Numerical Limits and Number Classification

1.17 Numerical Limits and Number Classification

When working with C's various numeric types, it's important to understand the range of values that each type can hold.

1.17.1 Value Ranges of Integer Types

The value ranges of the integer types are documented in the header file limits.h. The constants, listed in Table 1-25, indicate the largest and smallest values that can be represented by the given type.

Table 1-25. Limits of the integer types
Type	Minimum	Maximum	Maximum of the unsigned type
char	CHAR_MIN	CHAR_MAX	UCHAR_MAX
signed char	SCHAR_MIN	SCHAR_MAX
short	SHRT_MIN	SHRT_MAX	USHRT_MAX
int	INT_MIN	INT_MAX	UINT_MAX
long	LONG_MIN	LONG_MAX	ULONG_MAX
long long(*)	LLONG_MIN(*)	LLONG_MAX(*)	ULLONG_MAX(*)

If char is interpreted as signed, then CHAR_MIN is equal to SCHAR_MIN and CHAR_MAX is equal to SCHAR_MAX. If not, then CHAR_MIN is equal to 0 and CHAR_MAX is equal to UCHAR_MAX.

In addition to the constants listed in Table 1-25, limits.h also contains the following:

CHAR_BIT

The number of bits in a byte (usually 8)

MB_LEN_MAX

The maximum number of bytes in a multibyte character

In the header file stdint.h^(*), constants are also defined to document the minimum and maximum values of the types wchar_t, wint_t, size_t, ptrdiff_t, and sig_atomic_t, and of the fixed-width integer types, such as int_least32_t. The names of these constants are formed from the type names as follows: the type name is written all in capital letters, and the suffix _t is replaced by _MIN or _MAX. For example:

WCHAR_MIN  // Minimum value of wchar_t

INT_LEAST32_MAX  // Maximum value of int_least32_t

For the unsigned types only the ..._MAX constants are defined.

1.17.2 Range and Precision of Real Floating Types

The macros listed in Table 1-26 are defined in the header file float.h to represent the range and the precision of the types float, double, and long double. The macro names are formed using the prefixes FLT for float, DBL for double, and LDBL for long double. The macros FLT_RADIX and FLT_ROUNDS apply to all three floating types.

Table 1-26. Macros for floating types in float.h
Macro name	Purpose
FLT_RADIX	Base (or radix) of the exponential notation
FLT_ROUNDS	Indicates how rounding is performed on values that cannot be represented exactly: -1 = undetermined 0 = towards zero, 1 = towards the nearest representable value 2 = upwards 3 = downwards
FLT_MANT_DIG DBL_MANT_DIG LDBL_MANT_DIG	The number of digits in the mantissa to base FLT_RADIX
FLT_MIN_EXP DBL_MIN_EXP LDBL_MIN_EXP	Minimum value of the exponent to base FLT_RADIX
FLT_MAX_EXP DBL_MAX_EXP LDBL_MAX_EXP	Maximum value of the exponent to base FLT_RADIX

The macros listed in Table 1-26 document the range and precision of all real floating types. In actual programs, such information is most often needed for decimal (base 10) notation. Accordingly, you can use the macros for type float listed Table 1-27, and which are defined in float.h.

Table 1-27. Limits for the type float
Macro name	Purpose
FLT_DIG	Precision as a number of decimal digits
FLT_MIN_10_EXP	Minimum negative exponent to base 10
FLT_MAX_10_EXP	Maximum positive exponent to base 10
FLT_MIN	Minimum representable positive floating-point number
FLT_MAX	Maximum representable floating-point number
FLT_EPSILON	Minimum positive representable floating-point number x such that 1.0 + x != 1.0

Similar constants are also defined for the types double and long double. These have names beginning with DBL or LDBL in place of FLT.

ANSI C99 also introduces the macro DECIMAL_DIG, which indicates the precision of the largest floating type as a number of decimal digits.

1.17.3 Classification of Floating-Point Numbers

ANSI C99 defines five categories of floating-point numbers, listed in Table 1-28. A symbolic constant for each category is defined in the header file math.h.

Table 1-28. Floating-point categories
Macro name	Category
FP_ZERO	Floating-point numbers with the value 0
FP_NORMAL	Floating-point numbers in normalized representation
FP_SUBNORMAL[1] [1] Tiny numbers may be represented in subnormal notation.	Floating-point numbers in subnormal representation
FP_INFINITE	Floating-point numbers that represent an infinite value
FP_NAN	Not a Number (NAN): bit patterns that do not represent a valid floating-point number

^[1] Tiny numbers may be represented in subnormal notation.

The macros in Table 1-29 can be used to classify a real floating-point number x with respect to the categories in Table 1-28 without causing an error condition.

Table 1-29. Macros for floating-point number classification
Macro	Result
fpclassify(x)	Returns one of the constants described in Table 1-28 to indicate the category to which x belongs.
isfinite(x)	Returns "true" (i. e., a value other than 0) if the value of x is finite (0, normal, subnormal, not infinite, or NAN), otherwise 0.
isinf(x)	Returns "true" if x is an infinity, otherwise 0.
isnormal(x)	Returns "true" if the value of x is a normalized floating-point number not equal to 0. Returns 0 in all other cases.
isnan(x)	Returns "true" if x is "not a number" (NaN), otherwise 0.
signbit(x)	Returns "true" if x is negative (i. e., if the sign bit is set), otherwise 0

The following constants are also defined in math.h:

INFINITY

The maximum positive value of type float, used to represent infinity.

NAN (Not a Number)

A value of type float which is not a valid floating-point number.

NANs can be either quiet or signaling. If a signaling NAN occurs in the evaluation of an arithmetic expression, the exception status flag FE_INVALID in the floating point environment is set. This flag is not set when a quiet NAN occurs.

C implementations are not required to support the concept of NANs. If NANs are not supported, the constant NAN is not defined.