std::numeric_limits::tinyness_before(3)

NAME
std::numeric_limits::tinyness_before - std::numeric_limits::tinyness_before

Synopsis
   static const bool tinyness_before; (until C++11)
   static constexpr bool tinyness_before; (since C++11)
   The value of std::numeric_limits<T>::tinyness_before is true for all floating-point
   types T that test results of floating-point expressions for underflow before
   rounding.

Standard specializations
   T value of std::numeric_limits<T>::tinyness_before
   /* non-specialized */ false
   bool false
   char false
   signed char false
   unsigned char false
   wchar_t false
   char8_t (C++20) false
   char16_t (C++11) false
   char32_t (C++11) false
   short false
   unsigned short false
   int false
   unsigned int false
   long false
   unsigned long false
   long long (C++11) false
   unsigned long long (C++11) false
   float implementation-defined
   double implementation-defined
   long double implementation-defined

Notes
   Standard-compliant IEEE 754 floating-point implementations are required to detect
   the floating-point underflow, and have two alternative situations where this can be
   done
    1. Underflow occurs (and FE_UNDERFLOW may be raised) if a computation produces a
       result whose absolute value, computed as though both the exponent range and the
       precision were unbounded, is smaller than std::numeric_limits<T>::min(). Such
       implementation detects tinyness before rounding (e.g. UltraSparc, POWER).
    2. Underflow occurs (and FE_UNDERFLOW may be raised) if after the rounding of the
       result to the target floating-point type (that is, rounding to
       std::numeric_limits<T>::digits bits), the result's absolute value is smaller
       than std::numeric_limits<T>::min(). Formally, the absolute value of a nonzero
       result computed as though the exponent range were unbounded is smaller than
       std::numeric_limits<T>::min(). Such implementation detects tinyness after
       rounding (e.g. SuperSparc)

Example
   Multiplication of the largest subnormal number by the number one machine epsilon
   greater than 1.0 gives the tiny value 0x0.fffffffffffff8p-1022 before rounding, but
   normal value 1p-1022 after rounding. The implementation used to execute this test
   (IBM Power7) detects tinyness before rounding. // Run this code
 #include <iostream>
 #include <limits>
 #include <cmath>
 #include <cfenv>
 int main()
 {
     std::cout << "Tinyness before: " << std::boolalpha
               << std::numeric_limits<double>::tinyness_before << '\n';
     double denorm_max = std::nextafter(std::numeric_limits<double>::min(), 0);
     double multiplier = 1 + std::numeric_limits<double>::epsilon();
     std::feclearexcept(FE_ALL_EXCEPT);
     double result = denorm_max*multiplier; // Underflow only if tinyness_before
     if(std::fetestexcept(FE_UNDERFLOW))
         std::cout << "Underflow detected\n";
     std::cout << std::hexfloat << denorm_max << " x " << multiplier << " = "
               << result << '\n';
 }

Possible output:
 Tinyness before: true
 Underflow detected
 0xf.ffffffffffffp-1030 x 0x1.0000000000001p+0 = 0x1p-1022

See also
   has_denorm_loss identifies the floating-point types that detect loss of precision as
   [static] denormalization loss rather than inexact result
                   (public static member constant)
   has_denorm identifies the denormalization style used by the floating-point type
   [static] (public static member constant)