In computer science, subnormal numbers are the subset of denormalized numbers (sometimes called denormals) that fill the underflow gap around zero in floating-point arithmetic. Any non-zero number with magnitude smaller than the smallest positive normal number is subnormal, while denormal can also refer to numbers outside that range.
Terminology In some older documents (especially standards documents such as the initial release of IEEE 754 (1985) and the first standard version of the C language (1989)), "denormal" is used to refer exclusively to subnormal numbers. This usage persists in various standards documents, especially when discussing hardware that is incapable of representing any other denormalized numbers, but the discussion here uses the term "subnormal" in line with the 2008 revision of IEEE 754. In casual discussions, the terms subnormal and denormal are often used interchangeably, in part because there are no denormalized IEEE binary numbers outside the subnormal range. The term "number" is used rather loosely, to describe a particular sequence of digits, rather than a mathematical abstraction; see Floating-point arithmetic for details of how real numbers relate to floating-point representations. "Representation" rather than "number" may be used when clarity is required.
Definition Mathematical real numbers may be approximated by multiple floating-point representations. One representation is defined as normal, and others are defined as subnormal, denormal, or unnormal by their relationship to normal. In a normal floating-point value, there are no leading zeros in the significand (also commonly called mantissa); rather, leading zeros are removed by adjusting the exponent (for example, the number 0.0123 would be written as 1.23×10−2). Conversely, a denormalized floating-point value has a significand with a leading digit of zero. Of these, the subnormal numbers represent values which if normalized would have exponents below the smallest representable exponent (the exponent having a limited range). The significand (or mantissa) of an IEEE floating-point number is the part of a floating-point number that represents the significant digits. For a positive normalised number, it can be represented as m0.m1m2m3...mp−2mp−1 (where m represents a significant digit, and p is the precision) with non-zero m0. Notice that for a binary radix, the leading binary digit is always 1. In a subnormal number, since the exponent is the least that it can be, zero is the leading significant digit (0.m1m2m3...mp−2mp−1), allowing the representation of numbers closer to zero than the smallest normal number. A floating-point number may be recognized as subnormal whenever its exponent has the least possible value. By filling the underflow gap like this, significant digits are lost, but not as abruptly as when using the flush to zero on underflow approach (discarding all significant digits when underflow is reached). Hence the production of a subnormal number is sometimes called gradual underflow because it allows a calculation to lose precision slowly when the result is small. In IEEE 754-2008, subnormal numbers are supported in both binary and decimal formats. In binary interchange formats, subnormal numbers are encoded with a biased exponent of 0, but are interpreted with the value of the smallest allowed exponent, which is one greater (i.e., as if it were encoded as a 1). In decimal interchange formats they require no special encoding because the format supports unnormalized numbers directly. Mathematically speaking, the normalized floating-point numbers of a given sign are roughly logarithmically spaced, and as such any finite-sized normal float cannot include zero. The subnormal floats are a linearly spaced set of values, which span the gap between the negative and positive normal floats.
Background Subnormal numbers provide the guarantee that addition and subtraction of floating-point numbers never underflows; two nearby floating-point numbers always have a representable non-zero difference. Without gradual underflow, the subtraction a − b can underflow and produce zero even though the values are not equal. This can, in turn, lead to division by zero errors that cannot occur when gradual underflow is used. Subnormal numbers were implemented in the Intel 8087 while the IEEE 754 standard was being written. They were by far the most controversial feature in the K-C-S format proposal that was eventually adopted, but this implementation demonstrated that subnormal numbers could be supported in a practical implementation. Some implementations of floating-point units do not directly support subnormal numbers in hardware, but rather trap to some kind of software support. While this may be transparent to the user, it can result in calculations that produce or consume subnormal numbers being much slower than similar calculations on normal numbers.
IEEE In IEEE binary floating-point formats, subnormals are represented by having a zero exponent field with a non-zero significand field. In this standard, the existence of subnormal numbers allows the representations of some non-zero values with absolute value from 2 − 149 {\displaystyle 2^{-149}} to ( 2 − 126 − 2 − 149 ) {\displaystyle (2^{-126}-2^{-149})} for binary32, and from 2 − 1074 {\displaystyle 2^{-1074}} to ( 2 − 1022 − 2 − 1074 ) {\displaystyle (2^{-1022}-2^{-1074})} for binary64. No other denormalized numbers exist in the IEEE binary floating-point formats, but they do exist in some other formats, including the IEEE decimal floating-point formats.
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