Thread: floating point help!

I don't know how to calculate the smallest non-zero denorm.
okay maybe i know a little.

so the smallest denorm is this right?
s exp fraction
0 [000 0000 0] 000 0000 0000 0000 0000 0001

and the formula for floating points is this

1 * 2^-127 * (2^-23) + 1

how is this = 2^149???

That plus one should go next to the -23, I think. (I.e, when denormalized, the first bit is in front of the binary point, not behind it.)

Hmm,

According to this:

http://www.math.byu.edu/~schow/work/...atingPoint.htm

the smallest possible denorm is represented as:

0 00000001 00000...

but i fail to see why they would include a 1 in the exponent.
but i do see no why they wouldn't put a 1 in the fraction part

so if we calculate this

1 * 2 ^ (-126) * ( 0 + 1 [implicit 1] )

but now i fail to see why 2^-149 = 2^(-126)*2^(-23)

any suggestions?

Originally Posted by NoobieGecko

Hmm,

According to this:

http://www.math.byu.edu/~schow/work/...atingPoint.htm

the smallest possible denorm is represented as:

0 00000001 00000...

but i fail to see why they would include a 1 in the exponent.
but i do see no why they wouldn't put a 1 in the fraction part

That's the smallest normal number.

so if we calculate this

1 * 2 ^ (-126) * ( 0 + 1 [implicit 1] )

but now i fail to see why 2^-149 = 2^(-126)*2^(-23)

any suggestions?

Check any junior high or high school mathematics textbook.

i know how to add the powers, but of course 2^-149 = 2^(-126)*2^(-23)
let me fix the question,

the answer is 2^-149, but how does the above representation lead to that 2^-149??

or in binary
how is 0 00000001 0000 ---> 2^-149?

to get 2^-149 the representation could be

0 00000000 00000000000000000000001

without the implicit 1...
1 * 2^(-127) * 2^(-23) [without the implicit 1] = 2^(-149)

oh nevermnd about the 000000...1 000, it is in normal form :O
but how is the implicit 1 ignored on the second question?

That's what denormalized means, is that there is no implicit one.
So instead of 0000000000000000000000000000001 meaning
1.00000000000000000000001 times 2^(-127), it means
0.0000000000000000000001 times 2^(-127). Note that there's one less digit -- the first digit of a denormalized mantissa has to take the place of the implicit one that we had before. That is usually achieved by (1) ignoring the 1+ part, since it isn't there any more, and (2) changing the exponent bias to -126 instead of -127 and keeping the binary point in front, so that we would have
0.0000000000000000000001 times 2^(-126).

(BTW, if you don't happen to have a textbook with this in it, it's in wikipedia.)

thanks, that clears things up now, so every time i have to convert to denormalized i always have to ignore the implicit 1 and shift the decimal point to the right by 1 -- that should do it right.