IEEE vs. Microsoft Binary Format; Rounding Issues (Complete)

IEEE was chosen as the math package for QuickBasic version 4.00 and Microsoft Basic Compiler 6.00 to allow for mixed-language calling capabilities. This ability is a very desirable feature. In addition to this feature, IEEE also is more accurate than Microsoft Binary Format (MBF). Calculations are performed in an 80-bit temporary area rather than a 64-bit area. (Note, the Alternate-Math Libraries use a 64-bit temporary area.) The additional bits provide for more accurate calculations and decrease the possibility that the final result has been degraded by excessive roundoff errors. Keep in mind that precision errors are inherent in any binary floating-point math. Not all numbers can be accurately represented in a binary floating-point notation.

IEEE also can take advantage of a math coprocessor chip (such as the 8087, 80287, and 80387) for great speed. MBF cannot take advantage of a coprocessor.

MBF is accurate to 15 digits, while IEEE is accurate to 15 or 16 digits. Since the numbers are stored in different formats, the last digit may vary. MBF double-precision values are stored in the following format:

      -------------------------------------------------
     |              |    |                             |
     |8 Bit Exponent|Sign|   55 Bit Mantissa           |
     |              | Bit|                             |
      -------------------------------------------------

   IEEE double precision values are stored in the following format:

      -------------------------------------------------
     |    |                | |                         |
     |Sign| 11 Bit Exponent|1|  52 Bit Mantissa        |
     | Bit|                | |                         |
      -------------------------------------------------
                            ^
                            Implied Bit (always 1)

   You will notice that Microsoft Binary Format (MBF) has 4 more bits
   of precision in the mantissa. However, this does not mean that the
   value is any more accurate. Precision is the number of bits you are
   working with, while accuracy is how close you are to the real
   number. In most cases, the IEEE value will be more accurate because
   it was calculated in an 80-bit temporary. (When the IEEE standard
   was proposed, the main consideration for double precision values
   was range. As a minimum, the desire was that the product of any two
   32-bit numbers should not overflow the 64-bit format.)

Your rounding algorithm is correctly rounding the numbers, but the extra digit is occurring because of the inherent rounding errors and format differences. For example, 6.99999999999999D-2 is rounded to .07 but the internal IEEE representation of the value is 7.000000000000001D-2. (It is true that MBF displays the value as .07, but the difference in values is not considered as a problem. It is a difference between math packages.)

The STR$ function works correctly. The value placed in the string is the same as the value displayed on the screen with an unformatted PRINT. If the IEEE representation of .07 is 7.000000000000001D-2, then the STR$ will return 7.000000000000001D-2.

There are a few ways to generate the desired string. The method used depends on the range of numbers, other resources available, and programmer's preference. Listed below are three possible routines that can be used. Keep in mind that as soon as the string is converted back to a number, it will no longer be truncated.

Method 1 --------

If the range of numbers is between 2^32/100 and -2^32/100, the following method can be used:

FUNCTION round2$ (number#) n& = number# * 100# hold$ = LTRIM$(RTRIM$(STR$(n&)))

IF (MID$(hold$, 1, 1) = "-") THEN

      hold1$ = "-"
      hold$ = MID$(hold$, 2)

      hold1$ = ""

length = LEN(hold$) SELECT CASE length CASE 1

      hold1$ = hold1$ + ".0" + hold$

      hold1$ = hold1$ + "." + hold$

      hold1$ = hold1$ + LEFT$(hold$, LEN(hold$) - 2)
      hold1$ = hold1$ + "." + RIGHT$(hold$, 2)

The value being rounded is multiplied by 100# and the result is stored in a long integer. The long integer is converted to a string and the decimal point is inserted in the correct location.

Method 2 --------

This routine is much more complicated than the first method, though it handles a much larger range of values. The value being rounded is multiplied by 100# and this result must fit within the range of valid double precision numbers.

FUNCTION round$ (number#) STATIC number# = INT((number# + .005) * 100#) / 100# hold$ = STR$(number#) hold$ = RTRIM$(LTRIM$(hold$))

IF (MID$(hold$, 1, 1) = "-") THEN

     new$ = "-"
     hold$ = MID$(hold$, 2)

     new$ = ""

x = INSTR(hold$, "D") DecimalLocation = INSTR(hold$, ".")

IF (x) THEN 'scientific notation

     exponent = VAL(MID$(hold$, x + 1, LEN(hold$)))
     IF (exponent < 0) THEN
       new$ = new$ + "."
       new$ = new$ + STRING$(ABS(exponent) - 1, ASC("0"))
       round$ = new$ + MID$(hold$, 1, 1)
     ELSE
       new$ = new$ + MID$(hold$, 1, DecimalLocation - 1)
       num = LEN(hold$) - 6
       IF num < 0 THEN
         num = exponent
       ELSE
         num = exponent - num
      new$ = new$+MID$(hold$, DecimalLocation+1, x-DecimalLocation-1)
       END IF
       new$ = new$ + STRING$(num, ASC("0")) + ".00"
       round$ = new$
     END IF

   ELSE  'not scientific notation
     x = INSTR(hold$, ".") 'find decimal point
     IF (x) THEN
       IF MID$(hold$, x + 3, 1) = "9" THEN
         xx = VAL(MID$(hold$, x + 2, 1)) + 1
         hold1$ = LEFT$(hold$, x)
         IF xx = 10 THEN
     hold1$ = hold1$+LTRIM$(STR$(VAL(MID$(hold$, x + 1, 1)) + 1))+"0"
           round$ = new$ + hold1$
         ELSE
           hold1$ = hold1$ + MID$(hold$, x + 1, 1) + LTRIM$(STR$(xx))
           round$ = new$ + hold1$
         END IF
       ELSE
         round$ = new$ + LEFT$(hold$, x + 2)
       END IF
     ELSE
      round$ = new$ + hold$
     END IF
   END IF
   END FUNCTION

   Method 3
   --------

   This method requires the use of the Microsoft C Compiler 5.x. It
   uses the C library routine sprintf(). This routine takes formatted
   screen output and stores it in a string variable.

   C Routine:

   struct basic_string {
      int length;
       char *address;
       } ;

     void round(number,string)
     double *number;
     struct basic_string *string;
     {
     sprintf(string->address,"%.2f",*number);
     }

   Basic Program:

   DECLARE SUB Round CDECL (number#, answer$)
   CLS
   b# = .05#
   FOR i = 1 TO 10
        b# = b# + .01#
        answer$ = SPACE$(50)
        CALL Round(b#, answer$)
        PRINT b#, LTRIM$(RTRIM$(answer$))
        PRINT
        cnt = cnt + 4
        IF cnt > 40 THEN
           cnt = 0
           INPUT a$
        END IF
   NEXT i

   The same screen formatting can be accomplished with Basic's PRINT
   USING statement. However, Basic has no direct means of storing this
   information in a string. The information can be sent to a
   Sequential file and then read back into string variables.

   You can also write the information to the screen and read this
   information using the SCREEN function. The SCREEN function returns
   the ASCII value of the specified screen location. Consider the
   following example:

   x# = 7.000000000000001D-02
   CLS
   LOCATE 1, 1
   PRINT USING "#################.##"; x#
   FOR i = 1 TO 20
   num = SCREEN(1, i)
   SELECT CASE num
     CASE ASC(".")
       number$ = number$ + "."
     CASE ASC("-")
       number$ = "-"
     CASE ASC("0") TO ASC("9")
       number$ = number$ + CHR$(num)
     CASE ELSE
   END SELECT
   NEXT i
   PRINT number$

   The PRINT USING statement would display 17 spaces and then .07. The
   value of number$ would be .07.