Article ID: 42980 - Last Review: August 16, 2005 - Revision: 3.1

This article was previously published under Q42980

Floating-point mathematics is a complex topic that confuses many
programmers. The tutorial below should help you recognize programming
situations where floating-point errors are likely to occur and how to
avoid them. It should also allow you to recognize cases that are
caused by inherent floating-point math limitations as opposed to
actual compiler bugs.

The number 532.25 in decimal (base 10) means the following:

(5 * 10^2) + (3 * 10^1) + (2 * 10^0) + (2 * 10^-1) + (5 * 10^-2) 500 + 30 + 2 + 2/10 + 5/100 _________ = 532.25

In the binary number system (base 2), each column represents a power of 2 instead of 10. For example, the number 101.01 means the following:

(1 * 2^2) + (0 * 2^1) + (1 * 2^0) + (0 * 2^-1) + (1 * 2^-2) 4 + 0 + 1 + 0 + 1/4 _________ = 5.25 Decimal

1 Decimal = 1 Binary 2 Decimal = 10 Binary 22 Decimal = 10110 Binary, etc.

However, negative integers are represented using the two's complement scheme. To get the two's complement representation for a negative number, take the binary representation for the number's absolute value and then flip all the bits and add 1. For example:

4 Decimal = 0000 0000 0000 0100 1111 1111 1111 1011 Flip the Bits -4 = 1111 1111 1111 1100 Add 1

Note that -1 Decimal = 1111 1111 1111 1111 in Binary, which explains why Basic treats -1 as logical true (All bits = 1). This is a consequence of not having separate operators for bitwise and logical comparisons. Often in Basic, it is convenient to use the code fragment below when your program will be making many logical comparisons. This greatly aids readability.

CONST TRUE = -1 CONST FALSE = NOT TRUE

Note that adding any combination of two's complement numbers together using ordinary binary arithmetic produces the correct result.

For binary, in particular, only fractional numbers that can be represented in the form p/q, where q is an integer power of 2, can be expressed exactly, with a finite number of bits.

Even common decimal fractions, such as decimal 0.0001, cannot be represented exactly in binary. (0.0001 is a repeating binary fraction with a period of 104 bits!)

This explains why a simple example, such as the following

SUM = 0 FOR I% = 1 TO 10000 SUM = SUM + 0.0001 NEXT I% PRINT SUM ' Theoretically = 1.0.

will PRINT 1.000054 as output. The small error in representing 0.0001 in binary propagates to the sum.

For the same reason, you should always be very cautious when making comparisons on real numbers. The following example illustrates a common programming error:

item1# = 69.82# item2# = 69.20# + 0.62# IF item1# = item2# then print "Equality!"

This will NOT PRINT "Equality!" because 69.82 cannot be represented exactly in binary, which causes the value that results from the assignment to be SLIGHTLY different (in binary) than the value that is generated from the expression. In practice, you should always code such comparisons in such a way as to allow for some tolerance. For example:

IF (item1# < 69.83#) AND (item1# > 69.81#) then print "Equal"

This will PRINT "Equal".

- To allow Basic to use the Intel math coprocessors, which use IEEE format. The Intel 80x87 series coprocessors cannot work with Microsoft Binary Format numbers.
- To make interlanguage calling between Basic, C, Pascal, FORTRAN, and MASM much easier. Otherwise, conversion routines would have to be used to send numeric values from one language to another.
- To achieve consistency. IEEE is the accepted industry standard for C and FORTRAN compilers.

Sign Bits Exponent Bits Mantissa Bits --------- ------------- ------------- IEEE 1 11 52 + 1 (Implied) MBF 1 8 56

For more information on the differences between IEEE and MBF floating-point representation, query in the Microsoft Knowledge Base on the following words:

IEEE and floating and point and appnote

Note that IEEE has more bits dedicated to the exponent, which allows it to represent a wider range of values. MBF has more mantissa bits, which allows it to be more precise within its narrower range.

Also, keep in mind that the numbers that can be represented in IEEE are spread out over a very wide range. You can imagine them on a number line. There is a high density of representable numbers near 1.0 and -1.0 but fewer and fewer as you go towards 0 or infinity.

The goal of the IEEE standard, which is designed for engineering calculations, is to maximize accuracy (to get as close as possible to the actual number). Precision refers to the number of digits that you can represent. The IEEE standard attempts to balance the number of bits dedicated to the exponent with the number of bits used for the fractional part of the number, to keep both accuracy and precision within acceptable limits.

X = Fraction * 2^(exponent - bias)

[Fraction] is the normalized fractional part of the number, normalized because the exponent is adjusted so that the leading bit is always a 1. This way, it does not have to be stored, and you get one more bit of precision. This is why there is an implied bit. You can think of this like scientific notation, where you manipulate the exponent to have one digit to the left of the decimal point, except in binary, you can always manipulate the exponent so that the first bit is a 1, since there are only 1s and 0s.

[bias] is the bias value used to avoid having to store negative exponents.

The bias for single-precision numbers is 127 and 1023 (decimal) for double-precision numbers.

The values equal to all 0's and all 1's (binary) are reserved for representing special cases. There are other special cases as well, that indicate various error conditions.

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 1.

-2 = -1 * 2^1 = 1100 0000 0000 0000 ... 0000 0000 = C000 0000 hex

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 hex

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 hex

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, etc.).

1 = 1 * 2^0 = 0011 1111 1000 0000 ... 0000 0000 = 3F80 0000 hex

Same exponent as other powers of 2, mantissa is one less than
2 at 127, or 011 1111 1 in binary.

.75 = 1.5 * 2^-1 = 0011 1111 0100 0000 ... 0000 0000 = 3F40 0000 hex

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 hex

Exactly the same as 2 except that the bit which represents 1/4 is
set in the mantissa.

0.1 = 1.6 * 2^-4 = 0011 1101 1100 1100 ... 1100 1101 = 3DCC CCCD hex

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. This is 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 is similar
to the way that 1/3 is not exactly representable in decimal.)

0 = 1.0 * 2^-128 = all zeros -- a special case.

- Round-off error

This error results when all of the bits in a binary number cannot be used in a calculation.

Example: Adding 0.0001 to 0.9900 (Single Precision)

Decimal 0.0001 will be represented as:(1.)10100011011011100010111 * 2^(-14+Bias) (13 Leading 0s in Binary!)0.9900 will be represented as:(1.)11111010111000010100011 * 2^(-1+Bias)Now to actually add these numbers, the decimal (binary) points must be aligned. For this they must be Unnormalized. Here is the resulting addition:This is called a round-off error because some computers round when shifting for addition. Others simply truncate. Round-off errors are important to consider whenever you are adding or multiplying two very different values..000000000000011010001101 * 2^0 <- Only 11 of 23 Bits retained +.111111010111000010100011 * 2^0 ________________________________ .111111010111011100110000 * 2^0

- Subtracting two almost equal values
This will be normalized. Note that although the original numbers each had four significant digits, the result has only one significant digit.
.1235 -.1234 _____ .0001

- Overflow and underflow

This occurs when the result is too large or too small to be represented by the data type. - Quantizing error

This occurs with those numbers that cannot be represented in exact form by the floating-point standard. - Division by a very small number

This can trigger a "divide by zero" error or can produce bad results, as in the following example:In QuickBasic for MS-DOS, X now has the value 888887, instead of the correct answer, 900000.A = 112000000 B = 100000 C = 0.0009 X = A - B / C

- Output error

This type of error occurs when the output functions alter the values they are working with.

- Microsoft Visual Basic 2.0 Standard Edition
- Microsoft Visual Basic 3.0 Professional Edition
- Microsoft Visual Basic 4.0 Standard Edition
- Microsoft Visual Basic 1.0 Standard Edition
- Microsoft Visual Basic 2.0 Professional Edition
- Microsoft Visual Basic 4.0 Professional Edition
- Microsoft Visual Basic for MS-DOS
- Microsoft QuickBasic 4.0
- Microsoft QuickBASIC 4.0b
- Microsoft QuickBasic 4.5 for MS-DOS
- Microsoft BASIC Compiler 6.0
- Microsoft BASIC Compiler 6.0b
- Microsoft BASIC Professional Development System 7.0
- Microsoft Cinemania 97 Standard Edition

## Keywords: | KB42980 |

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