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float                       C Keyword                       float




Data type


Floating point  numbers are a  subset of the  real numbers.  Each
has a built-in radix point (or ``decimal point'') that shifts, or
``floats'', as  the value of the number  changes.  It consists of
the following:  one sign bit, which  indicates whether the number
is positive or  negative; bits that encode the number's _e_x_p_o_n_e_n_t;
and bits  that encode the  number's _f_r_a_c_t_i_o_n, or  the number upon
which the exponent works.   In general, the magnitude of the num-
ber  encoded depends  upon the  number of  bits in  the exponent,
whereas  its precision  depends upon  the number  of bits  in the
fraction.

The exponent  often uses a  bias.  This is  a value that  is sub-
tracted from the exponent to yield  the power of two by which the
fraction will be increased.

Floating point  numbers come in  two levels of  precision: single
precision, called  floats; and double  precision, called doubles.
With most microprocessors,  sizeof(float) returns four, which in-
dicates that  it is four  chars (bytes) long,  and sizeof(double)
returns eight.

Several  formats  are  used  to  encode floats,  including  IEEE,
DECVAX,  and BCD  (binary coded  decimal).  COHERENT  uses DECVAX
format throughout.

The following  describes DECVAX, IEEE, and  BCD formats, for your
information.

***** DECVAX Format *****

The 32 bits in a float  consist of one sign bit, an eight-bit ex-
ponent, and  a 24-bit  fraction, as  follows.  Note that  in this
diagram, `s' indicates  ``sign'', `e' indicates ``exponent'', and
`f` indicates ``fraction'':


        ZDDDDDDDDDDD?
        | seee eeee | Byte 4
        CDDDDDDDDDDD4
        | efff ffff | Byte 3
        CDDDDDDDDDDD4
        | ffff ffff | Byte 2
        CDDDDDDDDDDD4
        | ffff ffff | Byte 1
        @DDDDDDDDDDDY


The exponent has a bias of 129.

If the sign  bit is set to one, the  number is negative; if it is
set to zero,  then the number is positive.  If  the number is all


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float                       C Keyword                       float



zeroes, then  it equals  zero; an  exponent and fraction  of zero
plus a  sign of  one (``negative zero'')  is by definition  not a
number.  All other forms are numeric values.

The most significant bit in the fraction is always set to one and
is not stored.  It is usually called the ``hidden bit''.

The format  for doubles simply  adds another 32  fraction bits to
the end of the float representation, as follows:


        ZDDDDDDDDDDD?
        | seee eeee | Byte 8
        CDDDDDDDDDDD4
        | efff ffff | Byte 7
        CDDDDDDDDDDD4
        | ffff ffff | Byte 6
        CDDDDDDDDDDD4
        | ffff ffff | Byte 5
        CDDDDDDDDDDD4
        | ffff ffff | Byte 4
        CDDDDDDDDDDD4
        | ffff ffff | Byte 3
        CDDDDDDDDDDD4
        | ffff ffff | Byte 2
        CDDDDDDDDDDD4
        | ffff ffff | Byte 1
        @DDDDDDDDDDDY


***** IEEE Format *****

The IEEE  encoding of a float  is the same as  that in the DECVAX
format.   Note, however,  that the  exponent has  a bias  of 127,
rather than 129.

Unlike the  DECVAX format, IEEE format  assigns special values to
several  floating  point numbers.   Note  that  in the  following
description, a  tiny exponent  is one that  is all zeroes,  and a
huge exponent is one that is all ones:

*  A  tiny exponent with a fraction of  zero equals zero, regard-
   less of the setting of the sign bit.

*   A huge  exponent  with a  fraction of  zero equals  infinity,
   regardless of the setting of the sign bit.

*  A tiny exponent with a  fraction greater than zero is a denor-
   malized number,  i.e., a  number that  is less than  the least
   normalized number.

*   A huge  exponent with  a  fraction greater  than zero  is, by
   definition, not a number.   These values can be used to handle
   special conditions.



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float                       C Keyword                       float



An IEEE double, unlike DECVAX format, increases the number of ex-
ponent bits.  It consists of  a sign bit, an 11-bit exponent, and
a 53-bit fraction, as follows:


        ZDDDDDDDDDDD?
        | seee eeee | Byte 8
        CDDDDDDDDDDD4
        | eeee ffff | Byte 7
        CDDDDDDDDDDD4
        | ffff ffff | Byte 6
        CDDDDDDDDDDD4
        | ffff ffff | Byte 5
        CDDDDDDDDDDD4
        | ffff ffff | Byte 4
        CDDDDDDDDDDD4
        | ffff ffff | Byte 3
        CDDDDDDDDDDD4
        | ffff ffff | Byte 2
        CDDDDDDDDDDD4
        | ffff ffff | Byte 1
        @DDDDDDDDDDDY


The exponent has a bias of  1,023.  The rules of encoding are the
same as for floats.

***** BCD Format *****

The BCD  format (``binary  coded decimal'', also  called ``packed
decimal'') is  used to eliminate  rounding errors that  alter the
worth of  an account by a  fraction of a cent.   It consists of a
sign, an exponent, and a chain of four-bit numbers, each of which
is defined to hold the values zero through nine.

A BCD float has a sign bit, seven bits of exponent, and six four-
bit digits.  In the following diagrams, `d' indicates ``digit'':


        ZDDDDDDDDDDD?
        | seee eeee | Byte 4
        CDDDDDDDDDDD4
        | dddd dddd | Byte 3
        CDDDDDDDDDDD4
        | dddd dddd | Byte 2
        CDDDDDDDDDDD4
        | dddd dddd | Byte 1
        @DDDDDDDDDDDY


A BCD double has a sign bit, 11 bits of exponent, and 13 four-bit
digits, as follows:





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float                       C Keyword                       float



        ZDDDDDDDDDDD?
        | seee eeee | Byte 8
        CDDDDDDDDDDD4
        | eeee dddd | Byte 7
        CDDDDDDDDDDD4
        | dddd dddd | Byte 6
        CDDDDDDDDDDD4
        | dddd dddd | Byte 5
        CDDDDDDDDDDD4
        | dddd dddd | Byte 4
        CDDDDDDDDDDD4
        | dddd dddd | Byte 3
        CDDDDDDDDDDD4
        | dddd dddd | Byte 2
        CDDDDDDDDDDD4
        | dddd dddd | Byte 1
        @DDDDDDDDDDDY


Passing the hexadecimal numbers A through F in a digit yields un-
predictable results.

The following rules apply when handling BCD numbers:

*  A tiny exponent with a fraction of zero equals zero.

*  A tiny exponent with a fraction of non-zero indicates a denor-
   malized number.

*  A huge exponent with a fraction of zero indicates infinity.

*  A huge exponent with a fraction of non-zero is, by definition,
   not a number; these non-numbers are used to indicate errors.

***** See Also *****

C keywords, data formats, double
_T_h_e _A_r_t _o_f _C_o_m_p_u_t_e_r _P_r_o_g_r_a_m_m_i_n_g, vol. 2, page 180_f_f



















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