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Decimal Cases * * * () * () In programming, a floating point number is expressed as . In general, a floating-point number can be written as

where * M is the fraction mantissa or significand. * E is the exponent. * B is the base, in decimal case . Binary Cases As an example, a 32-bit word is used in MIPS computer to represent a floating-point number: 1 bit ..... 8 bits .............. 23 bits representing: * The implied base is 2 (not explicitly shown in the representation). * The exponent can be represented in signed 2's complement (but also see biased notation later). * The implied decimal point is between the exponent field E and the significand field M. * More bits in field E mean larger range of values representable. * More bits in field M mean higher precision. * Zero is represented by all bits equal to 0: Normalization To efficiently use the bits available for the significand, it is shifted to the left until all leading 0's disappear (as they make no contribution to the precision). The value can be kept unchanged by adjusting the exponent accordingly. Moreover, as the MSB of the significand is always 1, it does not need to be shown explicitly. The significand could be further shifted to the left by 1 bit to gain one more bit for precision. The first bit 1 before the decimal point is implicit. The actual value represented is

However, to avoid possible confusion, in the following the default normalization does not assume this implicit 1 unless otherwise specified. Zero is represented by all 0's and is not (and cannot be) normalized. Example: A binary number can be represented in 14-bit floating-point form in the following ways (1 sign bit, a 4-bit exponent field and a 9-bit significand field): * * * * * with an implied 1.0: By normalization, highest precision can be achieved. The bias depends on number of bits in the exponent field. If there are e bits in this field, the bias is , which lifts the representation (not the actual exponent) by half of the range to get rid of the negative parts represented by 2's complement. The range of actual exponents represented is still the same. With the biased exponent, the value represented by the notation is:

Note: * Zero exponent is represented by , the bias of the notation; * The range of exponents representable is from -126 to 127; * The exponent (with all zero significand) is reserved to represent infinities or not-a-number (NaN) which may occur when, e.g., a number is divided by zero; * The smallest exponent is reserved to represent denormalized numbers (smaller than which cannot be normalized) and zero, e.g., is represented by: Normalization: If the implied base is , the significand must be shifted multiple of q bits at a time so that the exponent can be correspondingly adjusted to keep the value unchanged. If at least one of the first q bits of the significand is 1, the representation is normalized. Obviously, the implied 1 can no longer be used. Examples: * Normalize . Note that the base is 4 (instead of 2)

Note that the significand has to be shifted to the left twobits at a time during normalization, because the smallest reduction of the exponent necessary to keep the value represented unchanged is 1, corresponding to dividing the value by 4. Similarly, if the implied base is , the significand has to be shifted 3 bits at a time. In general, if , normalization means to left shift the significand q bits at a time until there is at least one 1 in the highest q bits of the significand. Obviously the implied 1 can not be used. * Represent in biased notation with bits for exponent field. The bias is and implied base is 2.

The biased exponent is , and the notation is (without implied 1): or (with implied 1): * Find the value represented in this biased notation: The biased exponent is 17, the actual exponent is , the value is (without implied 1):

or (with implied 1):

Examples of IEEE 754: * -0.3125

The biased exponent is , * 1.0

The biased exponent is , * 37.5

The based exponent: , . * -78.25

The biased exponent: , * As the most negative exponent representable is -126, this value is a denorm which cannot be normalized: by GAURAV PANDEY & VIJAY MAHARA..........

AMRAPALI INSTITUTE...................

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Q: How is floating point representation done in computers?

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In a decimal floating point number representation it is always possible to use a power of 10 (exponent) to write a number so that it lies between 0 and 1, by changing the exponent. For example: 5 = 0.5 x 10 50 = 0.5 x (10 to the power 2) 0.05 = 0.5 x (10 to the power -1) This is called normalisation. In the binary representation of floating point numbers it is always possible to shift the number until it starts with a 1, provided you change the exponent at the same time. This is how computer memory works. If you do this, however, the 1 does not need to be stored (because it can always be added with a little extra processing). So in computers the number is often normalised, and the leading 1 omitted. But if the storage convention assumes this is done, then, of course, it must be done for every number stored in memory.

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If you are referring to normalization of floating point numbers, it is to maintain the most precision of the number possible. Leading zeros in floating point representation is lost precision, thus normalization removes the leading zeros by shifting left and adjusting the exponent. If the calculation was done in a hidden extended precision register (like IEEE 80-bit format) extra precision bits may be shifted in to the LSBs before restoring the result to a standard single or double precision register, reducing loss of precision.

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