XL: Floating-Point Arithmetic May Give Inaccurate Results

ID: Q78113

The information in this article applies to:

Microsoft Excel for Windows, versions 2.10d, 3.0, 3.0a, 4.0, 4.0a, 5.0, 5.0c
Microsoft Excel for Windows 95, version 7.0
Microsoft Excel 97 for Windows
Microsoft Excel for Windows NT, version 5.0
Microsoft Excel for the Macintosh, versions 2.20, 2.2a, 3.0, 3.0a, 4.0, 5.0, 5.0a
Microsoft Excel 98 Macintosh Edition
Microsoft Works for Windows, versions 2.0, 3.0, 4.0

SUMMARY

This article discusses how Microsoft Excel stores and calculates floating-point numbers. This may affect the results of some numbers or formulas due to rounding and/or data truncation.

Overview

Microsoft Excel was designed around the IEEE 754 specification with respect to storing and calculating floating-point numbers. IEEE is the Institute of Electrical and Electronics Engineers, an international body that, among other things, determines standards for computer software and hardware. The 754 specification is a very widely adopted specification that describes how floating-point numbers should be stored in a binary computer. It is popular because it allows floating-point numbers to be stored in a reasonable amount of space and calculations to occur relatively quickly. The 754 standard is used in the floating-point units and numeric data processors of nearly all of today's PC-based microprocessors that implement floating-point math, including the Intel, Motorola, Sun, and MIPS processors.

When storing numbers, a corresponding binary number can represent every number or fractional number. For example, the fraction 1/10 can be represented in a decimal number system as 0.1. However, the same number in binary format becomes the repeating binary decimal

0001100110011100110011 (and so on)

and can be infinitely repeated. This number cannot be represented in a finite (limited) amount of space. Therefore, this number is rounded down by approximately -2.8E-17 when it is stored.

However, there are some limitations of the IEEE 754 specification which fall into three general categories:

maximum/minimum limitations

precision

repeating binary numbers

MORE INFORMATION

Maximum/Minimum Limitations

All computers have a maximum and a minimum number that can be handled. Because the number of bits of memory in which the number is stored is finite, it follows that the maximum or minimum number that can be stored is also finite. For Excel, the maximum number that can be stored is 1.79769313486232E+308 and the minimum positive number that can be stored is 2.2250738585072E-308.

Cases Where We Adhere to IEEE 754

Underflow: Underflow occurs when a number is generated that is too small to be represented. In IEEE and Excel, the result is 0 (with the exception that IEEE has a concept of -0, and Excel does not).

Overflow: Overflow occurs when a number is too large to be represented. Excel uses its own special representation for this case (#NUM!).

Cases Where We Do Not Adhere to IEEE 754

Denormalized numbers: A denormalized number is indicated by an exponent of 0. In that case, the entire number is stored in the mantissa and the mantissa has no implicit leading 1. As a result, you lose precision, and the smaller the number, the more precision is lost. Numbers at the small end of this range have only one digit of precision.
Example: A normalized number has an implicit leading 1. For instance, if the mantissa represents 0011001, the normalized number becomes 10011001 because of the implied leading 1. A denormalized number does not have an implicit leading one, so in our example of 0011001, the denormalized number remains the same. In this case, the normalized number has eight significant digits (10011001) while the denormalized number has five significant digits (11001) with leading zeroes being insignificant. Denormalized numbers are basically a workaround to allow numbers smaller than the normal lower limit to be stored. Microsoft does not implement this optional portion of the specification because denormalized numbers by their very nature have a variable number of significant digits. This can allow significant error to enter into calculations.

Positive/Negative Infinities: Infinities occur when you divide by 0. Excel does not support infinities, rather, it gives a #DIV/0! error in these cases.

Not-a-Number (NaN): NaN is used to represent invalid operations (such as infinity/infinity, infinity-infinity, or the square root of -1). NaNs allow a program to continue past an invalid operation. Excel instead immediately generates an error such as #NUM! or #DIV/0!.

Precision

A floating-point number is stored in binary in three parts within a 65-bit range: the sign, the exponent, and the mantissa.

1 Sign Bit 11 Bit Exponent 1 Implied Bit 52 Bit Mantissa

The sign stores the sign of the number (positive or negative), the exponent stores the power of 2 to which the number is raised or lowered (the maximum/minimum power of 2 is +1,023 and -1,022), and the mantissa stores the actual number. The finite storage area for the mantissa limits how close two adjacent floating point numbers can be (that is, the precision).

The mantissa and the exponent are both stored as separate components. As a result, the amount of precision possible may vary depending on the size of the number (the mantissa) being manipulated. In the case of Excel, although Excel can store numbers from 1.79769313486232E308 to 2.2250738585072E-308, it can only do so within 15 digits of precision. This limitation is a direct result of strictly following the IEEE 754 specification and is not a limitation of Excel. This level of precision is found in other spreadsheet programs as well.

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. This is similar to 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, because 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 1,023 (decimal) for double-precision numbers. Excel stores numbers using double-precision.

Example Using Very Large Numbers

Enter the following into a new workbook:


   A1: 1.2E+200
   B1: 1E+100
   C1: =A1+B1

The resulting value in cell C1 would be 1.2E+200, the same value as cell A1. In fact if you compare cells A1 and C1 using the IF function, for example IF(A1=C1), the result will be TRUE. This is caused by the IEEE specification of storing only 15 significant digits of precision. To be able to store the calculation above, Excel would require at least 100 digits of precision.

Example Using Very Small Numbers

Enter the following into a new workbook:


   A1: 0.000123456789012345
   B1: 1
   C1: =A1+B1

The resulting value in cell C1 would be 1.00012345678901 instead of 1.000123456789012345. This is caused by the IEEE specification of storing only 15 significant digits of precision. To be able to store the calculation above, Excel would require at least 19 digits of precision.

Correcting Precision Errors

Excel offers two basic methods to compensate for rounding errors: the ROUND function and the Precision as displayed workbook option.

Method 1: The ROUND Function

The following example using the data above, uses the ROUND function to force a number to five digits. This allows you to successfully compare the result to another value.


   A1: 1.2E+200
   B1: 1E+100
   C1: =ROUND(A1+B1,5)

results in 1.00012


   D1: =IF(C1=1.00012, TRUE, FALSE)

results in the value TRUE

Method 2: Precision as Displayed

In some cases, you may be able to prevent rounding errors from affecting your work by using the Precision as displayed option. This option forces the value of each number in the worksheet to be the displayed value. To turn on this option, click Options on the Tools menu. On the Calculation tab, click to select the Precision as displayed check box.

For example, if you choose a number format showing two decimal places and then turn on the Precision as displayed option, all accuracy beyond two decimals will be lost when you save your workbook. This option affects the active workbook including all worksheets. You cannot undo this option and recover the lost data. It is recommended that you save your workbook prior to enabling this option.

Repeating Binary Numbers and Calculations With Results Close to Zero

Another confusing problem with storing floating point numbers in binary is that some numbers, which are finite, non-repeating numbers in decimal base 10, are infinite, repeating numbers in binary. The most common example of this is the value 0.1 and variants thereof. Although these numbers can be represented perfectly in base 10, the same number in binary format becomes the following repeating binary number when stored in the mantissa:

000110011001100110011 (and so on)

The IEEE 754 specification makes no special provision for any number; it stores what it can in the mantissa and truncates the rest. This results in an error of about -2.8E-17, or 0.000000000000000028 when stored.

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 is similar to why the fraction 1/3 cannot be exactly represented in decimal (a repeating 0.33333333333333333333).

This explains why a simple example in Microsoft Visual Basic for Applications


   Sub Main()
      MySum = 0
      For I% = 1 To 10000
         MySum = MySum + 0.0001
      Next I%
      Debug.Print MySum
   End Sub

will PRINT 0.999999999999996 as output. The small error in representing 0.0001 in binary propagates to the sum.

Example Adding a Negative Number

Enter the following into a new workbook:
A1: =(43.1-43.2)+1

Click cell A1. On the Format menu, click Cells. On the Number tab, click Scientific under Category. Set the Decimal places to 15.

Rather than displaying 0.9, Excel displays 0.899999999999999. Because (43.1-43.2) is calculated first, -0.1 is stored temporarily and the error from storing -0.1 is introduced into the calculation.

Example When a Value Reaches Zero

In Excel 95 or earlier, enter the following into a new workbook:
A1: =1.333+1.225-1.333-1.225

Click cell A1. On the Format menu, click Cells. On the Number tab, click Scientific under Category. Set the Decimal places to 15.

Rather than displaying 0, Excel 95 displays -2.22044604925031E-16.

Excel 97, however, introduced an optimization that attempts to correct for this problem. Should an addition or subtraction operation result in a value at or very close to zero, Excel 97 and later will compensate for any error introduced as a result of converting an operand to and from binary. The example above when performed in Excel 97 and later correctly displays 0 or 0.000000000000000E+00 in scientific notation.

For additional information, please see the following article in the Microsoft Knowledge Base:

Q172911 XL: Incorrect Result Raising 10 to Very Large/Very Small Power

For more information about floating-point numbers and the IEEE 754 specification, please see the following World Wide Web sites:

http://www.ieee.org

http://www.research.microsoft.com

Additional query words: XL98 XL97 XL7 XL5 XL4 XL3 2.0 2.01 2.1 2.10 2.2 2.20 2.21 3.0 4.0 5.00a 5.00c math error w_works


Keywords          : xlformat 
Version           : MACINTOSH:2.20,2.2a,3.0,3.0a,4.0,5.0,5.0a,98; WINDOWS:2.0,2.10d,3.0,3.0a,4.0,4.0a,5.0,5.0c,7.0,97; winnt:5.0
Platform          : MACINTOSH WINDOWS winnt 
Issue type        : kbinfo

Last Reviewed: February 5, 1999