xref: /freebsd/contrib/bc/manuals/bc/N.1.md (revision 681ce946)

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# NAME

bc - arbitrary-precision decimal arithmetic language and calculator

# SYNOPSIS

**bc** [**-ghilPqRsvVw**] [**-\-global-stacks**] [**-\-help**] [**-\-interactive**] [**-\-mathlib**] [**-\-no-prompt**] [**-\-no-read-prompt**] [**-\-quiet**] [**-\-standard**] [**-\-warn**] [**-\-version**] [**-e** *expr*] [**-\-expression**=*expr*...] [**-f** *file*...] [**-\-file**=*file*...] [*file*...]

# DESCRIPTION

bc(1) is an interactive processor for a language first standardized in 1991 by
POSIX. (The current standard is [here][1].) The language provides unlimited
precision decimal arithmetic and is somewhat C-like, but there are differences.
Such differences will be noted in this document.

After parsing and handling options, this bc(1) reads any files given on the
command line and executes them before reading from **stdin**.

This bc(1) is a drop-in replacement for *any* bc(1), including (and
especially) the GNU bc(1). It also has many extensions and extra features beyond
other implementations.

**Note**: If running this bc(1) on *any* script meant for another bc(1) gives a
parse error, it is probably because a word this bc(1) reserves as a keyword is
used as the name of a function, variable, or array. To fix that, use the
command-line option **-r** *keyword*, where *keyword* is the keyword that is
used as a name in the script. For more information, see the **OPTIONS** section.

If parsing scripts meant for other bc(1) implementations still does not work,
that is a bug and should be reported. See the **BUGS** section.

# OPTIONS

The following are the options that bc(1) accepts.

**-g**, **-\-global-stacks**

:   Turns the globals **ibase**, **obase**, **scale**, and **seed** into stacks.

    This has the effect that a copy of the current value of all four are pushed
    onto a stack for every function call, as well as popped when every function
    returns. This means that functions can assign to any and all of those
    globals without worrying that the change will affect other functions.
    Thus, a hypothetical function named **output(x,b)** that simply printed
    **x** in base **b** could be written like this:

        define void output(x, b) {
            obase=b
            x
        }

    instead of like this:

        define void output(x, b) {
            auto c
            c=obase
            obase=b
            x
            obase=c
        }

    This makes writing functions much easier.

    (**Note**: the function **output(x,b)** exists in the extended math library.
     See the **LIBRARY** section.)

    However, since using this flag means that functions cannot set **ibase**,
    **obase**, **scale**, or **seed** globally, functions that are made to do so
    cannot work anymore. There are two possible use cases for that, and each has
    a solution.

    First, if a function is called on startup to turn bc(1) into a number
    converter, it is possible to replace that capability with various shell
    aliases. Examples:

        alias d2o="bc -e ibase=A -e obase=8"
        alias h2b="bc -e ibase=G -e obase=2"

    Second, if the purpose of a function is to set **ibase**, **obase**,
    **scale**, or **seed** globally for any other purpose, it could be split
    into one to four functions (based on how many globals it sets) and each of
    those functions could return the desired value for a global.

    For functions that set **seed**, the value assigned to **seed** is not
    propagated to parent functions. This means that the sequence of
    pseudo-random numbers that they see will not be the same sequence of
    pseudo-random numbers that any parent sees. This is only the case once
    **seed** has been set.

    If a function desires to not affect the sequence of pseudo-random numbers
    of its parents, but wants to use the same **seed**, it can use the following
    line:

        seed = seed

    If the behavior of this option is desired for every run of bc(1), then users
    could make sure to define **BC_ENV_ARGS** and include this option (see the
    **ENVIRONMENT VARIABLES** section for more details).

    If **-s**, **-w**, or any equivalents are used, this option is ignored.

    This is a **non-portable extension**.

**-h**, **-\-help**

:   Prints a usage message and quits.

**-i**, **-\-interactive**

:   Forces interactive mode. (See the **INTERACTIVE MODE** section.)

    This is a **non-portable extension**.

**-L**, **-\-no-line-length**

:   Disables line length checking and prints numbers without backslashes and
    newlines. In other words, this option sets **BC_LINE_LENGTH** to **0** (see
    the **ENVIRONMENT VARIABLES** section).

    This is a **non-portable extension**.

**-l**, **-\-mathlib**

:   Sets **scale** (see the **SYNTAX** section) to **20** and loads the included
    math library and the extended math library before running any code,
    including any expressions or files specified on the command line.

    To learn what is in the libraries, see the **LIBRARY** section.

**-P**, **-\-no-prompt**

:   Disables the prompt in TTY mode. (The prompt is only enabled in TTY mode.
    See the **TTY MODE** section.) This is mostly for those users that do not
    want a prompt or are not used to having them in bc(1). Most of those users
    would want to put this option in **BC_ENV_ARGS** (see the
    **ENVIRONMENT VARIABLES** section).

    These options override the **BC_PROMPT** and **BC_TTY_MODE** environment
    variables (see the **ENVIRONMENT VARIABLES** section).

    This is a **non-portable extension**.

**-R**, **-\-no-read-prompt**

:   Disables the read prompt in TTY mode. (The read prompt is only enabled in
    TTY mode. See the **TTY MODE** section.) This is mostly for those users that
    do not want a read prompt or are not used to having them in bc(1). Most of
    those users would want to put this option in **BC_ENV_ARGS** (see the
    **ENVIRONMENT VARIABLES** section). This option is also useful in hash bang
    lines of bc(1) scripts that prompt for user input.

    This option does not disable the regular prompt because the read prompt is
    only used when the **read()** built-in function is called.

    These options *do* override the **BC_PROMPT** and **BC_TTY_MODE**
    environment variables (see the **ENVIRONMENT VARIABLES** section), but only
    for the read prompt.

    This is a **non-portable extension**.

**-r** *keyword*, **-\-redefine**=*keyword*

:   Redefines *keyword* in order to allow it to be used as a function, variable,
    or array name. This is useful when this bc(1) gives parse errors when
    parsing scripts meant for other bc(1) implementations.

    The keywords this bc(1) allows to be redefined are:

    * **abs**
    * **asciify**
    * **continue**
    * **divmod**
    * **else**
    * **halt**
    * **irand**
    * **last**
    * **limits**
    * **maxibase**
    * **maxobase**
    * **maxrand**
    * **maxscale**
    * **modexp**
    * **print**
    * **rand**
    * **read**
    * **seed**
	* **stream**

    If any of those keywords are used as a function, variable, or array name in
    a script, use this option with the keyword as the argument. If multiple are
    used, use this option for all of them; it can be used multiple times.

    Keywords are *not* redefined when parsing the builtin math library (see the
    **LIBRARY** section).

    It is a fatal error to redefine keywords mandated by the POSIX standard. It
    is a fatal error to attempt to redefine words that this bc(1) does not
    reserve as keywords.

**-q**, **-\-quiet**

:   This option is for compatibility with the [GNU bc(1)][2]; it is a no-op.
    Without this option, GNU bc(1) prints a copyright header. This bc(1) only
    prints the copyright header if one or more of the **-v**, **-V**, or
    **-\-version** options are given.

    This is a **non-portable extension**.

**-s**, **-\-standard**

:   Process exactly the language defined by the [standard][1] and error if any
    extensions are used.

    This is a **non-portable extension**.

**-v**, **-V**, **-\-version**

:   Print the version information (copyright header) and exit.

    This is a **non-portable extension**.

**-w**, **-\-warn**

:   Like **-s** and **-\-standard**, except that warnings (and not errors) are
    printed for non-standard extensions and execution continues normally.

    This is a **non-portable extension**.

**-z**, **-\-leading-zeroes**

:   Makes bc(1) print all numbers greater than **-1** and less than **1**, and
    not equal to **0**, with a leading zero.

    This can be set for individual numbers with the **plz(x)**, plznl(x)**,
    **pnlz(x)**, and **pnlznl(x)** functions in the extended math library (see
    the **LIBRARY** section).

    This is a **non-portable extension**.

**-e** *expr*, **-\-expression**=*expr*

:   Evaluates *expr*. If multiple expressions are given, they are evaluated in
    order. If files are given as well (see below), the expressions and files are
    evaluated in the order given. This means that if a file is given before an
    expression, the file is read in and evaluated first.

    If this option is given on the command-line (i.e., not in **BC_ENV_ARGS**,
    see the **ENVIRONMENT VARIABLES** section), then after processing all
    expressions and files, bc(1) will exit, unless **-** (**stdin**) was given
    as an argument at least once to **-f** or **-\-file**, whether on the
    command-line or in **BC_ENV_ARGS**. However, if any other **-e**,
    **-\-expression**, **-f**, or **-\-file** arguments are given after **-f-**
    or equivalent is given, bc(1) will give a fatal error and exit.

    This is a **non-portable extension**.

**-f** *file*, **-\-file**=*file*

:   Reads in *file* and evaluates it, line by line, as though it were read
    through **stdin**. If expressions are also given (see above), the
    expressions are evaluated in the order given.

    If this option is given on the command-line (i.e., not in **BC_ENV_ARGS**,
    see the **ENVIRONMENT VARIABLES** section), then after processing all
    expressions and files, bc(1) will exit, unless **-** (**stdin**) was given
    as an argument at least once to **-f** or **-\-file**. However, if any other
    **-e**, **-\-expression**, **-f**, or **-\-file** arguments are given after
    **-f-** or equivalent is given, bc(1) will give a fatal error and exit.

    This is a **non-portable extension**.

All long options are **non-portable extensions**.

# STDIN

If no files or expressions are given by the **-f**, **-\-file**, **-e**, or
**-\-expression** options, then bc(1) read from **stdin**.

However, there are a few caveats to this.

First, **stdin** is evaluated a line at a time. The only exception to this is if
the parse cannot complete. That means that starting a string without ending it
or starting a function, **if** statement, or loop without ending it will also
cause bc(1) to not execute.

Second, after an **if** statement, bc(1) doesn't know if an **else** statement
will follow, so it will not execute until it knows there will not be an **else**
statement.

# STDOUT

Any non-error output is written to **stdout**. In addition, if history (see the
**HISTORY** section) and the prompt (see the **TTY MODE** section) are enabled,
both are output to **stdout**.

**Note**: Unlike other bc(1) implementations, this bc(1) will issue a fatal
error (see the **EXIT STATUS** section) if it cannot write to **stdout**, so if
**stdout** is closed, as in **bc <file> >&-**, it will quit with an error. This
is done so that bc(1) can report problems when **stdout** is redirected to a
file.

If there are scripts that depend on the behavior of other bc(1) implementations,
it is recommended that those scripts be changed to redirect **stdout** to
**/dev/null**.

# STDERR

Any error output is written to **stderr**.

**Note**: Unlike other bc(1) implementations, this bc(1) will issue a fatal
error (see the **EXIT STATUS** section) if it cannot write to **stderr**, so if
**stderr** is closed, as in **bc <file> 2>&-**, it will quit with an error. This
is done so that bc(1) can exit with an error code when **stderr** is redirected
to a file.

If there are scripts that depend on the behavior of other bc(1) implementations,
it is recommended that those scripts be changed to redirect **stderr** to
**/dev/null**.

# SYNTAX

The syntax for bc(1) programs is mostly C-like, with some differences. This
bc(1) follows the [POSIX standard][1], which is a much more thorough resource
for the language this bc(1) accepts. This section is meant to be a summary and a
listing of all the extensions to the standard.

In the sections below, **E** means expression, **S** means statement, and **I**
means identifier.

Identifiers (**I**) start with a lowercase letter and can be followed by any
number (up to **BC_NAME_MAX-1**) of lowercase letters (**a-z**), digits
(**0-9**), and underscores (**\_**). The regex is **\[a-z\]\[a-z0-9\_\]\***.
Identifiers with more than one character (letter) are a
**non-portable extension**.

**ibase** is a global variable determining how to interpret constant numbers. It
is the "input" base, or the number base used for interpreting input numbers.
**ibase** is initially **10**. If the **-s** (**-\-standard**) and **-w**
(**-\-warn**) flags were not given on the command line, the max allowable value
for **ibase** is **36**. Otherwise, it is **16**. The min allowable value for
**ibase** is **2**. The max allowable value for **ibase** can be queried in
bc(1) programs with the **maxibase()** built-in function.

**obase** is a global variable determining how to output results. It is the
"output" base, or the number base used for outputting numbers. **obase** is
initially **10**. The max allowable value for **obase** is **BC_BASE_MAX** and
can be queried in bc(1) programs with the **maxobase()** built-in function. The
min allowable value for **obase** is **0**. If **obase** is **0**, values are
output in scientific notation, and if **obase** is **1**, values are output in
engineering notation. Otherwise, values are output in the specified base.

Outputting in scientific and engineering notations are **non-portable
extensions**.

The *scale* of an expression is the number of digits in the result of the
expression right of the decimal point, and **scale** is a global variable that
sets the precision of any operations, with exceptions. **scale** is initially
**0**. **scale** cannot be negative. The max allowable value for **scale** is
**BC_SCALE_MAX** and can be queried in bc(1) programs with the **maxscale()**
built-in function.

bc(1) has both *global* variables and *local* variables. All *local*
variables are local to the function; they are parameters or are introduced in
the **auto** list of a function (see the **FUNCTIONS** section). If a variable
is accessed which is not a parameter or in the **auto** list, it is assumed to
be *global*. If a parent function has a *local* variable version of a variable
that a child function considers *global*, the value of that *global* variable in
the child function is the value of the variable in the parent function, not the
value of the actual *global* variable.

All of the above applies to arrays as well.

The value of a statement that is an expression (i.e., any of the named
expressions or operands) is printed unless the lowest precedence operator is an
assignment operator *and* the expression is notsurrounded by parentheses.

The value that is printed is also assigned to the special variable **last**. A
single dot (**.**) may also be used as a synonym for **last**. These are
**non-portable extensions**.

Either semicolons or newlines may separate statements.

## Comments

There are two kinds of comments:

1.	Block comments are enclosed in **/\*** and **\*/**.
2.	Line comments go from **#** until, and not including, the next newline. This
	is a **non-portable extension**.

## Named Expressions

The following are named expressions in bc(1):

1.	Variables: **I**
2.	Array Elements: **I[E]**
3.	**ibase**
4.	**obase**
5.	**scale**
6.	**seed**
7.	**last** or a single dot (**.**)

Numbers 6 and 7 are **non-portable extensions**.

The meaning of **seed** is dependent on the current pseudo-random number
generator but is guaranteed to not change except for new major versions.

The *scale* and sign of the value may be significant.

If a previously used **seed** value is assigned to **seed** and used again, the
pseudo-random number generator is guaranteed to produce the same sequence of
pseudo-random numbers as it did when the **seed** value was previously used.

The exact value assigned to **seed** is not guaranteed to be returned if
**seed** is queried again immediately. However, if **seed** *does* return a
different value, both values, when assigned to **seed**, are guaranteed to
produce the same sequence of pseudo-random numbers. This means that certain
values assigned to **seed** will *not* produce unique sequences of pseudo-random
numbers. The value of **seed** will change after any use of the **rand()** and
**irand(E)** operands (see the *Operands* subsection below), except if the
parameter passed to **irand(E)** is **0**, **1**, or negative.

There is no limit to the length (number of significant decimal digits) or
*scale* of the value that can be assigned to **seed**.

Variables and arrays do not interfere; users can have arrays named the same as
variables. This also applies to functions (see the **FUNCTIONS** section), so a
user can have a variable, array, and function that all have the same name, and
they will not shadow each other, whether inside of functions or not.

Named expressions are required as the operand of **increment**/**decrement**
operators  and as the left side of **assignment** operators (see the *Operators*
subsection).

## Operands

The following are valid operands in bc(1):

1.	Numbers (see the *Numbers* subsection below).
2.	Array indices (**I[E]**).
3.	**(E)**: The value of **E** (used to change precedence).
4.	**sqrt(E)**: The square root of **E**. **E** must be non-negative.
5.	**length(E)**: The number of significant decimal digits in **E**. Returns
	**1** for **0** with no decimal places. If given a string, the length of the
	string is returned. Passing a string to **length(E)** is a **non-portable
	extension**.
6.	**length(I[])**: The number of elements in the array **I**. This is a
	**non-portable extension**.
7.	**scale(E)**: The *scale* of **E**.
8.	**abs(E)**: The absolute value of **E**. This is a **non-portable
	extension**.
9.	**modexp(E, E, E)**: Modular exponentiation, where the first expression is
	the base, the second is the exponent, and the third is the modulus. All
	three values must be integers. The second argument must be non-negative. The
	third argument must be non-zero. This is a **non-portable extension**.
10.	**divmod(E, E, I[])**: Division and modulus in one operation. This is for
	optimization. The first expression is the dividend, and the second is the
	divisor, which must be non-zero. The return value is the quotient, and the
	modulus is stored in index **0** of the provided array (the last argument).
	This is a **non-portable extension**.
11.	**asciify(E)**: If **E** is a string, returns a string that is the first
	letter of its argument. If it is a number, calculates the number mod **256**
	and returns that number as a one-character string. This is a **non-portable
	extension**.
12.	**I()**, **I(E)**, **I(E, E)**, and so on, where **I** is an identifier for
	a non-**void** function (see the *Void Functions* subsection of the
	**FUNCTIONS** section). The **E** argument(s) may also be arrays of the form
	**I[]**, which will automatically be turned into array references (see the
	*Array References* subsection of the **FUNCTIONS** section) if the
	corresponding parameter in the function definition is an array reference.
13.	**read()**: Reads a line from **stdin** and uses that as an expression. The
	result of that expression is the result of the **read()** operand. This is a
	**non-portable extension**.
14.	**maxibase()**: The max allowable **ibase**. This is a **non-portable
	extension**.
15.	**maxobase()**: The max allowable **obase**. This is a **non-portable
	extension**.
16.	**maxscale()**: The max allowable **scale**. This is a **non-portable
	extension**.
17.	**line_length()**: The line length set with **BC_LINE_LENGTH** (see the
	**ENVIRONMENT VARIABLES** section). This is a **non-portable extension**.
18.	**global_stacks()**: **0** if global stacks are not enabled with the **-g**
	or **-\-global-stacks** options, non-zero otherwise. See the **OPTIONS**
	section. This is a **non-portable extension**.
19.	**leading_zero()**: **0** if leading zeroes are not enabled with the **-z**
	or **--leading-zeroes** options, non-zero otherwise. See the **OPTIONS**
	section. This is a **non-portable extension**.
20.	**rand()**: A pseudo-random integer between **0** (inclusive) and
	**BC_RAND_MAX** (inclusive). Using this operand will change the value of
	**seed**. This is a **non-portable extension**.
21.	**irand(E)**: A pseudo-random integer between **0** (inclusive) and the
	value of **E** (exclusive). If **E** is negative or is a non-integer
	(**E**'s *scale* is not **0**), an error is raised, and bc(1) resets (see
	the **RESET** section) while **seed** remains unchanged. If **E** is larger
	than **BC_RAND_MAX**, the higher bound is honored by generating several
	pseudo-random integers, multiplying them by appropriate powers of
	**BC_RAND_MAX+1**, and adding them together. Thus, the size of integer that
	can be generated with this operand is unbounded. Using this operand will
	change the value of **seed**, unless the value of **E** is **0** or **1**.
	In that case, **0** is returned, and **seed** is *not* changed. This is a
	**non-portable extension**.
22.	**maxrand()**: The max integer returned by **rand()**. This is a
	**non-portable extension**.

The integers generated by **rand()** and **irand(E)** are guaranteed to be as
unbiased as possible, subject to the limitations of the pseudo-random number
generator.

**Note**: The values returned by the pseudo-random number generator with
**rand()** and **irand(E)** are guaranteed to *NOT* be cryptographically secure.
This is a consequence of using a seeded pseudo-random number generator. However,
they *are* guaranteed to be reproducible with identical **seed** values. This
means that the pseudo-random values from bc(1) should only be used where a
reproducible stream of pseudo-random numbers is *ESSENTIAL*. In any other case,
use a non-seeded pseudo-random number generator.

## Numbers

Numbers are strings made up of digits, uppercase letters, and at most **1**
period for a radix. Numbers can have up to **BC_NUM_MAX** digits. Uppercase
letters are equal to **9** + their position in the alphabet (i.e., **A** equals
**10**, or **9+1**). If a digit or letter makes no sense with the current value
of **ibase**, they are set to the value of the highest valid digit in **ibase**.

Single-character numbers (i.e., **A** alone) take the value that they would have
if they were valid digits, regardless of the value of **ibase**. This means that
**A** alone always equals decimal **10** and **Z** alone always equals decimal
**35**.

In addition, bc(1) accepts numbers in scientific notation. These have the form
**\<number\>e\<integer\>**. The exponent (the portion after the **e**) must be
an integer. An example is **1.89237e9**, which is equal to **1892370000**.
Negative exponents are also allowed, so **4.2890e-3** is equal to **0.0042890**.

Using scientific notation is an error or warning if the **-s** or **-w**,
respectively, command-line options (or equivalents) are given.

**WARNING**: Both the number and the exponent in scientific notation are
interpreted according to the current **ibase**, but the number is still
multiplied by **10\^exponent** regardless of the current **ibase**. For example,
if **ibase** is **16** and bc(1) is given the number string **FFeA**, the
resulting decimal number will be **2550000000000**, and if bc(1) is given the
number string **10e-4**, the resulting decimal number will be **0.0016**.

Accepting input as scientific notation is a **non-portable extension**.

## Operators

The following arithmetic and logical operators can be used. They are listed in
order of decreasing precedence. Operators in the same group have the same
precedence.

**++** **-\-**

:   Type: Prefix and Postfix

    Associativity: None

    Description: **increment**, **decrement**

**-** **!**

:   Type: Prefix

    Associativity: None

    Description: **negation**, **boolean not**

**\$**

:   Type: Postfix

    Associativity: None

    Description: **truncation**

**\@**

:   Type: Binary

    Associativity: Right

    Description: **set precision**

**\^**

:   Type: Binary

    Associativity: Right

    Description: **power**

**\*** **/** **%**

:   Type: Binary

    Associativity: Left

    Description: **multiply**, **divide**, **modulus**

**+** **-**

:   Type: Binary

    Associativity: Left

    Description: **add**, **subtract**

**\<\<** **\>\>**

:   Type: Binary

    Associativity: Left

    Description: **shift left**, **shift right**

**=** **\<\<=** **\>\>=** **+=** **-=** **\*=** **/=** **%=** **\^=** **\@=**

:   Type: Binary

    Associativity: Right

    Description: **assignment**

**==** **\<=** **\>=** **!=** **\<** **\>**

:   Type: Binary

    Associativity: Left

    Description: **relational**

**&&**

:   Type: Binary

    Associativity: Left

    Description: **boolean and**

**||**

:   Type: Binary

    Associativity: Left

    Description: **boolean or**

The operators will be described in more detail below.

**++** **-\-**

:   The prefix and postfix **increment** and **decrement** operators behave
    exactly like they would in C. They require a named expression (see the
    *Named Expressions* subsection) as an operand.

    The prefix versions of these operators are more efficient; use them where
    possible.

**-**

:   The **negation** operator returns **0** if a user attempts to negate any
    expression with the value **0**. Otherwise, a copy of the expression with
    its sign flipped is returned.

**!**

:   The **boolean not** operator returns **1** if the expression is **0**, or
    **0** otherwise.

    This is a **non-portable extension**.

**\$**

:   The **truncation** operator returns a copy of the given expression with all
    of its *scale* removed.

    This is a **non-portable extension**.

**\@**

:   The **set precision** operator takes two expressions and returns a copy of
    the first with its *scale* equal to the value of the second expression. That
    could either mean that the number is returned without change (if the
    *scale* of the first expression matches the value of the second
    expression), extended (if it is less), or truncated (if it is more).

    The second expression must be an integer (no *scale*) and non-negative.

    This is a **non-portable extension**.

**\^**

:   The **power** operator (not the **exclusive or** operator, as it would be in
    C) takes two expressions and raises the first to the power of the value of
    the second. The *scale* of the result is equal to **scale**.

    The second expression must be an integer (no *scale*), and if it is
    negative, the first value must be non-zero.

**\***

:   The **multiply** operator takes two expressions, multiplies them, and
    returns the product. If **a** is the *scale* of the first expression and
    **b** is the *scale* of the second expression, the *scale* of the result is
    equal to **min(a+b,max(scale,a,b))** where **min()** and **max()** return
    the obvious values.

**/**

:   The **divide** operator takes two expressions, divides them, and returns the
    quotient. The *scale* of the result shall be the value of **scale**.

    The second expression must be non-zero.

**%**

:   The **modulus** operator takes two expressions, **a** and **b**, and
    evaluates them by 1) Computing **a/b** to current **scale** and 2) Using the
    result of step 1 to calculate **a-(a/b)\*b** to *scale*
    **max(scale+scale(b),scale(a))**.

    The second expression must be non-zero.

**+**

:   The **add** operator takes two expressions, **a** and **b**, and returns the
    sum, with a *scale* equal to the max of the *scale*s of **a** and **b**.

**-**

:   The **subtract** operator takes two expressions, **a** and **b**, and
    returns the difference, with a *scale* equal to the max of the *scale*s of
    **a** and **b**.

**\<\<**

:   The **left shift** operator takes two expressions, **a** and **b**, and
    returns a copy of the value of **a** with its decimal point moved **b**
    places to the right.

    The second expression must be an integer (no *scale*) and non-negative.

    This is a **non-portable extension**.

**\>\>**

:   The **right shift** operator takes two expressions, **a** and **b**, and
    returns a copy of the value of **a** with its decimal point moved **b**
    places to the left.

    The second expression must be an integer (no *scale*) and non-negative.

    This is a **non-portable extension**.

**=** **\<\<=** **\>\>=** **+=** **-=** **\*=** **/=** **%=** **\^=** **\@=**

:   The **assignment** operators take two expressions, **a** and **b** where
    **a** is a named expression (see the *Named Expressions* subsection).

    For **=**, **b** is copied and the result is assigned to **a**. For all
    others, **a** and **b** are applied as operands to the corresponding
    arithmetic operator and the result is assigned to **a**.

    The **assignment** operators that correspond to operators that are
    extensions are themselves **non-portable extensions**.

**==** **\<=** **\>=** **!=** **\<** **\>**

:   The **relational** operators compare two expressions, **a** and **b**, and
    if the relation holds, according to C language semantics, the result is
    **1**. Otherwise, it is **0**.

    Note that unlike in C, these operators have a lower precedence than the
    **assignment** operators, which means that **a=b\>c** is interpreted as
    **(a=b)\>c**.

    Also, unlike the [standard][1] requires, these operators can appear anywhere
    any other expressions can be used. This allowance is a
    **non-portable extension**.

**&&**

:   The **boolean and** operator takes two expressions and returns **1** if both
    expressions are non-zero, **0** otherwise.

    This is *not* a short-circuit operator.

    This is a **non-portable extension**.

**||**

:   The **boolean or** operator takes two expressions and returns **1** if one
    of the expressions is non-zero, **0** otherwise.

    This is *not* a short-circuit operator.

    This is a **non-portable extension**.

## Statements

The following items are statements:

1.	**E**
2.	**{** **S** **;** ... **;** **S** **}**
3.	**if** **(** **E** **)** **S**
4.	**if** **(** **E** **)** **S** **else** **S**
5.	**while** **(** **E** **)** **S**
6.	**for** **(** **E** **;** **E** **;** **E** **)** **S**
7.	An empty statement
8.	**break**
9.	**continue**
10.	**quit**
11.	**halt**
12.	**limits**
13.	A string of characters, enclosed in double quotes
14.	**print** **E** **,** ... **,** **E**
15.	**stream** **E** **,** ... **,** **E**
16.	**I()**, **I(E)**, **I(E, E)**, and so on, where **I** is an identifier for
	a **void** function (see the *Void Functions* subsection of the
	**FUNCTIONS** section). The **E** argument(s) may also be arrays of the form
	**I[]**, which will automatically be turned into array references (see the
	*Array References* subsection of the **FUNCTIONS** section) if the
	corresponding parameter in the function definition is an array reference.

Numbers 4, 9, 11, 12, 14, 15, and 16 are **non-portable extensions**.

Also, as a **non-portable extension**, any or all of the expressions in the
header of a for loop may be omitted. If the condition (second expression) is
omitted, it is assumed to be a constant **1**.

The **break** statement causes a loop to stop iterating and resume execution
immediately following a loop. This is only allowed in loops.

The **continue** statement causes a loop iteration to stop early and returns to
the start of the loop, including testing the loop condition. This is only
allowed in loops.

The **if** **else** statement does the same thing as in C.

The **quit** statement causes bc(1) to quit, even if it is on a branch that will
not be executed (it is a compile-time command).

The **halt** statement causes bc(1) to quit, if it is executed. (Unlike **quit**
if it is on a branch of an **if** statement that is not executed, bc(1) does not
quit.)

The **limits** statement prints the limits that this bc(1) is subject to. This
is like the **quit** statement in that it is a compile-time command.

An expression by itself is evaluated and printed, followed by a newline.

Both scientific notation and engineering notation are available for printing the
results of expressions. Scientific notation is activated by assigning **0** to
**obase**, and engineering notation is activated by assigning **1** to
**obase**. To deactivate them, just assign a different value to **obase**.

Scientific notation and engineering notation are disabled if bc(1) is run with
either the **-s** or **-w** command-line options (or equivalents).

Printing numbers in scientific notation and/or engineering notation is a
**non-portable extension**.

## Strings

If strings appear as a statement by themselves, they are printed without a
trailing newline.

In addition to appearing as a lone statement by themselves, strings can be
assigned to variables and array elements. They can also be passed to functions
in variable parameters.

If any statement that expects a string is given a variable that had a string
assigned to it, the statement acts as though it had received a string.

If any math operation is attempted on a string or a variable or array element
that has been assigned a string, an error is raised, and bc(1) resets (see the
**RESET** section).

Assigning strings to variables and array elements and passing them to functions
are **non-portable extensions**.

## Print Statement

The "expressions" in a **print** statement may also be strings. If they are, there
are backslash escape sequences that are interpreted specially. What those
sequences are, and what they cause to be printed, are shown below:

**\\a**:   **\\a**

**\\b**:   **\\b**

**\\\\**:   **\\**

**\\e**:   **\\**

**\\f**:   **\\f**

**\\n**:   **\\n**

**\\q**:   **"**

**\\r**:   **\\r**

**\\t**:   **\\t**

Any other character following a backslash causes the backslash and character to
be printed as-is.

Any non-string expression in a print statement shall be assigned to **last**,
like any other expression that is printed.

## Stream Statement

The "expressions in a **stream** statement may also be strings.

If a **stream** statement is given a string, it prints the string as though the
string had appeared as its own statement. In other words, the **stream**
statement prints strings normally, without a newline.

If a **stream** statement is given a number, a copy of it is truncated and its
absolute value is calculated. The result is then printed as though **obase** is
**256** and each digit is interpreted as an 8-bit ASCII character, making it a
byte stream.

## Order of Evaluation

All expressions in a statment are evaluated left to right, except as necessary
to maintain order of operations. This means, for example, assuming that **i** is
equal to **0**, in the expression

    a[i++] = i++

the first (or 0th) element of **a** is set to **1**, and **i** is equal to **2**
at the end of the expression.

This includes function arguments. Thus, assuming **i** is equal to **0**, this
means that in the expression

    x(i++, i++)

the first argument passed to **x()** is **0**, and the second argument is **1**,
while **i** is equal to **2** before the function starts executing.

# FUNCTIONS

Function definitions are as follows:

```
define I(I,...,I){
	auto I,...,I
	S;...;S
	return(E)
}
```

Any **I** in the parameter list or **auto** list may be replaced with **I[]** to
make a parameter or **auto** var an array, and any **I** in the parameter list
may be replaced with **\*I[]** to make a parameter an array reference. Callers
of functions that take array references should not put an asterisk in the call;
they must be called with just **I[]** like normal array parameters and will be
automatically converted into references.

As a **non-portable extension**, the opening brace of a **define** statement may
appear on the next line.

As a **non-portable extension**, the return statement may also be in one of the
following forms:

1.	**return**
2.	**return** **(** **)**
3.	**return** **E**

The first two, or not specifying a **return** statement, is equivalent to
**return (0)**, unless the function is a **void** function (see the *Void
Functions* subsection below).

## Void Functions

Functions can also be **void** functions, defined as follows:

```
define void I(I,...,I){
	auto I,...,I
	S;...;S
	return
}
```

They can only be used as standalone expressions, where such an expression would
be printed alone, except in a print statement.

Void functions can only use the first two **return** statements listed above.
They can also omit the return statement entirely.

The word "void" is not treated as a keyword; it is still possible to have
variables, arrays, and functions named **void**. The word "void" is only
treated specially right after the **define** keyword.

This is a **non-portable extension**.

## Array References

For any array in the parameter list, if the array is declared in the form

```
*I[]
```

it is a **reference**. Any changes to the array in the function are reflected,
when the function returns, to the array that was passed in.

Other than this, all function arguments are passed by value.

This is a **non-portable extension**.

# LIBRARY

All of the functions below, including the functions in the extended math
library (see the *Extended Library* subsection below), are available when the
**-l** or **-\-mathlib** command-line flags are given, except that the extended
math library is not available when the **-s** option, the **-w** option, or
equivalents are given.

## Standard Library

The [standard][1] defines the following functions for the math library:

**s(x)**

:   Returns the sine of **x**, which is assumed to be in radians.

    This is a transcendental function (see the *Transcendental Functions*
    subsection below).

**c(x)**

:   Returns the cosine of **x**, which is assumed to be in radians.

    This is a transcendental function (see the *Transcendental Functions*
    subsection below).

**a(x)**

:   Returns the arctangent of **x**, in radians.

    This is a transcendental function (see the *Transcendental Functions*
    subsection below).

**l(x)**

:   Returns the natural logarithm of **x**.

    This is a transcendental function (see the *Transcendental Functions*
    subsection below).

**e(x)**

:   Returns the mathematical constant **e** raised to the power of **x**.

    This is a transcendental function (see the *Transcendental Functions*
    subsection below).

**j(x, n)**

:   Returns the bessel integer order **n** (truncated) of **x**.

    This is a transcendental function (see the *Transcendental Functions*
    subsection below).

## Extended Library

The extended library is *not* loaded when the **-s**/**-\-standard** or
**-w**/**-\-warn** options are given since they are not part of the library
defined by the [standard][1].

The extended library is a **non-portable extension**.

**p(x, y)**

:   Calculates **x** to the power of **y**, even if **y** is not an integer, and
    returns the result to the current **scale**.

    It is an error if **y** is negative and **x** is **0**.

    This is a transcendental function (see the *Transcendental Functions*
    subsection below).

**r(x, p)**

:   Returns **x** rounded to **p** decimal places according to the rounding mode
    [round half away from **0**][3].

**ceil(x, p)**

:   Returns **x** rounded to **p** decimal places according to the rounding mode
    [round away from **0**][6].

**f(x)**

:   Returns the factorial of the truncated absolute value of **x**.

**perm(n, k)**

:   Returns the permutation of the truncated absolute value of **n** of the
    truncated absolute value of **k**, if **k \<= n**. If not, it returns **0**.

**comb(n, k)**

:   Returns the combination of the truncated absolute value of **n** of the
    truncated absolute value of **k**, if **k \<= n**. If not, it returns **0**.

**l2(x)**

:   Returns the logarithm base **2** of **x**.

    This is a transcendental function (see the *Transcendental Functions*
    subsection below).

**l10(x)**

:   Returns the logarithm base **10** of **x**.

    This is a transcendental function (see the *Transcendental Functions*
    subsection below).

**log(x, b)**

:   Returns the logarithm base **b** of **x**.

    This is a transcendental function (see the *Transcendental Functions*
    subsection below).

**cbrt(x)**

:   Returns the cube root of **x**.

**root(x, n)**

:   Calculates the truncated value of **n**, **r**, and returns the **r**th root
    of **x** to the current **scale**.

    If **r** is **0** or negative, this raises an error and causes bc(1) to
    reset (see the **RESET** section). It also raises an error and causes bc(1)
    to reset if **r** is even and **x** is negative.

**gcd(a, b)**

:   Returns the greatest common divisor (factor) of the truncated absolute value
    of **a** and the truncated absolute value of **b**.

**lcm(a, b)**

:   Returns the least common multiple of the truncated absolute value of **a**
    and the truncated absolute value of **b**.

**pi(p)**

:   Returns **pi** to **p** decimal places.

    This is a transcendental function (see the *Transcendental Functions*
    subsection below).

**t(x)**

:   Returns the tangent of **x**, which is assumed to be in radians.

    This is a transcendental function (see the *Transcendental Functions*
    subsection below).

**a2(y, x)**

:   Returns the arctangent of **y/x**, in radians. If both **y** and **x** are
    equal to **0**, it raises an error and causes bc(1) to reset (see the
    **RESET** section). Otherwise, if **x** is greater than **0**, it returns
    **a(y/x)**. If **x** is less than **0**, and **y** is greater than or equal
    to **0**, it returns **a(y/x)+pi**. If **x** is less than **0**, and **y**
    is less than **0**, it returns **a(y/x)-pi**. If **x** is equal to **0**,
    and **y** is greater than **0**, it returns **pi/2**. If **x** is equal to
    **0**, and **y** is less than **0**, it returns **-pi/2**.

    This function is the same as the **atan2()** function in many programming
    languages.

    This is a transcendental function (see the *Transcendental Functions*
    subsection below).

**sin(x)**

:   Returns the sine of **x**, which is assumed to be in radians.

    This is an alias of **s(x)**.

    This is a transcendental function (see the *Transcendental Functions*
    subsection below).

**cos(x)**

:   Returns the cosine of **x**, which is assumed to be in radians.

    This is an alias of **c(x)**.

    This is a transcendental function (see the *Transcendental Functions*
    subsection below).

**tan(x)**

:   Returns the tangent of **x**, which is assumed to be in radians.

    If **x** is equal to **1** or **-1**, this raises an error and causes bc(1)
    to reset (see the **RESET** section).

    This is an alias of **t(x)**.

    This is a transcendental function (see the *Transcendental Functions*
    subsection below).

**atan(x)**

:   Returns the arctangent of **x**, in radians.

    This is an alias of **a(x)**.

    This is a transcendental function (see the *Transcendental Functions*
    subsection below).

**atan2(y, x)**

:   Returns the arctangent of **y/x**, in radians. If both **y** and **x** are
    equal to **0**, it raises an error and causes bc(1) to reset (see the
    **RESET** section). Otherwise, if **x** is greater than **0**, it returns
    **a(y/x)**. If **x** is less than **0**, and **y** is greater than or equal
    to **0**, it returns **a(y/x)+pi**. If **x** is less than **0**, and **y**
    is less than **0**, it returns **a(y/x)-pi**. If **x** is equal to **0**,
    and **y** is greater than **0**, it returns **pi/2**. If **x** is equal to
    **0**, and **y** is less than **0**, it returns **-pi/2**.

    This function is the same as the **atan2()** function in many programming
    languages.

    This is an alias of **a2(y, x)**.

    This is a transcendental function (see the *Transcendental Functions*
    subsection below).

**r2d(x)**

:   Converts **x** from radians to degrees and returns the result.

    This is a transcendental function (see the *Transcendental Functions*
    subsection below).

**d2r(x)**

:   Converts **x** from degrees to radians and returns the result.

    This is a transcendental function (see the *Transcendental Functions*
    subsection below).

**frand(p)**

:   Generates a pseudo-random number between **0** (inclusive) and **1**
    (exclusive) with the number of decimal digits after the decimal point equal
    to the truncated absolute value of **p**. If **p** is not **0**, then
    calling this function will change the value of **seed**. If **p** is **0**,
    then **0** is returned, and **seed** is *not* changed.

**ifrand(i, p)**

:   Generates a pseudo-random number that is between **0** (inclusive) and the
    truncated absolute value of **i** (exclusive) with the number of decimal
    digits after the decimal point equal to the truncated absolute value of
    **p**. If the absolute value of **i** is greater than or equal to **2**, and
    **p** is not **0**, then calling this function will change the value of
    **seed**; otherwise, **0** is returned and **seed** is not changed.

**srand(x)**

:   Returns **x** with its sign flipped with probability **0.5**. In other
    words, it randomizes the sign of **x**.

**brand()**

:   Returns a random boolean value (either **0** or **1**).

**band(a, b)**

:   Takes the truncated absolute value of both **a** and **b** and calculates
    and returns the result of the bitwise **and** operation between them.

    If you want to use signed two's complement arguments, use **s2u(x)** to
    convert.

**bor(a, b)**

:   Takes the truncated absolute value of both **a** and **b** and calculates
    and returns the result of the bitwise **or** operation between them.

    If you want to use signed two's complement arguments, use **s2u(x)** to
    convert.

**bxor(a, b)**

:   Takes the truncated absolute value of both **a** and **b** and calculates
    and returns the result of the bitwise **xor** operation between them.

    If you want to use signed two's complement arguments, use **s2u(x)** to
    convert.

**bshl(a, b)**

:   Takes the truncated absolute value of both **a** and **b** and calculates
    and returns the result of **a** bit-shifted left by **b** places.

    If you want to use signed two's complement arguments, use **s2u(x)** to
    convert.

**bshr(a, b)**

:   Takes the truncated absolute value of both **a** and **b** and calculates
    and returns the truncated result of **a** bit-shifted right by **b** places.

    If you want to use signed two's complement arguments, use **s2u(x)** to
    convert.

**bnotn(x, n)**

:   Takes the truncated absolute value of **x** and does a bitwise not as though
    it has the same number of bytes as the truncated absolute value of **n**.

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**bnot8(x)**

:   Does a bitwise not of the truncated absolute value of **x** as though it has
    **8** binary digits (1 unsigned byte).

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**bnot16(x)**

:   Does a bitwise not of the truncated absolute value of **x** as though it has
    **16** binary digits (2 unsigned bytes).

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**bnot32(x)**

:   Does a bitwise not of the truncated absolute value of **x** as though it has
    **32** binary digits (4 unsigned bytes).

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**bnot64(x)**

:   Does a bitwise not of the truncated absolute value of **x** as though it has
    **64** binary digits (8 unsigned bytes).

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**bnot(x)**

:   Does a bitwise not of the truncated absolute value of **x** as though it has
    the minimum number of power of two unsigned bytes.

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**brevn(x, n)**

:   Runs a bit reversal on the truncated absolute value of **x** as though it
    has the same number of 8-bit bytes as the truncated absolute value of **n**.

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**brev8(x)**

:   Runs a bit reversal on the truncated absolute value of **x** as though it
    has 8 binary digits (1 unsigned byte).

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**brev16(x)**

:   Runs a bit reversal on the truncated absolute value of **x** as though it
    has 16 binary digits (2 unsigned bytes).

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**brev32(x)**

:   Runs a bit reversal on the truncated absolute value of **x** as though it
    has 32 binary digits (4 unsigned bytes).

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**brev64(x)**

:   Runs a bit reversal on the truncated absolute value of **x** as though it
    has 64 binary digits (8 unsigned bytes).

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**brev(x)**

:   Runs a bit reversal on the truncated absolute value of **x** as though it
    has the minimum number of power of two unsigned bytes.

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**broln(x, p, n)**

:   Does a left bitwise rotatation of the truncated absolute value of **x**, as
    though it has the same number of unsigned 8-bit bytes as the truncated
    absolute value of **n**, by the number of places equal to the truncated
    absolute value of **p** modded by the **2** to the power of the number of
    binary digits in **n** 8-bit bytes.

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**brol8(x, p)**

:   Does a left bitwise rotatation of the truncated absolute value of **x**, as
    though it has **8** binary digits (**1** unsigned byte), by the number of
    places equal to the truncated absolute value of **p** modded by **2** to the
    power of **8**.

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**brol16(x, p)**

:   Does a left bitwise rotatation of the truncated absolute value of **x**, as
    though it has **16** binary digits (**2** unsigned bytes), by the number of
    places equal to the truncated absolute value of **p** modded by **2** to the
    power of **16**.

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**brol32(x, p)**

:   Does a left bitwise rotatation of the truncated absolute value of **x**, as
    though it has **32** binary digits (**2** unsigned bytes), by the number of
    places equal to the truncated absolute value of **p** modded by **2** to the
    power of **32**.

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**brol64(x, p)**

:   Does a left bitwise rotatation of the truncated absolute value of **x**, as
    though it has **64** binary digits (**2** unsigned bytes), by the number of
    places equal to the truncated absolute value of **p** modded by **2** to the
    power of **64**.

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**brol(x, p)**

:   Does a left bitwise rotatation of the truncated absolute value of **x**, as
    though it has the minimum number of power of two unsigned 8-bit bytes, by
    the number of places equal to the truncated absolute value of **p** modded
    by 2 to the power of the number of binary digits in the minimum number of
    8-bit bytes.

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**brorn(x, p, n)**

:   Does a right bitwise rotatation of the truncated absolute value of **x**, as
    though it has the same number of unsigned 8-bit bytes as the truncated
    absolute value of **n**, by the number of places equal to the truncated
    absolute value of **p** modded by the **2** to the power of the number of
    binary digits in **n** 8-bit bytes.

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**bror8(x, p)**

:   Does a right bitwise rotatation of the truncated absolute value of **x**, as
    though it has **8** binary digits (**1** unsigned byte), by the number of
    places equal to the truncated absolute value of **p** modded by **2** to the
    power of **8**.

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**bror16(x, p)**

:   Does a right bitwise rotatation of the truncated absolute value of **x**, as
    though it has **16** binary digits (**2** unsigned bytes), by the number of
    places equal to the truncated absolute value of **p** modded by **2** to the
    power of **16**.

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**bror32(x, p)**

:   Does a right bitwise rotatation of the truncated absolute value of **x**, as
    though it has **32** binary digits (**2** unsigned bytes), by the number of
    places equal to the truncated absolute value of **p** modded by **2** to the
    power of **32**.

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**bror64(x, p)**

:   Does a right bitwise rotatation of the truncated absolute value of **x**, as
    though it has **64** binary digits (**2** unsigned bytes), by the number of
    places equal to the truncated absolute value of **p** modded by **2** to the
    power of **64**.

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**bror(x, p)**

:   Does a right bitwise rotatation of the truncated absolute value of **x**, as
    though it has the minimum number of power of two unsigned 8-bit bytes, by
    the number of places equal to the truncated absolute value of **p** modded
    by 2 to the power of the number of binary digits in the minimum number of
    8-bit bytes.

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**bmodn(x, n)**

:   Returns the modulus of the truncated absolute value of **x** by **2** to the
    power of the multiplication of the truncated absolute value of **n** and
    **8**.

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**bmod8(x, n)**

:   Returns the modulus of the truncated absolute value of **x** by **2** to the
    power of **8**.

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**bmod16(x, n)**

:   Returns the modulus of the truncated absolute value of **x** by **2** to the
    power of **16**.

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**bmod32(x, n)**

:   Returns the modulus of the truncated absolute value of **x** by **2** to the
    power of **32**.

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**bmod64(x, n)**

:   Returns the modulus of the truncated absolute value of **x** by **2** to the
    power of **64**.

    If you want to a use signed two's complement argument, use **s2u(x)** to
    convert.

**bunrev(t)**

:   Assumes **t** is a bitwise-reversed number with an extra set bit one place
    more significant than the real most significant bit (which was the least
    significant bit in the original number). This number is reversed and
    returned without the extra set bit.

    This function is used to implement other bitwise functions; it is not meant
    to be used by users, but it can be.

**plz(x)**

:   If **x** is not equal to **0** and greater that **-1** and less than **1**,
    it is printed with a leading zero, regardless of the use of the **-z**
    option (see the **OPTIONS** section) and without a trailing newline.

    Otherwise, **x** is printed normally, without a trailing newline.

**plznl(x)**

:   If **x** is not equal to **0** and greater that **-1** and less than **1**,
    it is printed with a leading zero, regardless of the use of the **-z**
    option (see the **OPTIONS** section) and with a trailing newline.

    Otherwise, **x** is printed normally, with a trailing newline.

**pnlz(x)**

:   If **x** is not equal to **0** and greater that **-1** and less than **1**,
    it is printed without a leading zero, regardless of the use of the **-z**
    option (see the **OPTIONS** section) and without a trailing newline.

    Otherwise, **x** is printed normally, without a trailing newline.

**pnlznl(x)**

:   If **x** is not equal to **0** and greater that **-1** and less than **1**,
    it is printed without a leading zero, regardless of the use of the **-z**
    option (see the **OPTIONS** section) and with a trailing newline.

    Otherwise, **x** is printed normally, with a trailing newline.

**ubytes(x)**

:   Returns the numbers of unsigned integer bytes required to hold the truncated
    absolute value of **x**.

**sbytes(x)**

:   Returns the numbers of signed, two's-complement integer bytes required to
    hold the truncated value of **x**.

**s2u(x)**

:   Returns **x** if it is non-negative. If it *is* negative, then it calculates
    what **x** would be as a 2's-complement signed integer and returns the
    non-negative integer that would have the same representation in binary.

**s2un(x,n)**

:   Returns **x** if it is non-negative. If it *is* negative, then it calculates
    what **x** would be as a 2's-complement signed integer with **n** bytes and
    returns the non-negative integer that would have the same representation in
    binary. If **x** cannot fit into **n** 2's-complement signed bytes, it is
    truncated to fit.

**hex(x)**

:   Outputs the hexadecimal (base **16**) representation of **x**.

    This is a **void** function (see the *Void Functions* subsection of the
    **FUNCTIONS** section).

**binary(x)**

:   Outputs the binary (base **2**) representation of **x**.

    This is a **void** function (see the *Void Functions* subsection of the
    **FUNCTIONS** section).

**output(x, b)**

:   Outputs the base **b** representation of **x**.

    This is a **void** function (see the *Void Functions* subsection of the
    **FUNCTIONS** section).

**uint(x)**

:   Outputs the representation, in binary and hexadecimal, of **x** as an
    unsigned integer in as few power of two bytes as possible. Both outputs are
    split into bytes separated by spaces.

    If **x** is not an integer or is negative, an error message is printed
    instead, but bc(1) is not reset (see the **RESET** section).

    This is a **void** function (see the *Void Functions* subsection of the
    **FUNCTIONS** section).

**int(x)**

:   Outputs the representation, in binary and hexadecimal, of **x** as a signed,
    two's-complement integer in as few power of two bytes as possible. Both
    outputs are split into bytes separated by spaces.

    If **x** is not an integer, an error message is printed instead, but bc(1)
    is not reset (see the **RESET** section).

    This is a **void** function (see the *Void Functions* subsection of the
    **FUNCTIONS** section).

**uintn(x, n)**

:   Outputs the representation, in binary and hexadecimal, of **x** as an
    unsigned integer in **n** bytes. Both outputs are split into bytes separated
    by spaces.

    If **x** is not an integer, is negative, or cannot fit into **n** bytes, an
    error message is printed instead, but bc(1) is not reset (see the **RESET**
    section).

    This is a **void** function (see the *Void Functions* subsection of the
    **FUNCTIONS** section).

**intn(x, n)**

:   Outputs the representation, in binary and hexadecimal, of **x** as a signed,
    two's-complement integer in **n** bytes. Both outputs are split into bytes
    separated by spaces.

    If **x** is not an integer or cannot fit into **n** bytes, an error message
    is printed instead, but bc(1) is not reset (see the **RESET** section).

    This is a **void** function (see the *Void Functions* subsection of the
    **FUNCTIONS** section).

**uint8(x)**

:   Outputs the representation, in binary and hexadecimal, of **x** as an
    unsigned integer in **1** byte. Both outputs are split into bytes separated
    by spaces.

    If **x** is not an integer, is negative, or cannot fit into **1** byte, an
    error message is printed instead, but bc(1) is not reset (see the **RESET**
    section).

    This is a **void** function (see the *Void Functions* subsection of the
    **FUNCTIONS** section).

**int8(x)**

:   Outputs the representation, in binary and hexadecimal, of **x** as a signed,
    two's-complement integer in **1** byte. Both outputs are split into bytes
    separated by spaces.

    If **x** is not an integer or cannot fit into **1** byte, an error message
    is printed instead, but bc(1) is not reset (see the **RESET** section).

    This is a **void** function (see the *Void Functions* subsection of the
    **FUNCTIONS** section).

**uint16(x)**

:   Outputs the representation, in binary and hexadecimal, of **x** as an
    unsigned integer in **2** bytes. Both outputs are split into bytes separated
    by spaces.

    If **x** is not an integer, is negative, or cannot fit into **2** bytes, an
    error message is printed instead, but bc(1) is not reset (see the **RESET**
    section).

    This is a **void** function (see the *Void Functions* subsection of the
    **FUNCTIONS** section).

**int16(x)**

:   Outputs the representation, in binary and hexadecimal, of **x** as a signed,
    two's-complement integer in **2** bytes. Both outputs are split into bytes
    separated by spaces.

    If **x** is not an integer or cannot fit into **2** bytes, an error message
    is printed instead, but bc(1) is not reset (see the **RESET** section).

    This is a **void** function (see the *Void Functions* subsection of the
    **FUNCTIONS** section).

**uint32(x)**

:   Outputs the representation, in binary and hexadecimal, of **x** as an
    unsigned integer in **4** bytes. Both outputs are split into bytes separated
    by spaces.

    If **x** is not an integer, is negative, or cannot fit into **4** bytes, an
    error message is printed instead, but bc(1) is not reset (see the **RESET**
    section).

    This is a **void** function (see the *Void Functions* subsection of the
    **FUNCTIONS** section).

**int32(x)**

:   Outputs the representation, in binary and hexadecimal, of **x** as a signed,
    two's-complement integer in **4** bytes. Both outputs are split into bytes
    separated by spaces.

    If **x** is not an integer or cannot fit into **4** bytes, an error message
    is printed instead, but bc(1) is not reset (see the **RESET** section).

    This is a **void** function (see the *Void Functions* subsection of the
    **FUNCTIONS** section).

**uint64(x)**

:   Outputs the representation, in binary and hexadecimal, of **x** as an
    unsigned integer in **8** bytes. Both outputs are split into bytes separated
    by spaces.

    If **x** is not an integer, is negative, or cannot fit into **8** bytes, an
    error message is printed instead, but bc(1) is not reset (see the **RESET**
    section).

    This is a **void** function (see the *Void Functions* subsection of the
    **FUNCTIONS** section).

**int64(x)**

:   Outputs the representation, in binary and hexadecimal, of **x** as a signed,
    two's-complement integer in **8** bytes. Both outputs are split into bytes
    separated by spaces.

    If **x** is not an integer or cannot fit into **8** bytes, an error message
    is printed instead, but bc(1) is not reset (see the **RESET** section).

    This is a **void** function (see the *Void Functions* subsection of the
    **FUNCTIONS** section).

**hex_uint(x, n)**

:   Outputs the representation of the truncated absolute value of **x** as an
    unsigned integer in hexadecimal using **n** bytes. Not all of the value will
    be output if **n** is too small.

    This is a **void** function (see the *Void Functions* subsection of the
    **FUNCTIONS** section).

**binary_uint(x, n)**

:   Outputs the representation of the truncated absolute value of **x** as an
    unsigned integer in binary using **n** bytes. Not all of the value will be
    output if **n** is too small.

    This is a **void** function (see the *Void Functions* subsection of the
    **FUNCTIONS** section).

**output_uint(x, n)**

:   Outputs the representation of the truncated absolute value of **x** as an
    unsigned integer in the current **obase** (see the **SYNTAX** section) using
    **n** bytes. Not all of the value will be output if **n** is too small.

    This is a **void** function (see the *Void Functions* subsection of the
    **FUNCTIONS** section).

**output_byte(x, i)**

:   Outputs byte **i** of the truncated absolute value of **x**, where **0** is
    the least significant byte and **number_of_bytes - 1** is the most
    significant byte.

    This is a **void** function (see the *Void Functions* subsection of the
    **FUNCTIONS** section).

## Transcendental Functions

All transcendental functions can return slightly inaccurate results (up to 1
[ULP][4]). This is unavoidable, and [this article][5] explains why it is
impossible and unnecessary to calculate exact results for the transcendental
functions.

Because of the possible inaccuracy, I recommend that users call those functions
with the precision (**scale**) set to at least 1 higher than is necessary. If
exact results are *absolutely* required, users can double the precision
(**scale**) and then truncate.

The transcendental functions in the standard math library are:

* **s(x)**
* **c(x)**
* **a(x)**
* **l(x)**
* **e(x)**
* **j(x, n)**

The transcendental functions in the extended math library are:

* **l2(x)**
* **l10(x)**
* **log(x, b)**
* **pi(p)**
* **t(x)**
* **a2(y, x)**
* **sin(x)**
* **cos(x)**
* **tan(x)**
* **atan(x)**
* **atan2(y, x)**
* **r2d(x)**
* **d2r(x)**

# RESET

When bc(1) encounters an error or a signal that it has a non-default handler
for, it resets. This means that several things happen.

First, any functions that are executing are stopped and popped off the stack.
The behavior is not unlike that of exceptions in programming languages. Then
the execution point is set so that any code waiting to execute (after all
functions returned) is skipped.

Thus, when bc(1) resets, it skips any remaining code waiting to be executed.
Then, if it is interactive mode, and the error was not a fatal error (see the
**EXIT STATUS** section), it asks for more input; otherwise, it exits with the
appropriate return code.

Note that this reset behavior is different from the GNU bc(1), which attempts to
start executing the statement right after the one that caused an error.

# PERFORMANCE

Most bc(1) implementations use **char** types to calculate the value of **1**
decimal digit at a time, but that can be slow. This bc(1) does something
different.

It uses large integers to calculate more than **1** decimal digit at a time. If
built in a environment where **BC_LONG_BIT** (see the **LIMITS** section) is
**64**, then each integer has **9** decimal digits. If built in an environment
where **BC_LONG_BIT** is **32** then each integer has **4** decimal digits. This
value (the number of decimal digits per large integer) is called
**BC_BASE_DIGS**.

The actual values of **BC_LONG_BIT** and **BC_BASE_DIGS** can be queried with
the **limits** statement.

In addition, this bc(1) uses an even larger integer for overflow checking. This
integer type depends on the value of **BC_LONG_BIT**, but is always at least
twice as large as the integer type used to store digits.

# LIMITS

The following are the limits on bc(1):

**BC_LONG_BIT**

:   The number of bits in the **long** type in the environment where bc(1) was
    built. This determines how many decimal digits can be stored in a single
    large integer (see the **PERFORMANCE** section).

**BC_BASE_DIGS**

:   The number of decimal digits per large integer (see the **PERFORMANCE**
    section). Depends on **BC_LONG_BIT**.

**BC_BASE_POW**

:   The max decimal number that each large integer can store (see
    **BC_BASE_DIGS**) plus **1**. Depends on **BC_BASE_DIGS**.

**BC_OVERFLOW_MAX**

:   The max number that the overflow type (see the **PERFORMANCE** section) can
    hold. Depends on **BC_LONG_BIT**.

**BC_BASE_MAX**

:   The maximum output base. Set at **BC_BASE_POW**.

**BC_DIM_MAX**

:   The maximum size of arrays. Set at **SIZE_MAX-1**.

**BC_SCALE_MAX**

:   The maximum **scale**. Set at **BC_OVERFLOW_MAX-1**.

**BC_STRING_MAX**

:   The maximum length of strings. Set at **BC_OVERFLOW_MAX-1**.

**BC_NAME_MAX**

:   The maximum length of identifiers. Set at **BC_OVERFLOW_MAX-1**.

**BC_NUM_MAX**

:   The maximum length of a number (in decimal digits), which includes digits
    after the decimal point. Set at **BC_OVERFLOW_MAX-1**.

**BC_RAND_MAX**

:   The maximum integer (inclusive) returned by the **rand()** operand. Set at
    **2\^BC_LONG_BIT-1**.

Exponent

:   The maximum allowable exponent (positive or negative). Set at
    **BC_OVERFLOW_MAX**.

Number of vars

:   The maximum number of vars/arrays. Set at **SIZE_MAX-1**.

The actual values can be queried with the **limits** statement.

These limits are meant to be effectively non-existent; the limits are so large
(at least on 64-bit machines) that there should not be any point at which they
become a problem. In fact, memory should be exhausted before these limits should
be hit.

# ENVIRONMENT VARIABLES

bc(1) recognizes the following environment variables:

**POSIXLY_CORRECT**

:   If this variable exists (no matter the contents), bc(1) behaves as if
    the **-s** option was given.

**BC_ENV_ARGS**

:   This is another way to give command-line arguments to bc(1). They should be
    in the same format as all other command-line arguments. These are always
    processed first, so any files given in **BC_ENV_ARGS** will be processed
    before arguments and files given on the command-line. This gives the user
    the ability to set up "standard" options and files to be used at every
    invocation. The most useful thing for such files to contain would be useful
    functions that the user might want every time bc(1) runs.

    The code that parses **BC_ENV_ARGS** will correctly handle quoted arguments,
    but it does not understand escape sequences. For example, the string
    **"/home/gavin/some bc file.bc"** will be correctly parsed, but the string
    **"/home/gavin/some \"bc\" file.bc"** will include the backslashes.

    The quote parsing will handle either kind of quotes, **'** or **"**. Thus,
    if you have a file with any number of single quotes in the name, you can use
    double quotes as the outside quotes, as in **"some 'bc' file.bc"**, and vice
    versa if you have a file with double quotes. However, handling a file with
    both kinds of quotes in **BC_ENV_ARGS** is not supported due to the
    complexity of the parsing, though such files are still supported on the
    command-line where the parsing is done by the shell.

**BC_LINE_LENGTH**

:   If this environment variable exists and contains an integer that is greater
    than **1** and is less than **UINT16_MAX** (**2\^16-1**), bc(1) will output
    lines to that length, including the backslash (**\\**). The default line
    length is **70**.

    The special value of **0** will disable line length checking and print
    numbers without regard to line length and without backslashes and newlines.

**BC_BANNER**

:   If this environment variable exists and contains an integer, then a non-zero
    value activates the copyright banner when bc(1) is in interactive mode,
    while zero deactivates it.

    If bc(1) is not in interactive mode (see the **INTERACTIVE MODE** section),
    then this environment variable has no effect because bc(1) does not print
    the banner when not in interactive mode.

    This environment variable overrides the default, which can be queried with
    the **-h** or **-\-help** options.

**BC_SIGINT_RESET**

:   If bc(1) is not in interactive mode (see the **INTERACTIVE MODE** section),
    then this environment variable has no effect because bc(1) exits on
    **SIGINT** when not in interactive mode.

    However, when bc(1) is in interactive mode, then if this environment
    variable exists and contains an integer, a non-zero value makes bc(1) reset
    on **SIGINT**, rather than exit, and zero makes bc(1) exit. If this
    environment variable exists and is *not* an integer, then bc(1) will exit on
    **SIGINT**.

    This environment variable overrides the default, which can be queried with
    the **-h** or **-\-help** options.

**BC_TTY_MODE**

:   If TTY mode is *not* available (see the **TTY MODE** section), then this
    environment variable has no effect.

    However, when TTY mode is available, then if this environment variable
    exists and contains an integer, then a non-zero value makes bc(1) use TTY
    mode, and zero makes bc(1) not use TTY mode.

    This environment variable overrides the default, which can be queried with
    the **-h** or **-\-help** options.

**BC_PROMPT**

:   If TTY mode is *not* available (see the **TTY MODE** section), then this
    environment variable has no effect.

    However, when TTY mode is available, then if this environment variable
    exists and contains an integer, a non-zero value makes bc(1) use a prompt,
    and zero or a non-integer makes bc(1) not use a prompt. If this environment
    variable does not exist and **BC_TTY_MODE** does, then the value of the
    **BC_TTY_MODE** environment variable is used.

    This environment variable and the **BC_TTY_MODE** environment variable
    override the default, which can be queried with the **-h** or **-\-help**
    options.

**BC_EXPR_EXIT**

:   If any expressions or expression files are given on the command-line with
    **-e**, **-\-expression**, **-f**, or **-\-file**, then if this environment
    variable exists and contains an integer, a non-zero value makes bc(1) exit
    after executing the expressions and expression files, and a non-zero value
    makes bc(1) not exit.

    This environment variable overrides the default, which can be queried with
    the **-h** or **-\-help** options.

# EXIT STATUS

bc(1) returns the following exit statuses:

**0**

:   No error.

**1**

:   A math error occurred. This follows standard practice of using **1** for
    expected errors, since math errors will happen in the process of normal
    execution.

    Math errors include divide by **0**, taking the square root of a negative
    number, using a negative number as a bound for the pseudo-random number
    generator, attempting to convert a negative number to a hardware integer,
    overflow when converting a number to a hardware integer, overflow when
    calculating the size of a number, and attempting to use a non-integer where
    an integer is required.

    Converting to a hardware integer happens for the second operand of the power
    (**\^**), places (**\@**), left shift (**\<\<**), and right shift (**\>\>**)
    operators and their corresponding assignment operators.

**2**

:   A parse error occurred.

    Parse errors include unexpected **EOF**, using an invalid character, failing
    to find the end of a string or comment, using a token where it is invalid,
    giving an invalid expression, giving an invalid print statement, giving an
    invalid function definition, attempting to assign to an expression that is
    not a named expression (see the *Named Expressions* subsection of the
    **SYNTAX** section), giving an invalid **auto** list, having a duplicate
    **auto**/function parameter, failing to find the end of a code block,
    attempting to return a value from a **void** function, attempting to use a
    variable as a reference, and using any extensions when the option **-s** or
    any equivalents were given.

**3**

:   A runtime error occurred.

    Runtime errors include assigning an invalid number to any global (**ibase**,
    **obase**, or **scale**), giving a bad expression to a **read()** call,
    calling **read()** inside of a **read()** call, type errors, passing the
    wrong number of arguments to functions, attempting to call an undefined
    function, and attempting to use a **void** function call as a value in an
    expression.

**4**

:   A fatal error occurred.

    Fatal errors include memory allocation errors, I/O errors, failing to open
    files, attempting to use files that do not have only ASCII characters (bc(1)
    only accepts ASCII characters), attempting to open a directory as a file,
    and giving invalid command-line options.

The exit status **4** is special; when a fatal error occurs, bc(1) always exits
and returns **4**, no matter what mode bc(1) is in.

The other statuses will only be returned when bc(1) is not in interactive mode
(see the **INTERACTIVE MODE** section), since bc(1) resets its state (see the
**RESET** section) and accepts more input when one of those errors occurs in
interactive mode. This is also the case when interactive mode is forced by the
**-i** flag or **-\-interactive** option.

These exit statuses allow bc(1) to be used in shell scripting with error
checking, and its normal behavior can be forced by using the **-i** flag or
**-\-interactive** option.

# INTERACTIVE MODE

Per the [standard][1], bc(1) has an interactive mode and a non-interactive mode.
Interactive mode is turned on automatically when both **stdin** and **stdout**
are hooked to a terminal, but the **-i** flag and **-\-interactive** option can
turn it on in other situations.

In interactive mode, bc(1) attempts to recover from errors (see the **RESET**
section), and in normal execution, flushes **stdout** as soon as execution is
done for the current input. bc(1) may also reset on **SIGINT** instead of exit,
depending on the contents of, or default for, the **BC_SIGINT_RESET**
environment variable (see the **ENVIRONMENT VARIABLES** section).

# TTY MODE

If **stdin**, **stdout**, and **stderr** are all connected to a TTY, then "TTY
mode" is considered to be available, and thus, bc(1) can turn on TTY mode,
subject to some settings.

If there is the environment variable **BC_TTY_MODE** in the environment (see the
**ENVIRONMENT VARIABLES** section), then if that environment variable contains a
non-zero integer, bc(1) will turn on TTY mode when **stdin**, **stdout**, and
**stderr** are all connected to a TTY. If the **BC_TTY_MODE** environment
variable exists but is *not* a non-zero integer, then bc(1) will not turn TTY
mode on.

If the environment variable **BC_TTY_MODE** does *not* exist, the default
setting is used. The default setting can be queried with the **-h** or
**-\-help** options.

TTY mode is different from interactive mode because interactive mode is required
in the [bc(1) specification][1], and interactive mode requires only **stdin**
and **stdout** to be connected to a terminal.

## Command-Line History

Command-line history is only enabled if TTY mode is, i.e., that **stdin**,
**stdout**, and **stderr** are connected to a TTY and the **BC_TTY_MODE**
environment variable (see the **ENVIRONMENT VARIABLES** section) and its default
do not disable TTY mode. See the **COMMAND LINE HISTORY** section for more
information.

## Prompt

If TTY mode is available, then a prompt can be enabled. Like TTY mode itself, it
can be turned on or off with an environment variable: **BC_PROMPT** (see the
**ENVIRONMENT VARIABLES** section).

If the environment variable **BC_PROMPT** exists and is a non-zero integer, then
the prompt is turned on when **stdin**, **stdout**, and **stderr** are connected
to a TTY and the **-P** and **-\-no-prompt** options were not used. The read
prompt will be turned on under the same conditions, except that the **-R** and
**-\-no-read-prompt** options must also not be used.

However, if **BC_PROMPT** does not exist, the prompt can be enabled or disabled
with the **BC_TTY_MODE** environment variable, the **-P** and **-\-no-prompt**
options, and the **-R** and **-\-no-read-prompt** options. See the **ENVIRONMENT
VARIABLES** and **OPTIONS** sections for more details.

# SIGNAL HANDLING

Sending a **SIGINT** will cause bc(1) to do one of two things.

If bc(1) is not in interactive mode (see the **INTERACTIVE MODE** section), or
the **BC_SIGINT_RESET** environment variable (see the **ENVIRONMENT VARIABLES**
section), or its default, is either not an integer or it is zero, bc(1) will
exit.

However, if bc(1) is in interactive mode, and the **BC_SIGINT_RESET** or its
default is an integer and non-zero, then bc(1) will stop executing the current
input and reset (see the **RESET** section) upon receiving a **SIGINT**.

Note that "current input" can mean one of two things. If bc(1) is processing
input from **stdin** in interactive mode, it will ask for more input. If bc(1)
is processing input from a file in interactive mode, it will stop processing the
file and start processing the next file, if one exists, or ask for input from
**stdin** if no other file exists.

This means that if a **SIGINT** is sent to bc(1) as it is executing a file, it
can seem as though bc(1) did not respond to the signal since it will immediately
start executing the next file. This is by design; most files that users execute
when interacting with bc(1) have function definitions, which are quick to parse.
If a file takes a long time to execute, there may be a bug in that file. The
rest of the files could still be executed without problem, allowing the user to
continue.

**SIGTERM** and **SIGQUIT** cause bc(1) to clean up and exit, and it uses the
default handler for all other signals. The one exception is **SIGHUP**; in that
case, and only when bc(1) is in TTY mode (see the **TTY MODE** section), a
**SIGHUP** will cause bc(1) to clean up and exit.

# COMMAND LINE HISTORY

bc(1) supports interactive command-line editing.

If bc(1) can be in TTY mode (see the **TTY MODE** section), history can be
enabled. This means that command-line history can only be enabled when
**stdin**, **stdout**, and **stderr** are all connected to a TTY.

Like TTY mode itself, it can be turned on or off with the environment variable
**BC_TTY_MODE** (see the **ENVIRONMENT VARIABLES** section).

If history is enabled, previous lines can be recalled and edited with the arrow
keys.

**Note**: tabs are converted to 8 spaces.

# SEE ALSO

dc(1)

# STANDARDS

bc(1) is compliant with the [IEEE Std 1003.1-2017 (“POSIX.1-2017”)][1]
specification. The flags **-efghiqsvVw**, all long options, and the extensions
noted above are extensions to that specification.

Note that the specification explicitly says that bc(1) only accepts numbers that
use a period (**.**) as a radix point, regardless of the value of
**LC_NUMERIC**.

# BUGS

None are known. Report bugs at https://git.yzena.com/gavin/bc.

# AUTHORS

Gavin D. Howard <gavin@yzena.com> and contributors.

[1]: https://pubs.opengroup.org/onlinepubs/9699919799/utilities/bc.html
[2]: https://www.gnu.org/software/bc/
[3]: https://en.wikipedia.org/wiki/Rounding#Round_half_away_from_zero
[4]: https://en.wikipedia.org/wiki/Unit_in_the_last_place
[5]: https://people.eecs.berkeley.edu/~wkahan/LOG10HAF.TXT
[6]: https://en.wikipedia.org/wiki/Rounding#Rounding_away_from_zero