Floating point representation- Double precision

Question

The difference between 201 and next larger double precision number is 2^P.

If IEEE double precision format is used then the value of P is _________?

-- I know how number is stored using double precision, It is of 64 bits: 52 bits used for mantissa and 11 bits used for exponent and 1 bit used for sign. It uses excess 1023 code for biasing but i'm confused with this question. Please someone help me to visualize it.

Answer 1

The binary representation 201 is 1100 1001. The corresponding mantissa part of the floating point number is

The exponent is not shown. Instead, the binary point is shown (between the bits marked 7 and 8). The leading bit 1 is not actually stored in the IEEE format^$, so it is shown outside the 52 bits allocated for the mantissa.

The next double precision number after 201 can be obtained^#, by setting 1 in the least significant bit. That bit position is marked by the x in the figure.

Now take the difference of the two numbers and you will immediately get the result as a power of 2. That power is P. P is a negative integer.

To get an even better visualisation, rename the bits from 1 to 52 shown in the figure to the powers of 2 represented by those bits (i.e, 6, 5, 4, ... 0 \$\ \bullet\ \$ -1, -2, -3, ...).

^{$ There may be some exceptions.}

^{#This statement is not proved here. Any error in this solution will probably be in this assumption.}