3 SPDX-License-Identifier: BSD-2-Clause
5 Copyright (c) 2018-2021 Gavin D. Howard and contributors.
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10 * Redistributions of source code must retain the above copyright notice, this
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33 bc - arbitrary-precision decimal 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*...] [*file*...]
41 bc(1) is an interactive processor for a language first standardized in 1991 by
42 POSIX. (The current standard is [here][1].) The language provides unlimited
43 precision decimal arithmetic and is somewhat C-like, but there are differences.
44 Such differences will be noted in this document.
46 After parsing and handling options, this bc(1) reads any files given on the
47 command line and executes them before reading from **stdin**.
51 The following are the options that bc(1) accepts.
53 **-g**, **-\-global-stacks**
55 : Turns the globals **ibase**, **obase**, **scale**, and **seed** into stacks.
57 This has the effect that a copy of the current value of all four are pushed
58 onto a stack for every function call, as well as popped when every function
59 returns. This means that functions can assign to any and all of those
60 globals without worrying that the change will affect other functions.
61 Thus, a hypothetical function named **output(x,b)** that simply printed
62 **x** in base **b** could be written like this:
64 define void output(x, b) {
71 define void output(x, b) {
79 This makes writing functions much easier.
81 (**Note**: the function **output(x,b)** exists in the extended math library.
82 See the **LIBRARY** section.)
84 However, since using this flag means that functions cannot set **ibase**,
85 **obase**, **scale**, or **seed** globally, functions that are made to do so
86 cannot work anymore. There are two possible use cases for that, and each has
89 First, if a function is called on startup to turn bc(1) into a number
90 converter, it is possible to replace that capability with various shell
93 alias d2o="bc -e ibase=A -e obase=8"
94 alias h2b="bc -e ibase=G -e obase=2"
96 Second, if the purpose of a function is to set **ibase**, **obase**,
97 **scale**, or **seed** globally for any other purpose, it could be split
98 into one to four functions (based on how many globals it sets) and each of
99 those functions could return the desired value for a global.
101 For functions that set **seed**, the value assigned to **seed** is not
102 propagated to parent functions. This means that the sequence of
103 pseudo-random numbers that they see will not be the same sequence of
104 pseudo-random numbers that any parent sees. This is only the case once
105 **seed** has been set.
107 If a function desires to not affect the sequence of pseudo-random numbers
108 of its parents, but wants to use the same **seed**, it can use the following
113 If the behavior of this option is desired for every run of bc(1), then users
114 could make sure to define **BC_ENV_ARGS** and include this option (see the
115 **ENVIRONMENT VARIABLES** section for more details).
117 If **-s**, **-w**, or any equivalents are used, this option is ignored.
119 This is a **non-portable extension**.
123 : Prints a usage message and quits.
125 **-i**, **-\-interactive**
127 : Forces interactive mode. (See the **INTERACTIVE MODE** section.)
129 This is a **non-portable extension**.
131 **-l**, **-\-mathlib**
133 : Sets **scale** (see the **SYNTAX** section) to **20** and loads the included
134 math library and the extended math library before running any code,
135 including any expressions or files specified on the command line.
137 To learn what is in the libraries, see the **LIBRARY** section.
139 **-P**, **-\-no-prompt**
141 : Disables the prompt in TTY mode. (The prompt is only enabled in TTY mode.
142 See the **TTY MODE** section) This is mostly for those users that do not
143 want a prompt or are not used to having them in bc(1). Most of those users
144 would want to put this option in **BC_ENV_ARGS** (see the
145 **ENVIRONMENT VARIABLES** section).
147 This is a **non-portable extension**.
151 : This option is for compatibility with the [GNU bc(1)][2]; it is a no-op.
152 Without this option, GNU bc(1) prints a copyright header. This bc(1) only
153 prints the copyright header if one or more of the **-v**, **-V**, or
154 **-\-version** options are given.
156 This is a **non-portable extension**.
158 **-s**, **-\-standard**
160 : Process exactly the language defined by the [standard][1] and error if any
163 This is a **non-portable extension**.
165 **-v**, **-V**, **-\-version**
167 : Print the version information (copyright header) and exit.
169 This is a **non-portable extension**.
173 : Like **-s** and **-\-standard**, except that warnings (and not errors) are
174 printed for non-standard extensions and execution continues normally.
176 This is a **non-portable extension**.
178 **-e** *expr*, **-\-expression**=*expr*
180 : Evaluates *expr*. If multiple expressions are given, they are evaluated in
181 order. If files are given as well (see below), the expressions and files are
182 evaluated in the order given. This means that if a file is given before an
183 expression, the file is read in and evaluated first.
185 If this option is given on the command-line (i.e., not in **BC_ENV_ARGS**,
186 see the **ENVIRONMENT VARIABLES** section), then after processing all
187 expressions and files, bc(1) will exit, unless **-** (**stdin**) was given
188 as an argument at least once to **-f** or **-\-file**, whether on the
189 command-line or in **BC_ENV_ARGS**. However, if any other **-e**,
190 **-\-expression**, **-f**, or **-\-file** arguments are given after **-f-**
191 or equivalent is given, bc(1) will give a fatal error and exit.
193 This is a **non-portable extension**.
195 **-f** *file*, **-\-file**=*file*
197 : Reads in *file* and evaluates it, line by line, as though it were read
198 through **stdin**. If expressions are also given (see above), the
199 expressions are evaluated in the order given.
201 If this option is given on the command-line (i.e., not in **BC_ENV_ARGS**,
202 see the **ENVIRONMENT VARIABLES** section), then after processing all
203 expressions and files, bc(1) will exit, unless **-** (**stdin**) was given
204 as an argument at least once to **-f** or **-\-file**. However, if any other
205 **-e**, **-\-expression**, **-f**, or **-\-file** arguments are given after
206 **-f-** or equivalent is given, bc(1) will give a fatal error and exit.
208 This is a **non-portable extension**.
210 All long options are **non-portable extensions**.
214 Any non-error output is written to **stdout**. In addition, if history (see the
215 **HISTORY** section) and the prompt (see the **TTY MODE** section) are enabled,
216 both are output to **stdout**.
218 **Note**: Unlike other bc(1) implementations, this bc(1) will issue a fatal
219 error (see the **EXIT STATUS** section) if it cannot write to **stdout**, so if
220 **stdout** is closed, as in **bc <file> >&-**, it will quit with an error. This
221 is done so that bc(1) can report problems when **stdout** is redirected to a
224 If there are scripts that depend on the behavior of other bc(1) implementations,
225 it is recommended that those scripts be changed to redirect **stdout** to
230 Any error output is written to **stderr**.
232 **Note**: Unlike other bc(1) implementations, this bc(1) will issue a fatal
233 error (see the **EXIT STATUS** section) if it cannot write to **stderr**, so if
234 **stderr** is closed, as in **bc <file> 2>&-**, it will quit with an error. This
235 is done so that bc(1) can exit with an error code when **stderr** is redirected
238 If there are scripts that depend on the behavior of other bc(1) implementations,
239 it is recommended that those scripts be changed to redirect **stderr** to
244 The syntax for bc(1) programs is mostly C-like, with some differences. This
245 bc(1) follows the [POSIX standard][1], which is a much more thorough resource
246 for the language this bc(1) accepts. This section is meant to be a summary and a
247 listing of all the extensions to the standard.
249 In the sections below, **E** means expression, **S** means statement, and **I**
252 Identifiers (**I**) start with a lowercase letter and can be followed by any
253 number (up to **BC_NAME_MAX-1**) of lowercase letters (**a-z**), digits
254 (**0-9**), and underscores (**\_**). The regex is **\[a-z\]\[a-z0-9\_\]\***.
255 Identifiers with more than one character (letter) are a
256 **non-portable extension**.
258 **ibase** is a global variable determining how to interpret constant numbers. It
259 is the "input" base, or the number base used for interpreting input numbers.
260 **ibase** is initially **10**. If the **-s** (**-\-standard**) and **-w**
261 (**-\-warn**) flags were not given on the command line, the max allowable value
262 for **ibase** is **36**. Otherwise, it is **16**. The min allowable value for
263 **ibase** is **2**. The max allowable value for **ibase** can be queried in
264 bc(1) programs with the **maxibase()** built-in function.
266 **obase** is a global variable determining how to output results. It is the
267 "output" base, or the number base used for outputting numbers. **obase** is
268 initially **10**. The max allowable value for **obase** is **BC_BASE_MAX** and
269 can be queried in bc(1) programs with the **maxobase()** built-in function. The
270 min allowable value for **obase** is **0**. If **obase** is **0**, values are
271 output in scientific notation, and if **obase** is **1**, values are output in
272 engineering notation. Otherwise, values are output in the specified base.
274 Outputting in scientific and engineering notations are **non-portable
277 The *scale* of an expression is the number of digits in the result of the
278 expression right of the decimal point, and **scale** is a global variable that
279 sets the precision of any operations, with exceptions. **scale** is initially
280 **0**. **scale** cannot be negative. The max allowable value for **scale** is
281 **BC_SCALE_MAX** and can be queried in bc(1) programs with the **maxscale()**
284 bc(1) has both *global* variables and *local* variables. All *local*
285 variables are local to the function; they are parameters or are introduced in
286 the **auto** list of a function (see the **FUNCTIONS** section). If a variable
287 is accessed which is not a parameter or in the **auto** list, it is assumed to
288 be *global*. If a parent function has a *local* variable version of a variable
289 that a child function considers *global*, the value of that *global* variable in
290 the child function is the value of the variable in the parent function, not the
291 value of the actual *global* variable.
293 All of the above applies to arrays as well.
295 The value of a statement that is an expression (i.e., any of the named
296 expressions or operands) is printed unless the lowest precedence operator is an
297 assignment operator *and* the expression is notsurrounded by parentheses.
299 The value that is printed is also assigned to the special variable **last**. A
300 single dot (**.**) may also be used as a synonym for **last**. These are
301 **non-portable extensions**.
303 Either semicolons or newlines may separate statements.
307 There are two kinds of comments:
309 1. Block comments are enclosed in **/\*** and **\*/**.
310 2. Line comments go from **#** until, and not including, the next newline. This
311 is a **non-portable extension**.
315 The following are named expressions in bc(1):
318 2. Array Elements: **I[E]**
323 7. **last** or a single dot (**.**)
325 Numbers 6 and 7 are **non-portable extensions**.
327 The meaning of **seed** is dependent on the current pseudo-random number
328 generator but is guaranteed to not change except for new major versions.
330 The *scale* and sign of the value may be significant.
332 If a previously used **seed** value is assigned to **seed** and used again, the
333 pseudo-random number generator is guaranteed to produce the same sequence of
334 pseudo-random numbers as it did when the **seed** value was previously used.
336 The exact value assigned to **seed** is not guaranteed to be returned if
337 **seed** is queried again immediately. However, if **seed** *does* return a
338 different value, both values, when assigned to **seed**, are guaranteed to
339 produce the same sequence of pseudo-random numbers. This means that certain
340 values assigned to **seed** will *not* produce unique sequences of pseudo-random
341 numbers. The value of **seed** will change after any use of the **rand()** and
342 **irand(E)** operands (see the *Operands* subsection below), except if the
343 parameter passed to **irand(E)** is **0**, **1**, or negative.
345 There is no limit to the length (number of significant decimal digits) or
346 *scale* of the value that can be assigned to **seed**.
348 Variables and arrays do not interfere; users can have arrays named the same as
349 variables. This also applies to functions (see the **FUNCTIONS** section), so a
350 user can have a variable, array, and function that all have the same name, and
351 they will not shadow each other, whether inside of functions or not.
353 Named expressions are required as the operand of **increment**/**decrement**
354 operators and as the left side of **assignment** operators (see the *Operators*
359 The following are valid operands in bc(1):
361 1. Numbers (see the *Numbers* subsection below).
362 2. Array indices (**I[E]**).
363 3. **(E)**: The value of **E** (used to change precedence).
364 4. **sqrt(E)**: The square root of **E**. **E** must be non-negative.
365 5. **length(E)**: The number of significant decimal digits in **E**.
366 6. **length(I[])**: The number of elements in the array **I**. This is a
367 **non-portable extension**.
368 7. **scale(E)**: The *scale* of **E**.
369 8. **abs(E)**: The absolute value of **E**. This is a **non-portable
371 9. **I()**, **I(E)**, **I(E, E)**, and so on, where **I** is an identifier for
372 a non-**void** function (see the *Void Functions* subsection of the
373 **FUNCTIONS** section). The **E** argument(s) may also be arrays of the form
374 **I[]**, which will automatically be turned into array references (see the
375 *Array References* subsection of the **FUNCTIONS** section) if the
376 corresponding parameter in the function definition is an array reference.
377 10. **read()**: Reads a line from **stdin** and uses that as an expression. The
378 result of that expression is the result of the **read()** operand. This is a
379 **non-portable extension**.
380 11. **maxibase()**: The max allowable **ibase**. This is a **non-portable
382 12. **maxobase()**: The max allowable **obase**. This is a **non-portable
384 13. **maxscale()**: The max allowable **scale**. This is a **non-portable
386 14. **rand()**: A pseudo-random integer between **0** (inclusive) and
387 **BC_RAND_MAX** (inclusive). Using this operand will change the value of
388 **seed**. This is a **non-portable extension**.
389 15. **irand(E)**: A pseudo-random integer between **0** (inclusive) and the
390 value of **E** (exclusive). If **E** is negative or is a non-integer
391 (**E**'s *scale* is not **0**), an error is raised, and bc(1) resets (see
392 the **RESET** section) while **seed** remains unchanged. If **E** is larger
393 than **BC_RAND_MAX**, the higher bound is honored by generating several
394 pseudo-random integers, multiplying them by appropriate powers of
395 **BC_RAND_MAX+1**, and adding them together. Thus, the size of integer that
396 can be generated with this operand is unbounded. Using this operand will
397 change the value of **seed**, unless the value of **E** is **0** or **1**.
398 In that case, **0** is returned, and **seed** is *not* changed. This is a
399 **non-portable extension**.
400 16. **maxrand()**: The max integer returned by **rand()**. This is a
401 **non-portable extension**.
403 The integers generated by **rand()** and **irand(E)** are guaranteed to be as
404 unbiased as possible, subject to the limitations of the pseudo-random number
407 **Note**: The values returned by the pseudo-random number generator with
408 **rand()** and **irand(E)** are guaranteed to *NOT* be cryptographically secure.
409 This is a consequence of using a seeded pseudo-random number generator. However,
410 they *are* guaranteed to be reproducible with identical **seed** values. This
411 means that the pseudo-random values from bc(1) should only be used where a
412 reproducible stream of pseudo-random numbers is *ESSENTIAL*. In any other case,
413 use a non-seeded pseudo-random number generator.
417 Numbers are strings made up of digits, uppercase letters, and at most **1**
418 period for a radix. Numbers can have up to **BC_NUM_MAX** digits. Uppercase
419 letters are equal to **9** + their position in the alphabet (i.e., **A** equals
420 **10**, or **9+1**). If a digit or letter makes no sense with the current value
421 of **ibase**, they are set to the value of the highest valid digit in **ibase**.
423 Single-character numbers (i.e., **A** alone) take the value that they would have
424 if they were valid digits, regardless of the value of **ibase**. This means that
425 **A** alone always equals decimal **10** and **Z** alone always equals decimal
428 In addition, bc(1) accepts numbers in scientific notation. These have the form
429 **\<number\>e\<integer\>**. The exponent (the portion after the **e**) must be
430 an integer. An example is **1.89237e9**, which is equal to **1892370000**.
431 Negative exponents are also allowed, so **4.2890e-3** is equal to **0.0042890**.
433 Using scientific notation is an error or warning if the **-s** or **-w**,
434 respectively, command-line options (or equivalents) are given.
436 **WARNING**: Both the number and the exponent in scientific notation are
437 interpreted according to the current **ibase**, but the number is still
438 multiplied by **10\^exponent** regardless of the current **ibase**. For example,
439 if **ibase** is **16** and bc(1) is given the number string **FFeA**, the
440 resulting decimal number will be **2550000000000**, and if bc(1) is given the
441 number string **10e-4**, the resulting decimal number will be **0.0016**.
443 Accepting input as scientific notation is a **non-portable extension**.
447 The following arithmetic and logical operators can be used. They are listed in
448 order of decreasing precedence. Operators in the same group have the same
453 : Type: Prefix and Postfix
457 Description: **increment**, **decrement**
465 Description: **negation**, **boolean not**
473 Description: **truncation**
481 Description: **set precision**
489 Description: **power**
497 Description: **multiply**, **divide**, **modulus**
505 Description: **add**, **subtract**
513 Description: **shift left**, **shift right**
515 **=** **\<\<=** **\>\>=** **+=** **-=** **\*=** **/=** **%=** **\^=** **\@=**
521 Description: **assignment**
523 **==** **\<=** **\>=** **!=** **\<** **\>**
529 Description: **relational**
537 Description: **boolean and**
545 Description: **boolean or**
547 The operators will be described in more detail below.
551 : The prefix and postfix **increment** and **decrement** operators behave
552 exactly like they would in C. They require a named expression (see the
553 *Named Expressions* subsection) as an operand.
555 The prefix versions of these operators are more efficient; use them where
560 : The **negation** operator returns **0** if a user attempts to negate any
561 expression with the value **0**. Otherwise, a copy of the expression with
562 its sign flipped is returned.
566 : The **boolean not** operator returns **1** if the expression is **0**, or
569 This is a **non-portable extension**.
573 : The **truncation** operator returns a copy of the given expression with all
574 of its *scale* removed.
576 This is a **non-portable extension**.
580 : The **set precision** operator takes two expressions and returns a copy of
581 the first with its *scale* equal to the value of the second expression. That
582 could either mean that the number is returned without change (if the
583 *scale* of the first expression matches the value of the second
584 expression), extended (if it is less), or truncated (if it is more).
586 The second expression must be an integer (no *scale*) and non-negative.
588 This is a **non-portable extension**.
592 : The **power** operator (not the **exclusive or** operator, as it would be in
593 C) takes two expressions and raises the first to the power of the value of
594 the second. The *scale* of the result is equal to **scale**.
596 The second expression must be an integer (no *scale*), and if it is
597 negative, the first value must be non-zero.
601 : The **multiply** operator takes two expressions, multiplies them, and
602 returns the product. If **a** is the *scale* of the first expression and
603 **b** is the *scale* of the second expression, the *scale* of the result is
604 equal to **min(a+b,max(scale,a,b))** where **min()** and **max()** return
609 : The **divide** operator takes two expressions, divides them, and returns the
610 quotient. The *scale* of the result shall be the value of **scale**.
612 The second expression must be non-zero.
616 : The **modulus** operator takes two expressions, **a** and **b**, and
617 evaluates them by 1) Computing **a/b** to current **scale** and 2) Using the
618 result of step 1 to calculate **a-(a/b)\*b** to *scale*
619 **max(scale+scale(b),scale(a))**.
621 The second expression must be non-zero.
625 : The **add** operator takes two expressions, **a** and **b**, and returns the
626 sum, with a *scale* equal to the max of the *scale*s of **a** and **b**.
630 : The **subtract** operator takes two expressions, **a** and **b**, and
631 returns the difference, with a *scale* equal to the max of the *scale*s of
636 : The **left shift** operator takes two expressions, **a** and **b**, and
637 returns a copy of the value of **a** with its decimal point moved **b**
640 The second expression must be an integer (no *scale*) and non-negative.
642 This is a **non-portable extension**.
646 : The **right shift** operator takes two expressions, **a** and **b**, and
647 returns a copy of the value of **a** with its decimal point moved **b**
650 The second expression must be an integer (no *scale*) and non-negative.
652 This is a **non-portable extension**.
654 **=** **\<\<=** **\>\>=** **+=** **-=** **\*=** **/=** **%=** **\^=** **\@=**
656 : The **assignment** operators take two expressions, **a** and **b** where
657 **a** is a named expression (see the *Named Expressions* subsection).
659 For **=**, **b** is copied and the result is assigned to **a**. For all
660 others, **a** and **b** are applied as operands to the corresponding
661 arithmetic operator and the result is assigned to **a**.
663 The **assignment** operators that correspond to operators that are
664 extensions are themselves **non-portable extensions**.
666 **==** **\<=** **\>=** **!=** **\<** **\>**
668 : The **relational** operators compare two expressions, **a** and **b**, and
669 if the relation holds, according to C language semantics, the result is
670 **1**. Otherwise, it is **0**.
672 Note that unlike in C, these operators have a lower precedence than the
673 **assignment** operators, which means that **a=b\>c** is interpreted as
676 Also, unlike the [standard][1] requires, these operators can appear anywhere
677 any other expressions can be used. This allowance is a
678 **non-portable extension**.
682 : The **boolean and** operator takes two expressions and returns **1** if both
683 expressions are non-zero, **0** otherwise.
685 This is *not* a short-circuit operator.
687 This is a **non-portable extension**.
691 : The **boolean or** operator takes two expressions and returns **1** if one
692 of the expressions is non-zero, **0** otherwise.
694 This is *not* a short-circuit operator.
696 This is a **non-portable extension**.
700 The following items are statements:
703 2. **{** **S** **;** ... **;** **S** **}**
704 3. **if** **(** **E** **)** **S**
705 4. **if** **(** **E** **)** **S** **else** **S**
706 5. **while** **(** **E** **)** **S**
707 6. **for** **(** **E** **;** **E** **;** **E** **)** **S**
708 7. An empty statement
714 13. A string of characters, enclosed in double quotes
715 14. **print** **E** **,** ... **,** **E**
716 15. **I()**, **I(E)**, **I(E, E)**, and so on, where **I** is an identifier for
717 a **void** function (see the *Void Functions* subsection of the
718 **FUNCTIONS** section). The **E** argument(s) may also be arrays of the form
719 **I[]**, which will automatically be turned into array references (see the
720 *Array References* subsection of the **FUNCTIONS** section) if the
721 corresponding parameter in the function definition is an array reference.
723 Numbers 4, 9, 11, 12, 14, and 15 are **non-portable extensions**.
725 Also, as a **non-portable extension**, any or all of the expressions in the
726 header of a for loop may be omitted. If the condition (second expression) is
727 omitted, it is assumed to be a constant **1**.
729 The **break** statement causes a loop to stop iterating and resume execution
730 immediately following a loop. This is only allowed in loops.
732 The **continue** statement causes a loop iteration to stop early and returns to
733 the start of the loop, including testing the loop condition. This is only
736 The **if** **else** statement does the same thing as in C.
738 The **quit** statement causes bc(1) to quit, even if it is on a branch that will
739 not be executed (it is a compile-time command).
741 The **halt** statement causes bc(1) to quit, if it is executed. (Unlike **quit**
742 if it is on a branch of an **if** statement that is not executed, bc(1) does not
745 The **limits** statement prints the limits that this bc(1) is subject to. This
746 is like the **quit** statement in that it is a compile-time command.
748 An expression by itself is evaluated and printed, followed by a newline.
750 Both scientific notation and engineering notation are available for printing the
751 results of expressions. Scientific notation is activated by assigning **0** to
752 **obase**, and engineering notation is activated by assigning **1** to
753 **obase**. To deactivate them, just assign a different value to **obase**.
755 Scientific notation and engineering notation are disabled if bc(1) is run with
756 either the **-s** or **-w** command-line options (or equivalents).
758 Printing numbers in scientific notation and/or engineering notation is a
759 **non-portable extension**.
763 The "expressions" in a **print** statement may also be strings. If they are, there
764 are backslash escape sequences that are interpreted specially. What those
765 sequences are, and what they cause to be printed, are shown below:
779 Any other character following a backslash causes the backslash and character to
782 Any non-string expression in a print statement shall be assigned to **last**,
783 like any other expression that is printed.
785 ## Order of Evaluation
787 All expressions in a statment are evaluated left to right, except as necessary
788 to maintain order of operations. This means, for example, assuming that **i** is
789 equal to **0**, in the expression
793 the first (or 0th) element of **a** is set to **1**, and **i** is equal to **2**
794 at the end of the expression.
796 This includes function arguments. Thus, assuming **i** is equal to **0**, this
797 means that in the expression
801 the first argument passed to **x()** is **0**, and the second argument is **1**,
802 while **i** is equal to **2** before the function starts executing.
806 Function definitions are as follows:
816 Any **I** in the parameter list or **auto** list may be replaced with **I[]** to
817 make a parameter or **auto** var an array, and any **I** in the parameter list
818 may be replaced with **\*I[]** to make a parameter an array reference. Callers
819 of functions that take array references should not put an asterisk in the call;
820 they must be called with just **I[]** like normal array parameters and will be
821 automatically converted into references.
823 As a **non-portable extension**, the opening brace of a **define** statement may
824 appear on the next line.
826 As a **non-portable extension**, the return statement may also be in one of the
830 2. **return** **(** **)**
833 The first two, or not specifying a **return** statement, is equivalent to
834 **return (0)**, unless the function is a **void** function (see the *Void
835 Functions* subsection below).
839 Functions can also be **void** functions, defined as follows:
842 define void I(I,...,I){
849 They can only be used as standalone expressions, where such an expression would
850 be printed alone, except in a print statement.
852 Void functions can only use the first two **return** statements listed above.
853 They can also omit the return statement entirely.
855 The word "void" is not treated as a keyword; it is still possible to have
856 variables, arrays, and functions named **void**. The word "void" is only
857 treated specially right after the **define** keyword.
859 This is a **non-portable extension**.
863 For any array in the parameter list, if the array is declared in the form
869 it is a **reference**. Any changes to the array in the function are reflected,
870 when the function returns, to the array that was passed in.
872 Other than this, all function arguments are passed by value.
874 This is a **non-portable extension**.
878 All of the functions below, including the functions in the extended math
879 library (see the *Extended Library* subsection below), are available when the
880 **-l** or **-\-mathlib** command-line flags are given, except that the extended
881 math library is not available when the **-s** option, the **-w** option, or
882 equivalents are given.
886 The [standard][1] defines the following functions for the math library:
890 : Returns the sine of **x**, which is assumed to be in radians.
892 This is a transcendental function (see the *Transcendental Functions*
897 : Returns the cosine of **x**, which is assumed to be in radians.
899 This is a transcendental function (see the *Transcendental Functions*
904 : Returns the arctangent of **x**, in radians.
906 This is a transcendental function (see the *Transcendental Functions*
911 : Returns the natural logarithm of **x**.
913 This is a transcendental function (see the *Transcendental Functions*
918 : Returns the mathematical constant **e** raised to the power of **x**.
920 This is a transcendental function (see the *Transcendental Functions*
925 : Returns the bessel integer order **n** (truncated) of **x**.
927 This is a transcendental function (see the *Transcendental Functions*
932 The extended library is *not* loaded when the **-s**/**-\-standard** or
933 **-w**/**-\-warn** options are given since they are not part of the library
934 defined by the [standard][1].
936 The extended library is a **non-portable extension**.
940 : Calculates **x** to the power of **y**, even if **y** is not an integer, and
941 returns the result to the current **scale**.
943 It is an error if **y** is negative and **x** is **0**.
945 This is a transcendental function (see the *Transcendental Functions*
950 : Returns **x** rounded to **p** decimal places according to the rounding mode
951 [round half away from **0**][3].
955 : Returns **x** rounded to **p** decimal places according to the rounding mode
956 [round away from **0**][6].
960 : Returns the factorial of the truncated absolute value of **x**.
964 : Returns the permutation of the truncated absolute value of **n** of the
965 truncated absolute value of **k**, if **k \<= n**. If not, it returns **0**.
969 : Returns the combination of the truncated absolute value of **n** of the
970 truncated absolute value of **k**, if **k \<= n**. If not, it returns **0**.
974 : Returns the logarithm base **2** of **x**.
976 This is a transcendental function (see the *Transcendental Functions*
981 : Returns the logarithm base **10** of **x**.
983 This is a transcendental function (see the *Transcendental Functions*
988 : Returns the logarithm base **b** of **x**.
990 This is a transcendental function (see the *Transcendental Functions*
995 : Returns the cube root of **x**.
999 : Calculates the truncated value of **n**, **r**, and returns the **r**th root
1000 of **x** to the current **scale**.
1002 If **r** is **0** or negative, this raises an error and causes bc(1) to
1003 reset (see the **RESET** section). It also raises an error and causes bc(1)
1004 to reset if **r** is even and **x** is negative.
1008 : Returns **pi** to **p** decimal places.
1010 This is a transcendental function (see the *Transcendental Functions*
1015 : Returns the tangent of **x**, which is assumed to be in radians.
1017 This is a transcendental function (see the *Transcendental Functions*
1022 : Returns the arctangent of **y/x**, in radians. If both **y** and **x** are
1023 equal to **0**, it raises an error and causes bc(1) to reset (see the
1024 **RESET** section). Otherwise, if **x** is greater than **0**, it returns
1025 **a(y/x)**. If **x** is less than **0**, and **y** is greater than or equal
1026 to **0**, it returns **a(y/x)+pi**. If **x** is less than **0**, and **y**
1027 is less than **0**, it returns **a(y/x)-pi**. If **x** is equal to **0**,
1028 and **y** is greater than **0**, it returns **pi/2**. If **x** is equal to
1029 **0**, and **y** is less than **0**, it returns **-pi/2**.
1031 This function is the same as the **atan2()** function in many programming
1034 This is a transcendental function (see the *Transcendental Functions*
1039 : Returns the sine of **x**, which is assumed to be in radians.
1041 This is an alias of **s(x)**.
1043 This is a transcendental function (see the *Transcendental Functions*
1048 : Returns the cosine of **x**, which is assumed to be in radians.
1050 This is an alias of **c(x)**.
1052 This is a transcendental function (see the *Transcendental Functions*
1057 : Returns the tangent of **x**, which is assumed to be in radians.
1059 If **x** is equal to **1** or **-1**, this raises an error and causes bc(1)
1060 to reset (see the **RESET** section).
1062 This is an alias of **t(x)**.
1064 This is a transcendental function (see the *Transcendental Functions*
1069 : Returns the arctangent of **x**, in radians.
1071 This is an alias of **a(x)**.
1073 This is a transcendental function (see the *Transcendental Functions*
1078 : Returns the arctangent of **y/x**, in radians. If both **y** and **x** are
1079 equal to **0**, it raises an error and causes bc(1) to reset (see the
1080 **RESET** section). Otherwise, if **x** is greater than **0**, it returns
1081 **a(y/x)**. If **x** is less than **0**, and **y** is greater than or equal
1082 to **0**, it returns **a(y/x)+pi**. If **x** is less than **0**, and **y**
1083 is less than **0**, it returns **a(y/x)-pi**. If **x** is equal to **0**,
1084 and **y** is greater than **0**, it returns **pi/2**. If **x** is equal to
1085 **0**, and **y** is less than **0**, it returns **-pi/2**.
1087 This function is the same as the **atan2()** function in many programming
1090 This is an alias of **a2(y, x)**.
1092 This is a transcendental function (see the *Transcendental Functions*
1097 : Converts **x** from radians to degrees and returns the result.
1099 This is a transcendental function (see the *Transcendental Functions*
1104 : Converts **x** from degrees to radians and returns the result.
1106 This is a transcendental function (see the *Transcendental Functions*
1111 : Generates a pseudo-random number between **0** (inclusive) and **1**
1112 (exclusive) with the number of decimal digits after the decimal point equal
1113 to the truncated absolute value of **p**. If **p** is not **0**, then
1114 calling this function will change the value of **seed**. If **p** is **0**,
1115 then **0** is returned, and **seed** is *not* changed.
1119 : Generates a pseudo-random number that is between **0** (inclusive) and the
1120 truncated absolute value of **i** (exclusive) with the number of decimal
1121 digits after the decimal point equal to the truncated absolute value of
1122 **p**. If the absolute value of **i** is greater than or equal to **2**, and
1123 **p** is not **0**, then calling this function will change the value of
1124 **seed**; otherwise, **0** is returned and **seed** is not changed.
1128 : Returns **x** with its sign flipped with probability **0.5**. In other
1129 words, it randomizes the sign of **x**.
1133 : Returns a random boolean value (either **0** or **1**).
1137 : Returns the numbers of unsigned integer bytes required to hold the truncated
1138 absolute value of **x**.
1142 : Returns the numbers of signed, two's-complement integer bytes required to
1143 hold the truncated value of **x**.
1147 : Outputs the hexadecimal (base **16**) representation of **x**.
1149 This is a **void** function (see the *Void Functions* subsection of the
1150 **FUNCTIONS** section).
1154 : Outputs the binary (base **2**) representation of **x**.
1156 This is a **void** function (see the *Void Functions* subsection of the
1157 **FUNCTIONS** section).
1161 : Outputs the base **b** representation of **x**.
1163 This is a **void** function (see the *Void Functions* subsection of the
1164 **FUNCTIONS** section).
1168 : Outputs the representation, in binary and hexadecimal, of **x** as an
1169 unsigned integer in as few power of two bytes as possible. Both outputs are
1170 split into bytes separated by spaces.
1172 If **x** is not an integer or is negative, an error message is printed
1173 instead, but bc(1) is not reset (see the **RESET** section).
1175 This is a **void** function (see the *Void Functions* subsection of the
1176 **FUNCTIONS** section).
1180 : Outputs the representation, in binary and hexadecimal, of **x** as a signed,
1181 two's-complement integer in as few power of two bytes as possible. Both
1182 outputs are split into bytes separated by spaces.
1184 If **x** is not an integer, an error message is printed instead, but bc(1)
1185 is not reset (see the **RESET** section).
1187 This is a **void** function (see the *Void Functions* subsection of the
1188 **FUNCTIONS** section).
1192 : Outputs the representation, in binary and hexadecimal, of **x** as an
1193 unsigned integer in **n** bytes. Both outputs are split into bytes separated
1196 If **x** is not an integer, is negative, or cannot fit into **n** bytes, an
1197 error message is printed instead, but bc(1) is not reset (see the **RESET**
1200 This is a **void** function (see the *Void Functions* subsection of the
1201 **FUNCTIONS** section).
1205 : Outputs the representation, in binary and hexadecimal, of **x** as a signed,
1206 two's-complement integer in **n** bytes. Both outputs are split into bytes
1207 separated by spaces.
1209 If **x** is not an integer or cannot fit into **n** bytes, an error message
1210 is printed instead, but bc(1) is not reset (see the **RESET** section).
1212 This is a **void** function (see the *Void Functions* subsection of the
1213 **FUNCTIONS** section).
1217 : Outputs the representation, in binary and hexadecimal, of **x** as an
1218 unsigned integer in **1** byte. Both outputs are split into bytes separated
1221 If **x** is not an integer, is negative, or cannot fit into **1** byte, an
1222 error message is printed instead, but bc(1) is not reset (see the **RESET**
1225 This is a **void** function (see the *Void Functions* subsection of the
1226 **FUNCTIONS** section).
1230 : Outputs the representation, in binary and hexadecimal, of **x** as a signed,
1231 two's-complement integer in **1** byte. Both outputs are split into bytes
1232 separated by spaces.
1234 If **x** is not an integer or cannot fit into **1** byte, an error message
1235 is printed instead, but bc(1) is not reset (see the **RESET** section).
1237 This is a **void** function (see the *Void Functions* subsection of the
1238 **FUNCTIONS** section).
1242 : Outputs the representation, in binary and hexadecimal, of **x** as an
1243 unsigned integer in **2** bytes. Both outputs are split into bytes separated
1246 If **x** is not an integer, is negative, or cannot fit into **2** bytes, an
1247 error message is printed instead, but bc(1) is not reset (see the **RESET**
1250 This is a **void** function (see the *Void Functions* subsection of the
1251 **FUNCTIONS** section).
1255 : Outputs the representation, in binary and hexadecimal, of **x** as a signed,
1256 two's-complement integer in **2** bytes. Both outputs are split into bytes
1257 separated by spaces.
1259 If **x** is not an integer or cannot fit into **2** bytes, an error message
1260 is printed instead, but bc(1) is not reset (see the **RESET** section).
1262 This is a **void** function (see the *Void Functions* subsection of the
1263 **FUNCTIONS** section).
1267 : Outputs the representation, in binary and hexadecimal, of **x** as an
1268 unsigned integer in **4** bytes. Both outputs are split into bytes separated
1271 If **x** is not an integer, is negative, or cannot fit into **4** bytes, an
1272 error message is printed instead, but bc(1) is not reset (see the **RESET**
1275 This is a **void** function (see the *Void Functions* subsection of the
1276 **FUNCTIONS** section).
1280 : Outputs the representation, in binary and hexadecimal, of **x** as a signed,
1281 two's-complement integer in **4** bytes. Both outputs are split into bytes
1282 separated by spaces.
1284 If **x** is not an integer or cannot fit into **4** bytes, an error message
1285 is printed instead, but bc(1) is not reset (see the **RESET** section).
1287 This is a **void** function (see the *Void Functions* subsection of the
1288 **FUNCTIONS** section).
1292 : Outputs the representation, in binary and hexadecimal, of **x** as an
1293 unsigned integer in **8** bytes. Both outputs are split into bytes separated
1296 If **x** is not an integer, is negative, or cannot fit into **8** bytes, an
1297 error message is printed instead, but bc(1) is not reset (see the **RESET**
1300 This is a **void** function (see the *Void Functions* subsection of the
1301 **FUNCTIONS** section).
1305 : Outputs the representation, in binary and hexadecimal, of **x** as a signed,
1306 two's-complement integer in **8** bytes. Both outputs are split into bytes
1307 separated by spaces.
1309 If **x** is not an integer or cannot fit into **8** bytes, an error message
1310 is printed instead, but bc(1) is not reset (see the **RESET** section).
1312 This is a **void** function (see the *Void Functions* subsection of the
1313 **FUNCTIONS** section).
1317 : Outputs the representation of the truncated absolute value of **x** as an
1318 unsigned integer in hexadecimal using **n** bytes. Not all of the value will
1319 be output if **n** is too small.
1321 This is a **void** function (see the *Void Functions* subsection of the
1322 **FUNCTIONS** section).
1324 **binary_uint(x, n)**
1326 : Outputs the representation of the truncated absolute value of **x** as an
1327 unsigned integer in binary using **n** bytes. Not all of the value will be
1328 output if **n** is too small.
1330 This is a **void** function (see the *Void Functions* subsection of the
1331 **FUNCTIONS** section).
1333 **output_uint(x, n)**
1335 : Outputs the representation of the truncated absolute value of **x** as an
1336 unsigned integer in the current **obase** (see the **SYNTAX** section) using
1337 **n** bytes. Not all of the value will be output if **n** is too small.
1339 This is a **void** function (see the *Void Functions* subsection of the
1340 **FUNCTIONS** section).
1342 **output_byte(x, i)**
1344 : Outputs byte **i** of the truncated absolute value of **x**, where **0** is
1345 the least significant byte and **number_of_bytes - 1** is the most
1348 This is a **void** function (see the *Void Functions* subsection of the
1349 **FUNCTIONS** section).
1351 ## Transcendental Functions
1353 All transcendental functions can return slightly inaccurate results (up to 1
1354 [ULP][4]). This is unavoidable, and [this article][5] explains why it is
1355 impossible and unnecessary to calculate exact results for the transcendental
1358 Because of the possible inaccuracy, I recommend that users call those functions
1359 with the precision (**scale**) set to at least 1 higher than is necessary. If
1360 exact results are *absolutely* required, users can double the precision
1361 (**scale**) and then truncate.
1363 The transcendental functions in the standard math library are:
1372 The transcendental functions in the extended math library are:
1390 When bc(1) encounters an error or a signal that it has a non-default handler
1391 for, it resets. This means that several things happen.
1393 First, any functions that are executing are stopped and popped off the stack.
1394 The behavior is not unlike that of exceptions in programming languages. Then
1395 the execution point is set so that any code waiting to execute (after all
1396 functions returned) is skipped.
1398 Thus, when bc(1) resets, it skips any remaining code waiting to be executed.
1399 Then, if it is interactive mode, and the error was not a fatal error (see the
1400 **EXIT STATUS** section), it asks for more input; otherwise, it exits with the
1401 appropriate return code.
1403 Note that this reset behavior is different from the GNU bc(1), which attempts to
1404 start executing the statement right after the one that caused an error.
1408 Most bc(1) implementations use **char** types to calculate the value of **1**
1409 decimal digit at a time, but that can be slow. This bc(1) does something
1412 It uses large integers to calculate more than **1** decimal digit at a time. If
1413 built in a environment where **BC_LONG_BIT** (see the **LIMITS** section) is
1414 **64**, then each integer has **9** decimal digits. If built in an environment
1415 where **BC_LONG_BIT** is **32** then each integer has **4** decimal digits. This
1416 value (the number of decimal digits per large integer) is called
1419 The actual values of **BC_LONG_BIT** and **BC_BASE_DIGS** can be queried with
1420 the **limits** statement.
1422 In addition, this bc(1) uses an even larger integer for overflow checking. This
1423 integer type depends on the value of **BC_LONG_BIT**, but is always at least
1424 twice as large as the integer type used to store digits.
1428 The following are the limits on bc(1):
1432 : The number of bits in the **long** type in the environment where bc(1) was
1433 built. This determines how many decimal digits can be stored in a single
1434 large integer (see the **PERFORMANCE** section).
1438 : The number of decimal digits per large integer (see the **PERFORMANCE**
1439 section). Depends on **BC_LONG_BIT**.
1443 : The max decimal number that each large integer can store (see
1444 **BC_BASE_DIGS**) plus **1**. Depends on **BC_BASE_DIGS**.
1448 : The max number that the overflow type (see the **PERFORMANCE** section) can
1449 hold. Depends on **BC_LONG_BIT**.
1453 : The maximum output base. Set at **BC_BASE_POW**.
1457 : The maximum size of arrays. Set at **SIZE_MAX-1**.
1461 : The maximum **scale**. Set at **BC_OVERFLOW_MAX-1**.
1465 : The maximum length of strings. Set at **BC_OVERFLOW_MAX-1**.
1469 : The maximum length of identifiers. Set at **BC_OVERFLOW_MAX-1**.
1473 : The maximum length of a number (in decimal digits), which includes digits
1474 after the decimal point. Set at **BC_OVERFLOW_MAX-1**.
1478 : The maximum integer (inclusive) returned by the **rand()** operand. Set at
1479 **2\^BC_LONG_BIT-1**.
1483 : The maximum allowable exponent (positive or negative). Set at
1484 **BC_OVERFLOW_MAX**.
1488 : The maximum number of vars/arrays. Set at **SIZE_MAX-1**.
1490 The actual values can be queried with the **limits** statement.
1492 These limits are meant to be effectively non-existent; the limits are so large
1493 (at least on 64-bit machines) that there should not be any point at which they
1494 become a problem. In fact, memory should be exhausted before these limits should
1497 # ENVIRONMENT VARIABLES
1499 bc(1) recognizes the following environment variables:
1503 : If this variable exists (no matter the contents), bc(1) behaves as if
1504 the **-s** option was given.
1508 : This is another way to give command-line arguments to bc(1). They should be
1509 in the same format as all other command-line arguments. These are always
1510 processed first, so any files given in **BC_ENV_ARGS** will be processed
1511 before arguments and files given on the command-line. This gives the user
1512 the ability to set up "standard" options and files to be used at every
1513 invocation. The most useful thing for such files to contain would be useful
1514 functions that the user might want every time bc(1) runs.
1516 The code that parses **BC_ENV_ARGS** will correctly handle quoted arguments,
1517 but it does not understand escape sequences. For example, the string
1518 **"/home/gavin/some bc file.bc"** will be correctly parsed, but the string
1519 **"/home/gavin/some \"bc\" file.bc"** will include the backslashes.
1521 The quote parsing will handle either kind of quotes, **'** or **"**. Thus,
1522 if you have a file with any number of single quotes in the name, you can use
1523 double quotes as the outside quotes, as in **"some 'bc' file.bc"**, and vice
1524 versa if you have a file with double quotes. However, handling a file with
1525 both kinds of quotes in **BC_ENV_ARGS** is not supported due to the
1526 complexity of the parsing, though such files are still supported on the
1527 command-line where the parsing is done by the shell.
1531 : If this environment variable exists and contains an integer that is greater
1532 than **1** and is less than **UINT16_MAX** (**2\^16-1**), bc(1) will output
1533 lines to that length, including the backslash (**\\**). The default line
1538 bc(1) returns the following exit statuses:
1546 : A math error occurred. This follows standard practice of using **1** for
1547 expected errors, since math errors will happen in the process of normal
1550 Math errors include divide by **0**, taking the square root of a negative
1551 number, using a negative number as a bound for the pseudo-random number
1552 generator, attempting to convert a negative number to a hardware integer,
1553 overflow when converting a number to a hardware integer, and attempting to
1554 use a non-integer where an integer is required.
1556 Converting to a hardware integer happens for the second operand of the power
1557 (**\^**), places (**\@**), left shift (**\<\<**), and right shift (**\>\>**)
1558 operators and their corresponding assignment operators.
1562 : A parse error occurred.
1564 Parse errors include unexpected **EOF**, using an invalid character, failing
1565 to find the end of a string or comment, using a token where it is invalid,
1566 giving an invalid expression, giving an invalid print statement, giving an
1567 invalid function definition, attempting to assign to an expression that is
1568 not a named expression (see the *Named Expressions* subsection of the
1569 **SYNTAX** section), giving an invalid **auto** list, having a duplicate
1570 **auto**/function parameter, failing to find the end of a code block,
1571 attempting to return a value from a **void** function, attempting to use a
1572 variable as a reference, and using any extensions when the option **-s** or
1573 any equivalents were given.
1577 : A runtime error occurred.
1579 Runtime errors include assigning an invalid number to **ibase**, **obase**,
1580 or **scale**; give a bad expression to a **read()** call, calling **read()**
1581 inside of a **read()** call, type errors, passing the wrong number of
1582 arguments to functions, attempting to call an undefined function, and
1583 attempting to use a **void** function call as a value in an expression.
1587 : A fatal error occurred.
1589 Fatal errors include memory allocation errors, I/O errors, failing to open
1590 files, attempting to use files that do not have only ASCII characters (bc(1)
1591 only accepts ASCII characters), attempting to open a directory as a file,
1592 and giving invalid command-line options.
1594 The exit status **4** is special; when a fatal error occurs, bc(1) always exits
1595 and returns **4**, no matter what mode bc(1) is in.
1597 The other statuses will only be returned when bc(1) is not in interactive mode
1598 (see the **INTERACTIVE MODE** section), since bc(1) resets its state (see the
1599 **RESET** section) and accepts more input when one of those errors occurs in
1600 interactive mode. This is also the case when interactive mode is forced by the
1601 **-i** flag or **-\-interactive** option.
1603 These exit statuses allow bc(1) to be used in shell scripting with error
1604 checking, and its normal behavior can be forced by using the **-i** flag or
1605 **-\-interactive** option.
1609 Per the [standard][1], bc(1) has an interactive mode and a non-interactive mode.
1610 Interactive mode is turned on automatically when both **stdin** and **stdout**
1611 are hooked to a terminal, but the **-i** flag and **-\-interactive** option can
1612 turn it on in other cases.
1614 In interactive mode, bc(1) attempts to recover from errors (see the **RESET**
1615 section), and in normal execution, flushes **stdout** as soon as execution is
1616 done for the current input.
1620 If **stdin**, **stdout**, and **stderr** are all connected to a TTY, bc(1) turns
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.
1658 bc(1) is compliant with the [IEEE Std 1003.1-2017 (“POSIX.1-2017”)][1]
1659 specification. The flags **-efghiqsvVw**, all long options, and the extensions
1660 noted above are extensions to that specification.
1662 Note that the specification explicitly says that bc(1) only accepts numbers that
1663 use a period (**.**) as a radix point, regardless of the value of
1668 None are known. Report bugs at https://git.yzena.com/gavin/bc.
1672 Gavin D. Howard <gavin@yzena.com> and contributors.
1674 [1]: https://pubs.opengroup.org/onlinepubs/9699919799/utilities/bc.html
1675 [2]: https://www.gnu.org/software/bc/
1676 [3]: https://en.wikipedia.org/wiki/Rounding#Round_half_away_from_zero
1677 [4]: https://en.wikipedia.org/wiki/Unit_in_the_last_place
1678 [5]: https://people.eecs.berkeley.edu/~wkahan/LOG10HAF.TXT
1679 [6]: https://en.wikipedia.org/wiki/Rounding#Rounding_away_from_zero