Editing Arithmetic underflow (section)

==Underflow gap==
The interval between −''fminN'' and ''fminN'', where ''fminN'' is the smallest positive normal floating-point value, is called the underflow gap. This is because the size of this interval is many orders of magnitude larger than the distance between adjacent normal floating-point values just outside the gap. For instance, if the floating-point datatype can represent 20&nbsp;[[bit]]s, the underflow gap is 2<sup>21</sup> times larger than the absolute distance between adjacent floating-point values just outside the gap.<ref>{{cite book |last1=Sun Microsystems |title=Numerical Computation Guide |date=2005 |publisher=Oracle |url=https://docs.oracle.com/cd/E19422-01/819-3693/ |accessdate=21 April 2018}}</ref>

In older designs, the underflow gap had just one usable value, zero. When an underflow occurred, the true result was replaced by zero (either directly by the hardware, or by system software handling the primary underflow condition). This replacement is called "flush to zero".

The 1984 edition of [[IEEE 754]] introduced [[subnormal numbers]]. The subnormal numbers (including zero) fill the underflow gap with values where the absolute distance between adjacent values is the same as for adjacent values just outside the underflow gap. This enables "gradual underflow", where a nearest subnormal value is used, just as a nearest normal value is used when possible. Even when using gradual underflow, the nearest value may be zero.<ref>{{cite journal |last1=Demmel |first1=James |title=Underflow and the Reliability of Numerical Software |journal=SIAM Journal on Scientific and Statistical Computing |date=1984 |volume=5 |issue=4 |pages=887–919 |doi=10.1137/0905062}}</ref>

The absolute distance between adjacent floating-point values just outside the gap is called the [[machine epsilon]], typically characterized by the largest value whose sum with the value 1 will result in the answer with value 1 in that floating-point scheme.<ref>{{cite book |last1=Heath |first1=Michael T. |title=Scientific Computing |date=2002 |publisher=McGraw-Hill |location=New York |isbn=0-07-239910-4 |page=20 |edition=2nd}}</ref> This is the maximum value of <math>\epsilon</math> that satisfies <math>\operatorname{fl}(1 + \epsilon) = \operatorname{fl}(1)</math>, where <math>\operatorname{fl}</math> is a function which converts the real value into the floating-point representation. While the machine epsilon is not to be confused with the underflow level (assuming subnormal numbers), it is closely related. The machine epsilon is dependent on the number of bits which make up the [[significand]], whereas the underflow level depends on the number of digits which make up the exponent field. In most floating-point systems, the underflow level is smaller than the machine epsilon.