Question

2. Now, consider the smallest single precision floating point number, 21". What wll happen the "gap" at that number if you change the type to double precision? DO NOT discuss the smallest double precision number -just the largest single precision represented as a double type.

Answer 1

Double precision floating point : Double-precision floating-point format is a format which opts 64 bytes memory. The significands of IEEE binary floating-point numbers have a limited number of bits, called the precision; single-precision has 24 bits, and double-precision has 53 bits.

The number has significand, which contains the significant digits of the number, and a power of two, which places the “floating” radix point.

The gap count is calculated by it’s precision.

For 1 bit precision one 1-bit significand: 1

For 2 bit precision - two 2-bit significands: 1.0, 1.1

The number of significant is same as the number of gaps : 2^p-1

there is a gap from the highest significand to the next power of two.

All binary floating numbers with power of 2 and exponent e, these are in the interval of [2^e, 2^e+1).

Interval is calculated by pre and post difference.

2^e+1 – 2^e = 2^e.

So for P bits

2^e/2^p-1 = 2^e-(p-1) = 2^e+1-p

Gap size is 2e+1-p

In the given p is taken as -149

Keep in place of p with -149.

2^e+1-(-149)

=2^e+150

^{= e taken as 10 generally}

2^160.