lecture 8 eecs2021e chapter 3 part ii floating point
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sure! i'll provide an overview of floating-point representation as it might be covered in a computer science course like eecs 2021e, particularly focusing on chapter 3, part ii of a relevant textbook or lecture notes. this will include the concepts of floating-point numbers, their representation, and a code example to illustrate how they work.
floating point representation
overview
floating-point representation is a way to represent real numbers in a format that can accommodate a wide range of values. it allows computers to handle very large or very small numbers efficiently, which is especially important in scientific computing and graphics.
structure of floating-point numbers
a floating-point number is typically represented in the following format:
where:
`s` is the sign bit (0 for positive, 1 for negative).
`m` is the significand or mantissa, which represents the precision bits of the number.
`e` is the exponent, which scales the number.
for example, in the ieee 754 standard (the most widely used standard for floating-point computation), a single precision (32-bit) floating-point number is structured as follows:
1 bit for the sign (s)
8 bits for the exponent (e)
23 bits for the mantissa (m)
steps to represent a floating-point number
1. *convert the number to binary.*
2. *normalize the binary number.*
3. *determine the sign bit.*
4. *calculate the exponent and bias it.*
5. *extract the mantissa.*
6. *pack the bits together to form the final representation.*
example: representing the number 6.75 in ieee 754 format
1. *convert to binary:*
the integer part `6` in binary is `110`.
the fractional part `.75` in binary is `.11` (since 0.75 = 0.5 + 0.25).
therefore, `6.75` in binary is `110.11`.
2. *normalize:*
normalize `110.11` to `1.1011 x 2^2`.
3. *determine the sign bit:*
since `6.75` is positive, the sign bit `s` is `0`.
4. *calculate the exponent:*
the exponent is `2`. in ieee 754, we ...
#FloatingPoint #EECS2021E #Lecture8
floating point representation
IEEE 754
precision
rounding modes
normalization
denormalized numbers
exponent bias
binary format
overflow
underflow
arithmetic operations
rounding errors
mantissa
floating-point arithmetic
computational accuracy
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