Quiet Not a Number, the sign bit is meaningless. The 8087 and 80287 treat this as a Signaling Not a Number.

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. 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. 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 log 2 ⁡ ( 2 ( − 1 ) s ⋅ E ⋅ M ) = ( − 1 ) s ⋅ E ⋅ log 2 ⁡ ( 2 ) + log 2 ⁡ ( M ) = ± E + log 2 ⁡ ( M ) {\displaystyle \log _{2}(2 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:

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. 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 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 ]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: 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 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] long float: type to which float values are promoted for computations (and might be as small as float or as big as long double. 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. 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. 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.

Further fun facts:

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

It may be worth noticing that the C language standard is intentionally vague in defining the type for exactly this issue you're seeing. 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 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]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. 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. 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.