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 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. 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 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). Calculations can be completed a little faster if all bits of the significand are present in the register.

Pseudo Denormal. The 80387 and later properly interpret this value but will not generate it. The value is (−1) s × m × 2 −16382 Quiet Not a Number, the sign bit is meaningless. The 8087 and 80287 treat this as a Signaling Not a Number. 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. 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.

Floating-Point Reference Sheet for Intel® Architecture

Taking the log of this representation of a double-precision number and simplifying results in the following: 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). 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:

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. 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 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.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] 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