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A notable example of the need for a minimum of 64bits of precision in the significand of the extended precision format is the need to avoid precision loss when performing exponentiation on double-precision values. [26] [27] [28] [c] The x86 floating-point units do not provide an instruction that directly performs exponentiation. Instead they provide a set of instructions that a program can use in sequence to perform exponentiation using the equation:
Extended precision refers to floating-point number formats that provide greater precision than the basic floating-point formats. [1] Extended precision formats support a basic format by minimizing roundoff and overflow errors in intermediate values of expressions on the base format. In contrast to extended precision, arbitrary-precision arithmetic refers to implementations of much larger numeric types (with a storage count that usually is not a power of two) using special software (or, rarely, hardware). The IA32, x86-64, and Itanium processors support what is by far the most influential format on this standard, the Intel 80-bit (64 bit significand) "double extended" format, described in the next section.Why they'd chose to use old SW/HW instead of modern ones needs to be evaluated on a case by case basis, but often the reason boils down to one of these three: In contrast to the single and double-precision formats, this format does not utilize an implicit/ hidden bit. Rather, bit 63 contains the integer part of the significand and bits 62-0 hold the fractional part. Bit 63 will be 1 on all normalized numbers. There were several advantages to this design when the 8087 was being developed: The 8087 had 80-bit registers so that if the inputs to your computation had 64-bit accuracy, the outputs would also have 64-bit accuracy. long float: type to which float values are promoted for computations (and might be as small as float or as big as long double.
Floating-point Indefinite, the result of invalid calculations such as square root of a negative number, logarithm of a negative number, 0/0, infinity / infinity, infinity times 0, and others when the processor has been configured to not generate exceptions for invalid operands. The sign bit is meaningless. This is a special case of a Quiet Not a Number.
Pseudo-Infinity. The sign bit gives the sign of the infinity. The 8087 and 80287 treat this as Infinity. The 80387 and later treat this as an invalid operand. The IBM System/360 supports a 32-bit "short" floating-point format and a 64-bit "long" floating-point format. [4] The 360/85 and follow-on System/370 add support for a 128-bit "extended" format. [5] These formats are still supported in the current design, where they are now called the " hexadecimal floating-point" (HFP) formats. Writing for Legacy is a thing. There are industries still using Windows XP for their QA software/hardware and some banks still run SW written in COBOL for Mainframes in the 70's. In the following table, " s" is the value of the sign bit (0 means positive, 1 means negative), " e" is the value of the exponent field interpreted as a positive integer, and " m" is the significand interpreted as a positive binary number where the binary point is located between bits 63 and 62. The " m" field is the combination of the integer and fraction parts in the above diagram.
Unnormal. Only generated on the 8087 and 80287. The 80387 and later treat this as an invalid operand. The value is (−1) s × m × 2 e−16383 where s is the sign of the exponent (either 0 or 1), E is the unbiased exponent, which is an integer that ranges from 0 to 1023, and M is the significand which is a 53-bit value that falls in the range 1 ≤ M< 2. Negative numbers and zero can be ignored because the logarithm of these values is undefined. For purposes of this discussion M does not have 53bits of precision because it is constrained to be greater than or equal to one i.e. the hidden bit does not count towards the precision (Note that in situations where M is less than 1, the value is actually a de-normal and therefore may have already suffered precision loss. This situation is beyond the scope of this article). The FPA10 math coprocessor for early ARM processors also supports this extended precision type (similar to the Intel format although padded to a 96-bit format with 16zero bits inserted between the sign and the exponent fields), but without correct rounding. [11] In addition to supporting IEEE single and double precision numbers, it also supported an 80-bit extended precision number. Some C compilers (e.g. clang) mapped this to the long double type in C It may be worth noticing that the C language standard is intentionally vague in defining the type for exactly this issue you're seeing.
Floating-Point Reference Sheet for Intel® Architecture
Calculations can be completed a little faster if all bits of the significand are present in the register.
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