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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. 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. It may be worth noticing that the C language standard is intentionally vague in defining the type for exactly this issue you're seeing. Taking the log of this representation of a double-precision number and simplifying results in the following: 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-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. log 2 ( 2 ( − 1 ) s ⋅ E ⋅ M ) = ( − 1 ) s ⋅ E ⋅ log 2 ( 2 ) + log 2 ( M ) = ± E + log 2 ( M ) {\displaystyle \log _{2}(2 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. double must have greater or equal precision as float. At no point it says one must be 64-bit and the other 32-bit precision. The 80-bit floating-point format was widely available by 1984, [25] after the development of C, Fortran and similar computer languages, which initially offered only the common 32- and 64-bit floating-point sizes. On the x86 design most C compilers now support 80-bit extended precision via the long double type, and this was specified in the C99 / C11 standards (IEC 60559 floating-point arithmetic (Annex F)). Compilers on x86 for other languages often support extended precision as well, sometimes via nonstandard extensions: for example, Turbo Pascal offers an Extended type, and several Fortran compilers have a REAL*10 type (analogous to REAL*4 and REAL*8). Such compilers also typically include extended-precision mathematical subroutines, such as square root and trigonometric functions, in their standard libraries.
Further fun facts:
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] The x86 extended precision format is an 80-bit format first implemented in the Intel 8087 math coprocessor and is supported by all processors that are based on the x86 design that incorporate a floating-point unit (FPU).
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. It is also worth remembering that the x87 FPU had no ability to store the 80 bits into memory. Those extra 16 bits only lived in registers and were lost once they spill into memory. Its usefulness has always been limited. 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. Pseudo Not a Number. The sign bit is meaningless. The 8087 and 80287 treat this as a Signaling Not a Number. The 80387 and later treat this as an invalid operand.
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] I would then suggest having a means of explicitly passing types other than double to functions, but say that expressions that don't explicitly force the type of a floating-point value passed to a variadic function would by default be converted to double. And since it's a "legacy" instruction set, modern CPUs don't tend to optimize x87 instructions very well. If you don't need float80, then you have the option to do all your x86 floating point with SSE, which is a much more "normal" architecture with a random-access register file ( xmm), and ignore the x87 altogether. SSE is supported by all x86-64 CPUs, and by all 32-bit x86 CPUs from the last 20 years or so. 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: 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 Denormal. The 80387 and later properly interpret this value but will not generate it. The value is (−1) s × m × 2 −16382 The IBM 1130, sold in 1965, [2] offered two floating-point formats: A 32-bit "standard precision" format and a 40-bit "extended precision" format. Standard precision format contains a 24-bit two's complement significand while extended precision utilizes a 32-bit two's complement significand. The latter format makes full use of the CPU's 32-bit integer operations. The characteristic in both formats is an 8-bit field containing the power of two biased by 128. Floating-point arithmetic operations are performed by software, and double precision is not supported at all. The extended format occupies three 16-bit words, with the extra space simply ignored. [3] 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 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.The x87 and Motorola68881 80-bit formats meet the requirements of the IEEE 754 double extended format, [12] as does the IEEE754 128-bit format. This means that, for x64 applications, there is no guarantee the x87 instruction set will be present. 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.
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