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The representation has a limited precision. For example, only 15 decimal digits can be represented with a 64-bit real. If a very small floating-point number is added to a large one, the result is just the large one. The small number was too small to even show up in 15 or 16 digits of resolution, and the computer effectively discards it.
Double-precision floating-point format (sometimes called FP64 or float64) is a floating-point number format, usually occupying 64 bits in computer memory; it represents a wide range of numeric values by using a floating radix point. Double precision may be chosen when the range or precision of single precision would be insufficient.
A decimal data type could be implemented as either a floating-point number or as a fixed-point number. In the fixed-point case, the denominator would be set to a fixed power of ten. In the floating-point case, a variable exponent would represent the power of ten to which the mantissa of the number is multiplied.
For example, the decimal numbers 0.1 and 0.01 cannot be represented exactly as binary floating-point numbers. In the IEEE 754 binary32 format with its 24-bit significand, the result of attempting to square the approximation to 0.1 is neither 0.01 nor the representable number closest to it.
The Q notation is a way to specify the parameters of a binary fixed point number format. For example, in Q notation, the number format denoted by Q8.8 means that the fixed point numbers in this format have 8 bits for the integer part and 8 bits for the fraction part. A number of other notations have been used for the same purpose.
Single-precision floating-point format (sometimes called FP32 or float32) is a computer number format, usually occupying 32 bits in computer memory; it represents a wide dynamic range of numeric values by using a floating radix point.
The figure below shows the absolute precision for both formats over a range of values. This figure can be used to select an appropriate format given the expected value of a number and the required precision. Precision of binary32 and binary64 in the range 10 −12 to 10 12. An example of a layout for 32-bit floating point is
Note that there are other possible scenarios of format conversions to or from bfloat16. For example, int16 and bfloat16. 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 ...