wclawson said:Homework Statement
Floating Point representation:
Given Xmax/Xmin=2^4076, what is the size of the mantissa? How many numbers can exist?
If mantissa is 0.xxx instead of 1.xxx, how many numbers can exist?
Homework Equations
(-1)S x 1.M x BE
The Attempt at a Solution
I'm not sure, but is this the correct method to use?
{(-1)-S'(1.M')(BE'-E'')+(-1)-S''(1.M'')}BE''
Mark44 said:Do you have a typo in your given information about Xmax/Xmin? The exponent should be 4096, I believe. 4096 = 212, which means 12 bits for the exponent of your floating point number.
You didn't mention how many bits there are overall in your floating point number, but the typical sizes are 32 bits, 64 bits, and sometimes 80 bits. If 1 bit is used for the sign, and 12 bits for the exponent, how many bits are left for the mantissa?
If the mantissa is assumed to be 1.xxx rather than 0.xxx, how many bits do you save by doing this?
No, it doesn't. That bit is assumed to be 1, and this gives you an extra bit for the mantissa.wclawson said:Yes, thank you for noticing my mistake it should be 4096.
So a FP number with 32 bits would have 32-12-1=19 bits, but does that include the 1 preceding the M (1.M)?
Yes. There's no effect on the range, as xmin and xmax are still the same. It gives you a little more precision though.wclawson said:By having 0.xx instead of .xx, does that mean that only half the amount of numbers represented with 1.xx can be used? What is it's effect of number range?
Mark44 said:No, it doesn't. That bit is assumed to be 1, and this gives you an extra bit for the mantissa.
Yes. There's no effect on the range, as xmin and xmax are still the same. It gives you a little more precision though.
I think this is right.wclawson said:Ok, I see. Let me make sure I have this straight.
Since there are 12 bits for E, that means the bias is 2047.
It wouldn't be 1.9999... The representation is binary, not decimal. It would be 1.1111...111, with 20 1's after the binary point. (Remember, that leading one to the left of the binary point is assumed.)wclawson said:Therefore the largest number possible to display is 22047 times 1.9999... (from the mantissa), right?
Mark44 said:I think this is right.
It wouldn't be 1.9999... The representation is binary, not decimal. It would be 1.1111...111, with 20 1's after the binary point. (Remember, that leading one to the left of the binary point is assumed.)
The size of a mantissa in computer architecture refers to the number of bits used to represent the fractional part of a floating-point number. This size can vary depending on the specific computer architecture and can range from 23 bits to 52 bits.
The size of a mantissa is important because it determines the precision of floating-point numbers. A larger mantissa allows for more accurate representation of numbers, but also requires more memory and processing power.
The size of a mantissa is directly related to the range of values that can be represented. A larger mantissa allows for a wider range of numbers to be represented, including very small and very large numbers.
Yes, the size of a mantissa can be changed in some computer architectures. However, this usually requires special hardware or software and can impact the overall performance of the system.
The size of a mantissa and the size of an exponent are both important components of a floating-point number, but they serve different purposes. The mantissa represents the precision of the number, while the exponent represents the magnitude of the number.