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The arithmetical difference between two consecutive representable floating-point numbers which have the same exponent is called a unit in the last place (ULP). For example, if there is no representable number lying between the representable numbers 1.45a70c22 hex and 1.45a70c24 hex , the ULP is 2×16 −8 , or 2 −31 .
The floating-point difference is computed exactly because the numbers are close—the Sterbenz lemma guarantees this, even in case of underflow when gradual underflow is supported. Despite this, the difference of the original numbers is e = −1; s = 4.877000, which differs more than 20% from the difference e = −1; s = 4.000000 of the ...
Because floating-point numbers have limited precision, only a subset of real or rational numbers are exactly representable; other numbers can be represented only approximately. Many languages have both a single precision (often called float) and a double precision type (often called double).
All integers with seven or fewer decimal digits, and any 2 n for a whole number −149 ≤ n ≤ 127, can be converted exactly into an IEEE 754 single-precision floating-point value. In the IEEE 754 standard, the 32-bit base-2 format is officially referred to as binary32; it was called single in IEEE 754-1985.
The number representations described above are called normalized, meaning that the implicit leading binary digit is a 1. To reduce the loss of precision when an underflow occurs, IEEE 754 includes the ability to represent fractions smaller than are possible in the normalized representation, by making the implicit leading digit a 0.
The element pc requires ten blocks of memory of the size of pointer to char (usually 40 or 80 bytes on common platforms), but element pa is only one pointer (size 4 or 8 bytes), and the data it refers to is an array of ten bytes (sizeof * pa == 10).
In contrast with a record, which could be defined to contain a float and an integer, a union may only contain one subtype at a time. A tagged union (also called a variant , variant record, discriminated union, or disjoint union) contains an additional field indicating its current type for enhanced type safety.
E.g. binary128 has approximately the same precision as a 34 digit decimal number. log 10 MAXVAL is a measure of the range of the encoding. Its integer part is the largest exponent shown on the output of a value in scientific notation with one leading digit in the significand before the decimal point (e.g. 1.698·10 38 is near the largest value ...