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About bc

bc is an arbitrary-precision language for performing math calculations.

Description

bc is a language that supports arbitrary-precision numbers, meaning that it delivers accurate results regardless of how large (or very very small) the numbers are.

It has an interactive mode, accepting input from the terminal and providing calculations on request. As a language, its syntax is similar to the C programming language. A standard math library is available using a 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.

Newer versions of bc contain 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 newer version of the bc language; where a particular function is an extension of the standard, it is noted accordingly below.

Syntax

bc [ -hlwsqv ] [long-options] [ file ... ]

Options

-h, --help Print a help message 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 message.
-v, --version Print the version number and copyright information, and exit.

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. (Some versions of bc truncate results from divide and multiply operations.)

There are two attributes of numbers, the length and the 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.

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 lowercase. (Full alphanumeric 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.

All variables may have values assigned to them, and can be 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 placed 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-line. The end-of-line character is not part of the comment and is processed normally.

Expressions

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 16. 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. (Note: They must be capital letters. Lowercase letters are reserved for variable names.) Single-digit numbers always have the value of the digit regardless of the value of ibase. For multi-digit numbers, bc 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 variable 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 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.

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 power of 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.
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 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 runtime 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 is encountered and there are 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. 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.
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 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 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 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. Newer versions 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, "<---", "0; }
define void px (x) { print "--->", x, "<---", "0; }
py(1)
--->1<---
0
px(1)
--->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, in radians.
  • c (x): The cosine of x, in radians.
  • a (x): The arctangent of x, in 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 sh, the following code will assign the value of pi to the shell variable pi. Here, a refers to the arctangent function, which is part of the math library loaded with the -l option:

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:
     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"
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); }

Related commands

dc — An arbitrary precision arithmetic package.
awk — Interpreter for the AWK text processing programming language.