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(Complete) Tutorial to Understand IEEE Floating-Point Errors
The information in this article applies to:
- Microsoft QuickBasic for MS-DOS, versions 3.0 (QB87.EXE
coprocessor version only), 4.0, 4.0b, and 4.5
- Microsoft Basic Compiler for MS-DOS and MS OS/2, versions 6.0 and
6.0b
- Microsoft Basic Professional Development System (PDS) for MS-DOS
and MS OS/2, versions 7.0 and 7.1
SUMMARY
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.
MORE INFORMATION
Decimal and Binary Number Systems
Normally, we count things in base 10. The base is completely arbitrary.
The only reason that people have traditionally used base 10 is that they
have 10 fingers, which have made handy counting tools.
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
How Integers Are Represented in PCs
Because there is no fractional part to an integer, its machine representation
is much simpler than it is for floating-point values.
Normal integers on personal computers (PCs) are 2 bytes (16 bits) long with
the most significant bit indicating the sign.
Long integers are 4 bytes long.
Positive values are straightforward binary numbers.
For example:
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.
Floating-Point Complications
Every decimal integer can be exactly represented by a binary integer; however,
this is not true for fractional numbers.
In fact, every number that is irrational in base 10 will also be irrational
in any system with a base smaller than 10.
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".
IEEE Format Numbers
QuickBasic for MS-DOS, version 3.0 was shipped with an
MBF (Microsoft Binary Floating Point) version and an
IEEE (Institute of Electrical and Electronics Engineers) version for
machines with a math coprocessor. QuickBasic for MS-DOS,
versions 4.0 and later only use IEEE.
Microsoft chose the IEEE standard to represent floating-point values
in current versions of Basic for the following three primary reasons:
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.
The following is a quick comparison of IEEE and MBF representations for a
double-precision number:
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.
General Floating-Point Concepts
It is very important to realize that any binary floating-point system can
represent only a finite number of floating-point values in exact form.
All other values must be approximated by the closest representable value.
The IEEE standard specifies the method for rounding values to the "closest"
representable value.
QuickBasic for MS-DOS supports the standard and rounds according to
the IEEE rules.
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.
IEEE Details
Floating-point numbers are represented in the following form,
where [exponent] is the binary exponent:
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.
Single-Precision Examples
2 = 1 * 2^1 = 0100 0000 0000 0000 ... 0000 0000 = 4000 0000 hex
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.
Other Common Floating-Point Errors
The following are common floating-point errors:
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:
.000000000000011010001101 * 2^0 <- Only 11 of 23 Bits retained
+.111111010111000010100011 * 2^0
________________________________
.111111010111011100110000 * 2^0
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.
Subtracting two almost equal values
.1235
-.1234
_____
.0001
This will be normalized.
Note that although the original numbers each had four significant digits,
the result has only one significant digit.
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:
A = 112000000
B = 100000
C = 0.0009
X = A - B / C
In QuickBasic for MS-DOS, X now has the value 888887,
instead of the correct answer, 900000.
Output error
This type of error occurs when the output functions alter the values they
are working with.
73's from Lodewijk PA3BNX@PI8OSS as long as PI8SHB has no backbone net.
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