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INFO: IEEE Floating-Point Representation and MS Languages


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Summary

The following information discusses how real*4 (single precision) and real*8 (double precision) numbers are stored internally by Microsoft languages that use the IEEE floating-point format.

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There are three internal varieties of real numbers. Microsoft is consistent with the IEEE numeric standards. Real*4 and real*8 are used in all of our languages. Real*10 is used with MASM and in the 16-bit versions of the C compiler. In 16-bit applications, Real* 10 is the internal floating-point format used in the numerical processors and coprocessors for the Intel 80x86 processor family and also in the emulator math package which emulates the Intel processors.

In FORTRAN, real*4 is declared using the words "REAL" or "REAL*4." The words "DOUBLE PRECISION" or "REAL*8" are used to declare a real*8 number. Real*10 variables cannot be specified in FORTRAN.

In C, real*4 is declared using the word "float." Real*8 is declared using the word "double". Real*10 is declared using the words "long double".

In MASM, real*4 is declared with the "DD" directive, real*8 is declared with the "DQ" directive, and real*10 is declared with the "DT" directive.

The values are stored as follows:
   real*4  sign bit, 8  bit exponent, 23 bit mantissa
   real*8  sign bit, 11 bit exponent, 52 bit mantissa
   real*10 sign bit, 15 bit exponent, 64 bit mantissa
				
In real*4 and real*8 formats, there is an assumed leading 1 in the mantissa that is not stored in memory, so the mantissas are actually 24 or 53 bits, even though only 23 or 52 bits are stored. The real*10 format actually stores this bit.

The exponents are biased by half of their possible value. This means you subtract this bias from the stored exponent to get the actual exponent. If the stored exponent is less than the bias, it is actually a negative exponent.

The exponents are biased as follows:
   8-bit  (real*4)  exponents are biased by 127
   11-bit (real*8)  exponents are biased by 1023
   15-bit (real*10) exponents are biased by 16383
				
These exponents are not powers of ten; they are powers of two, that is, 8-bit stored exponents can be up to 127. 2**127 is roughly equivalent to 10**38, which is the actual limit of real*4.

The mantissa is stored as a binary fraction of the form 1.XXX... . This fraction has a value greater than or equal to 1 and less than 2. Note that real numbers are always stored in normalized form, that is, the mantissa is left-shifted such that the high-order bit of the mantissa is always 1. Because this bit is always 1, it is assumed (not stored) in the real*4 and real*8 formats. The binary (not decimal) point is assumed to be just to the right of the leading 1.

The format, then, for the various sizes is as follows:
              BYTE 1    BYTE 2    BYTE 3    BYTE 4   ...  BYTE n
   real*4    SXXX XXXX XMMM MMMM MMMM MMMM MMMM MMMM
   real*8    SXXX XXXX XXXX MMMM MMMM MMMM MMMM MMMM ... MMMM MMMM
   real*10   SXXX XXXX XXXX XXXX 1MMM MMMM MMMM MMMM ... MMMM MMMM
				
S represents the sign bit, the X's are the exponent bits, and the M's are the mantissa bits. Note that the leftmost bit is assumed in real*4 and real*8 formats, but present as "1" in BYTE 3 of the real*10 format.

To shift the binary point properly, you first un-bias the exponent and then move the binary point to the right or left the appropriate number of bits.

The following are some examples in real*4 format:
                    SXXX XXXX XMMM MMMM ... MMMM MMMM
2   =  1  * 2**1  = 0100 0000 0000 0000 ... 0000 0000 = 4000 0000
				
Note the sign bit is zero, and the stored exponent is 128, or 100 0000 0 in binary, which is 127 plus 1. The stored mantissa is (1.) 000 0000 ... 0000 0000, which has an implied leading 1 and binary point, so the actual mantissa is one.
-2  = -1  * 2**1  = 1100 0000 0000 0000 ... 0000 0000 = C000 0000
				
Same as +2 except that the sign bit is set. This is true for all IEEE format floating-point numbers.
 4  =  1  * 2**2  = 0100 0000 1000 0000 ... 0000 0000 = 4080 0000
				
Same mantissa, exponent increases by one (biased value is 129, or 100 0000 1 in binary.
 6  = 1.5 * 2**2  = 0100 0000 1100 0000 ... 0000 0000 = 40C0 0000
				
Same exponent, mantissa is larger by half--it's (1.) 100 0000 ... 0000 0000, which, since this is a binary fraction, is 1 1/2 (the values of the fractional digits are 1/2, 1/4, 1/8, and so forth.).
 1  = 1   * 2**0  = 0011 1111 1000 0000 ... 0000 0000 = 3F80 0000
				
Same exponent as other powers of two, mantissa is one less than two at 127, or 011 1111 1 in binary.
.75 = 1.5 * 2**-1 = 0011 1111 0100 0000 ... 0000 0000 = 3F40 0000
				
The biased exponent is 126, 011 1111 0 in binary, and the mantissa is (1.) 100 0000 ... 0000 0000, which is 1 1/2.
2.5 = 1.25 * 2**1 = 0100 0000 0010 0000 ... 0000 0000 = 4020 0000
				
Exactly the same as two except that the bit that represents 1/4 is set in the mantissa.
0.1 = 1.6 * 2**-4 = 0011 1101 1100 1100 ... 1100 1101 = 3DCC CCCD
				
1/10 is a repeating fraction in binary. The mantissa is just shy of 1.6, and the biased exponent says that 1.6 is to be divided by 16 (it is 011 1101 1 in binary, which is 123 in decimal). The true exponent is 123 - 127 = -4, which means that the factor by which to multiply is 2**-4 = 1/16. Note that the stored mantissa is rounded up in the last bit -- an attempt to represent the unrepresentable number as accurately as possible. (The reason that 1/10 and 1/100 are not exactly representable in binary similar to the reason that 1/3 is not exactly representable in decimal.)
 0  = 1.0 * 2**-128 = all zero's--a special case.
				

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Keywords: kbinfo, kblangc, kblangfortran, KB36068

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Article Info
Article ID : 36068
Revision : 2
Created on : 12/15/2003
Published on : 12/15/2003
Exists online : False
Views : 404