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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, the last digit, in binary form, is equal ...
Exponents range from −1022 to +1023 because exponents of −1023 (all 0s) and +1024 (all 1s) are reserved for special numbers. The 53-bit significand precision gives from 15 to 17 significant decimal digits precision (2 −53 ≈ 1.11 × 10 −16). If a decimal string with at most 15 significant digits is converted to the IEEE 754 double ...
In the example from "Double rounding" section, rounding 9.46 to one decimal gives 9.4, which rounding to integer in turn gives 9. With binary arithmetic, this rounding is also called "round to odd" (not to be confused with "round half to odd"). For example, when rounding to 1/4 (0.01 in binary), x = 2.0 ⇒ result is 2 (10.00 in binary)
For example, the decimal number 123456789 cannot be exactly represented if only eight decimal digits of precision are available (it would be rounded to one of the two straddling representable values, 12345678 × 10 1 or 12345679 × 10 1), the same applies to non-terminating digits (. 5 to be rounded to either .55555555 or .55555556).
36 decimal digits for binary128. For other binary formats, the required number of decimal digits is [h] + ⌈ ⌉, where p is the number of significant bits in the binary format, e.g. 237 bits for binary256. When using a decimal floating-point format, the decimal representation will be preserved using: 7 decimal digits for decimal32, 16 ...
The addition of the two numbers is: 0.0256*10^2 2.3400*10^2 + _____ 2.3656*10^2 After padding the second number (i.e., ) with two s, the bit after is the guard digit, and the bit after is the round digit
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3.14159 26535 89793 23846 is π rounded to 20 decimal places 2.71828 18284 59045 23536 is e rounded to 20 decimal places. In some programming languages, it is possible to group the digits in the program's source code to make it easier to read; see Integer literal: Digit separators.