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Q28855: CINT and Integer Assignments Round x.5 to Nearest Even Integer

Article: Q28855
Product(s): See article
Version(s): 3.00 4.00 4.00b 4.50
Operating System(s): MS-DOS
Keyword(s): ENDUSER | B_BasicCom | mspl13_basic
Last Modified: 25-APR-1989

When a numeric expression ending in .5 is assigned to an integer
variable, a compiled BASIC program will round the expression to the
nearest even integer. For example, .5 converts to 0, 1.5 converts to
2, 2.5 converts to 2, and 3.5 converts to 4.

This rounding to the nearest even integer occurs for the CINT function
and for an integer division assigned to an integer variable. This
behavior is a feature of the IEEE Floating Point Standard.

This behavior occurs in any BASIC application that uses the IEEE
Floating Point format, including the following products:

1. Microsoft QuickBASIC Versions 4.00, 4.00b, and 4.50

2. The coprocessor version of QuickBASIC Version 3.00: QB87.EXE

3. The Microsoft BASIC Compiler Versions 6.00 and 6.00b for MS-DOS and
   MS OS/2 (when compiled with the BC /FPi option, where "i" stands
   for IEEE)

This type of integer rounding complies with the following IEEE
standard:

   "If the difference between the unrounded operand and the rounded
   result is exactly one half, the rounded result is even" (Section
   5.5 of the "IEEE Standard for Binary Floating-Point Arithmetic")

The purpose of this behavior is to prevent an average upward (or
downward) bias as various calculations are rounded. If the number was
always rounded up, there would be an upward bias in calculations.
Rounding to the nearest even number averages out; therefore, no
rounding bias occurs.

The compilers listed above store and manipulate numbers using IEEE
format. This exclusive use of IEEE format for real numbers is
necessary to enable compiled BASIC to CALL routines written in
FORTRAN, Pascal, and C, all of which use IEEE format.

B.EXE (noncoprocessor) Version 3.00 and QuickBASIC Versions 1.00,
1.01, 1.02, 2.00, and 2.01 all store numbers in Microsoft Binary
Floating Point format. Note that QB87.EXE, the coprocessor version of
QuickBASIC Version 3.00, uses IEEE format. QuickBASIC Versions 1.x,
2.00, and 2.01 do not support IEEE (or 8087, 80287, or 80387)
coprocessors.

Microsoft Binary format uses a standard different from that of IEEE
for converting between floating point and integers. In particular,
numbers with "5" as the least significant digit are always rounded up
to the next integer. The result does not have to be an even number.
Thus, the Microsoft Binary format has an upward rounding bias.

Note that QuickBASIC Versions 3.00 and earlier cannot make
interlanguage CALLs to FORTRAN, Pascal, or C.

The following are two examples of the above rounding behavior:

1. The following is an example of always rounding expressions ending
   in .5 to an even number by integer assignment:

   DEFINT A-Z
   INPUT "Type a whole number (1,2,3,4,5,6,...)",INUM
   IRESULT=INUM/2
   PRINT "If INUM/2 ends in .5, it rounds/truncates to even number:"
   PRINT IRESULT

2. The following is an example of rounding of the CINT() function:

      a=.5
      b=1.0
      c=1.5
      d=2.0
      e=2.5
      cls
      print "CINT (0.5) = "; CINT(A)
      PRINT "CINT (1.0) = "; CINT(B)
      PRINT "CINT (1.5) = "; CINT(C)
      PRINT "CINT (2.0) = "; CINT(D)
      PRINT "CINT (2.5) = "; CINT(E)

OUTPUT FROM:      B.EXE 4.00    |    B.EXE 3.00, 2.01, 2.00

CINT (0.5) =           0        |         1
CINT (1.0) =           1        |         1
CINT (1.5) =           2        |         2
CINT (2.0) =           2        |         2
CINT (2.5) =           2        |         3

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