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IEEE – 754 Single-Precision
IEEE – 754:
IEEE 754 uses implicit normalization. Where radix point is put after the first non-zero digit. It is composed of an implicit (hidden) 1 leading the mantissa bits. This representation tends to maximize the quantity of the representable numbers.
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IEEE 754 Standards:
1. Single-Precision (32-bits)
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2. Double-Precision (64-bits)
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1. Single-Precision (32-bits)
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Normal Case
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Special Case
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Same thing different form
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In IEEE floating-point representation, certain mantissa exponent combinations do not represent any number. (In single Precision format)
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IEEE-754 standard was a bias of +127, instead of +128:
If excess -128 formats are used,
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and the range of numbers with 8 bits in 2’S complement form is -128 to +127.
If largest exponent 127 is taken, then biased exponent becomes (127+128) = 255, i.e. E = 255. But, in IEEE notation, if E = 255 then it represent an invalid representation, since,
(a) E = 255, M = 0 represents ±∞ (infinity) and
(b) E = 255, M ≠ 0 represents NAN.
So, IEEE was +127 and +1023 as a bias for single precision and double precision respectively.
Range of Exponent of Single Precision
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Maximum mantissa
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Second maximum mantissa
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We will get the maximum number by maximum mantissa and maximum exponent
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Second maximum number (second highest mantissa and highest exponent)
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Maximum Difference
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Minimum Number
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Second minimum positive number
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Question:
.0101 represents the number in IEEE 754 single precision.
Answer:
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