The term arithmetic underflow (also floating point underflow, or just underflow) is a condition in a
computer program where the result of a calculation is a number of more precise absolute value than the computer can actually represent in
memory on its
central processing unit (CPU).
Arithmetic underflow can occur when the true result of a
floating point operation
In computing, floating point operations per second (FLOPS, flops or flop/s) is a measure of computer performance, useful in fields of scientific computations that require floating-point calculations. For such cases, it is a more accurate meas ...
is smaller in magnitude (that is, closer to zero) than the smallest value representable as a
normal floating point number in the target
datatype
In computer science and computer programming, a data type (or simply type) is a set of possible values and a set of allowed operations on it. A data type tells the compiler or interpreter how the programmer intends to use the data. Most progra ...
. Underflow can in part be regarded as negative
overflow of the
exponent
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
of the floating point value. For example, if the exponent part can represent values from −128 to 127, then a result with a value less than −128 may cause underflow.
Storing values that are too low in an
integer variable (e.g., attempting to store −1 in an
unsigned
Unsigned can refer to:
* An unsigned artist is a musical artist or group not attached or signed to a record label
** Unsigned Music Awards, ceremony noting achievements of unsigned artists
** Unsigned band web, online community
* Similarly, the ...
integer) is properly referred to as
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of ...
, or more broadly, ''integer wraparound''. The term ''underflow'' normally refers to floating point numbers only, which is a separate issue. It is not possible in most floating-point designs to store a too-low value, as usually they are signed and have a negative
infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
value.
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
bit
The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represented a ...
s, the underflow gap is 2
21 times larger than the absolute distance between adjacent floating point values just outside the gap.
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
The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard addressed many problems found i ...
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.
The absolute distance between adjacent floating point values just outside the gap is called the
machine epsilon
Machine epsilon or machine precision is an upper bound on the relative approximation error due to rounding in floating point arithmetic. This value characterizes computer arithmetic in the field of numerical analysis, and by extension in the subjec ...
, 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.
This can be written as
, where
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
The significand (also mantissa or coefficient, sometimes also argument, or ambiguously fraction or characteristic) is part of a number in scientific notation or in floating-point representation, consisting of its significant digits. Depending on ...
, 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.
Handling of underflow
The occurrence of an underflow may set a ("sticky") status bit, raise an exception, at the hardware level generate an interrupt, or may cause some combination of these effects.
As specified in
IEEE 754
The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard addressed many problems found i ...
, the underflow condition is only signaled if there is also a loss of precision. Typically this is determined as the final result being inexact.
However, if the user is
trapping on underflow, this may happen regardless of consideration for loss of precision. The default handling in IEEE 754 for underflow (as well as other exceptions) is to record as a floating point status that underflow has occurred. This is specified for the application-programming level, but often also interpreted as how to handle it at the hardware level.
See also
*
Denormal number
*
Floating-point arithmetic
In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be ...
*
IEEE 754
The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard addressed many problems found i ...
*
Integer overflow
*
Logarithmic number system
*
Machine epsilon
Machine epsilon or machine precision is an upper bound on the relative approximation error due to rounding in floating point arithmetic. This value characterizes computer arithmetic in the field of numerical analysis, and by extension in the subjec ...
*
Normal number (computing)
References
{{DEFAULTSORT:Arithmetic Underflow
Computer arithmetic