3 SPDX-License-Identifier: BSD-2-Clause
5 Copyright (c) 2018-2021 Gavin D. Howard and contributors.
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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 : This option is a no-op.
143 This is a **non-portable extension**.
147 : This option is for compatibility with the [GNU bc(1)][2]; it is a no-op.
148 Without this option, GNU bc(1) prints a copyright header. This bc(1) only
149 prints the copyright header if one or more of the **-v**, **-V**, or
150 **-\-version** options are given.
152 This is a **non-portable extension**.
154 **-s**, **-\-standard**
156 : Process exactly the language defined by the [standard][1] and error if any
159 This is a **non-portable extension**.
161 **-v**, **-V**, **-\-version**
163 : Print the version information (copyright header) and exit.
165 This is a **non-portable extension**.
169 : Like **-s** and **-\-standard**, except that warnings (and not errors) are
170 printed for non-standard extensions and execution continues normally.
172 This is a **non-portable extension**.
174 **-e** *expr*, **-\-expression**=*expr*
176 : Evaluates *expr*. If multiple expressions are given, they are evaluated in
177 order. If files are given as well (see below), the expressions and files are
178 evaluated in the order given. This means that if a file is given before an
179 expression, the file is read in and evaluated first.
181 If this option is given on the command-line (i.e., not in **BC_ENV_ARGS**,
182 see the **ENVIRONMENT VARIABLES** section), then after processing all
183 expressions and files, bc(1) will exit, unless **-** (**stdin**) was given
184 as an argument at least once to **-f** or **-\-file**, whether on the
185 command-line or in **BC_ENV_ARGS**. However, if any other **-e**,
186 **-\-expression**, **-f**, or **-\-file** arguments are given after **-f-**
187 or equivalent is given, bc(1) will give a fatal error and exit.
189 This is a **non-portable extension**.
191 **-f** *file*, **-\-file**=*file*
193 : Reads in *file* and evaluates it, line by line, as though it were read
194 through **stdin**. If expressions are also given (see above), the
195 expressions are evaluated in the order given.
197 If this option is given on the command-line (i.e., not in **BC_ENV_ARGS**,
198 see the **ENVIRONMENT VARIABLES** section), then after processing all
199 expressions and files, bc(1) will exit, unless **-** (**stdin**) was given
200 as an argument at least once to **-f** or **-\-file**. However, if any other
201 **-e**, **-\-expression**, **-f**, or **-\-file** arguments are given after
202 **-f-** or equivalent is given, bc(1) will give a fatal error and exit.
204 This is a **non-portable extension**.
206 All long options are **non-portable extensions**.
210 Any non-error output is written to **stdout**. In addition, if history (see the
211 **HISTORY** section) and the prompt (see the **TTY MODE** section) are enabled,
212 both are output 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. This
407 means that the pseudo-random values from bc(1) should only be used where a
408 reproducible stream of pseudo-random numbers is *ESSENTIAL*. In any other case,
409 use a non-seeded pseudo-random number generator.
413 Numbers are strings made up of digits, uppercase letters, and at most **1**
414 period for a radix. Numbers can have up to **BC_NUM_MAX** digits. Uppercase
415 letters are equal to **9** + their position in the alphabet (i.e., **A** equals
416 **10**, or **9+1**). If a digit or letter makes no sense with the current value
417 of **ibase**, they are set to the value of the highest valid digit in **ibase**.
419 Single-character numbers (i.e., **A** alone) take the value that they would have
420 if they were valid digits, regardless of the value of **ibase**. This means that
421 **A** alone always equals decimal **10** and **Z** alone always equals decimal
424 In addition, bc(1) accepts numbers in scientific notation. These have the form
425 **\<number\>e\<integer\>**. The exponent (the portion after the **e**) must be
426 an integer. An example is **1.89237e9**, which is equal to **1892370000**.
427 Negative exponents are also allowed, so **4.2890e-3** is equal to **0.0042890**.
429 Using scientific notation is an error or warning if the **-s** or **-w**,
430 respectively, command-line options (or equivalents) are given.
432 **WARNING**: Both the number and the exponent in scientific notation are
433 interpreted according to the current **ibase**, but the number is still
434 multiplied by **10\^exponent** regardless of the current **ibase**. For example,
435 if **ibase** is **16** and bc(1) is given the number string **FFeA**, the
436 resulting decimal number will be **2550000000000**, and if bc(1) is given the
437 number string **10e-4**, the resulting decimal number will be **0.0016**.
439 Accepting input as scientific notation is a **non-portable extension**.
443 The following arithmetic and logical operators can be used. They are listed in
444 order of decreasing precedence. Operators in the same group have the same
449 : Type: Prefix and Postfix
453 Description: **increment**, **decrement**
461 Description: **negation**, **boolean not**
469 Description: **truncation**
477 Description: **set precision**
485 Description: **power**
493 Description: **multiply**, **divide**, **modulus**
501 Description: **add**, **subtract**
509 Description: **shift left**, **shift right**
511 **=** **\<\<=** **\>\>=** **+=** **-=** **\*=** **/=** **%=** **\^=** **\@=**
517 Description: **assignment**
519 **==** **\<=** **\>=** **!=** **\<** **\>**
525 Description: **relational**
533 Description: **boolean and**
541 Description: **boolean or**
543 The operators will be described in more detail below.
547 : The prefix and postfix **increment** and **decrement** operators behave
548 exactly like they would in C. They require a named expression (see the
549 *Named Expressions* subsection) as an operand.
551 The prefix versions of these operators are more efficient; use them where
556 : The **negation** operator returns **0** if a user attempts to negate any
557 expression with the value **0**. Otherwise, a copy of the expression with
558 its sign flipped is returned.
562 : The **boolean not** operator returns **1** if the expression is **0**, or
565 This is a **non-portable extension**.
569 : The **truncation** operator returns a copy of the given expression with all
570 of its *scale* removed.
572 This is a **non-portable extension**.
576 : The **set precision** operator takes two expressions and returns a copy of
577 the first with its *scale* equal to the value of the second expression. That
578 could either mean that the number is returned without change (if the
579 *scale* of the first expression matches the value of the second
580 expression), extended (if it is less), or truncated (if it is more).
582 The second expression must be an integer (no *scale*) and non-negative.
584 This is a **non-portable extension**.
588 : The **power** operator (not the **exclusive or** operator, as it would be in
589 C) takes two expressions and raises the first to the power of the value of
590 the second. The *scale* of the result is equal to **scale**.
592 The second expression must be an integer (no *scale*), and if it is
593 negative, the first value must be non-zero.
597 : The **multiply** operator takes two expressions, multiplies them, and
598 returns the product. If **a** is the *scale* of the first expression and
599 **b** is the *scale* of the second expression, the *scale* of the result is
600 equal to **min(a+b,max(scale,a,b))** where **min()** and **max()** return
605 : The **divide** operator takes two expressions, divides them, and returns the
606 quotient. The *scale* of the result shall be the value of **scale**.
608 The second expression must be non-zero.
612 : The **modulus** operator takes two expressions, **a** and **b**, and
613 evaluates them by 1) Computing **a/b** to current **scale** and 2) Using the
614 result of step 1 to calculate **a-(a/b)\*b** to *scale*
615 **max(scale+scale(b),scale(a))**.
617 The second expression must be non-zero.
621 : The **add** operator takes two expressions, **a** and **b**, and returns the
622 sum, with a *scale* equal to the max of the *scale*s of **a** and **b**.
626 : The **subtract** operator takes two expressions, **a** and **b**, and
627 returns the difference, with a *scale* equal to the max of the *scale*s of
632 : The **left shift** operator takes two expressions, **a** and **b**, and
633 returns a copy of the value of **a** with its decimal point moved **b**
636 The second expression must be an integer (no *scale*) and non-negative.
638 This is a **non-portable extension**.
642 : The **right shift** operator takes two expressions, **a** and **b**, and
643 returns a copy of the value of **a** with its decimal point moved **b**
646 The second expression must be an integer (no *scale*) and non-negative.
648 This is a **non-portable extension**.
650 **=** **\<\<=** **\>\>=** **+=** **-=** **\*=** **/=** **%=** **\^=** **\@=**
652 : The **assignment** operators take two expressions, **a** and **b** where
653 **a** is a named expression (see the *Named Expressions* subsection).
655 For **=**, **b** is copied and the result is assigned to **a**. For all
656 others, **a** and **b** are applied as operands to the corresponding
657 arithmetic operator and the result is assigned to **a**.
659 The **assignment** operators that correspond to operators that are
660 extensions are themselves **non-portable extensions**.
662 **==** **\<=** **\>=** **!=** **\<** **\>**
664 : The **relational** operators compare two expressions, **a** and **b**, and
665 if the relation holds, according to C language semantics, the result is
666 **1**. Otherwise, it is **0**.
668 Note that unlike in C, these operators have a lower precedence than the
669 **assignment** operators, which means that **a=b\>c** is interpreted as
672 Also, unlike the [standard][1] requires, these operators can appear anywhere
673 any other expressions can be used. This allowance is a
674 **non-portable extension**.
678 : The **boolean and** operator takes two expressions and returns **1** if both
679 expressions are non-zero, **0** otherwise.
681 This is *not* a short-circuit operator.
683 This is a **non-portable extension**.
687 : The **boolean or** operator takes two expressions and returns **1** if one
688 of the expressions is non-zero, **0** otherwise.
690 This is *not* a short-circuit operator.
692 This is a **non-portable extension**.
696 The following items are statements:
699 2. **{** **S** **;** ... **;** **S** **}**
700 3. **if** **(** **E** **)** **S**
701 4. **if** **(** **E** **)** **S** **else** **S**
702 5. **while** **(** **E** **)** **S**
703 6. **for** **(** **E** **;** **E** **;** **E** **)** **S**
704 7. An empty statement
710 13. A string of characters, enclosed in double quotes
711 14. **print** **E** **,** ... **,** **E**
712 15. **I()**, **I(E)**, **I(E, E)**, and so on, where **I** is an identifier for
713 a **void** function (see the *Void Functions* subsection of the
714 **FUNCTIONS** section). The **E** argument(s) may also be arrays of the form
715 **I[]**, which will automatically be turned into array references (see the
716 *Array References* subsection of the **FUNCTIONS** section) if the
717 corresponding parameter in the function definition is an array reference.
719 Numbers 4, 9, 11, 12, 14, and 15 are **non-portable extensions**.
721 Also, as a **non-portable extension**, any or all of the expressions in the
722 header of a for loop may be omitted. If the condition (second expression) is
723 omitted, it is assumed to be a constant **1**.
725 The **break** statement causes a loop to stop iterating and resume execution
726 immediately following a loop. This is only allowed in loops.
728 The **continue** statement causes a loop iteration to stop early and returns to
729 the start of the loop, including testing the loop condition. This is only
732 The **if** **else** statement does the same thing as in C.
734 The **quit** statement causes bc(1) to quit, even if it is on a branch that will
735 not be executed (it is a compile-time command).
737 The **halt** statement causes bc(1) to quit, if it is executed. (Unlike **quit**
738 if it is on a branch of an **if** statement that is not executed, bc(1) does not
741 The **limits** statement prints the limits that this bc(1) is subject to. This
742 is like the **quit** statement in that it is a compile-time command.
744 An expression by itself is evaluated and printed, followed by a newline.
746 Both scientific notation and engineering notation are available for printing the
747 results of expressions. Scientific notation is activated by assigning **0** to
748 **obase**, and engineering notation is activated by assigning **1** to
749 **obase**. To deactivate them, just assign a different value to **obase**.
751 Scientific notation and engineering notation are disabled if bc(1) is run with
752 either the **-s** or **-w** command-line options (or equivalents).
754 Printing numbers in scientific notation and/or engineering notation is a
755 **non-portable extension**.
759 The "expressions" in a **print** statement may also be strings. If they are, there
760 are backslash escape sequences that are interpreted specially. What those
761 sequences are, and what they cause to be printed, are shown below:
775 Any other character following a backslash causes the backslash and character to
778 Any non-string expression in a print statement shall be assigned to **last**,
779 like any other expression that is printed.
781 ## Order of Evaluation
783 All expressions in a statment are evaluated left to right, except as necessary
784 to maintain order of operations. This means, for example, assuming that **i** is
785 equal to **0**, in the expression
789 the first (or 0th) element of **a** is set to **1**, and **i** is equal to **2**
790 at the end of the expression.
792 This includes function arguments. Thus, assuming **i** is equal to **0**, this
793 means that in the expression
797 the first argument passed to **x()** is **0**, and the second argument is **1**,
798 while **i** is equal to **2** before the function starts executing.
802 Function definitions are as follows:
812 Any **I** in the parameter list or **auto** list may be replaced with **I[]** to
813 make a parameter or **auto** var an array, and any **I** in the parameter list
814 may be replaced with **\*I[]** to make a parameter an array reference. Callers
815 of functions that take array references should not put an asterisk in the call;
816 they must be called with just **I[]** like normal array parameters and will be
817 automatically converted into references.
819 As a **non-portable extension**, the opening brace of a **define** statement may
820 appear on the next line.
822 As a **non-portable extension**, the return statement may also be in one of the
826 2. **return** **(** **)**
829 The first two, or not specifying a **return** statement, is equivalent to
830 **return (0)**, unless the function is a **void** function (see the *Void
831 Functions* subsection below).
835 Functions can also be **void** functions, defined as follows:
838 define void I(I,...,I){
845 They can only be used as standalone expressions, where such an expression would
846 be printed alone, except in a print statement.
848 Void functions can only use the first two **return** statements listed above.
849 They can also omit the return statement entirely.
851 The word "void" is not treated as a keyword; it is still possible to have
852 variables, arrays, and functions named **void**. The word "void" is only
853 treated specially right after the **define** keyword.
855 This is a **non-portable extension**.
859 For any array in the parameter list, if the array is declared in the form
865 it is a **reference**. Any changes to the array in the function are reflected,
866 when the function returns, to the array that was passed in.
868 Other than this, all function arguments are passed by value.
870 This is a **non-portable extension**.
874 All of the functions below, including the functions in the extended math
875 library (see the *Extended Library* subsection below), are available when the
876 **-l** or **-\-mathlib** command-line flags are given, except that the extended
877 math library is not available when the **-s** option, the **-w** option, or
878 equivalents are given.
882 The [standard][1] defines the following functions for the math library:
886 : Returns the sine of **x**, which is assumed to be in radians.
888 This is a transcendental function (see the *Transcendental Functions*
893 : Returns the cosine of **x**, which is assumed to be in radians.
895 This is a transcendental function (see the *Transcendental Functions*
900 : Returns the arctangent of **x**, in radians.
902 This is a transcendental function (see the *Transcendental Functions*
907 : Returns the natural logarithm of **x**.
909 This is a transcendental function (see the *Transcendental Functions*
914 : Returns the mathematical constant **e** raised to the power of **x**.
916 This is a transcendental function (see the *Transcendental Functions*
921 : Returns the bessel integer order **n** (truncated) of **x**.
923 This is a transcendental function (see the *Transcendental Functions*
928 The extended library is *not* loaded when the **-s**/**-\-standard** or
929 **-w**/**-\-warn** options are given since they are not part of the library
930 defined by the [standard][1].
932 The extended library is a **non-portable extension**.
936 : Calculates **x** to the power of **y**, even if **y** is not an integer, and
937 returns the result to the current **scale**.
939 It is an error if **y** is negative and **x** is **0**.
941 This is a transcendental function (see the *Transcendental Functions*
946 : Returns **x** rounded to **p** decimal places according to the rounding mode
947 [round half away from **0**][3].
951 : Returns **x** rounded to **p** decimal places according to the rounding mode
952 [round away from **0**][6].
956 : Returns the factorial of the truncated absolute value of **x**.
960 : Returns the permutation 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 combination of the truncated absolute value of **n** of the
966 truncated absolute value of **k**, if **k \<= n**. If not, it returns **0**.
970 : Returns the logarithm base **2** of **x**.
972 This is a transcendental function (see the *Transcendental Functions*
977 : Returns the logarithm base **10** of **x**.
979 This is a transcendental function (see the *Transcendental Functions*
984 : Returns the logarithm base **b** of **x**.
986 This is a transcendental function (see the *Transcendental Functions*
991 : Returns the cube root of **x**.
995 : Calculates the truncated value of **n**, **r**, and returns the **r**th root
996 of **x** to the current **scale**.
998 If **r** is **0** or negative, this raises an error and causes bc(1) to
999 reset (see the **RESET** section). It also raises an error and causes bc(1)
1000 to reset if **r** is even and **x** is negative.
1004 : Returns **pi** to **p** decimal places.
1006 This is a transcendental function (see the *Transcendental Functions*
1011 : Returns the tangent of **x**, which is assumed to be in radians.
1013 This is a transcendental function (see the *Transcendental Functions*
1018 : Returns the arctangent of **y/x**, in radians. If both **y** and **x** are
1019 equal to **0**, it raises an error and causes bc(1) to reset (see the
1020 **RESET** section). Otherwise, if **x** is greater than **0**, it returns
1021 **a(y/x)**. If **x** is less than **0**, and **y** is greater than or equal
1022 to **0**, it returns **a(y/x)+pi**. If **x** is less than **0**, and **y**
1023 is less than **0**, it returns **a(y/x)-pi**. If **x** is equal to **0**,
1024 and **y** is greater than **0**, it returns **pi/2**. If **x** is equal to
1025 **0**, and **y** is less than **0**, it returns **-pi/2**.
1027 This function is the same as the **atan2()** function in many programming
1030 This is a transcendental function (see the *Transcendental Functions*
1035 : Returns the sine of **x**, which is assumed to be in radians.
1037 This is an alias of **s(x)**.
1039 This is a transcendental function (see the *Transcendental Functions*
1044 : Returns the cosine of **x**, which is assumed to be in radians.
1046 This is an alias of **c(x)**.
1048 This is a transcendental function (see the *Transcendental Functions*
1053 : Returns the tangent of **x**, which is assumed to be in radians.
1055 If **x** is equal to **1** or **-1**, this raises an error and causes bc(1)
1056 to reset (see the **RESET** section).
1058 This is an alias of **t(x)**.
1060 This is a transcendental function (see the *Transcendental Functions*
1065 : Returns the arctangent of **x**, in radians.
1067 This is an alias of **a(x)**.
1069 This is a transcendental function (see the *Transcendental Functions*
1074 : Returns the arctangent of **y/x**, in radians. If both **y** and **x** are
1075 equal to **0**, it raises an error and causes bc(1) to reset (see the
1076 **RESET** section). Otherwise, if **x** is greater than **0**, it returns
1077 **a(y/x)**. If **x** is less than **0**, and **y** is greater than or equal
1078 to **0**, it returns **a(y/x)+pi**. If **x** is less than **0**, and **y**
1079 is less than **0**, it returns **a(y/x)-pi**. If **x** is equal to **0**,
1080 and **y** is greater than **0**, it returns **pi/2**. If **x** is equal to
1081 **0**, and **y** is less than **0**, it returns **-pi/2**.
1083 This function is the same as the **atan2()** function in many programming
1086 This is an alias of **a2(y, x)**.
1088 This is a transcendental function (see the *Transcendental Functions*
1093 : Converts **x** from radians to degrees and returns the result.
1095 This is a transcendental function (see the *Transcendental Functions*
1100 : Converts **x** from degrees to radians and returns the result.
1102 This is a transcendental function (see the *Transcendental Functions*
1107 : Generates a pseudo-random number between **0** (inclusive) and **1**
1108 (exclusive) with the number of decimal digits after the decimal point equal
1109 to the truncated absolute value of **p**. If **p** is not **0**, then
1110 calling this function will change the value of **seed**. If **p** is **0**,
1111 then **0** is returned, and **seed** is *not* changed.
1115 : Generates a pseudo-random number that is between **0** (inclusive) and the
1116 truncated absolute value of **i** (exclusive) with the number of decimal
1117 digits after the decimal point equal to the truncated absolute value of
1118 **p**. If the absolute value of **i** is greater than or equal to **2**, and
1119 **p** is not **0**, then calling this function will change the value of
1120 **seed**; otherwise, **0** is returned and **seed** is not changed.
1124 : Returns **x** with its sign flipped with probability **0.5**. In other
1125 words, it randomizes the sign of **x**.
1129 : Returns a random boolean value (either **0** or **1**).
1133 : Returns the numbers of unsigned integer bytes required to hold the truncated
1134 absolute value of **x**.
1138 : Returns the numbers of signed, two's-complement integer bytes required to
1139 hold the truncated value of **x**.
1143 : Outputs the hexadecimal (base **16**) representation of **x**.
1145 This is a **void** function (see the *Void Functions* subsection of the
1146 **FUNCTIONS** section).
1150 : Outputs the binary (base **2**) representation of **x**.
1152 This is a **void** function (see the *Void Functions* subsection of the
1153 **FUNCTIONS** section).
1157 : Outputs the base **b** representation of **x**.
1159 This is a **void** function (see the *Void Functions* subsection of the
1160 **FUNCTIONS** section).
1164 : Outputs the representation, in binary and hexadecimal, of **x** as an
1165 unsigned integer in as few power of two bytes as possible. Both outputs are
1166 split into bytes separated by spaces.
1168 If **x** is not an integer or is negative, an error message is printed
1169 instead, but bc(1) is not reset (see the **RESET** section).
1171 This is a **void** function (see the *Void Functions* subsection of the
1172 **FUNCTIONS** section).
1176 : Outputs the representation, in binary and hexadecimal, of **x** as a signed,
1177 two's-complement integer in as few power of two bytes as possible. Both
1178 outputs are split into bytes separated by spaces.
1180 If **x** is not an integer, an error message is printed instead, but bc(1)
1181 is not reset (see the **RESET** section).
1183 This is a **void** function (see the *Void Functions* subsection of the
1184 **FUNCTIONS** section).
1188 : Outputs the representation, in binary and hexadecimal, of **x** as an
1189 unsigned integer in **n** bytes. Both outputs are split into bytes separated
1192 If **x** is not an integer, is negative, or cannot fit into **n** bytes, an
1193 error message is printed instead, but bc(1) is not reset (see the **RESET**
1196 This is a **void** function (see the *Void Functions* subsection of the
1197 **FUNCTIONS** section).
1201 : Outputs the representation, in binary and hexadecimal, of **x** as a signed,
1202 two's-complement integer in **n** bytes. Both outputs are split into bytes
1203 separated by spaces.
1205 If **x** is not an integer or cannot fit into **n** bytes, an error message
1206 is printed instead, but bc(1) is not reset (see the **RESET** section).
1208 This is a **void** function (see the *Void Functions* subsection of the
1209 **FUNCTIONS** section).
1213 : Outputs the representation, in binary and hexadecimal, of **x** as an
1214 unsigned integer in **1** byte. Both outputs are split into bytes separated
1217 If **x** is not an integer, is negative, or cannot fit into **1** byte, an
1218 error message is printed instead, but bc(1) is not reset (see the **RESET**
1221 This is a **void** function (see the *Void Functions* subsection of the
1222 **FUNCTIONS** section).
1226 : Outputs the representation, in binary and hexadecimal, of **x** as a signed,
1227 two's-complement integer in **1** byte. Both outputs are split into bytes
1228 separated by spaces.
1230 If **x** is not an integer or cannot fit into **1** byte, an error message
1231 is printed instead, but bc(1) is not reset (see the **RESET** section).
1233 This is a **void** function (see the *Void Functions* subsection of the
1234 **FUNCTIONS** section).
1238 : Outputs the representation, in binary and hexadecimal, of **x** as an
1239 unsigned integer in **2** bytes. Both outputs are split into bytes separated
1242 If **x** is not an integer, is negative, or cannot fit into **2** bytes, an
1243 error message is printed instead, but bc(1) is not reset (see the **RESET**
1246 This is a **void** function (see the *Void Functions* subsection of the
1247 **FUNCTIONS** section).
1251 : Outputs the representation, in binary and hexadecimal, of **x** as a signed,
1252 two's-complement integer in **2** bytes. Both outputs are split into bytes
1253 separated by spaces.
1255 If **x** is not an integer or cannot fit into **2** bytes, an error message
1256 is printed instead, but bc(1) is not reset (see the **RESET** section).
1258 This is a **void** function (see the *Void Functions* subsection of the
1259 **FUNCTIONS** section).
1263 : Outputs the representation, in binary and hexadecimal, of **x** as an
1264 unsigned integer in **4** bytes. Both outputs are split into bytes separated
1267 If **x** is not an integer, is negative, or cannot fit into **4** bytes, an
1268 error message is printed instead, but bc(1) is not reset (see the **RESET**
1271 This is a **void** function (see the *Void Functions* subsection of the
1272 **FUNCTIONS** section).
1276 : Outputs the representation, in binary and hexadecimal, of **x** as a signed,
1277 two's-complement integer in **4** bytes. Both outputs are split into bytes
1278 separated by spaces.
1280 If **x** is not an integer or cannot fit into **4** bytes, an error message
1281 is printed instead, but bc(1) is not reset (see the **RESET** section).
1283 This is a **void** function (see the *Void Functions* subsection of the
1284 **FUNCTIONS** section).
1288 : Outputs the representation, in binary and hexadecimal, of **x** as an
1289 unsigned integer in **8** bytes. Both outputs are split into bytes separated
1292 If **x** is not an integer, is negative, or cannot fit into **8** bytes, an
1293 error message is printed instead, but bc(1) is not reset (see the **RESET**
1296 This is a **void** function (see the *Void Functions* subsection of the
1297 **FUNCTIONS** section).
1301 : Outputs the representation, in binary and hexadecimal, of **x** as a signed,
1302 two's-complement integer in **8** bytes. Both outputs are split into bytes
1303 separated by spaces.
1305 If **x** is not an integer or cannot fit into **8** bytes, an error message
1306 is printed instead, but bc(1) is not reset (see the **RESET** section).
1308 This is a **void** function (see the *Void Functions* subsection of the
1309 **FUNCTIONS** section).
1313 : Outputs the representation of the truncated absolute value of **x** as an
1314 unsigned integer in hexadecimal using **n** bytes. Not all of the value will
1315 be output if **n** is too small.
1317 This is a **void** function (see the *Void Functions* subsection of the
1318 **FUNCTIONS** section).
1320 **binary_uint(x, n)**
1322 : Outputs the representation of the truncated absolute value of **x** as an
1323 unsigned integer in binary using **n** bytes. Not all of the value will be
1324 output if **n** is too small.
1326 This is a **void** function (see the *Void Functions* subsection of the
1327 **FUNCTIONS** section).
1329 **output_uint(x, n)**
1331 : Outputs the representation of the truncated absolute value of **x** as an
1332 unsigned integer in the current **obase** (see the **SYNTAX** section) using
1333 **n** bytes. Not all of the value will be output if **n** is too small.
1335 This is a **void** function (see the *Void Functions* subsection of the
1336 **FUNCTIONS** section).
1338 **output_byte(x, i)**
1340 : Outputs byte **i** of the truncated absolute value of **x**, where **0** is
1341 the least significant byte and **number_of_bytes - 1** is the most
1344 This is a **void** function (see the *Void Functions* subsection of the
1345 **FUNCTIONS** section).
1347 ## Transcendental Functions
1349 All transcendental functions can return slightly inaccurate results (up to 1
1350 [ULP][4]). This is unavoidable, and [this article][5] explains why it is
1351 impossible and unnecessary to calculate exact results for the transcendental
1354 Because of the possible inaccuracy, I recommend that users call those functions
1355 with the precision (**scale**) set to at least 1 higher than is necessary. If
1356 exact results are *absolutely* required, users can double the precision
1357 (**scale**) and then truncate.
1359 The transcendental functions in the standard math library are:
1368 The transcendental functions in the extended math library are:
1386 When bc(1) encounters an error or a signal that it has a non-default handler
1387 for, it resets. This means that several things happen.
1389 First, any functions that are executing are stopped and popped off the stack.
1390 The behavior is not unlike that of exceptions in programming languages. Then
1391 the execution point is set so that any code waiting to execute (after all
1392 functions returned) is skipped.
1394 Thus, when bc(1) resets, it skips any remaining code waiting to be executed.
1395 Then, if it is interactive mode, and the error was not a fatal error (see the
1396 **EXIT STATUS** section), it asks for more input; otherwise, it exits with the
1397 appropriate return code.
1399 Note that this reset behavior is different from the GNU bc(1), which attempts to
1400 start executing the statement right after the one that caused an error.
1404 Most bc(1) implementations use **char** types to calculate the value of **1**
1405 decimal digit at a time, but that can be slow. This bc(1) does something
1408 It uses large integers to calculate more than **1** decimal digit at a time. If
1409 built in a environment where **BC_LONG_BIT** (see the **LIMITS** section) is
1410 **64**, then each integer has **9** decimal digits. If built in an environment
1411 where **BC_LONG_BIT** is **32** then each integer has **4** decimal digits. This
1412 value (the number of decimal digits per large integer) is called
1415 The actual values of **BC_LONG_BIT** and **BC_BASE_DIGS** can be queried with
1416 the **limits** statement.
1418 In addition, this bc(1) uses an even larger integer for overflow checking. This
1419 integer type depends on the value of **BC_LONG_BIT**, but is always at least
1420 twice as large as the integer type used to store digits.
1424 The following are the limits on bc(1):
1428 : The number of bits in the **long** type in the environment where bc(1) was
1429 built. This determines how many decimal digits can be stored in a single
1430 large integer (see the **PERFORMANCE** section).
1434 : The number of decimal digits per large integer (see the **PERFORMANCE**
1435 section). Depends on **BC_LONG_BIT**.
1439 : The max decimal number that each large integer can store (see
1440 **BC_BASE_DIGS**) plus **1**. Depends on **BC_BASE_DIGS**.
1444 : The max number that the overflow type (see the **PERFORMANCE** section) can
1445 hold. Depends on **BC_LONG_BIT**.
1449 : The maximum output base. Set at **BC_BASE_POW**.
1453 : The maximum size of arrays. Set at **SIZE_MAX-1**.
1457 : The maximum **scale**. Set at **BC_OVERFLOW_MAX-1**.
1461 : The maximum length of strings. Set at **BC_OVERFLOW_MAX-1**.
1465 : The maximum length of identifiers. Set at **BC_OVERFLOW_MAX-1**.
1469 : The maximum length of a number (in decimal digits), which includes digits
1470 after the decimal point. Set at **BC_OVERFLOW_MAX-1**.
1474 : The maximum integer (inclusive) returned by the **rand()** operand. Set at
1475 **2\^BC_LONG_BIT-1**.
1479 : The maximum allowable exponent (positive or negative). Set at
1480 **BC_OVERFLOW_MAX**.
1484 : The maximum number of vars/arrays. Set at **SIZE_MAX-1**.
1486 The actual values can be queried with the **limits** statement.
1488 These limits are meant to be effectively non-existent; the limits are so large
1489 (at least on 64-bit machines) that there should not be any point at which they
1490 become a problem. In fact, memory should be exhausted before these limits should
1493 # ENVIRONMENT VARIABLES
1495 bc(1) recognizes the following environment variables:
1499 : If this variable exists (no matter the contents), bc(1) behaves as if
1500 the **-s** option was given.
1504 : This is another way to give command-line arguments to bc(1). They should be
1505 in the same format as all other command-line arguments. These are always
1506 processed first, so any files given in **BC_ENV_ARGS** will be processed
1507 before arguments and files given on the command-line. This gives the user
1508 the ability to set up "standard" options and files to be used at every
1509 invocation. The most useful thing for such files to contain would be useful
1510 functions that the user might want every time bc(1) runs.
1512 The code that parses **BC_ENV_ARGS** will correctly handle quoted arguments,
1513 but it does not understand escape sequences. For example, the string
1514 **"/home/gavin/some bc file.bc"** will be correctly parsed, but the string
1515 **"/home/gavin/some \"bc\" file.bc"** will include the backslashes.
1517 The quote parsing will handle either kind of quotes, **'** or **"**. Thus,
1518 if you have a file with any number of single quotes in the name, you can use
1519 double quotes as the outside quotes, as in **"some 'bc' file.bc"**, and vice
1520 versa if you have a file with double quotes. However, handling a file with
1521 both kinds of quotes in **BC_ENV_ARGS** is not supported due to the
1522 complexity of the parsing, though such files are still supported on the
1523 command-line where the parsing is done by the shell.
1527 : If this environment variable exists and contains an integer that is greater
1528 than **1** and is less than **UINT16_MAX** (**2\^16-1**), bc(1) will output
1529 lines to that length, including the backslash (**\\**). The default line
1534 bc(1) returns the following exit statuses:
1542 : A math error occurred. This follows standard practice of using **1** for
1543 expected errors, since math errors will happen in the process of normal
1546 Math errors include divide by **0**, taking the square root of a negative
1547 number, using a negative number as a bound for the pseudo-random number
1548 generator, attempting to convert a negative number to a hardware integer,
1549 overflow when converting a number to a hardware integer, and attempting to
1550 use a non-integer where an integer is required.
1552 Converting to a hardware integer happens for the second operand of the power
1553 (**\^**), places (**\@**), left shift (**\<\<**), and right shift (**\>\>**)
1554 operators and their corresponding assignment operators.
1558 : A parse error occurred.
1560 Parse errors include unexpected **EOF**, using an invalid character, failing
1561 to find the end of a string or comment, using a token where it is invalid,
1562 giving an invalid expression, giving an invalid print statement, giving an
1563 invalid function definition, attempting to assign to an expression that is
1564 not a named expression (see the *Named Expressions* subsection of the
1565 **SYNTAX** section), giving an invalid **auto** list, having a duplicate
1566 **auto**/function parameter, failing to find the end of a code block,
1567 attempting to return a value from a **void** function, attempting to use a
1568 variable as a reference, and using any extensions when the option **-s** or
1569 any equivalents were given.
1573 : A runtime error occurred.
1575 Runtime errors include assigning an invalid number to **ibase**, **obase**,
1576 or **scale**; give a bad expression to a **read()** call, calling **read()**
1577 inside of a **read()** call, type errors, passing the wrong number of
1578 arguments to functions, attempting to call an undefined function, and
1579 attempting to use a **void** function call as a value in an expression.
1583 : A fatal error occurred.
1585 Fatal errors include memory allocation errors, I/O errors, failing to open
1586 files, attempting to use files that do not have only ASCII characters (bc(1)
1587 only accepts ASCII characters), attempting to open a directory as a file,
1588 and giving invalid command-line options.
1590 The exit status **4** is special; when a fatal error occurs, bc(1) always exits
1591 and returns **4**, no matter what mode bc(1) is in.
1593 The other statuses will only be returned when bc(1) is not in interactive mode
1594 (see the **INTERACTIVE MODE** section), since bc(1) resets its state (see the
1595 **RESET** section) and accepts more input when one of those errors occurs in
1596 interactive mode. This is also the case when interactive mode is forced by the
1597 **-i** flag or **-\-interactive** option.
1599 These exit statuses allow bc(1) to be used in shell scripting with error
1600 checking, and its normal behavior can be forced by using the **-i** flag or
1601 **-\-interactive** option.
1605 Per the [standard][1], bc(1) has an interactive mode and a non-interactive mode.
1606 Interactive mode is turned on automatically when both **stdin** and **stdout**
1607 are hooked to a terminal, but the **-i** flag and **-\-interactive** option can
1608 turn it on in other cases.
1610 In interactive mode, bc(1) attempts to recover from errors (see the **RESET**
1611 section), and in normal execution, flushes **stdout** as soon as execution is
1612 done for the current input.
1616 If **stdin**, **stdout**, and **stderr** are all connected to a TTY, bc(1) turns
1619 TTY mode is different from interactive mode because interactive mode is required
1620 in the [bc(1) specification][1], and interactive mode requires only **stdin**
1621 and **stdout** to be connected to a terminal.
1625 Sending a **SIGINT** will cause bc(1) to stop execution of the current input. If
1626 bc(1) is in TTY mode (see the **TTY MODE** section), it will reset (see the
1627 **RESET** section). Otherwise, it will clean up and exit.
1629 Note that "current input" can mean one of two things. If bc(1) is processing
1630 input from **stdin** in TTY mode, it will ask for more input. If bc(1) is
1631 processing input from a file in TTY mode, it will stop processing the file and
1632 start processing the next file, if one exists, or ask for input from **stdin**
1633 if no other file exists.
1635 This means that if a **SIGINT** is sent to bc(1) as it is executing a file, it
1636 can seem as though bc(1) did not respond to the signal since it will immediately
1637 start executing the next file. This is by design; most files that users execute
1638 when interacting with bc(1) have function definitions, which are quick to parse.
1639 If a file takes a long time to execute, there may be a bug in that file. The
1640 rest of the files could still be executed without problem, allowing the user to
1643 **SIGTERM** and **SIGQUIT** cause bc(1) to clean up and exit, and it uses the
1644 default handler for all other signals.
1648 This bc(1) ships with support for adding error messages for different locales
1649 and thus, supports **LC_MESSAGES**.
1657 bc(1) is compliant with the [IEEE Std 1003.1-2017 (“POSIX.1-2017”)][1]
1658 specification. The flags **-efghiqsvVw**, all long options, and the extensions
1659 noted above are extensions to that specification.
1661 Note that the specification explicitly says that bc(1) only accepts numbers that
1662 use a period (**.**) as a radix point, regardless of the value of
1665 This bc(1) supports error messages for different locales, and thus, it supports
1670 None are known. Report bugs at https://git.yzena.com/gavin/bc.
1674 Gavin D. Howard <gavin@yzena.com> and contributors.
1676 [1]: https://pubs.opengroup.org/onlinepubs/9699919799/utilities/bc.html
1677 [2]: https://www.gnu.org/software/bc/
1678 [3]: https://en.wikipedia.org/wiki/Rounding#Round_half_away_from_zero
1679 [4]: https://en.wikipedia.org/wiki/Unit_in_the_last_place
1680 [5]: https://people.eecs.berkeley.edu/~wkahan/LOG10HAF.TXT
1681 [6]: https://en.wikipedia.org/wiki/Rounding#Rounding_away_from_zero