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The representation has a limited precision. For example, only 15 decimal digits can be represented with a 64-bit real. If a very small floating-point number is added to a large one, the result is just the large one. The small number was too small to even show up in 15 or 16 digits of resolution, and the computer effectively discards it.
Subnormal numbers and zeros (which are the floating-point numbers smaller in magnitude than the least positive normal number) are represented with the biased exponent value 0, giving the implicit leading bit the value 0. Thus only 23 fraction bits of the significand appear in the memory format, but the total precision is 24 bits (equivalent to ...
With the 52 bits of the fraction (F) significand appearing in the memory format, the total precision is therefore 53 bits (approximately 16 decimal digits, 53 log 10 (2) ≈ 15.955). The bits are laid out as follows: The real value assumed by a given 64-bit double-precision datum with a given biased exponent and a 52-bit fraction is
Download as PDF; Printable version; In other projects ... supposing by induction that the decimal fraction ... often a 64-bit representation with around 16 decimal ...
Decimal fractions are commonly expressed using decimal notation in which the implied denominator is determined by the number of digits to the right of a decimal separator, the appearance of which (e.g., a period, an interpunct (·), a comma) depends on the locale (for examples, see Decimal separator). Thus, for 0.75 the numerator is 75 and the ...
The remaining rightmost digits are padded with zeros, yielding a 24-bit fraction of .0111 0110 1010 0000 0000 0000 2. The normalized value moved the radix point two hexadecimal digits to the left, yielding a multiplier and exponent of 16 +2. A bias of +64 is added to the exponent (+2), yielding +66, which is 100 0010 2.