Tutorial: Floating-Point Binary

The two most common floating-point binary storage formats used by Intel processors were created for Intel and later standardized by the IEEE organization:

Both formats use essentially the same method for storing floating-point binary numbers, so we will use the Short Real as an example in this tutorial. The bits in an IEEE Short Real are arranged as follows, with the most significant bit (MSB) on the left:

Fig. 1

The Sign

The sign of a binary floating-point number is represented by a single bit. A 1 bit indicates a negative number, and a 0 bit indicates a positive number.

The Mantissa

It is useful to consider the way decimal floating-point numbers represent their mantissa. Using -3.154 x 10⁵ as an example, the sign is negative, the mantissa is 3.154, and the exponent is 5. The fractional portion of the mantissa is the sum of each digit multiplied by a power of 10:

    .154 =  1/10 + 5/100 + 4/1000

A binary floating-point number is similar. For example, in the number +11.1011 x 2³, the sign is positive, the mantissa is 11.1011, and the exponent is 3. The fractional portion of the mantissa is the sum of successive powers of 2. In our example, it is expressed as:

    .1011 = 1/2 + 0/4 + 1/8 + 1/16

Or, you can calculate this value as 1011 divided by 2⁴. In decimal terms, this is eleven divided by sixteen, or 0.6875. Combined with the left-hand side of 11.1011, the decimal value of the number is 3.6875. Here are additional examples:

The last entry in this table shows the smallest fraction that can be stored in a 23-bit mantissa. The following table shows a few simple examples of binary floating-point numbers alongside their equivalent decimal fractions and decimal values:

The Exponent

IEEE Short Real exponents are stored as 8-bit unsigned integers with a bias of 127. Let's use the number 1.101 x 2⁵ as an example. The exponent (5) is added to 127 and the sum (132) is binary 10000100. Here are some examples of exponents, first shown in decimal, then adjusted, and finally in unsigned binary:

The binary exponent is unsigned, and therefore cannot be negative. The largest possible exponent is 128-- when added to 127, it produces 255, the largest unsigned value represented by 8 bits. The approximate range is from 1.0 x 2^-127 to 1.0 x 2⁺¹²⁸. 

Normalizing the Mantissa

Before a floating-point binary number can be stored correctly, its mantissa must be normalized. The process is basically the same as when normalizing a floating-point decimal number. For example, decimal 1234.567 is normalized as 1.234567 x 10³ by moving the decimal point so that only one digit appears before the decimal. The exponent expresses the number of positions the decimal point was moved left (positive exponent) or moved right (negative exponent).

Similarly, the floating-point binary value 1101.101 is normalized as 1.101101 x 2³ by moving the decimal point 3 positions to the left, and multiplying by 2³. Here are some examples of normalizations:

You may have noticed that in a normalized mantissa, the digit 1 always appears to the left of the decimal point. In fact, the leading 1 is omitted from the mantissa in the IEEE storage format because it is redundant.

Creating the IEEE Bit Representation

We can now combine the sign, exponent, and normalized mantissa into the binary IEEE short real representation. Using Figure 1 as a reference, the value 1.101 x 2⁰ is stored as sign = 0 (positive), mantissa = 101, and exponent = 01111111 (the exponent value is added to 127). The "1" to the left of the decimal point is dropped from the mantissa. Here are  more examples:

Converting Decimal Fractions to Binary Reals

If a decimal fraction can be easily represented as a sum of fractions in the form (1/2 + 1/4 + 1/8 + ... ), it is fairly easy to discover the corresponding binary real. Here are a few simple examples

Of course, the real world is never so simple. A fraction such as 1/5 (0.2) must be represented by a sum of fractions whose denominators are powers of 2. Here is the output from a program that subtracts each succesive fraction from 0.2 and shows each remainder. In fact, an exact value is not found after creating the 23 mantissa bits. The result, however, is accurate to 7 digits. The blank lines are for fractions that were too large to be subtracted from the remaining value of the number. Bit 1, for example, was equal to .5 (1/2), which could not be subtracted from 0.2.

     starting:   0.200000000000

 1
 2
 3   subtracting 0.125000000000
     remainder = 0.075000000000
 4   subtracting 0.062500000000
     remainder = 0.012500000000
 5
 6
 7   subtracting 0.007812500000
     remainder = 0.004687500000
 8   subtracting 0.003906250000
     remainder = 0.000781250000
 9
10
11   subtracting 0.000488281250
     remainder = 0.000292968750
12   subtracting 0.000244140625
     remainder = 0.000048828125
13
14
15   subtracting 0.000030517578
     remainder = 0.000018310547

16   subtracting 0.000015258789
     remainder = 0.000003051758
17
18
19   subtracting 0.000001907349
     remainder = 0.000001144409
20   subtracting 0.000000953674
     remainder = 0.000000190735
21
22
23   subtracting 0.000000119209
     remainder = 0.000000071526


Mantissa: .00110011001100110011001