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The term arithmetic underflow (also floating-point underflow, or just underflow) is a condition in a
computer program A computer program is a sequence or set of instructions in a programming language for a computer to Execution (computing), execute. It is one component of software, which also includes software documentation, documentation and other intangibl ...
where the result of a calculation is a number of more precise absolute value than the computer can actually represent in
memory Memory is the faculty of the mind by which data or information is encoded, stored, and retrieved when needed. It is the retention of information over time for the purpose of influencing future action. If past events could not be remembe ...
on its
central processing unit A central processing unit (CPU), also called a central processor, main processor, or just processor, is the primary Processor (computing), processor in a given computer. Its electronic circuitry executes Instruction (computing), instructions ...
(CPU). Arithmetic underflow can occur when the true result of a floating-point operation 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 collection or grouping of data values, usually specified by a set of possible values, a set of allowed operations on these values, and/or a representation of these ...
. Underflow can in part be regarded as negative overflow of the
exponent In mathematics, exponentiation, denoted , is an operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, i ...
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.


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  bits, the underflow gap is 221 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 originally established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard #Design rationale, add ...
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, 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 is the maximum value of \epsilon that satisfies \operatorname(1 + \epsilon) = \operatorname(1), where \operatorname 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 coefficient, sometimes argument, or more ambiguously mantissa, fraction, or characteristic) is the first (left) part of a number in scientific notation or related concepts in floating-point representation, consisting of its s ...
, 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 originally established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard #Design rationale, add ...
, 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
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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 on subsets of real numbers formed by a ''significand'' (a Sign (mathematics), signed sequence of a fixed number of digits in some Radix, base) multiplied by an integer power of that ba ...
*
IEEE 754 The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic originally established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard #Design rationale, add ...
*
Integer overflow In computer programming, an integer overflow occurs when an arithmetic operation on integers attempts to create a numeric value that is outside of the range that can be represented with a given number of digits – either higher than the maximu ...
* Logarithmic number system * Machine epsilon * Normal number (computing)


References

{{DEFAULTSORT:Arithmetic Underflow Computer arithmetic