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It may be worth noticing that the C language standard is intentionally vague in defining the type for exactly this issue you're seeing. 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). 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. 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. 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.
The IEEE 754 floating-point standard recommends that implementations provide extended precision formats. The standard specifies the minimum requirements for an extended format but does not specify an encoding. [7] The encoding is the implementor's choice. [8] This means that, for x64 applications, there is no guarantee the x87 instruction set will be present.
Floating-Point Reference Sheet for Intel® Architecture
Taking the log of this representation of a double-precision number and simplifying results in the following: Quiet Not a Number, the sign bit is meaningless. The 8087 and 80287 treat this as a Signaling Not a Number. 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). 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.
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 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 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. 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. 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 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 ] 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. log 2 ( 2 ( − 1 ) s ⋅ E ⋅ M ) = ( − 1 ) s ⋅ E ⋅ log 2 ( 2 ) + log 2 ( M ) = ± E + log 2 ( M ) {\displaystyle \log _{2}(2 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).
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