These variables will be discussed in further detail where appropriate.


Comments in bc(1) 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 cannot 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.


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(1) 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 16 (F). Assigning a value outside this range to ibase will result in a value of 2 or 16. Input numbers may contain the characters 0-9 and A-F. They must be uppercase; lowercase letters are variable names. Single-digit numbers always have the value of the digit regardless of the value of ibase (that is, A = 10.) For multidigit numbers, bc(1) changes all input digits greater or equal to ibase to the value of ibase-1. This makes the number FFF 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 that can be represented.

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

and an array variable is specified as:

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.
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"; it is computed as follows:
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. For example:
scale(a^b) = min(scale(a)*b, max(scale, scale(a))).
Note 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
except 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(1) requires that relational expressions be used only in if, while, and for statements, and that only one relational test may be done in them.) The relational operators are as follows:

expr1 < expr2
The result is 1 if expr1 is strictly less than expr2.
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(1) 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 areas follows:

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 Association
|| left associative
&& left associative
! nonassociative
Relational (<,>,==,<=,>=) left associative
= right associative
+, - left associative
*, /, % left associative
^ right associative
unary - nonassociative
++, -- nonassociative

This precedence was chosen so that POSIX-compliant bc(1) programs will run correctly. This causes the use of the relational and logical operators to demonstrate 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". In bc(1)this assigns the value 3 to the variable "a" and then compares 3 to 5. It is best to use parenthesis when using relational and logical operators with the assignment operators.

Several more special expressions are also provided in bc(1). 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 as follows:

The value of the length() function is the number of significant digits in the expression.
The read(2) function (an extension) will read a number from the standard input, regardless of where the function occurs. Be aware, however, that 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 requires 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.
The value of the scale() function is the number of digits after the decimal point in the expression.
The value of the sqrt(3) function is the square root of the expression. If the expression is negative, a run time error is generated.


As in most algebraic languages, statements provide the sequencing of expression evaluation. In bc(1) statements are executed "as soon as possible." Execution happens when a newline is encountered and there is one or more complete statements. Due to this immediate execution, newlines are very important in bc(1). 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(1) as white space instead of a newline. A statement list is a series of statements separated by semicolons and newlines. The following is a list of bc(1) statements and what they do (things enclosed in brackets ([]) are optional parts of the statement):

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. 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), or bases 2 through 16; the usual method of writing numbers is used. For bases greater than 16, bc(1) uses a multicharacter digit method of printing the numbers, where each higher base digit is printed as a base 10 number. The multicharacter 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(1), 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.)
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 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(1) are: 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.
if(expression) then 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 at which 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(1) requires all three expressions.) The following is equivalent code for the for statement:
while (expression2) {
This statement causes a forced exit of the most recent enclosing while statement or for statement.
The continue statement (an extension) causes the most recent enclosing for statement to start the next iteration.
The halt statement (an extension) is an executed statement that causes the bc(1) processor to quit only when it is executed. For example, "if (0 == 1) halt" will not cause bc(1) to terminate because the halt is not executed.
Return the value 0 from a function. (See the section on functions.)
Return the value of the expression from a function. (See the section on functions.)


These statements are not statements in the traditional sense. They are not executed statements. Their function is performed at "compile" time.
Print the local limits enforced by the local version of bc(1). This is an extension.
When the quit statement is read, the bc(1) processor is terminated, regardless of where the quit statement is found. For example, "if (0 == 1) quit" will cause bc(1) to terminate.
Print a longer warranty notice. This is an extension.


Functions provide a method of defining a computation that can be executed later. Functions in bc(1) 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:


Parameters are numbers or arrays (an extension). In the function definition, zero or more parameters are defined by listing their names separated by commas. Numbers are only call-by-value parameters. Arrays are only call-by-variable. 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 variable 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 as follows (the semicolon is optional):

auto name, ... ;

Each name is the name of an auto variable. Arrays may be specified by using the same notation that is 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 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 in that 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. Because auto variables and parameters are pushed onto a stack, bc(1) supports recursive functions.

The function body is a list of bc(1) statements. Again, statements are separated by semicolons or newlines. To end the function and return a value, use a return statement. You do not need an explicit return. There is one implied, returning 0, at the end of every function. There are two versions of the return statement:

Returns the value 0 to the calling expression.
return (expression )
Computes the value of the expression and returns that value to the calling expression.

Functions also change the usage of the variable ibase. All constants in the function body are converted using the value of base at the time of the function call. Changes of ibase are ignored during the execution of the function except for the standard function read(2), which will always use the current value of ibase for conversion of numbers.


If bc(1) is invoked with the -l option, it loads a math library and sets the default scale 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:

The sine of x in radians.
The cosine of x in radians.
The arctangent of x.
The natural logarithm of x.
The exponential function of raising e to the value x.
The Bessel function of integer order n of x.


In sh(1), 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(1).

scale = 20
/* Uses the fact that e^x = (e^(x/2))^2
   When x is small enough, use the series:
	 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 code uses the extended features of bc(1) 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.

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"
while (1) {
  "current balance = "; bal
  "transaction? "; trans = read()
  if (trans == 0) break;
  bal -= trans
  bal /= 1

The following is the definition of the recursive factorial function.

define f (x) {
  if (x <= 1) return (1);
  return (f(x-1) * x);


This version of bc(1) 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 that 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.

Extensions are a major source of differences, where a feature is extended to add more functionality, and additions where new features are added.

To disable extensions and make this bc(1) behave as specified by POSIX, use the -s option. 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_.
Traditional and POSIX bc(1) have single-letter names for functions, variables and arrays. They have been extended to be multicharacter names that start with a letter and may contain letters, numbers and the underscore character.
Strings are not allowed to contain NUL characters. POSIX says all characters must be included in strings.
POSIX bc(1) does not have a last variable. Some implementations of bc(1) use the dot (.) in a similar way.
POSIX bc(1) 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(1) does not have an else clause.
for statement
POSIX bc(1) requires all expressions to be present in the forstatement.
&&, ||, !
POSIX bc(1) does not have the logical operators.
read() function
POSIX bc(1) does not have a read(2) function.
print statement
POSIX bc(1) does not have a print statement.
continue statement
POSIX bc(1) does not have a continue statement.
Array parameters
POSIX bc(1) does not have array parameters. Other implementations of bc(1) may have call by value array parameters.
=+, =-, =*, =/, =%, =^
POSIX bc(1) 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(1) 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(1).
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.
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(1) 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 noninteractive session, the SIGINT signal will terminate the entire run of bc(1).


The following are the limits currently in place for this bc(1) processor. Some of them may have been changed by an installation. Use the limits statement to see the actual values.

The maximum output base is currently set at 999. The maximum input base is 16.
This is currently an arbitrary limit of 65535 as distributed. Your installation may be different.
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.
The limit on the number of characters in a string is {INT_MAX} characters.
The value of the exponent in the raise operation (^) is limited to {LONG_MAX}.
The multiply routine may yield incorrect results if a number has more than LONG_MAX / 90 total digits. For 32-bit longs, this number is 23,860,929 digits.
Code size
Each function and the "main" program are limited to 16384 bytes of compiled byte code each. This limit ({BC_MAX_SEGS}) can be easily changed to have more than 16 segments of 1024 bytes.
Variable names
The current limit on the number of unique names is 32767 for each of simple variables, array, and functions.


In most installations, bc(1) is completely self contained. Where executable size is of importance, or the C compiler does not deal well with very long strings, bc(1) will read the standard math library from the file /share/libmath.b. On other operating systems, the package may be installed in /usr/local/lib/libmath.b or /lib/libmath.b.)


If any file on the command line cannot be opened, bc(1) will report that the file is unavailable and terminate. Also, there are compile and run-time diagnostics that should be self explanatory.


This topic documents GNU bc(1) version 1.03.


Error recovery is not very good yet.


Philip A. Nelson phil@cs.wwu.edu


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.