@node Integer Properties
@section Integer Properties

@c Copyright (C) 2011--2021 Free Software Foundation, Inc.

@c Permission is granted to copy, distribute and/or modify this document
@c under the terms of the GNU Free Documentation License, Version 1.3 or
@c any later version published by the Free Software Foundation; with no
@c Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts.  A
@c copy of the license is at <https://www.gnu.org/licenses/fdl-1.3.en.html>.

@c Written by Paul Eggert.

@cindex integer properties

The @code{intprops} module consists of an include file @code{<intprops.h>}
that defines several macros useful for testing properties of integer
types.

@cindex integer overflow
@cindex overflow, integer

Integer overflow is a common source of problems in programs written in
C and other languages.  In some cases, such as signed integer
arithmetic in C programs, the resulting behavior is undefined, and
practical platforms do not always behave as if integers wrap around
reliably.  In other cases, such as unsigned integer arithmetic in C,
the resulting behavior is well-defined, but programs may still
misbehave badly after overflow occurs.

Many techniques have been proposed to attack these problems.  These
include precondition testing, wraparound behavior where signed integer
arithmetic is guaranteed to be modular, saturation semantics where
overflow reliably yields an extreme value, undefined behavior
sanitizers where overflow is guaranteed to trap, and various static
analysis techniques.

Gnulib supports wraparound arithmetic and precondition testing, as
these are relatively easy to support portably and efficiently.  There
are two families of precondition tests: the first, for integer types,
is easier to use, while the second, for integer ranges, has a simple
and straightforward portable implementation.

Like other Gnulib modules, the implementation of the @code{intprops}
module assumes that integers use a two's complement representation,
but it does not assume that signed integer arithmetic wraps around.
@xref{Other portability assumptions}.

@menu
* Arithmetic Type Properties::  Determining properties of arithmetic types.
* Integer Bounds::              Bounds on integer values and representations.
* Checking Integer Overflow::   Checking for overflow while computing integers.
* Wraparound Arithmetic::       Well-defined behavior on integer overflow.
* Integer Type Overflow::       General integer overflow checking.
* Integer Range Overflow::      Integer overflow checking if bounds are known.
@end menu

@node Arithmetic Type Properties
@subsection Arithmetic Type Properties

@findex TYPE_IS_INTEGER
@code{TYPE_IS_INTEGER (@var{t})} is an arithmetic constant
expression that is 1 if the arithmetic type @var{t} is an integer type.
@code{_Bool} counts as an integer type.

@findex TYPE_SIGNED
@code{TYPE_SIGNED (@var{t})} is an arithmetic constant expression
that is 1 if the real type @var{t} is a signed integer type or a
floating type.  If @var{t} is an integer type, @code{TYPE_SIGNED (@var{t})}
is an integer constant expression.

@findex EXPR_SIGNED
@code{EXPR_SIGNED (@var{e})} is 1 if the real expression @var{e}
has a signed integer type or a floating type.  If @var{e} is an
integer constant expression or an arithmetic constant expression,
@code{EXPR_SIGNED (@var{e})} is likewise.  The expression
@var{e} is not evaluated, and @code{EXPR_SIGNED
(@var{e})} is typically optimized to a constant.

Example usage:

@example
#include <intprops.h>
#include <time.h>

enum
@{
  time_t_is_signed_integer =
    TYPE_IS_INTEGER (time_t) && TYPE_SIGNED (time_t)
@};

int
CLOCKS_PER_SEC_is_signed (void)
@{
  return EXPR_SIGNED (CLOCKS_PER_SEC);
@}
@end example

@node Integer Bounds
@subsection Integer Bounds

@cindex integer bounds

@findex INT_BUFSIZE_BOUND
@code{INT_BUFSIZE_BOUND (@var{t})} is an integer constant
expression that is a bound on the size of the string representing an
integer type or expression @var{t} in decimal notation, including the
terminating null character and any leading @code{-} character.  For
example, if @code{INT_BUFSIZE_BOUND (int)} is 12, any value of type
@code{int} can be represented in 12 bytes or less, including the
terminating null.  The bound is not necessarily tight.

Example usage:

@example
#include <intprops.h>
#include <stdio.h>
int
int_strlen (int i)
@{
  char buf[INT_BUFSIZE_BOUND (int)];
  return sprintf (buf, "%d", i);
@}
@end example

@findex INT_STRLEN_BOUND
@code{INT_STRLEN_BOUND (@var{t})} is an integer constant
expression that is a bound on the length of the string representing an
integer type or expression @var{t} in decimal notation, including any
leading @code{-} character.  This is one less than
@code{INT_BUFSIZE_BOUND (@var{t})}.

@findex TYPE_MINIMUM
@findex TYPE_MAXIMUM
@code{TYPE_MINIMUM (@var{t})} and @code{TYPE_MAXIMUM (@var{t})} are
integer constant expressions equal to the minimum and maximum
values of the integer type @var{t}.  These expressions are of the type
@var{t} (or more precisely, the type @var{t} after integer
promotions).

Example usage:

@example
#include <stdint.h>
#include <sys/types.h>
#include <intprops.h>
int
in_off_t_range (intmax_t a)
@{
  return TYPE_MINIMUM (off_t) <= a && a <= TYPE_MAXIMUM (off_t);
@}
@end example

@node Checking Integer Overflow
@subsection Checking Integer Overflow

@cindex integer overflow checking

Signed integer arithmetic has undefined behavior on overflow in C@.
Although almost all modern computers use two's complement signed
arithmetic that is well-defined to wrap around, C compilers routinely
optimize assuming that signed integer overflow cannot occur, which
means that a C program cannot easily get at the underlying machine
arithmetic.  For example:

@example
if ((a + b < b) == (a < 0))
  a += b;
else
  printf ("overflow\n");
@end example

@noindent
might not work as expected if @code{a} and @code{b} are signed,
because a compiler can assume that signed overflow cannot occur and
treat the entire @code{if} expression as if it were true.  And even if
@code{a} is unsigned, the expression might not work as expected if
@code{b} is negative or is wider than @code{a}.

The following macros work around this problem by returning an overflow
indication while computing the sum, difference, or product of two
integers.  For example, if @code{i} is of type @code{int},
@code{INT_ADD_OK (INT_MAX - 1, 1, &i)} sets @code{i} to
@code{INT_MAX} and returns true, whereas @code{INT_ADD_OK (INT_MAX, 1,
&i)} returns false.

Example usage:

@example
#include <intprops.h>
#include <stdio.h>

/* Compute A * B, reporting whether overflow occurred.  */
void
print_product (long int a, long int b)
@{
  long int r;
  if (INT_MULTIPLY_OK (a, b, r))
    printf ("result is %ld\n", r);
  else
    printf ("overflow\n");
@}
@end example

These macros work for both signed and unsigned integers, so they can
be used with integer types like @code{time_t} that may or may not be
signed, depending on the platform.

These macros have the following restrictions:

@itemize @bullet
@item
Their first two arguments must be integer expressions.

@item
Their last argument must be a non-null pointer to an integer.

@item
They may evaluate their arguments zero or multiple times, so the
arguments should not have side effects.

@item
They are not necessarily constant expressions, even if all their
arguments are constant expressions.
@end itemize

@table @code
@item INT_ADD_OK (@var{a}, @var{b}, @var{r})
@findex INT_ADD_OK
Compute the sum of @var{a} and @var{b}.  If it fits into
@code{*@var{r}}, store it there and return true.  Otherwise return
false, possibly modifying @code{*@var{r}} to an unspecified value.
See above for restrictions.

@item INT_SUBTRACT_OK (@var{a}, @var{b}, @var{r})
@findex INT_SUBTRACT_OK
Compute the difference between @var{a} and @var{b}.  If it fits into
@code{*@var{r}}, store it there and return true.  Otherwise return
false, possibly modifying @code{*@var{r}} to an unspecified value.
See above for restrictions.

@item INT_MULTIPLY_OK (@var{a}, @var{b}, @var{r})
@findex INT_MULTIPLY_OK
Compute the product of @var{a} and @var{b}.  If it fits into
@code{*@var{r}}, store it there and return true.  Otherwise return
false, possibly modifying @code{*@var{r}} to an unspecified value.
See above for restrictions.
@end table

Other macros are available if you need wrapped-around results when
overflow occurs (@pxref{Wraparound Arithmetic}), or if you need to
check for overflow in operations other than addition, subtraction, and
multiplication (@pxref{Integer Type Overflow}).

@node Wraparound Arithmetic
@subsection Wraparound Arithmetic with Integers

@cindex wraparound integer arithmetic

Signed integer arithmetic has undefined behavior on overflow in C@.
Although almost all modern computers use two's complement signed
arithmetic that is well-defined to wrap around, C compilers routinely
optimize assuming that signed integer overflow cannot occur, which
means that a C program cannot easily get at the underlying machine
arithmetic.  For example, on a typical machine with 32-bit two's
complement @code{int} the expression @code{INT_MAX + 1} does not
necessarily yield @code{INT_MIN}, because the compiler may do
calculations with a 64-bit register, or may generate code that
traps on signed integer overflow.

The following macros work around this problem by storing the
wraparound value, i.e., the low-order bits of the correct answer, and
by returning an overflow indication.  For example, if @code{i} is of
type @code{int}, @code{INT_ADD_WRAPV (INT_MAX, 1, &i)} sets @code{i}
to @code{INT_MIN} and returns 1 on a two's complement machine.
@xref{Integer Type Overflow}.

Example usage:

@example
#include <intprops.h>
#include <stdio.h>

/* Print the low order bits of A * B,
   reporting whether overflow occurred.  */
void
print_product (long int a, long int b)
@{
  long int r;
  int overflow = INT_MULTIPLY_WRAPV (a, b, &r);
  printf ("result is %ld (%s)\n", r,
          (overflow
           ? "after overflow"
           : "no overflow"));
@}
@end example

These macros work for both signed and unsigned integers, so they can
be used with integer types like @code{time_t} that may or may not be
signed, depending on the platform.

These macros have the following restrictions:

@itemize @bullet
@item
Their first two arguments must be integer expressions.

@item
Their last argument must be a non-null pointer to an integer.

@item
They may evaluate their arguments zero or multiple times, so the
arguments should not have side effects.

@item
They are not necessarily constant expressions, even if all their
arguments are constant expressions.
@end itemize

@table @code
@item INT_ADD_WRAPV (@var{a}, @var{b}, @var{r})
@findex INT_ADD_WRAPV
Store the low-order bits of the sum of @var{a} and @var{b} into
@code{*@var{r}}.  Return true if overflow occurred, false if the
low-order bits are the mathematically-correct sum.  See above for
restrictions.

@item INT_SUBTRACT_WRAPV (@var{a}, @var{b}, @var{r})
@findex INT_SUBTRACT_WRAPV
Store the low-order bits of the difference between @var{a} and @var{b}
into @code{*@var{r}}.  Return true if overflow occurred, false if the
low-order bits are the mathematically-correct difference.  See above
for restrictions.

@item INT_MULTIPLY_WRAPV (@var{a}, @var{b}, @var{r})
@findex INT_MULTIPLY_WRAPV
Store the low-order bits of the product of @var{a} and @var{b} into
@code{*@var{r}}.  Return true if overflow occurred, false if the
low-order bits are the mathematically-correct product.  See above for
restrictions.
@end table

Other macros are available if you do not need wrapped-around results
when overflow occurs (@pxref{Checking Integer Overflow}), or if you
need to check for overflow in operations other than addition,
subtraction, and multiplication (@pxref{Integer Type Overflow}).

@node Integer Type Overflow
@subsection Integer Type Overflow

@cindex integer type overflow
@cindex overflow, integer type

Although unsigned integer arithmetic wraps around modulo a power of
two, signed integer arithmetic has undefined behavior on overflow in
C@.  Almost all modern computers use two's complement signed
arithmetic that is well-defined to wrap around, but C compilers
routinely optimize based on the assumption that signed integer
overflow cannot occur, which means that a C program cannot easily get
at the underlying machine behavior.  For example, the signed integer
expression @code{(a + b < b) != (a < 0)} is not a reliable test for
whether @code{a + b} overflows, because a compiler can assume that
signed overflow cannot occur and treat the entire expression as if it
were false.

These macros yield 1 if the corresponding C operators might not yield
numerically correct answers due to arithmetic overflow of an integer
type.  They work correctly on all known practical hosts, and do not
rely on undefined behavior due to signed arithmetic overflow.  They
are integer constant expressions if their arguments are.  They
are typically easier to use than the integer range overflow macros
(@pxref{Integer Range Overflow}), and they support more operations and
evaluation contexts than the wraparound macros (@pxref{Wraparound
Arithmetic}).

Example usage:

@example
#include <intprops.h>
#include <limits.h>
#include <stdio.h>

/* Print A * B if in range, an overflow
   indicator otherwise.  */
void
print_product (long int a, long int b)
@{
  if (INT_MULTIPLY_OVERFLOW (a, b))
    printf ("multiply would overflow");
  else
    printf ("product is %ld", a * b);
@}

/* Does the product of two ints always fit
   in a long int?  */
enum @{
  INT_PRODUCTS_FIT_IN_LONG
    = ! (INT_MULTIPLY_OVERFLOW
         ((long int) INT_MIN, INT_MIN))
@};
@end example

@noindent
These macros have the following restrictions:

@itemize @bullet
@item
Their arguments must be integer expressions.

@item
They may evaluate their arguments zero or multiple times, so the
arguments should not have side effects.
@end itemize

@noindent
These macros are tuned for their last argument being a constant.

@table @code
@item INT_ADD_OVERFLOW (@var{a}, @var{b})
@findex INT_ADD_OVERFLOW
Yield 1 if @code{@var{a} + @var{b}} would overflow.  See above for
restrictions.

@item INT_SUBTRACT_OVERFLOW (@var{a}, @var{b})
@findex INT_SUBTRACT_OVERFLOW
Yield 1 if @code{@var{a} - @var{b}} would overflow.  See above for
restrictions.

@item INT_NEGATE_OVERFLOW (@var{a})
@findex INT_NEGATE_OVERFLOW
Yields 1 if @code{-@var{a}} would overflow.  See above for restrictions.

@item INT_MULTIPLY_OVERFLOW (@var{a}, @var{b})
@findex INT_MULTIPLY_OVERFLOW
Yield 1 if @code{@var{a} * @var{b}} would overflow.  See above for
restrictions.

@item INT_DIVIDE_OVERFLOW (@var{a}, @var{b})
@findex INT_DIVIDE_OVERFLOW
Yields 1 if @code{@var{a} / @var{b}} would overflow.  See above for
restrictions.  Division overflow can happen on two's complement hosts
when dividing the most negative integer by @minus{}1.  This macro does
not check for division by zero.

@item INT_REMAINDER_OVERFLOW (@var{a}, @var{b})
@findex INT_REMAINDER_OVERFLOW
Yield 1 if @code{@var{a} % @var{b}} would overflow.  See above for
restrictions.  Remainder overflow can happen on two's complement hosts
when dividing the most negative integer by @minus{}1; although the
mathematical result is always 0, in practice some implementations
trap, so this counts as an overflow.  This macro does not check for
division by zero.

@item INT_LEFT_SHIFT_OVERFLOW (@var{a}, @var{b})
@findex INT_LEFT_SHIFT_OVERFLOW
Yield 1 if @code{@var{a} << @var{b}} would overflow.  See above for
restrictions.  The C standard says that behavior is undefined for
shifts unless 0@leq{}@var{b}<@var{w} where @var{w} is @var{a}'s word
width, and that when @var{a} is negative then @code{@var{a} <<
@var{b}} has undefined behavior, but this macro does not check these
other restrictions.
@end table

@node Integer Range Overflow
@subsection Integer Range Overflow

@cindex integer range overflow
@cindex overflow, integer range

These macros yield 1 if the corresponding C operators might not yield
numerically correct answers due to arithmetic overflow.  They do not
rely on undefined or implementation-defined behavior.  They are
integer constant expressions if their arguments are.  Their
implementations are simple and straightforward, but they are typically
harder to use than the integer type overflow macros.  @xref{Integer
Type Overflow}.

Although the implementation of these macros is similar to that
suggested in the SEI CERT C Secure Coding Standard,
in its two sections
``@url{https://www.securecoding.cert.org/confluence/display/c/INT30-C.+Ensure+that+unsigned+integer+operations+do+not+wrap,
INT30-C@. Ensure that unsigned integer operations do not wrap}'' and
``@url{https://www.securecoding.cert.org/confluence/display/c/INT32-C.+Ensure+that+operations+on+signed+integers+do+not+result+in+overflow,
INT32-C@. Ensure that operations on signed integers do not result in
overflow}'', Gnulib's implementation was derived independently of
CERT's suggestions.

Example usage:

@example
#include <intprops.h>
#include <limits.h>
#include <stdio.h>

void
print_product (long int a, long int b)
@{
  if (INT_MULTIPLY_RANGE_OVERFLOW (a, b, LONG_MIN, LONG_MAX))
    printf ("multiply would overflow");
  else
    printf ("product is %ld", a * b);
@}

/* Does the product of two ints always fit
   in a long int?  */
enum @{
  INT_PRODUCTS_FIT_IN_LONG
    = ! (INT_MULTIPLY_RANGE_OVERFLOW
         ((long int) INT_MIN, (long int) INT_MIN,
          LONG_MIN, LONG_MAX))
@};
@end example

@noindent
These macros have the following restrictions:

@itemize @bullet
@item
Their arguments must be integer expressions.

@item
They may evaluate their arguments zero or multiple times, so
the arguments should not have side effects.

@item
The arithmetic arguments (including the @var{min} and @var{max}
arguments) must be of the same integer type after the usual arithmetic
conversions, and the type must have minimum value @var{min} and
maximum @var{max}.  Unsigned values should use a zero @var{min} of the
proper type, for example, @code{(unsigned int) 0}.
@end itemize

@noindent
These macros are tuned for constant @var{min} and @var{max}.  For
commutative operations such as @code{@var{a} + @var{b}}, they are also
tuned for constant @var{b}.

@table @code
@item INT_ADD_RANGE_OVERFLOW (@var{a}, @var{b}, @var{min}, @var{max})
@findex INT_ADD_RANGE_OVERFLOW
Yield 1 if @code{@var{a} + @var{b}} would overflow in
[@var{min},@var{max}] integer arithmetic.  See above for restrictions.

@item INT_SUBTRACT_RANGE_OVERFLOW (@var{a}, @var{b}, @var{min}, @var{max})
@findex INT_SUBTRACT_RANGE_OVERFLOW
Yield 1 if @code{@var{a} - @var{b}} would overflow in
[@var{min},@var{max}] integer arithmetic.  See above for restrictions.

@item INT_NEGATE_RANGE_OVERFLOW (@var{a}, @var{min}, @var{max})
@findex INT_NEGATE_RANGE_OVERFLOW
Yield 1 if @code{-@var{a}} would overflow in [@var{min},@var{max}]
integer arithmetic.  See above for restrictions.

@item INT_MULTIPLY_RANGE_OVERFLOW (@var{a}, @var{b}, @var{min}, @var{max})
@findex INT_MULTIPLY_RANGE_OVERFLOW
Yield 1 if @code{@var{a} * @var{b}} would overflow in
[@var{min},@var{max}] integer arithmetic.  See above for restrictions.

@item INT_DIVIDE_RANGE_OVERFLOW (@var{a}, @var{b}, @var{min}, @var{max})
@findex INT_DIVIDE_RANGE_OVERFLOW
Yield 1 if @code{@var{a} / @var{b}} would overflow in
[@var{min},@var{max}] integer arithmetic.  See above for restrictions.
Division overflow can happen on two's complement hosts when dividing
the most negative integer by @minus{}1.  This macro does not check for
division by zero.

@item INT_REMAINDER_RANGE_OVERFLOW (@var{a}, @var{b}, @var{min}, @var{max})
@findex INT_REMAINDER_RANGE_OVERFLOW
Yield 1 if @code{@var{a} % @var{b}} would overflow in
[@var{min},@var{max}] integer arithmetic.  See above for restrictions.
Remainder overflow can happen on two's complement hosts when dividing
the most negative integer by @minus{}1; although the mathematical
result is always 0, in practice some implementations trap, so this
counts as an overflow.  This macro does not check for division by
zero.

@item INT_LEFT_SHIFT_RANGE_OVERFLOW (@var{a}, @var{b}, @var{min}, @var{max})
@findex INT_LEFT_SHIFT_RANGE_OVERFLOW
Yield 1 if @code{@var{a} << @var{b}} would overflow in
[@var{min},@var{max}] integer arithmetic.  See above for restrictions.
Here, @var{min} and @var{max} are for @var{a} only, and @var{b} need
not be of the same type as the other arguments.  The C standard says
that behavior is undefined for shifts unless 0@leq{}@var{b}<@var{w}
where @var{w} is @var{a}'s word width, and that when @var{a} is negative
then @code{@var{a} << @var{b}} has undefined behavior, but this macro
does not check these other restrictions.
@end table
