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The exponents 000 16 and 7ff 16 have a special meaning: . 00000000000 2 =000 16 is used to represent a signed zero (if F = 0) and subnormal numbers (if F ≠ 0); and; 11111111111 2 =7ff 16 is used to represent ∞ (if F = 0) and NaNs (if F ≠ 0),
This is pretty efficient, because 2 10 = 1024, is only little more than needed to still contain all numbers from 0 to 999. Both alternatives provide exactly the same set of representable numbers: 16 digits of significand and 3 × 2 8 = 768 possible decimal exponent values.
For numbers with a base-2 exponent part of 0, i.e. numbers with an absolute value higher than or equal to 1 but lower than 2, an ULP is exactly 2 −23 or about 10 −7 in single precision, and exactly 2 −53 or about 10 −16 in double precision. The mandated behavior of IEEE-compliant hardware is that the result be within one-half of a ULP.
As the magnitude of the value decreases, the amount of extra precision also decreases. Therefore, the smallest number in the normalized range is narrower than double precision. The smallest number with full precision is 1000...0 2 (106 zeros) × 2 −1074, or 1.000...0 2 (106 zeros) × 2 −968. Numbers whose magnitude is smaller than 2 −1021 ...
Be aware that the bit numbering used here for e.g. b 9 … b 0 is in opposite direction than that used in the document for the IEEE 754 standard b 0 … b 9, add. the decimal digits are numbered 0-base here while in opposite direction and 1-based in the IEEE 754 paper. The bits on white background are not counting for the value, but signal how ...
In CORBA (from specification of 3.0, which uses "ANSI/IEEE Standard 754-1985" as its reference), "the long double data type represents an IEEE double-extended floating-point number, which has an exponent of at least 15 bits in length and a signed fraction of at least 64 bits", with GIOP/IIOP CDR, whose floating-point types "exactly follow the ...
A simple method to add floating-point numbers is to first represent them with the same exponent. In the example below, the second number is shifted right by 3 digits. We proceed with the usual addition method: The following example is decimal, which simply means the base is 10. 123456.7 = 1.234567 × 10 5 101.7654 = 1.017654 × 10 2 = 0. ...
Here we start with 0 in single precision (binary32) and repeatedly add 1 until the operation does not change the value. Since the significand for a single-precision number contains 24 bits, the first integer that is not exactly representable is 2 24 +1, and this value rounds to 2 24 in round to nearest, ties to even.