3 SPDX-License-Identifier: BSD-2-Clause
5 Copyright (c) 2018-2020 Gavin D. Howard and contributors.
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8 modification, are permitted provided that the following conditions are met:
10 * Redistributions of source code must retain the above copyright notice, this
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33 bc - arbitrary-precision arithmetic language and calculator
37 **bc** [**-ghilPqsvVw**] [**--global-stacks**] [**--help**] [**--interactive**] [**--mathlib**] [**--no-prompt**] [**--quiet**] [**--standard**] [**--warn**] [**--version**] [**-e** *expr*] [**--expression**=*expr*...] [**-f** *file*...] [**-file**=*file*...]
42 bc(1) is an interactive processor for a language first standardized in 1991 by
43 POSIX. (The current standard is [here][1].) The language provides unlimited
44 precision decimal arithmetic and is somewhat C-like, but there are differences.
45 Such differences will be noted in this document.
47 After parsing and handling options, this bc(1) reads any files given on the
48 command line and executes them before reading from **stdin**.
50 This bc(1) is a drop-in replacement for *any* bc(1), including (and
51 especially) the GNU bc(1). It also has many extensions and extra features beyond
52 other implementations.
56 The following are the options that bc(1) accepts.
58 **-g**, **--global-stacks**
60 : Turns the globals **ibase**, **obase**, **scale**, and **seed** into stacks.
62 This has the effect that a copy of the current value of all four are pushed
63 onto a stack for every function call, as well as popped when every function
64 returns. This means that functions can assign to any and all of those
65 globals without worrying that the change will affect other functions.
66 Thus, a hypothetical function named **output(x,b)** that simply printed
67 **x** in base **b** could be written like this:
69 define void output(x, b) {
76 define void output(x, b) {
84 This makes writing functions much easier.
86 (**Note**: the function **output(x,b)** exists in the extended math library.
87 See the **LIBRARY** section.)
89 However, since using this flag means that functions cannot set **ibase**,
90 **obase**, **scale**, or **seed** globally, functions that are made to do so
91 cannot work anymore. There are two possible use cases for that, and each has
94 First, if a function is called on startup to turn bc(1) into a number
95 converter, it is possible to replace that capability with various shell
98 alias d2o="bc -e ibase=A -e obase=8"
99 alias h2b="bc -e ibase=G -e obase=2"
101 Second, if the purpose of a function is to set **ibase**, **obase**,
102 **scale**, or **seed** globally for any other purpose, it could be split
103 into one to four functions (based on how many globals it sets) and each of
104 those functions could return the desired value for a global.
106 For functions that set **seed**, the value assigned to **seed** is not
107 propagated to parent functions. This means that the sequence of
108 pseudo-random numbers that they see will not be the same sequence of
109 pseudo-random numbers that any parent sees. This is only the case once
110 **seed** has been set.
112 If a function desires to not affect the sequence of pseudo-random numbers
113 of its parents, but wants to use the same **seed**, it can use the following
118 If the behavior of this option is desired for every run of bc(1), then users
119 could make sure to define **BC_ENV_ARGS** and include this option (see the
120 **ENVIRONMENT VARIABLES** section for more details).
122 If **-s**, **-w**, or any equivalents are used, this option is ignored.
124 This is a **non-portable extension**.
128 : Prints a usage message and quits.
130 **-i**, **--interactive**
132 : Forces interactive mode. (See the **INTERACTIVE MODE** section.)
134 This is a **non-portable extension**.
136 **-l**, **--mathlib**
138 : Sets **scale** (see the **SYNTAX** section) to **20** and loads the included
139 math library and the extended math library before running any code,
140 including any expressions or files specified on the command line.
142 To learn what is in the libraries, see the **LIBRARY** section.
144 **-P**, **--no-prompt**
146 : Disables the prompt in TTY mode. (The prompt is only enabled in TTY mode.
147 See the **TTY MODE** section) This is mostly for those users that do not
148 want a prompt or are not used to having them in bc(1). Most of those users
149 would want to put this option in **BC_ENV_ARGS** (see the
150 **ENVIRONMENT VARIABLES** section).
152 This is a **non-portable extension**.
156 : Do not print copyright header. bc(1) will also suppress the header in
157 non-interactive mode.
159 This is mostly for compatibility with the [GNU bc(1)][2].
161 This is a **non-portable extension**.
163 **-s**, **--standard**
165 : Process exactly the language defined by the [standard][1] and error if any
168 This is a **non-portable extension**.
170 **-v**, **-V**, **--version**
172 : Print the version information (copyright header) and exit.
174 This is a **non-portable extension**.
178 : Like **-s** and **--standard**, except that warnings (and not errors) are
179 printed for non-standard extensions and execution continues normally.
181 This is a **non-portable extension**.
183 **-e** *expr*, **--expression**=*expr*
185 : Evaluates *expr*. If multiple expressions are given, they are evaluated in
186 order. If files are given as well (see below), the expressions and files are
187 evaluated in the order given. This means that if a file is given before an
188 expression, the file is read in and evaluated first.
190 In other bc(1) implementations, this option causes the program to execute
191 the expressions and then exit. This bc(1) does not, unless the
192 **BC_EXPR_EXIT** is defined (see the **ENVIRONMENT VARIABLES** section).
194 This is a **non-portable extension**.
196 **-f** *file*, **--file**=*file*
198 : Reads in *file* and evaluates it, line by line, as though it were read
199 through **stdin**. If expressions are also given (see above), the
200 expressions are evaluated in the order given.
202 In other bc(1) implementations, this option causes the program to execute
203 the files and then exit. This bc(1) does not, unless the
204 **BC_EXPR_EXIT** is defined (see the **ENVIRONMENT VARIABLES** section).
206 This is a **non-portable extension**.
208 All long options are **non-portable extensions**.
212 Any non-error output is written to **stdout**.
214 **Note**: Unlike other bc(1) implementations, this bc(1) will issue a fatal
215 error (see the **EXIT STATUS** section) if it cannot write to **stdout**, so if
216 **stdout** is closed, as in **bc <file> >&-**, it will quit with an error. This
217 is done so that bc(1) can report problems when **stdout** is redirected to a
220 If there are scripts that depend on the behavior of other bc(1) implementations,
221 it is recommended that those scripts be changed to redirect **stdout** to
226 Any error output is written to **stderr**.
228 **Note**: Unlike other bc(1) implementations, this bc(1) will issue a fatal
229 error (see the **EXIT STATUS** section) if it cannot write to **stderr**, so if
230 **stderr** is closed, as in **bc <file> 2>&-**, it will quit with an error. This
231 is done so that bc(1) can exit with an error code when **stderr** is redirected
234 If there are scripts that depend on the behavior of other bc(1) implementations,
235 it is recommended that those scripts be changed to redirect **stderr** to
240 The syntax for bc(1) programs is mostly C-like, with some differences. This
241 bc(1) follows the [POSIX standard][1], which is a much more thorough resource
242 for the language this bc(1) accepts. This section is meant to be a summary and a
243 listing of all the extensions to the standard.
245 In the sections below, **E** means expression, **S** means statement, and **I**
248 Identifiers (**I**) start with a lowercase letter and can be followed by any
249 number (up to **BC_NAME_MAX-1**) of lowercase letters (**a-z**), digits
250 (**0-9**), and underscores (**\_**). The regex is **\[a-z\]\[a-z0-9\_\]\***.
251 Identifiers with more than one character (letter) are a
252 **non-portable extension**.
254 **ibase** is a global variable determining how to interpret constant numbers. It
255 is the "input" base, or the number base used for interpreting input numbers.
256 **ibase** is initially **10**. If the **-s** (**--standard**) and **-w**
257 (**--warn**) flags were not given on the command line, the max allowable value
258 for **ibase** is **36**. Otherwise, it is **16**. The min allowable value for
259 **ibase** is **2**. The max allowable value for **ibase** can be queried in
260 bc(1) programs with the **maxibase()** built-in function.
262 **obase** is a global variable determining how to output results. It is the
263 "output" base, or the number base used for outputting numbers. **obase** is
264 initially **10**. The max allowable value for **obase** is **BC_BASE_MAX** and
265 can be queried in bc(1) programs with the **maxobase()** built-in function. The
266 min allowable value for **obase** is **0**. If **obase** is **0**, values are
267 output in scientific notation, and if **obase** is **1**, values are output in
268 engineering notation. Otherwise, values are output in the specified base.
270 Outputting in scientific and engineering notations are **non-portable
273 The *scale* of an expression is the number of digits in the result of the
274 expression right of the decimal point, and **scale** is a global variable that
275 sets the precision of any operations, with exceptions. **scale** is initially
276 **0**. **scale** cannot be negative. The max allowable value for **scale** is
277 **BC_SCALE_MAX** and can be queried in bc(1) programs with the **maxscale()**
280 bc(1) has both *global* variables and *local* variables. All *local*
281 variables are local to the function; they are parameters or are introduced in
282 the **auto** list of a function (see the **FUNCTIONS** section). If a variable
283 is accessed which is not a parameter or in the **auto** list, it is assumed to
284 be *global*. If a parent function has a *local* variable version of a variable
285 that a child function considers *global*, the value of that *global* variable in
286 the child function is the value of the variable in the parent function, not the
287 value of the actual *global* variable.
289 All of the above applies to arrays as well.
291 The value of a statement that is an expression (i.e., any of the named
292 expressions or operands) is printed unless the lowest precedence operator is an
293 assignment operator *and* the expression is notsurrounded by parentheses.
295 The value that is printed is also assigned to the special variable **last**. A
296 single dot (**.**) may also be used as a synonym for **last**. These are
297 **non-portable extensions**.
299 Either semicolons or newlines may separate statements.
303 There are two kinds of comments:
305 1. Block comments are enclosed in **/\*** and **\*/**.
306 2. Line comments go from **#** until, and not including, the next newline. This
307 is a **non-portable extension**.
311 The following are named expressions in bc(1):
314 2. Array Elements: **I[E]**
319 7. **last** or a single dot (**.**)
321 Numbers 6 and 7 are **non-portable extensions**.
323 The meaning of **seed** is dependent on the current pseudo-random number
324 generator but is guaranteed to not change except for new major versions.
326 The *scale* and sign of the value may be significant.
328 If a previously used **seed** value is assigned to **seed** and used again, the
329 pseudo-random number generator is guaranteed to produce the same sequence of
330 pseudo-random numbers as it did when the **seed** value was previously used.
332 The exact value assigned to **seed** is not guaranteed to be returned if
333 **seed** is queried again immediately. However, if **seed** *does* return a
334 different value, both values, when assigned to **seed**, are guaranteed to
335 produce the same sequence of pseudo-random numbers. This means that certain
336 values assigned to **seed** will *not* produce unique sequences of pseudo-random
337 numbers. The value of **seed** will change after any use of the **rand()** and
338 **irand(E)** operands (see the *Operands* subsection below), except if the
339 parameter passed to **irand(E)** is **0**, **1**, or negative.
341 There is no limit to the length (number of significant decimal digits) or
342 *scale* of the value that can be assigned to **seed**.
344 Variables and arrays do not interfere; users can have arrays named the same as
345 variables. This also applies to functions (see the **FUNCTIONS** section), so a
346 user can have a variable, array, and function that all have the same name, and
347 they will not shadow each other, whether inside of functions or not.
349 Named expressions are required as the operand of **increment**/**decrement**
350 operators and as the left side of **assignment** operators (see the *Operators*
355 The following are valid operands in bc(1):
357 1. Numbers (see the *Numbers* subsection below).
358 2. Array indices (**I[E]**).
359 3. **(E)**: The value of **E** (used to change precedence).
360 4. **sqrt(E)**: The square root of **E**. **E** must be non-negative.
361 5. **length(E)**: The number of significant decimal digits in **E**.
362 6. **length(I[])**: The number of elements in the array **I**. This is a
363 **non-portable extension**.
364 7. **scale(E)**: The *scale* of **E**.
365 8. **abs(E)**: The absolute value of **E**. This is a **non-portable
367 9. **I()**, **I(E)**, **I(E, E)**, and so on, where **I** is an identifier for
368 a non-**void** function (see the *Void Functions* subsection of the
369 **FUNCTIONS** section). The **E** argument(s) may also be arrays of the form
370 **I[]**, which will automatically be turned into array references (see the
371 *Array References* subsection of the **FUNCTIONS** section) if the
372 corresponding parameter in the function definition is an array reference.
373 10. **read()**: Reads a line from **stdin** and uses that as an expression. The
374 result of that expression is the result of the **read()** operand. This is a
375 **non-portable extension**.
376 11. **maxibase()**: The max allowable **ibase**. This is a **non-portable
378 12. **maxobase()**: The max allowable **obase**. This is a **non-portable
380 13. **maxscale()**: The max allowable **scale**. This is a **non-portable
382 14. **rand()**: A pseudo-random integer between **0** (inclusive) and
383 **BC_RAND_MAX** (inclusive). Using this operand will change the value of
384 **seed**. This is a **non-portable extension**.
385 15. **irand(E)**: A pseudo-random integer between **0** (inclusive) and the
386 value of **E** (exclusive). If **E** is negative or is a non-integer
387 (**E**'s *scale* is not **0**), an error is raised, and bc(1) resets (see
388 the **RESET** section) while **seed** remains unchanged. If **E** is larger
389 than **BC_RAND_MAX**, the higher bound is honored by generating several
390 pseudo-random integers, multiplying them by appropriate powers of
391 **BC_RAND_MAX+1**, and adding them together. Thus, the size of integer that
392 can be generated with this operand is unbounded. Using this operand will
393 change the value of **seed**, unless the value of **E** is **0** or **1**.
394 In that case, **0** is returned, and **seed** is *not* changed. This is a
395 **non-portable extension**.
396 16. **maxrand()**: The max integer returned by **rand()**. This is a
397 **non-portable extension**.
399 The integers generated by **rand()** and **irand(E)** are guaranteed to be as
400 unbiased as possible, subject to the limitations of the pseudo-random number
403 **Note**: The values returned by the pseudo-random number generator with
404 **rand()** and **irand(E)** are guaranteed to *NOT* be cryptographically secure.
405 This is a consequence of using a seeded pseudo-random number generator. However,
406 they *are* guaranteed to be reproducible with identical **seed** values.
410 Numbers are strings made up of digits, uppercase letters, and at most **1**
411 period for a radix. Numbers can have up to **BC_NUM_MAX** digits. Uppercase
412 letters are equal to **9** + their position in the alphabet (i.e., **A** equals
413 **10**, or **9+1**). If a digit or letter makes no sense with the current value
414 of **ibase**, they are set to the value of the highest valid digit in **ibase**.
416 Single-character numbers (i.e., **A** alone) take the value that they would have
417 if they were valid digits, regardless of the value of **ibase**. This means that
418 **A** alone always equals decimal **10** and **Z** alone always equals decimal
421 In addition, bc(1) accepts numbers in scientific notation. These have the form
422 **\<number\>e\<integer\>**. The power (the portion after the **e**) must be an
423 integer. An example is **1.89237e9**, which is equal to **1892370000**. Negative
424 exponents are also allowed, so **4.2890e-3** is equal to **0.0042890**.
426 Using scientific notation is an error or warning if the **-s** or **-w**,
427 respectively, command-line options (or equivalents) are given.
429 **WARNING**: Both the number and the exponent in scientific notation are
430 interpreted according to the current **ibase**, but the number is still
431 multiplied by **10\^exponent** regardless of the current **ibase**. For example,
432 if **ibase** is **16** and bc(1) is given the number string **FFeA**, the
433 resulting decimal number will be **2550000000000**, and if bc(1) is given the
434 number string **10e-4**, the resulting decimal number will be **0.0016**.
436 Accepting input as scientific notation is a **non-portable extension**.
440 The following arithmetic and logical operators can be used. They are listed in
441 order of decreasing precedence. Operators in the same group have the same
446 : Type: Prefix and Postfix
450 Description: **increment**, **decrement**
458 Description: **negation**, **boolean not**
466 Description: **truncation**
474 Description: **set precision**
482 Description: **power**
490 Description: **multiply**, **divide**, **modulus**
498 Description: **add**, **subtract**
506 Description: **shift left**, **shift right**
508 **=** **\<\<=** **\>\>=** **+=** **-=** **\*=** **/=** **%=** **\^=** **\@=**
514 Description: **assignment**
516 **==** **\<=** **\>=** **!=** **\<** **\>**
522 Description: **relational**
530 Description: **boolean and**
538 Description: **boolean or**
540 The operators will be described in more detail below.
544 : The prefix and postfix **increment** and **decrement** operators behave
545 exactly like they would in C. They require a named expression (see the
546 *Named Expressions* subsection) as an operand.
548 The prefix versions of these operators are more efficient; use them where
553 : The **negation** operator returns **0** if a user attempts to negate any
554 expression with the value **0**. Otherwise, a copy of the expression with
555 its sign flipped is returned.
559 : The **boolean not** operator returns **1** if the expression is **0**, or
562 This is a **non-portable extension**.
566 : The **truncation** operator returns a copy of the given expression with all
567 of its *scale* removed.
569 This is a **non-portable extension**.
573 : The **set precision** operator takes two expressions and returns a copy of
574 the first with its *scale* equal to the value of the second expression. That
575 could either mean that the number is returned without change (if the
576 *scale* of the first expression matches the value of the second
577 expression), extended (if it is less), or truncated (if it is more).
579 The second expression must be an integer (no *scale*) and non-negative.
581 This is a **non-portable extension**.
585 : The **power** operator (not the **exclusive or** operator, as it would be in
586 C) takes two expressions and raises the first to the power of the value of
589 The second expression must be an integer (no *scale*), and if it is
590 negative, the first value must be non-zero.
594 : The **multiply** operator takes two expressions, multiplies them, and
595 returns the product. If **a** is the *scale* of the first expression and
596 **b** is the *scale* of the second expression, the *scale* of the result is
597 equal to **min(a+b,max(scale,a,b))** where **min()** and **max()** return
602 : The **divide** operator takes two expressions, divides them, and returns the
603 quotient. The *scale* of the result shall be the value of **scale**.
605 The second expression must be non-zero.
609 : The **modulus** operator takes two expressions, **a** and **b**, and
610 evaluates them by 1) Computing **a/b** to current **scale** and 2) Using the
611 result of step 1 to calculate **a-(a/b)\*b** to *scale*
612 **max(scale+scale(b),scale(a))**.
614 The second expression must be non-zero.
618 : The **add** operator takes two expressions, **a** and **b**, and returns the
619 sum, with a *scale* equal to the max of the *scale*s of **a** and **b**.
623 : The **subtract** operator takes two expressions, **a** and **b**, and
624 returns the difference, with a *scale* equal to the max of the *scale*s of
629 : The **left shift** operator takes two expressions, **a** and **b**, and
630 returns a copy of the value of **a** with its decimal point moved **b**
633 The second expression must be an integer (no *scale*) and non-negative.
635 This is a **non-portable extension**.
639 : The **right shift** operator takes two expressions, **a** and **b**, and
640 returns a copy of the value of **a** with its decimal point moved **b**
643 The second expression must be an integer (no *scale*) and non-negative.
645 This is a **non-portable extension**.
647 **=** **\<\<=** **\>\>=** **+=** **-=** **\*=** **/=** **%=** **\^=** **\@=**
649 : The **assignment** operators take two expressions, **a** and **b** where
650 **a** is a named expression (see the *Named Expressions* subsection).
652 For **=**, **b** is copied and the result is assigned to **a**. For all
653 others, **a** and **b** are applied as operands to the corresponding
654 arithmetic operator and the result is assigned to **a**.
656 The **assignment** operators that correspond to operators that are
657 extensions are themselves **non-portable extensions**.
659 **==** **\<=** **\>=** **!=** **\<** **\>**
661 : The **relational** operators compare two expressions, **a** and **b**, and
662 if the relation holds, according to C language semantics, the result is
663 **1**. Otherwise, it is **0**.
665 Note that unlike in C, these operators have a lower precedence than the
666 **assignment** operators, which means that **a=b\>c** is interpreted as
669 Also, unlike the [standard][1] requires, these operators can appear anywhere
670 any other expressions can be used. This allowance is a
671 **non-portable extension**.
675 : The **boolean and** operator takes two expressions and returns **1** if both
676 expressions are non-zero, **0** otherwise.
678 This is *not* a short-circuit operator.
680 This is a **non-portable extension**.
684 : The **boolean or** operator takes two expressions and returns **1** if one
685 of the expressions is non-zero, **0** otherwise.
687 This is *not* a short-circuit operator.
689 This is a **non-portable extension**.
693 The following items are statements:
696 2. **{** **S** **;** ... **;** **S** **}**
697 3. **if** **(** **E** **)** **S**
698 4. **if** **(** **E** **)** **S** **else** **S**
699 5. **while** **(** **E** **)** **S**
700 6. **for** **(** **E** **;** **E** **;** **E** **)** **S**
701 7. An empty statement
707 13. A string of characters, enclosed in double quotes
708 14. **print** **E** **,** ... **,** **E**
709 15. **I()**, **I(E)**, **I(E, E)**, and so on, where **I** is an identifier for
710 a **void** function (see the *Void Functions* subsection of the
711 **FUNCTIONS** section). The **E** argument(s) may also be arrays of the form
712 **I[]**, which will automatically be turned into array references (see the
713 *Array References* subsection of the **FUNCTIONS** section) if the
714 corresponding parameter in the function definition is an array reference.
716 Numbers 4, 9, 11, 12, 14, and 15 are **non-portable extensions**.
718 Also, as a **non-portable extension**, any or all of the expressions in the
719 header of a for loop may be omitted. If the condition (second expression) is
720 omitted, it is assumed to be a constant **1**.
722 The **break** statement causes a loop to stop iterating and resume execution
723 immediately following a loop. This is only allowed in loops.
725 The **continue** statement causes a loop iteration to stop early and returns to
726 the start of the loop, including testing the loop condition. This is only
729 The **if** **else** statement does the same thing as in C.
731 The **quit** statement causes bc(1) to quit, even if it is on a branch that will
732 not be executed (it is a compile-time command).
734 The **halt** statement causes bc(1) to quit, if it is executed. (Unlike **quit**
735 if it is on a branch of an **if** statement that is not executed, bc(1) does not
738 The **limits** statement prints the limits that this bc(1) is subject to. This
739 is like the **quit** statement in that it is a compile-time command.
741 An expression by itself is evaluated and printed, followed by a newline.
743 Both scientific notation and engineering notation are available for printing the
744 results of expressions. Scientific notation is activated by assigning **0** to
745 **obase**, and engineering notation is activated by assigning **1** to
746 **obase**. To deactivate them, just assign a different value to **obase**.
748 Scientific notation and engineering notation are disabled if bc(1) is run with
749 either the **-s** or **-w** command-line options (or equivalents).
751 Printing numbers in scientific notation and/or engineering notation is a
752 **non-portable extension**.
756 The "expressions" in a **print** statement may also be strings. If they are, there
757 are backslash escape sequences that are interpreted specially. What those
758 sequences are, and what they cause to be printed, are shown below:
772 Any other character following a backslash causes the backslash and character to
775 Any non-string expression in a print statement shall be assigned to **last**,
776 like any other expression that is printed.
778 ## Order of Evaluation
780 All expressions in a statment are evaluated left to right, except as necessary
781 to maintain order of operations. This means, for example, assuming that **i** is
782 equal to **0**, in the expression
786 the first (or 0th) element of **a** is set to **1**, and **i** is equal to **2**
787 at the end of the expression.
789 This includes function arguments. Thus, assuming **i** is equal to **0**, this
790 means that in the expression
794 the first argument passed to **x()** is **0**, and the second argument is **1**,
795 while **i** is equal to **2** before the function starts executing.
799 Function definitions are as follows:
809 Any **I** in the parameter list or **auto** list may be replaced with **I[]** to
810 make a parameter or **auto** var an array, and any **I** in the parameter list
811 may be replaced with **\*I[]** to make a parameter an array reference. Callers
812 of functions that take array references should not put an asterisk in the call;
813 they must be called with just **I[]** like normal array parameters and will be
814 automatically converted into references.
816 As a **non-portable extension**, the opening brace of a **define** statement may
817 appear on the next line.
819 As a **non-portable extension**, the return statement may also be in one of the
823 2. **return** **(** **)**
826 The first two, or not specifying a **return** statement, is equivalent to
827 **return (0)**, unless the function is a **void** function (see the *Void
828 Functions* subsection below).
832 Functions can also be **void** functions, defined as follows:
835 define void I(I,...,I){
842 They can only be used as standalone expressions, where such an expression would
843 be printed alone, except in a print statement.
845 Void functions can only use the first two **return** statements listed above.
846 They can also omit the return statement entirely.
848 The word "void" is not treated as a keyword; it is still possible to have
849 variables, arrays, and functions named **void**. The word "void" is only
850 treated specially right after the **define** keyword.
852 This is a **non-portable extension**.
856 For any array in the parameter list, if the array is declared in the form
862 it is a **reference**. Any changes to the array in the function are reflected,
863 when the function returns, to the array that was passed in.
865 Other than this, all function arguments are passed by value.
867 This is a **non-portable extension**.
871 All of the functions below, including the functions in the extended math
872 library (see the *Extended Library* subsection below), are available when the
873 **-l** or **--mathlib** command-line flags are given, except that the extended
874 math library is not available when the **-s** option, the **-w** option, or
875 equivalents are given.
879 The [standard][1] defines the following functions for the math library:
883 : Returns the sine of **x**, which is assumed to be in radians.
885 This is a transcendental function (see the *Transcendental Functions*
890 : Returns the cosine of **x**, which is assumed to be in radians.
892 This is a transcendental function (see the *Transcendental Functions*
897 : Returns the arctangent of **x**, in radians.
899 This is a transcendental function (see the *Transcendental Functions*
904 : Returns the natural logarithm of **x**.
906 This is a transcendental function (see the *Transcendental Functions*
911 : Returns the mathematical constant **e** raised to the power of **x**.
913 This is a transcendental function (see the *Transcendental Functions*
918 : Returns the bessel integer order **n** (truncated) of **x**.
920 This is a transcendental function (see the *Transcendental Functions*
925 The extended library is *not* loaded when the **-s**/**--standard** or
926 **-w**/**--warn** options are given since they are not part of the library
927 defined by the [standard][1].
929 The extended library is a **non-portable extension**.
933 : Calculates **x** to the power of **y**, even if **y** is not an integer, and
934 returns the result to the current **scale**.
936 This is a transcendental function (see the *Transcendental Functions*
941 : Returns **x** rounded to **p** decimal places according to the rounding mode
942 [round half away from **0**][3].
946 : Returns **x** rounded to **p** decimal places according to the rounding mode
947 [round away from **0**][6].
951 : Returns the factorial of the truncated absolute value of **x**.
955 : Returns the permutation of the truncated absolute value of **n** of the
956 truncated absolute value of **k**, if **k \<= n**. If not, it returns **0**.
960 : Returns the combination of the truncated absolute value of **n** of the
961 truncated absolute value of **k**, if **k \<= n**. If not, it returns **0**.
965 : Returns the logarithm base **2** of **x**.
967 This is a transcendental function (see the *Transcendental Functions*
972 : Returns the logarithm base **10** of **x**.
974 This is a transcendental function (see the *Transcendental Functions*
979 : Returns the logarithm base **b** of **x**.
981 This is a transcendental function (see the *Transcendental Functions*
986 : Returns the cube root of **x**.
990 : Calculates the truncated value of **n**, **r**, and returns the **r**th root
991 of **x** to the current **scale**.
993 If **r** is **0** or negative, this raises an error and causes bc(1) to
994 reset (see the **RESET** section). It also raises an error and causes bc(1)
995 to reset if **r** is even and **x** is negative.
999 : Returns **pi** to **p** decimal places.
1001 This is a transcendental function (see the *Transcendental Functions*
1006 : Returns the tangent of **x**, which is assumed to be in radians.
1008 This is a transcendental function (see the *Transcendental Functions*
1013 : Returns the arctangent of **y/x**, in radians. If both **y** and **x** are
1014 equal to **0**, it raises an error and causes bc(1) to reset (see the
1015 **RESET** section). Otherwise, if **x** is greater than **0**, it returns
1016 **a(y/x)**. If **x** is less than **0**, and **y** is greater than or equal
1017 to **0**, it returns **a(y/x)+pi**. If **x** is less than **0**, and **y**
1018 is less than **0**, it returns **a(y/x)-pi**. If **x** is equal to **0**,
1019 and **y** is greater than **0**, it returns **pi/2**. If **x** is equal to
1020 **0**, and **y** is less than **0**, it returns **-pi/2**.
1022 This function is the same as the **atan2()** function in many programming
1025 This is a transcendental function (see the *Transcendental Functions*
1030 : Returns the sine of **x**, which is assumed to be in radians.
1032 This is an alias of **s(x)**.
1034 This is a transcendental function (see the *Transcendental Functions*
1039 : Returns the cosine of **x**, which is assumed to be in radians.
1041 This is an alias of **c(x)**.
1043 This is a transcendental function (see the *Transcendental Functions*
1048 : Returns the tangent of **x**, which is assumed to be in radians.
1050 If **x** is equal to **1** or **-1**, this raises an error and causes bc(1)
1051 to reset (see the **RESET** section).
1053 This is an alias of **t(x)**.
1055 This is a transcendental function (see the *Transcendental Functions*
1060 : Returns the arctangent of **x**, in radians.
1062 This is an alias of **a(x)**.
1064 This is a transcendental function (see the *Transcendental Functions*
1069 : Returns the arctangent of **y/x**, in radians. If both **y** and **x** are
1070 equal to **0**, it raises an error and causes bc(1) to reset (see the
1071 **RESET** section). Otherwise, if **x** is greater than **0**, it returns
1072 **a(y/x)**. If **x** is less than **0**, and **y** is greater than or equal
1073 to **0**, it returns **a(y/x)+pi**. If **x** is less than **0**, and **y**
1074 is less than **0**, it returns **a(y/x)-pi**. If **x** is equal to **0**,
1075 and **y** is greater than **0**, it returns **pi/2**. If **x** is equal to
1076 **0**, and **y** is less than **0**, it returns **-pi/2**.
1078 This function is the same as the **atan2()** function in many programming
1081 This is an alias of **a2(y, x)**.
1083 This is a transcendental function (see the *Transcendental Functions*
1088 : Converts **x** from radians to degrees and returns the result.
1090 This is a transcendental function (see the *Transcendental Functions*
1095 : Converts **x** from degrees to radians and returns the result.
1097 This is a transcendental function (see the *Transcendental Functions*
1102 : Generates a pseudo-random number between **0** (inclusive) and **1**
1103 (exclusive) with the number of decimal digits after the decimal point equal
1104 to the truncated absolute value of **p**. If **p** is not **0**, then
1105 calling this function will change the value of **seed**. If **p** is **0**,
1106 then **0** is returned, and **seed** is *not* changed.
1110 : Generates a pseudo-random number that is between **0** (inclusive) and the
1111 truncated absolute value of **i** (exclusive) with the number of decimal
1112 digits after the decimal point equal to the truncated absolute value of
1113 **p**. If the absolute value of **i** is greater than or equal to **2**, and
1114 **p** is not **0**, then calling this function will change the value of
1115 **seed**; otherwise, **0** is returned and **seed** is not changed.
1119 : Returns **x** with its sign flipped with probability **0.5**. In other
1120 words, it randomizes the sign of **x**.
1124 : Returns a random boolean value (either **0** or **1**).
1128 : Returns the numbers of unsigned integer bytes required to hold the truncated
1129 absolute value of **x**.
1133 : Returns the numbers of signed, two's-complement integer bytes required to
1134 hold the truncated value of **x**.
1138 : Outputs the hexadecimal (base **16**) representation of **x**.
1140 This is a **void** function (see the *Void Functions* subsection of the
1141 **FUNCTIONS** section).
1145 : Outputs the binary (base **2**) representation of **x**.
1147 This is a **void** function (see the *Void Functions* subsection of the
1148 **FUNCTIONS** section).
1152 : Outputs the base **b** representation of **x**.
1154 This is a **void** function (see the *Void Functions* subsection of the
1155 **FUNCTIONS** section).
1159 : Outputs the representation, in binary and hexadecimal, of **x** as an
1160 unsigned integer in as few power of two bytes as possible. Both outputs are
1161 split into bytes separated by spaces.
1163 If **x** is not an integer or is negative, an error message is printed
1164 instead, but bc(1) is not reset (see the **RESET** section).
1166 This is a **void** function (see the *Void Functions* subsection of the
1167 **FUNCTIONS** section).
1171 : Outputs the representation, in binary and hexadecimal, of **x** as a signed,
1172 two's-complement integer in as few power of two bytes as possible. Both
1173 outputs are split into bytes separated by spaces.
1175 If **x** is not an integer, an error message is printed instead, but bc(1)
1176 is not reset (see the **RESET** section).
1178 This is a **void** function (see the *Void Functions* subsection of the
1179 **FUNCTIONS** section).
1183 : Outputs the representation, in binary and hexadecimal, of **x** as an
1184 unsigned integer in **n** bytes. Both outputs are split into bytes separated
1187 If **x** is not an integer, is negative, or cannot fit into **n** bytes, an
1188 error message is printed instead, but bc(1) is not reset (see the **RESET**
1191 This is a **void** function (see the *Void Functions* subsection of the
1192 **FUNCTIONS** section).
1196 : Outputs the representation, in binary and hexadecimal, of **x** as a signed,
1197 two's-complement integer in **n** bytes. Both outputs are split into bytes
1198 separated by spaces.
1200 If **x** is not an integer or cannot fit into **n** bytes, an error message
1201 is printed instead, but bc(1) is not reset (see the **RESET** section).
1203 This is a **void** function (see the *Void Functions* subsection of the
1204 **FUNCTIONS** section).
1208 : Outputs the representation, in binary and hexadecimal, of **x** as an
1209 unsigned integer in **1** byte. Both outputs are split into bytes separated
1212 If **x** is not an integer, is negative, or cannot fit into **1** byte, an
1213 error message is printed instead, but bc(1) is not reset (see the **RESET**
1216 This is a **void** function (see the *Void Functions* subsection of the
1217 **FUNCTIONS** section).
1221 : Outputs the representation, in binary and hexadecimal, of **x** as a signed,
1222 two's-complement integer in **1** byte. Both outputs are split into bytes
1223 separated by spaces.
1225 If **x** is not an integer or cannot fit into **1** byte, an error message
1226 is printed instead, but bc(1) is not reset (see the **RESET** section).
1228 This is a **void** function (see the *Void Functions* subsection of the
1229 **FUNCTIONS** section).
1233 : Outputs the representation, in binary and hexadecimal, of **x** as an
1234 unsigned integer in **2** bytes. Both outputs are split into bytes separated
1237 If **x** is not an integer, is negative, or cannot fit into **2** bytes, an
1238 error message is printed instead, but bc(1) is not reset (see the **RESET**
1241 This is a **void** function (see the *Void Functions* subsection of the
1242 **FUNCTIONS** section).
1246 : Outputs the representation, in binary and hexadecimal, of **x** as a signed,
1247 two's-complement integer in **2** bytes. Both outputs are split into bytes
1248 separated by spaces.
1250 If **x** is not an integer or cannot fit into **2** bytes, an error message
1251 is printed instead, but bc(1) is not reset (see the **RESET** section).
1253 This is a **void** function (see the *Void Functions* subsection of the
1254 **FUNCTIONS** section).
1258 : Outputs the representation, in binary and hexadecimal, of **x** as an
1259 unsigned integer in **4** bytes. Both outputs are split into bytes separated
1262 If **x** is not an integer, is negative, or cannot fit into **4** bytes, an
1263 error message is printed instead, but bc(1) is not reset (see the **RESET**
1266 This is a **void** function (see the *Void Functions* subsection of the
1267 **FUNCTIONS** section).
1271 : Outputs the representation, in binary and hexadecimal, of **x** as a signed,
1272 two's-complement integer in **4** bytes. Both outputs are split into bytes
1273 separated by spaces.
1275 If **x** is not an integer or cannot fit into **4** bytes, an error message
1276 is printed instead, but bc(1) is not reset (see the **RESET** section).
1278 This is a **void** function (see the *Void Functions* subsection of the
1279 **FUNCTIONS** section).
1283 : Outputs the representation, in binary and hexadecimal, of **x** as an
1284 unsigned integer in **8** bytes. Both outputs are split into bytes separated
1287 If **x** is not an integer, is negative, or cannot fit into **8** bytes, an
1288 error message is printed instead, but bc(1) is not reset (see the **RESET**
1291 This is a **void** function (see the *Void Functions* subsection of the
1292 **FUNCTIONS** section).
1296 : Outputs the representation, in binary and hexadecimal, of **x** as a signed,
1297 two's-complement integer in **8** bytes. Both outputs are split into bytes
1298 separated by spaces.
1300 If **x** is not an integer or cannot fit into **8** bytes, an error message
1301 is printed instead, but bc(1) is not reset (see the **RESET** section).
1303 This is a **void** function (see the *Void Functions* subsection of the
1304 **FUNCTIONS** section).
1308 : Outputs the representation of the truncated absolute value of **x** as an
1309 unsigned integer in hexadecimal using **n** bytes. Not all of the value will
1310 be output if **n** is too small.
1312 This is a **void** function (see the *Void Functions* subsection of the
1313 **FUNCTIONS** section).
1315 **binary_uint(x, n)**
1317 : Outputs the representation of the truncated absolute value of **x** as an
1318 unsigned integer in binary using **n** bytes. Not all of the value will be
1319 output if **n** is too small.
1321 This is a **void** function (see the *Void Functions* subsection of the
1322 **FUNCTIONS** section).
1324 **output_uint(x, n)**
1326 : Outputs the representation of the truncated absolute value of **x** as an
1327 unsigned integer in the current **obase** (see the **SYNTAX** section) using
1328 **n** bytes. Not all of the value will be output if **n** is too small.
1330 This is a **void** function (see the *Void Functions* subsection of the
1331 **FUNCTIONS** section).
1333 **output_byte(x, i)**
1335 : Outputs byte **i** of the truncated absolute value of **x**, where **0** is
1336 the least significant byte and **number_of_bytes - 1** is the most
1339 This is a **void** function (see the *Void Functions* subsection of the
1340 **FUNCTIONS** section).
1342 ## Transcendental Functions
1344 All transcendental functions can return slightly inaccurate results (up to 1
1345 [ULP][4]). This is unavoidable, and [this article][5] explains why it is
1346 impossible and unnecessary to calculate exact results for the transcendental
1349 Because of the possible inaccuracy, I recommend that users call those functions
1350 with the precision (**scale**) set to at least 1 higher than is necessary. If
1351 exact results are *absolutely* required, users can double the precision
1352 (**scale**) and then truncate.
1354 The transcendental functions in the standard math library are:
1363 The transcendental functions in the extended math library are:
1381 When bc(1) encounters an error or a signal that it has a non-default handler
1382 for, it resets. This means that several things happen.
1384 First, any functions that are executing are stopped and popped off the stack.
1385 The behavior is not unlike that of exceptions in programming languages. Then
1386 the execution point is set so that any code waiting to execute (after all
1387 functions returned) is skipped.
1389 Thus, when bc(1) resets, it skips any remaining code waiting to be executed.
1390 Then, if it is interactive mode, and the error was not a fatal error (see the
1391 **EXIT STATUS** section), it asks for more input; otherwise, it exits with the
1392 appropriate return code.
1394 Note that this reset behavior is different from the GNU bc(1), which attempts to
1395 start executing the statement right after the one that caused an error.
1399 Most bc(1) implementations use **char** types to calculate the value of **1**
1400 decimal digit at a time, but that can be slow. This bc(1) does something
1403 It uses large integers to calculate more than **1** decimal digit at a time. If
1404 built in a environment where **BC_LONG_BIT** (see the **LIMITS** section) is
1405 **64**, then each integer has **9** decimal digits. If built in an environment
1406 where **BC_LONG_BIT** is **32** then each integer has **4** decimal digits. This
1407 value (the number of decimal digits per large integer) is called
1410 The actual values of **BC_LONG_BIT** and **BC_BASE_DIGS** can be queried with
1411 the **limits** statement.
1413 In addition, this bc(1) uses an even larger integer for overflow checking. This
1414 integer type depends on the value of **BC_LONG_BIT**, but is always at least
1415 twice as large as the integer type used to store digits.
1419 The following are the limits on bc(1):
1423 : The number of bits in the **long** type in the environment where bc(1) was
1424 built. This determines how many decimal digits can be stored in a single
1425 large integer (see the **PERFORMANCE** section).
1429 : The number of decimal digits per large integer (see the **PERFORMANCE**
1430 section). Depends on **BC_LONG_BIT**.
1434 : The max decimal number that each large integer can store (see
1435 **BC_BASE_DIGS**) plus **1**. Depends on **BC_BASE_DIGS**.
1439 : The max number that the overflow type (see the **PERFORMANCE** section) can
1440 hold. Depends on **BC_LONG_BIT**.
1444 : The maximum output base. Set at **BC_BASE_POW**.
1448 : The maximum size of arrays. Set at **SIZE_MAX-1**.
1452 : The maximum **scale**. Set at **BC_OVERFLOW_MAX-1**.
1456 : The maximum length of strings. Set at **BC_OVERFLOW_MAX-1**.
1460 : The maximum length of identifiers. Set at **BC_OVERFLOW_MAX-1**.
1464 : The maximum length of a number (in decimal digits), which includes digits
1465 after the decimal point. Set at **BC_OVERFLOW_MAX-1**.
1469 : The maximum integer (inclusive) returned by the **rand()** operand. Set at
1470 **2\^BC_LONG_BIT-1**.
1474 : The maximum allowable exponent (positive or negative). Set at
1475 **BC_OVERFLOW_MAX**.
1479 : The maximum number of vars/arrays. Set at **SIZE_MAX-1**.
1481 The actual values can be queried with the **limits** statement.
1483 These limits are meant to be effectively non-existent; the limits are so large
1484 (at least on 64-bit machines) that there should not be any point at which they
1485 become a problem. In fact, memory should be exhausted before these limits should
1488 # ENVIRONMENT VARIABLES
1490 bc(1) recognizes the following environment variables:
1494 : If this variable exists (no matter the contents), bc(1) behaves as if
1495 the **-s** option was given.
1499 : This is another way to give command-line arguments to bc(1). They should be
1500 in the same format as all other command-line arguments. These are always
1501 processed first, so any files given in **BC_ENV_ARGS** will be processed
1502 before arguments and files given on the command-line. This gives the user
1503 the ability to set up "standard" options and files to be used at every
1504 invocation. The most useful thing for such files to contain would be useful
1505 functions that the user might want every time bc(1) runs.
1507 The code that parses **BC_ENV_ARGS** will correctly handle quoted arguments,
1508 but it does not understand escape sequences. For example, the string
1509 **"/home/gavin/some bc file.bc"** will be correctly parsed, but the string
1510 **"/home/gavin/some \"bc\" file.bc"** will include the backslashes.
1512 The quote parsing will handle either kind of quotes, **'** or **"**. Thus,
1513 if you have a file with any number of single quotes in the name, you can use
1514 double quotes as the outside quotes, as in **"some 'bc' file.bc"**, and vice
1515 versa if you have a file with double quotes. However, handling a file with
1516 both kinds of quotes in **BC_ENV_ARGS** is not supported due to the
1517 complexity of the parsing, though such files are still supported on the
1518 command-line where the parsing is done by the shell.
1522 : If this environment variable exists and contains an integer that is greater
1523 than **1** and is less than **UINT16_MAX** (**2\^16-1**), bc(1) will output
1524 lines to that length, including the backslash (**\\**). The default line
1529 : If this variable exists (no matter the contents), bc(1) will exit
1530 immediately after executing expressions and files given by the **-e** and/or
1531 **-f** command-line options (and any equivalents).
1535 bc(1) returns the following exit statuses:
1543 : A math error occurred. This follows standard practice of using **1** for
1544 expected errors, since math errors will happen in the process of normal
1547 Math errors include divide by **0**, taking the square root of a negative
1548 number, using a negative number as a bound for the pseudo-random number
1549 generator, attempting to convert a negative number to a hardware integer,
1550 overflow when converting a number to a hardware integer, and attempting to
1551 use a non-integer where an integer is required.
1553 Converting to a hardware integer happens for the second operand of the power
1554 (**\^**), places (**\@**), left shift (**\<\<**), and right shift (**\>\>**)
1555 operators and their corresponding assignment operators.
1559 : A parse error occurred.
1561 Parse errors include unexpected **EOF**, using an invalid character, failing
1562 to find the end of a string or comment, using a token where it is invalid,
1563 giving an invalid expression, giving an invalid print statement, giving an
1564 invalid function definition, attempting to assign to an expression that is
1565 not a named expression (see the *Named Expressions* subsection of the
1566 **SYNTAX** section), giving an invalid **auto** list, having a duplicate
1567 **auto**/function parameter, failing to find the end of a code block,
1568 attempting to return a value from a **void** function, attempting to use a
1569 variable as a reference, and using any extensions when the option **-s** or
1570 any equivalents were given.
1574 : A runtime error occurred.
1576 Runtime errors include assigning an invalid number to **ibase**, **obase**,
1577 or **scale**; give a bad expression to a **read()** call, calling **read()**
1578 inside of a **read()** call, type errors, passing the wrong number of
1579 arguments to functions, attempting to call an undefined function, and
1580 attempting to use a **void** function call as a value in an expression.
1584 : A fatal error occurred.
1586 Fatal errors include memory allocation errors, I/O errors, failing to open
1587 files, attempting to use files that do not have only ASCII characters (bc(1)
1588 only accepts ASCII characters), attempting to open a directory as a file,
1589 and giving invalid command-line options.
1591 The exit status **4** is special; when a fatal error occurs, bc(1) always exits
1592 and returns **4**, no matter what mode bc(1) is in.
1594 The other statuses will only be returned when bc(1) is not in interactive mode
1595 (see the **INTERACTIVE MODE** section), since bc(1) resets its state (see the
1596 **RESET** section) and accepts more input when one of those errors occurs in
1597 interactive mode. This is also the case when interactive mode is forced by the
1598 **-i** flag or **--interactive** option.
1600 These exit statuses allow bc(1) to be used in shell scripting with error
1601 checking, and its normal behavior can be forced by using the **-i** flag or
1602 **--interactive** option.
1606 Per the [standard][1], bc(1) has an interactive mode and a non-interactive mode.
1607 Interactive mode is turned on automatically when both **stdin** and **stdout**
1608 are hooked to a terminal, but the **-i** flag and **--interactive** option can
1609 turn it on in other cases.
1611 In interactive mode, bc(1) attempts to recover from errors (see the **RESET**
1612 section), and in normal execution, flushes **stdout** as soon as execution is
1613 done for the current input.
1617 If **stdin**, **stdout**, and **stderr** are all connected to a TTY, bc(1) turns
1620 TTY mode is required for history to be enabled (see the **COMMAND LINE HISTORY**
1621 section). It is also required to enable special handling for **SIGINT** signals.
1623 The prompt is enabled in TTY mode.
1625 TTY mode is different from interactive mode because interactive mode is required
1626 in the [bc(1) specification][1], and interactive mode requires only **stdin**
1627 and **stdout** to be connected to a terminal.
1631 Sending a **SIGINT** will cause bc(1) to stop execution of the current input. If
1632 bc(1) is in TTY mode (see the **TTY MODE** section), it will reset (see the
1633 **RESET** section). Otherwise, it will clean up and exit.
1635 Note that "current input" can mean one of two things. If bc(1) is processing
1636 input from **stdin** in TTY mode, it will ask for more input. If bc(1) is
1637 processing input from a file in TTY mode, it will stop processing the file and
1638 start processing the next file, if one exists, or ask for input from **stdin**
1639 if no other file exists.
1641 This means that if a **SIGINT** is sent to bc(1) as it is executing a file, it
1642 can seem as though bc(1) did not respond to the signal since it will immediately
1643 start executing the next file. This is by design; most files that users execute
1644 when interacting with bc(1) have function definitions, which are quick to parse.
1645 If a file takes a long time to execute, there may be a bug in that file. The
1646 rest of the files could still be executed without problem, allowing the user to
1649 **SIGTERM** and **SIGQUIT** cause bc(1) to clean up and exit, and it uses the
1650 default handler for all other signals. The one exception is **SIGHUP**; in that
1651 case, when bc(1) is in TTY mode, a **SIGHUP** will cause bc(1) to clean up and
1654 # COMMAND LINE HISTORY
1656 bc(1) supports interactive command-line editing. If bc(1) is in TTY mode (see
1657 the **TTY MODE** section), history is enabled. Previous lines can be recalled
1658 and edited with the arrow keys.
1660 **Note**: tabs are converted to 8 spaces.
1664 This bc(1) ships with support for adding error messages for different locales
1665 and thus, supports **LC_MESSAGES**.
1673 bc(1) is compliant with the [IEEE Std 1003.1-2017 (“POSIX.1-2017”)][1]
1674 specification. The flags **-efghiqsvVw**, all long options, and the extensions
1675 noted above are extensions to that specification.
1677 Note that the specification explicitly says that bc(1) only accepts numbers that
1678 use a period (**.**) as a radix point, regardless of the value of
1681 This bc(1) supports error messages for different locales, and thus, it supports
1686 None are known. Report bugs at https://git.yzena.com/gavin/bc.
1690 Gavin D. Howard <yzena.tech@gmail.com> and contributors.
1692 [1]: https://pubs.opengroup.org/onlinepubs/9699919799/utilities/bc.html
1693 [2]: https://www.gnu.org/software/bc/
1694 [3]: https://en.wikipedia.org/wiki/Rounding#Round_half_away_from_zero
1695 [4]: https://en.wikipedia.org/wiki/Unit_in_the_last_place
1696 [5]: https://people.eecs.berkeley.edu/~wkahan/LOG10HAF.TXT
1697 [6]: https://en.wikipedia.org/wiki/Rounding#Rounding_away_from_zero