Editing Floating-point arithmetic (section)

== Range of floating-point numbers ==
A floating-point number consists of two [[Fixed-point arithmetic|fixed-point]] components, whose range depends exclusively on the number of bits or digits in their representation. Whereas components linearly depend on their range, the floating-point range linearly depends on the significand range and exponentially on the range of exponent component, which attaches outstandingly wider range to the number.

On a typical computer system, a ''[[Double-precision floating-point format|double-precision]]'' (64-bit) binary floating-point number has a coefficient of 53 bits (including 1 implied bit), an exponent of 11 bits, and 1 sign bit. Since 2<sup>10</sup> = 1024, the complete range of the positive normal floating-point numbers in this format is from 2<sup>−1022</sup>&nbsp;≈&nbsp;2&nbsp;×&nbsp;10<sup>−308</sup> to approximately 2<sup>1024</sup>&nbsp;≈&nbsp;2&nbsp;×&nbsp;10<sup>308</sup>.

The number of normal floating-point numbers in a system (''B'', ''P'', ''L'', ''U'') where
* ''B'' is the base of the system,
* ''P'' is the precision of the significand (in base ''B''),
* ''L'' is the smallest exponent of the system,
* ''U'' is the largest exponent of the system,
is <math>2 \left(B - 1\right) \left(B^{P-1}\right) \left(U - L + 1\right)</math>.

There is a smallest positive normal floating-point number,
: Underflow level = UFL = <math>B^L</math>,

which has a 1 as the leading digit and 0 for the remaining digits of the significand, and the smallest possible value for the exponent.

There is a largest floating-point number,
: Overflow level = OFL = <math>\left(1 - B^{-P}\right)\left(B^{U + 1}\right)</math>,

which has ''B'' − 1 as the value for each digit of the significand and the largest possible value for the exponent.

In addition, there are representable values strictly between −UFL and UFL. Namely, [[signed zero|positive and negative zeros]], as well as [[subnormal number]]s.