17 Language support library [support]

17.3 Implementation properties [support.limits]

17.3.5 Class template numeric_­limits [numeric.limits]

17.3.5.1 General [numeric.limits.general]

1
#
The numeric_­limits class template provides a C++ program with information about various properties of the implementation's representation of the arithmetic types.
namespace std { template<class T> class numeric_limits { public: static constexpr bool is_specialized = false; static constexpr T min() noexcept { return T(); } static constexpr T max() noexcept { return T(); } static constexpr T lowest() noexcept { return T(); } static constexpr int digits = 0; static constexpr int digits10 = 0; static constexpr int max_digits10 = 0; static constexpr bool is_signed = false; static constexpr bool is_integer = false; static constexpr bool is_exact = false; static constexpr int radix = 0; static constexpr T epsilon() noexcept { return T(); } static constexpr T round_error() noexcept { return T(); } static constexpr int min_exponent = 0; static constexpr int min_exponent10 = 0; static constexpr int max_exponent = 0; static constexpr int max_exponent10 = 0; static constexpr bool has_infinity = false; static constexpr bool has_quiet_NaN = false; static constexpr bool has_signaling_NaN = false; static constexpr float_denorm_style has_denorm = denorm_absent; static constexpr bool has_denorm_loss = false; static constexpr T infinity() noexcept { return T(); } static constexpr T quiet_NaN() noexcept { return T(); } static constexpr T signaling_NaN() noexcept { return T(); } static constexpr T denorm_min() noexcept { return T(); } static constexpr bool is_iec559 = false; static constexpr bool is_bounded = false; static constexpr bool is_modulo = false; static constexpr bool traps = false; static constexpr bool tinyness_before = false; static constexpr float_round_style round_style = round_toward_zero; }; }
2
#
For all members declared static constexpr in the numeric_­limits template, specializations shall define these values in such a way that they are usable as constant expressions.
3
#
For the numeric_­limits primary template, all data members are value-initialized and all member functions return a value-initialized object.
[Note 1:
This means all members have zero or false values unless numeric_­limits is specialized for a type.
— end note]
4
#
Specializations shall be provided for each arithmetic type, both floating-point and integer, including bool.
The member is_­specialized shall be true for all such specializations of numeric_­limits.
5
#
The value of each member of a specialization of numeric_­limits on a cv-qualified type cv T shall be equal to the value of the corresponding member of the specialization on the unqualified type T.
6
#
Non-arithmetic standard types, such as complex<T>, shall not have specializations.

17.3.5.2 numeric_­limits members [numeric.limits.members]

1
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Each member function defined in this subclause is signal-safe.
🔗
static constexpr T min() noexcept;
2
#
Minimum finite value.189
3
#
For floating-point types with subnormal numbers, returns the minimum positive normalized value.
4
#
Meaningful for all specializations in which is_­bounded != false, or is_­bounded == false && is_­signed == false.
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static constexpr T max() noexcept;
5
#
Maximum finite value.190
6
#
Meaningful for all specializations in which is_­bounded != false.
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static constexpr T lowest() noexcept;
7
#
A finite value x such that there is no other finite value y where y < x.191
8
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Meaningful for all specializations in which is_­bounded != false.
🔗
static constexpr int digits;
9
#
Number of radix digits that can be represented without change.
10
#
For integer types, the number of non-sign bits in the representation.
11
#
For floating-point types, the number of radix digits in the mantissa.192
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static constexpr int digits10;
12
#
Number of base 10 digits that can be represented without change.193
13
#
Meaningful for all specializations in which is_­bounded != false.
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static constexpr int max_digits10;
14
#
Number of base 10 digits required to ensure that values which differ are always differentiated.
15
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Meaningful for all floating-point types.
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static constexpr bool is_signed;
16
#
true if the type is signed.
17
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Meaningful for all specializations.
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static constexpr bool is_integer;
18
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true if the type is integer.
19
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Meaningful for all specializations.
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static constexpr bool is_exact;
20
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true if the type uses an exact representation.
All integer types are exact, but not all exact types are integer.
For example, rational and fixed-exponent representations are exact but not integer.
21
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Meaningful for all specializations.
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static constexpr int radix;
22
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For floating-point types, specifies the base or radix of the exponent representation (often 2).194
23
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For integer types, specifies the base of the representation.195
24
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Meaningful for all specializations.
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static constexpr T epsilon() noexcept;
25
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Machine epsilon: the difference between 1 and the least value greater than 1 that is representable.196
26
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Meaningful for all floating-point types.
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static constexpr T round_error() noexcept;
27
#
Measure of the maximum rounding error.197
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static constexpr int min_exponent;
28
#
Minimum negative integer such that radix raised to the power of one less than that integer is a normalized floating-point number.198
29
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Meaningful for all floating-point types.
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static constexpr int min_exponent10;
30
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Minimum negative integer such that 10 raised to that power is in the range of normalized floating-point numbers.199
31
#
Meaningful for all floating-point types.
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static constexpr int max_exponent;
32
#
Maximum positive integer such that radix raised to the power one less than that integer is a representable finite floating-point number.200
33
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Meaningful for all floating-point types.
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static constexpr int max_exponent10;
34
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Maximum positive integer such that 10 raised to that power is in the range of representable finite floating-point numbers.201
35
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Meaningful for all floating-point types.
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static constexpr bool has_infinity;
36
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true if the type has a representation for positive infinity.
37
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Meaningful for all floating-point types.
38
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Shall be true for all specializations in which is_­iec559 != false.
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static constexpr bool has_quiet_NaN;
39
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true if the type has a representation for a quiet (non-signaling) “Not a Number”.202
40
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Meaningful for all floating-point types.
41
#
Shall be true for all specializations in which is_­iec559 != false.
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static constexpr bool has_signaling_NaN;
42
#
true if the type has a representation for a signaling “Not a Number”.203
43
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Meaningful for all floating-point types.
44
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Shall be true for all specializations in which is_­iec559 != false.
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static constexpr float_denorm_style has_denorm;
45
#
denorm_­present if the type allows subnormal values (variable number of exponent bits)204, denorm_­absent if the type does not allow subnormal values, and denorm_­indeterminate if it is indeterminate at compile time whether the type allows subnormal values.
46
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Meaningful for all floating-point types.
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static constexpr bool has_denorm_loss;
47
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true if loss of accuracy is detected as a denormalization loss, rather than as an inexact result.205
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static constexpr T infinity() noexcept;
48
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Representation of positive infinity, if available.206
49
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Meaningful for all specializations for which has_­infinity != false.
Required in specializations for which is_­iec559 != false.
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static constexpr T quiet_NaN() noexcept;
50
#
Representation of a quiet “Not a Number”, if available.207
51
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Meaningful for all specializations for which has_­quiet_­NaN != false.
Required in specializations for which is_­iec559 != false.
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static constexpr T signaling_NaN() noexcept;
52
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Representation of a signaling “Not a Number”, if available.208
53
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Meaningful for all specializations for which has_­signaling_­NaN != false.
Required in specializations for which is_­iec559 != false.
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static constexpr T denorm_min() noexcept;
54
#
Minimum positive subnormal value.209
55
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Meaningful for all floating-point types.
56
#
In specializations for which has_­denorm == false, returns the minimum positive normalized value.
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static constexpr bool is_iec559;
57
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true if and only if the type adheres to ISO/IEC/IEEE 60559.210
58
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Meaningful for all floating-point types.
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static constexpr bool is_bounded;
59
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true if the set of values representable by the type is finite.211
[Note 1:
All fundamental types ([basic.fundamental]) are bounded.
This member would be false for arbitrary precision types.
— end note]
60
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Meaningful for all specializations.
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static constexpr bool is_modulo;
61
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true if the type is modulo.212
A type is modulo if, for any operation involving +, -, or * on values of that type whose result would fall outside the range [min(), max()], the value returned differs from the true value by an integer multiple of max() - min() + 1.
62
#
[Example 1:
is_­modulo is false for signed integer types ([basic.fundamental]) unless an implementation, as an extension to this document, defines signed integer overflow to wrap.
— end example]
63
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Meaningful for all specializations.
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static constexpr bool traps;
64
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true if, at the start of the program, there exists a value of the type that would cause an arithmetic operation using that value to trap.213
65
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Meaningful for all specializations.
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static constexpr bool tinyness_before;
66
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true if tinyness is detected before rounding.214
67
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Meaningful for all floating-point types.
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static constexpr float_round_style round_style;
68
#
The rounding style for the type.215
69
#
Meaningful for all floating-point types.
Specializations for integer types shall return round_­toward_­zero.
189)
Equivalent to CHAR_­MIN, SHRT_­MIN, FLT_­MIN, DBL_­MIN, etc.
 ⮥
190)
Equivalent to CHAR_­MAX, SHRT_­MAX, FLT_­MAX, DBL_­MAX, etc.
 ⮥
191)
lowest() is necessary because not all floating-point representations have a smallest (most negative) value that is the negative of the largest (most positive) finite value.
 ⮥
192)
Equivalent to FLT_­MANT_­DIG, DBL_­MANT_­DIG, LDBL_­MANT_­DIG.
 ⮥
193)
Equivalent to FLT_­DIG, DBL_­DIG, LDBL_­DIG.
 ⮥
194)
Equivalent to FLT_­RADIX.
 ⮥
195)
Distinguishes types with bases other than 2 (e.g. BCD).
 ⮥
196)
Equivalent to FLT_­EPSILON, DBL_­EPSILON, LDBL_­EPSILON.
 ⮥
197)
Rounding error is described in LIA-1 Section 5.2.4 and Annex C Rationale Section C.5.2.4 — Rounding and rounding constants.
 ⮥
198)
Equivalent to FLT_­MIN_­EXP, DBL_­MIN_­EXP, LDBL_­MIN_­EXP.
 ⮥
199)
Equivalent to FLT_­MIN_­10_­EXP, DBL_­MIN_­10_­EXP, LDBL_­MIN_­10_­EXP.
 ⮥
200)
Equivalent to FLT_­MAX_­EXP, DBL_­MAX_­EXP, LDBL_­MAX_­EXP.
 ⮥
201)
Equivalent to FLT_­MAX_­10_­EXP, DBL_­MAX_­10_­EXP, LDBL_­MAX_­10_­EXP.
 ⮥
202)
Required by LIA-1.
 ⮥
203)
Required by LIA-1.
 ⮥
204)
Required by LIA-1.
 ⮥
205)
See ISO/IEC/IEEE 60559.
 ⮥
206)
Required by LIA-1.
 ⮥
207)
Required by LIA-1.
 ⮥
208)
Required by LIA-1.
 ⮥
209)
Required by LIA-1.
 ⮥
210)
ISO/IEC/IEEE 60559:2011 is the same as IEEE 754-2008.
 ⮥
211)
Required by LIA-1.
 ⮥
212)
Required by LIA-1.
 ⮥
213)
Required by LIA-1.
 ⮥
214)
Refer to ISO/IEC/IEEE 60559.
Required by LIA-1.
 ⮥
215)
Equivalent to FLT_­ROUNDS.
Required by LIA-1.
 ⮥

17.3.5.3 numeric_­limits specializations [numeric.special]

1
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All members shall be provided for all specializations.
However, many values are only required to be meaningful under certain conditions (for example, epsilon() is only meaningful if is_­integer is false).
Any value that is not “meaningful” shall be set to 0 or false.
2
#
[Example 1: namespace std { template<> class numeric_limits<float> { public: static constexpr bool is_specialized = true; static constexpr float min() noexcept { return 1.17549435E-38F; } static constexpr float max() noexcept { return 3.40282347E+38F; } static constexpr float lowest() noexcept { return -3.40282347E+38F; } static constexpr int digits = 24; static constexpr int digits10 = 6; static constexpr int max_digits10 = 9; static constexpr bool is_signed = true; static constexpr bool is_integer = false; static constexpr bool is_exact = false; static constexpr int radix = 2; static constexpr float epsilon() noexcept { return 1.19209290E-07F; } static constexpr float round_error() noexcept { return 0.5F; } static constexpr int min_exponent = -125; static constexpr int min_exponent10 = - 37; static constexpr int max_exponent = +128; static constexpr int max_exponent10 = + 38; static constexpr bool has_infinity = true; static constexpr bool has_quiet_NaN = true; static constexpr bool has_signaling_NaN = true; static constexpr float_denorm_style has_denorm = denorm_absent; static constexpr bool has_denorm_loss = false; static constexpr float infinity() noexcept { return value; } static constexpr float quiet_NaN() noexcept { return value; } static constexpr float signaling_NaN() noexcept { return value; } static constexpr float denorm_min() noexcept { return min(); } static constexpr bool is_iec559 = true; static constexpr bool is_bounded = true; static constexpr bool is_modulo = false; static constexpr bool traps = true; static constexpr bool tinyness_before = true; static constexpr float_round_style round_style = round_to_nearest; }; } — end example]
3
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The specialization for bool shall be provided as follows: namespace std { template<> class numeric_limits<bool> { public: static constexpr bool is_specialized = true; static constexpr bool min() noexcept { return false; } static constexpr bool max() noexcept { return true; } static constexpr bool lowest() noexcept { return false; } static constexpr int digits = 1; static constexpr int digits10 = 0; static constexpr int max_digits10 = 0; static constexpr bool is_signed = false; static constexpr bool is_integer = true; static constexpr bool is_exact = true; static constexpr int radix = 2; static constexpr bool epsilon() noexcept { return 0; } static constexpr bool round_error() noexcept { return 0; } static constexpr int min_exponent = 0; static constexpr int min_exponent10 = 0; static constexpr int max_exponent = 0; static constexpr int max_exponent10 = 0; static constexpr bool has_infinity = false; static constexpr bool has_quiet_NaN = false; static constexpr bool has_signaling_NaN = false; static constexpr float_denorm_style has_denorm = denorm_absent; static constexpr bool has_denorm_loss = false; static constexpr bool infinity() noexcept { return 0; } static constexpr bool quiet_NaN() noexcept { return 0; } static constexpr bool signaling_NaN() noexcept { return 0; } static constexpr bool denorm_min() noexcept { return 0; } static constexpr bool is_iec559 = false; static constexpr bool is_bounded = true; static constexpr bool is_modulo = false; static constexpr bool traps = false; static constexpr bool tinyness_before = false; static constexpr float_round_style round_style = round_toward_zero; }; }