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Conversely, precision can be lost when converting representations from integer to floating-point, since a floating-point type may be unable to exactly represent all possible values of some integer type. For example, float might be an IEEE 754 single precision type, which cannot represent the integer 16777217 exactly, while a 32-bit integer type ...
Here we can show how to convert a base-10 real number into an IEEE 754 binary32 format using the following outline: Consider a real number with an integer and a fraction part such as 12.375; Convert and normalize the integer part into binary; Convert the fraction part using the following technique as shown here
The sign bit determines the sign of the number (including when this number is zero, which is signed). The exponent field is an 11-bit unsigned integer from 0 to 2047, in biased form: an exponent value of 1023 represents the actual zero. Exponents range from −1022 to +1023 because exponents of −1023 (all 0s) and +1024 (all 1s) are reserved ...
Conversions to integer are not intuitive: converting (63.0/9.0) to integer yields 7, but converting (0.63/0.09) may yield 6. This is because conversions generally truncate rather than round. Floor and ceiling functions may produce answers which are off by one from the intuitively expected value.
A byte is a bit string containing the number of bits needed to represent a character. On most modern computers, this is an eight bit string. Because the definition of a byte is related to the number of bits composing a character, some older computers have used a different bit length for their byte. [2]
The most common representation of a positive integer is a string of bits, using the binary numeral system. The order of the memory bytes storing the bits varies; see endianness . The width , precision , or bitness [ 3 ] of an integral type is the number of bits in its representation.
This gives from 33 to 36 significant decimal digits precision. If a decimal string with at most 33 significant digits is converted to the IEEE 754 quadruple-precision format, giving a normal number, and then converted back to a decimal string with the same number of digits, the final result should match the original string.
In the table above, integer values are exact, whereas values in decimal notation (e.g. 1.0) are rounded values. The minimum exponents listed are for normal numbers; the special subnormal number representation allows even smaller (in magnitude