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If a decimal string with at most 15 significant digits is converted to the IEEE 754 double-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.
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).
FLT_DIG, DBL_DIG, LDBL_DIG – number of decimal digits that can be represented without losing precision by float, double, long double, respectively; FLT_EPSILON, DBL_EPSILON, LDBL_EPSILON – difference between 1.0 and the next representable value of float, double, long double, respectively
The two characters commonly used for this purpose are the hyphen ("-") and the underscore ("_"); e.g., the two-word name "two words" would be represented as "two-words" or "two_words". The hyphen is used by nearly all programmers writing COBOL (1959), Forth (1970), and Lisp (1958); it is also common in Unix for commands and packages, and is ...
In computing, half precision (sometimes called FP16 or float16) is a binary floating-point computer number format that occupies 16 bits (two bytes in modern computers) in computer memory. It is intended for storage of floating-point values in applications where higher precision is not essential, in particular image processing and neural networks .
The two-symbol system used is often "0" and "1" from the binary number system. The binary code assigns a pattern of binary digits, also known as bits , to each character, instruction, etc. For example, a binary string of eight bits (which is also called a byte) can represent any of 256 possible values and can, therefore, represent a wide ...
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.
This is different from the way rounding is usually done in signed integer division (which rounds towards 0). This discrepancy has led to bugs in a number of compilers. [8] For example, in the x86 instruction set, the SAR instruction (arithmetic right shift) divides a signed number by a power of two, rounding towards negative infinity. [9]