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Quiet Not a Number, the sign bit is meaningless. The 8087 and 80287 treat this as a Signaling Not a Number. log 2 ( 2 ( − 1 ) s ⋅ E ⋅ M ) = ( − 1 ) s ⋅ E ⋅ log 2 ( 2 ) + log 2 ( M ) = ± E + log 2 ( M ) {\displaystyle \log _{2}(2 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: Infinity. The sign bit gives the sign of the infinity. The 8087 and 80287 treat this as a Signaling Not a Number. The 8087 and 80287 coprocessors used the pseudo-infinity representation for infinities.
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. Intel created a series of floating-point coprocessors for the x86 called the x87. 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, but others (e.g. MSVC) didn't. 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 way floating-point arithmetic was supposed to work, when IEEE 754 and the 8087 were designed, is that when you compute something like w ← a + bx + cyz, all of the intermediate values are computed at a higher precision than the inputs and outputs. This is similar to the best practice for hand calculation. People sometimes ask "if I'm calculating a result to 3 sig figs, should I round all of the intermediates to 3 sig figs also?" and the answer to that is no—not if you can avoid it. Keeping extra digits around helps to avoid cumulative accuracy loss from roundoff. Taking the log of this representation of a double-precision number and simplifying results in the following:
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] 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. On many embedded platforms without floating-point units, computations using a 32-bit or 64-bit mantissa without an "implied 1" would be faster, more precise, and in just about every way better than those IEEE-754 64-bit double-precision values. Unfortunately, the way the C Standard added long double broke a key aspect of the language: that all floating-point values passed to variadic functions be converted to a common type. Pseudo Denormal. The 80387 and later properly interpret this value but will not generate it. The value is (−1) s × m × 2 −16382 This 80-bit format uses one bit for the sign of the significand, 15 bits for the exponent field (i.e. the same range as the 128-bit quadruple precision IEEE 754 format) and 64 bits for the significand. The exponent field is biased by 16383, meaning that 16383 has to be subtracted from the value in the exponent field to compute the actual power of 2. [20] An exponent field value of 32767 (all fifteen bits 1) is reserved so as to enable the representation of special states such as infinity and Not a Number. If the exponent field is zero, the value is a denormal number and the exponent of 2 is −16382. [21]
Many languages have no built-in support for this type. The most recent example I know of that does is Swift, which has a Float80 type only available when compiling for Intel processors. (Swift also has CLongDouble which represents the exact type that the C compiler takes long double to mean, which is sometimes the same thing as Double.) The only time I've seen Float80 or long double used in practice is to use the increased precision to emulate a fused multiply-add instruction on older processors that don't support it, or very rarely to avoid loss of precision when converting from a 64-bit integer. Extra digits make it easier for ordinary mortals to write floating-point calculations that won't go wrong for hard-to-analyze reasons. The more extra precision you have, the more you can imagine that your custom formulas (a + bx + cyz) will behave similarly to library functions (sin x) that were designed by experts to have a error of no more than 1 ulp over their whole domain. You don't get guaranteed accuracy, but you do get more reliable accuracy. The Microsoft BASIC port for the 6502 CPU, such as in adaptations like Commodore BASIC, AppleSoft BASIC, KIM-1 BASIC or MicroTAN BASIC, supports an extended 40-bit variant of the floating-point format Microsoft Binary Format (MBF) since 1977. [6] IEEE 754 extended precision formats [ edit ]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. 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.
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