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The "decimal" data type of the C# and Python programming languages, and the decimal formats of the IEEE 754-2008 standard, are designed to avoid the problems of binary floating-point representations when applied to human-entered exact decimal values, and make the arithmetic always behave as expected when numbers are printed in decimal.
In SQL Server versions 6.5 and earlier, database administrators (DBAs) used SQL Server Transfer Manager and Bulk Copy Program, included with SQL Server, to transfer data. These tools had significant shortcomings, and many [ quantify ] DBAs used third-party tools such as Pervasive Data Integrator to transfer data more flexibly and easily.
Rounding to a specified power is very different from rounding to a specified multiple; for example, it is common in computing to need to round a number to a whole power of 2. The steps, in general, to round a positive number x to a power of some positive number b other than 1, are:
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.
Numbers with a magnitude strictly larger than k are rounded to the corresponding infinity. [18] "Round to nearest, ties to even" is the default for binary floating point and the recommended default for decimal. "Round to nearest, ties to away" is only required for decimal implementations. [19]
Implicit type conversion, also known as coercion or type juggling, is an automatic type conversion by the compiler. Some programming languages allow compilers to provide coercion; others require it. In a mixed-type expression, data of one or more subtypes can be converted to a supertype as needed at runtime so that
largest number less than one 0 01111 0000000000: 3c00: 2 0 × (1 + 0 / 1024 ) = 1: one 0 01111 0000000001: 3c01: 2 0 × (1 + 1 / 1024 ) ≈ 1.00097656: smallest number larger than one 0 11110 1111111111: 7bff: 2 15 × (1 + 1023 / 1024 ) = 65504: largest normal number 0 11111 0000000000: 7c00: ∞: infinity 1 00000 ...
From binary32 to bfloat16. When bfloat16 was first introduced as a storage format, [15] the conversion from IEEE 754 binary32 (32-bit floating point) to bfloat16 is truncation (round toward 0). Later on, when it becomes the input of matrix multiplication units, the conversion can have various rounding mechanisms depending on the hardware platforms.