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float FormatA floatingpoint number is expressed as the product of three parts: the sign, the mantissa, and an exponent. For example: float value = sign × 1.mantissa × 2^{exponentbias} Where
Floatingpoint numbers are stored in normalized form which maximizes the quantity of numbers that can be represented. Normalized numbers have a binary point ('.') after the first nonzero digit. This is how the mantissa is able to hold 24 binary digits in only 23 bits. Denormalized floatingpoint numbers are used to represent values smaller than what can be represented by normalized values. The drawback is that the precision decreases with smaller values. Denormalized floatingpoint values are represented as follows: float value = sign × 0.mantissa × 2^{126} Where
If the stored exponent is zero and the mantissa is nonzero the floatingpoint value is a denormalized number. For denormalized numbers, the exponent is treated as if a 1 were stored. Hence, the actual exponent is 126 (1 minus 127). Floatingpoint numbers are stored using the following 32bit format:
Where
Using the above format, the floatingpoint number 12.5 is stored as a hexadecimal value of 0xC1480000. In memory, this value appears as follows:
It is fairly simple to convert floatingpoint numbers to and from their hexadecimal storage equivalents. The following example demonstrates how this is done for the value 12.5 shown above. The floatingpoint storage representation is not an intuitive format. To convert this to a floatingpoint number, the bits must be separated as specified in the floatingpoint number storage format table shown above. For example:
From this illustration, you can determine the following:
There is an understood binary point at the left of the mantissa that is always preceded by a 1. This digit is omitted from the stored form of the floatingpoint number. Adding 1 and the binary point to the beginning of the mantissa gives the following value: 1.10010000000000000000000 To adjust the mantissa for the exponent, move the decimal point to the left for negative exponent values or right for positive exponent values. Since the exponent is three, the mantissa is adjusted as follows: 1100.10000000000000000000 The result is a binary floatingpoint number. Binary digits to the left of the decimal point represent the power of two corresponding to their position. For example, 1100 represents (1 × 2^{3}) + (1 × 2^{2}) + (0 × 2^{1}) + (0 × 2^{0}), which is 12. Binary digits to the right of the decimal point also represent the power of two corresponding to their position. However, the powers are negative. For example, .100... represents (1 × 2^{1}) + (0 × 2^{2}) + (0 × 2^{3}) + ... which equals .5. The sum of these values is 12.5. Because the sign bit was set, this number should be negative. So, the hexadecimal value 0xC1480000 is 12.5.  

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