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#### `std::numeric_limits<>` functions

##### `max` function

Function `(std::numeric_limits<T>::max)()` returns the largest finite value that can be represented by the type T. If there is no such value (and `numeric_limits<T>::bounded` is `false`) then returns `T()`.

For fundamental (built-in) types there is usually a corresponding MACRO value TYPE_MAX, where TYPE is CHAR, INT, FLOAT etc.

Other types, including those provided by a typedef, for example `INT64_T_MAX` for `int64_t`, may provide a macro definition.

To cater for situations where no `numeric_limits` specialization is available (for example because the precision of the type varies at runtime), packaged versions of this (and other functions) are provided using

    #include <boost/math/tools/precision.hpp>

    T = boost::math::tools::max_value<T>();

Of course, these simply use `(std::numeric_limits<T>::max)()` if available, but otherwise 'do something sensible'.

##### lowest function

Since C++11: `std::numeric_limits<T>::lowest()` is

- For integral types, the same as function `min()`.
- For floating-point types, generally the negative of `max()` (but implementation-dependent).

    -(std::numeric_limits<double>::max)() == std::numeric_limits<double>::lowest();

##### `min` function

Function `(std::numeric_limits<T>::min)()` returns the minimum finite value that can be represented by the type T.

For fundamental (built-in) types, there is usually a corresponding MACRO value TYPE_MIN, where TYPE is CHAR, INT, FLOAT etc.

Other types, including those provided by a `typedef`, for example, `INT64_T_MIN` for `int64_t`, may provide a macro definition.

For floating-point types, it is more fully defined as the minimum positive normalized value.

See `std::numeric_limits<T>::denorm_min()` for the smallest denormalized value, provided

    std::numeric_limits<T>::has_denorm == std::denorm_present

To cater for situations where no `numeric_limits` specialization is available (for example because the precision of the type varies at runtime), packaged versions of this (and other functions) are provided using

    #include <boost/math/tools/precision.hpp>

    T = boost::math::tools::min_value<T>();

Of course, these simply use `std::numeric_limits<T>::min()` if available.

##### denorm_min function

Function `std::numeric_limits<T>::denorm_min()` returns the smallest denormalized value, provided

    std::numeric_limits<T>::has_denorm == std::denorm_present

    std::cout.precision(std::numeric_limits<double>::max_digits10);
    if (std::numeric_limits<double>::has_denorm == std::denorm_present)
    {
      double d = std::numeric_limits<double>::denorm_min();

        std::cout << d << std::endl; //  4.9406564584124654e-324

        int exponent;

        double significand = frexp(d, &exponent);
        std::cout << "exponent = " << std::hex << exponent << std::endl; //  fffffbcf
        std::cout << "significand = " << std::hex << significand << std::endl; // 0.50000000000000000
    }
    else
    {
      std::cout << "No denormalization. " << std::endl;
    }

The exponent is effectively reduced from -308 to -324 (though it remains encoded as zero and leading zeros appear in the significand, thereby losing precision until the significand reaches zero).

##### round_error

Function `std::numeric_limits<T>::round_error()` returns the maximum error (in units of Unit in the last place (ULP)) that can be caused by any basic arithmetic operation.

    round_style == std::round_indeterminate;

The rounding style is indeterminable at compile time.

For floating-point types, when rounding is to nearest, only half a bit is lost by rounding, and `round_error == 0.5`. In contrast when rounding is towards zero, or plus/minus infinity, we can loose up to one bit from rounding, and `round_error == 1`.

For integer types, rounding always to zero, so at worst almost one bit can be rounded, so `round_error == 1`.

`round_error()` can be used with `std::numeric_limits<T>::epsilon()` to estimate the maximum potential error caused by rounding. For typical floating-point types, `round_error() = 1/2`, so half epsilon is the maximum potential error.

    double round_err = std::numeric_limits<double>::epsilon() // 2.2204460492503131e-016
                     * std::numeric_limits<double>::round_error(); // 1/2
    std::cout << round_err << std::endl; // 1.1102230246251565e-016

There are, of course, many occasions when much bigger loss of precision occurs, for example, caused by Loss of significance or cancellation error or very many iterations.

##### epsilon

Function `std::numeric_limits<T>::epsilon()` is meaningful only for non-integral types.

It returns the difference between `1.0` and the next value representable by the floating-point type T. So it is a one least-significant-bit change in this floating-point value.

For `double` (`float_64t`) it is `2.2204460492503131e-016` showing all possibly significant 17 decimal digits.

    std::cout.precision(std::numeric_limits<double>::max_digits10);
    double d = 1.;
    double eps = std::numeric_limits<double>::epsilon();
    double dpeps = d+eps;
    std::cout << std::showpoint // Ensure all trailing zeros are shown.
      << d << "\n"           // 1.0000000000000000
      << dpeps << std::endl; // 2.2204460492503131e-016
    std::cout << dpeps - d   // 1.0000000000000002
      << std::endl;

We can explicitly increment by one bit using the function `boost::math::float_next()` and the result is the same as adding `epsilon`.

    double one = 1.;
    double nad = boost::math::float_next(one);
    std::cout << nad << "\n"  //  1.0000000000000002
      << nad - one // 2.2204460492503131e-016
      << std::endl;

Adding any smaller value, like half `epsilon`, will have no effect on this value.

    std::cout.precision(std::numeric_limits<double>::max_digits10);
    double d = 1.;
    double eps = std::numeric_limits<double>::epsilon();
    double dpeps = d + eps/2;

    std::cout << std::showpoint // Ensure all trailing zeros are shown.
      << dpeps << "\n"       // 1.0000000000000000
      << eps/2 << std::endl; // 1.1102230246251565e-016
    std::cout << dpeps - d   // 0.00000000000000000
      << std::endl;

So this cancellation error leaves the values equal, despite adding half `epsilon`.

To achieve greater portability over platform and floating-point type, Boost.Math and Boost.Multiprecision provide a package of functions that 'do something sensible' if the standard `numeric_limits` is not available. To use these `#include <boost/math/tools/precision.hpp>`.

A tolerance might be defined using this version of epsilon thus:

    RealType tolerance = boost::math::tools::epsilon<RealType>() * 2;

###### Tolerance for Floating-point Comparisons

Machine epsilon ε is very useful to compute a tolerance when comparing floating-point values, a much more difficult task than is commonly imagined.

The C++ standard specifies `std::numeric_limits<>::epsilon()` and Boost.Multiprecision implements this (where possible) for its program-defined types analogous to the fundamental (built-in) floating-point types like `double` `float`.

For more information than you probably want (but still need) see What Every Computer Scientist Should Know About Floating-Point Arithmetic

The naive test comparing the absolute difference between two values and a tolerance does not give useful results if the values are too large or too small.

So Boost.Test uses an algorithm first devised by Knuth for reliably checking if floating-point values are close enough.

See Donald. E. Knuth. The art of computer programming (vol II). Copyright 1998 Addison-Wesley Longman, Inc., 0-201-89684-2. Addison-Wesley Professional; 3rd edition. (The relevant equations are in paragraph 4.2.2, Eq. 36 and 37.)

See Boost.Math floating_point comparison for more details.

See also:

Alberto Squassia, Comparing floats

Alberto Squassia, Comparing floats code

Boost.Test Floating-Point_Comparison

For example, if we want a tolerance that might suit about 9 arithmetical operations, say sqrt(9) = 3, we could define:

    T tolerance =  3 * std::numeric_limits<T>::epsilon();

This is very widely used in Boost.Math testing with Boost.Test's macro `BOOST_CHECK_CLOSE_FRACTION`

    T expected = 1.0;
    T calculated = 1.0 + std::numeric_limits<T>::epsilon();

    BOOST_CHECK_CLOSE_FRACTION(expected, calculated, tolerance);

used thus:

    cd ./test
    BOOST_CHECK_CLOSE_FRACTION(expected, calculated, tolerance);

(There is also a version BOOST_CHECK_CLOSE using tolerance as a percentage rather than a fraction; usually the fraction version is simpler to use).

    using boost::multiprecision::number;
    using boost::multiprecision::cpp_dec_float;
    using boost::multiprecision::et_off;

    typedef number<cpp_dec_float<50>, et_off > cpp_dec_float_50; // 50 decimal digits.

| Note
that Boost.Test does not yet allow floating-point comparisons with expression templates on, so the default expression template parameter has been replaced by `et_off`.

    cpp_dec_float_50 tolerance =  3 * std::numeric_limits<cpp_dec_float_50>::epsilon();
    cpp_dec_float_50 expected = boost::math::constants::two_pi<cpp_dec_float_50>();
    cpp_dec_float_50 calculated = 2 * boost::math::constants::pi<cpp_dec_float_50>();

    BOOST_CHECK_CLOSE_FRACTION(expected, calculated, tolerance);

##### Infinity - positive and negative

For floating-point types only, for which `std::numeric_limits<T>::has_infinity == true`, function `std::numeric_limits<T>::infinity()` provides an implementation-defined representation for ∞.

The 'representation' is a particular bit pattern reserved for infinity. For IEEE754 system (for which `std::numeric_limits<T>::is_iec559 == true`) positive and negative infinity are assigned bit patterns for all defined floating-point types.

Confusingly, the string resulting from outputting this representation, is also implementation-defined. And the string that can be input to generate the representation is also implementation-defined.

For example, the output is `1.#INF` on Microsoft systems, but `inf` on most *nix platforms.

This implementation-defined-ness has hampered use of infinity (and NaNs) but Boost.Math and Boost.Multiprecision work hard to provide a sensible representation for all floating-point types, not just the fundamental (built-in) types, which with the use of suitable facets to define the input and output strings, makes it possible to use these useful features portably and including Boost.Serialization.

##### Not-A-Number NaN

###### Quiet_NaN

For floating-point types only, for which `std::numeric_limits<T>::has_quiet_NaN == true`, function `std::numeric_limits<T>::quiet_NaN()` provides an implementation-defined representation for NaN.

NaNs are values to indicate that the result of an assignment or computation is meaningless. A typical example is `0/0` but there are many others.

NaNs may also be used, to represent missing values: for example, these could, by convention, be ignored in calculations of statistics like means.

Many of the problems with a representation for Not-A-Number has hampered portable use, similar to those with infinity.

NaN can be used with binary multiprecision types like `cpp_bin_float_quad`:

    using boost::multiprecision::cpp_bin_float_quad;

    if (std::numeric_limits<cpp_bin_float_quad>::has_quiet_NaN == true)
    {
      cpp_bin_float_quad NaN =  std::numeric_limits<cpp_bin_float_quad>::quiet_NaN();
      std::cout << "cpp_bin_float_quad NaN is "  << NaN << std::endl; //   cpp_bin_float_quad NaN is nan

      cpp_bin_float_quad expected = NaN;
      cpp_bin_float_quad calculated = 2 * NaN;
      // Comparisons of NaN's always fail:
      bool b = expected == calculated;
      std::cout << b << std::endl;
      BOOST_CHECK_NE(expected, expected);
      BOOST_CHECK_NE(expected, calculated);
    }
    else
    {
      std::cout << "Type " << typeid(cpp_bin_float_quad).name() << " does not have NaNs!" << std::endl;
    }

But using Boost.Math and suitable facets can permit portable use of both NaNs and positive and negative infinity.

See boost:/libs/math/example/nonfinite_facet_sstream.cpp and we also need

    #include <boost/math/special_functions/nonfinite_num_facets.hpp>

Then we can equally well use a multiprecision type cpp_bin_float_quad:

    using boost::multiprecision::cpp_bin_float_quad;

    typedef cpp_bin_float_quad T;

    using boost::math::nonfinite_num_put;
    using boost::math::nonfinite_num_get;
    {
      std::locale old_locale;
      std::locale tmp_locale(old_locale, new nonfinite_num_put<char>);
      std::locale new_locale(tmp_locale, new nonfinite_num_get<char>);
      std::stringstream ss;
      ss.imbue(new_locale);
      T inf = std::numeric_limits<T>::infinity();
      ss << inf; // Write out.
     BOOST_MP_ASSERT(ss.str() == "inf");
      T r;
      ss >> r; // Read back in.
      BOOST_MP_ASSERT(inf == r); // Confirms that the floating-point values really are identical.
      std::cout << "infinity output was " << ss.str() << std::endl;
      std::cout << "infinity input was " << r << std::endl;
    }

    infinity output was inf
    infinity input was inf

Similarly we can do the same with NaN (except that we cannot use `assert` (because any comparisons with NaN always return false).

    {
      std::locale old_locale;
      std::locale tmp_locale(old_locale, new nonfinite_num_put<char>);
      std::locale new_locale(tmp_locale, new nonfinite_num_get<char>);
      std::stringstream ss;
      ss.imbue(new_locale);
      T n;
      T NaN = std::numeric_limits<T>::quiet_NaN();
      ss << NaN; // Write out.
      BOOST_MP_ASSERT(ss.str() == "nan");
      std::cout << "NaN output was " << ss.str() << std::endl;
      ss >> n; // Read back in.
      std::cout << "NaN input was " << n << std::endl;
    }

    NaN output was nan
    NaN input was nan

###### Signaling NaN

For floating-point types only, for which `std::numeric_limits<T>::has_signaling_NaN == true`, function `std::numeric_limits<T>::signaling_NaN()` provides an implementation-defined representation for NaN that causes a hardware trap. It should be noted however, that at least one implementation of this function causes a hardware trap to be triggered simply by calling `std::numeric_limits<T>::signaling_NaN()`, and not only by using the value returned.