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Here the 'IEEE 754 double value' resulting of the 15 bit figure is 3.330560653658221E-15, which is rounded by Excel for the 'user interface' to 15 digits 3.33056065365822E-15, and then displayed with 30 decimals digits gets one 'fake zero' added, thus the 'binary' and 'decimal' values in the sample are identical only in display, the values ...
The IEEE standard uses round-to-nearest. Round-by-chop: The base-expansion of is truncated after the ()-th digit. This rounding rule is biased because it always moves the result toward zero. Round-to-nearest: () is set to the nearest floating-point number to . When there is a tie, the floating-point number whose last stored digit is even (also ...
Some programming languages (or compilers for them) provide a built-in (primitive) or library decimal data type to represent non-repeating decimal fractions like 0.3 and −1.17 without rounding, and to do arithmetic on them. Examples are the decimal.Decimal or num7.Num type of Python, and analogous types provided by other languages.
Huberto M. Sierra noted in his 1956 patent "Floating Decimal Point Arithmetic Control Means for Calculator": [1] Thus under some conditions, the major portion of the significant data digits may lie beyond the capacity of the registers. Therefore, the result obtained may have little meaning if not totally erroneous.
This variant of the round-to-nearest method is also called convergent rounding, statistician's rounding, Dutch rounding, Gaussian rounding, odd–even rounding, [6] or bankers' rounding. [ 7 ] This is the default rounding mode used in IEEE 754 operations for results in binary floating-point formats.
For example, sometimes instead of rounding off a numerical value obtained from a calculation, some of the digits might just be removed i.e. truncated See also [ edit ]
Alternative rounding options are also available. IEEE 754 specifies the following rounding modes: round to nearest, where ties round to the nearest even digit in the required position (the default and by far the most common mode) round to nearest, where ties round away from zero (optional for binary floating-point and commonly used in decimal)
However, for negative numbers truncation does not round in the same direction as the floor function: truncation always rounds toward zero, the function rounds towards negative infinity. For a given number x ∈ R − {\displaystyle x\in \mathbb {R} _{-}} , the function ceil {\displaystyle \operatorname {ceil} } is used instead