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# 2.87 Floating-Point II

**Problem**:

Consider a 16-bit floating-point representation based on the IEEE floating-point format, with one sign bit, seven exponenet bits (k=7), and eight fraction bits (n=8). The exponent bias is 2^6-1=63.

Fill in the table that follows for each of the numbers given, with the following instructions for each column:

Hex: The four hexadecimal digits describing the encoded form.

M: The value of the significand. This should be a number of the form x or x/y, where x is an integer, and y is an integral power of 2. Examples include: 0, 67/64, and 1/256.

E: The integer value of the exponent.

V: The numeric value represented. Use the notation x or x\*2^z, where x and z are integers.

As an example, to represent the number 7/8, we would have s=0, M=7/4, and E=-1. Our number would therefore have an exponent field of `0x3E` (decimal value 63-1=62) and a signifcand field `0xC0` (binary 11000000), giving a hex representation `3EC0`.

You need not fill in entries marked "--".

| Desc                                  | Hex    | M         | E   | V           | D                 |
| ------------------------------------- | ------ | --------- | --- | ----------- | ----------------- |
| -0                                    | 0x8000 | 0         | -14 | -0          | -0.0              |
| Smallest value >2                     | 0x4001 | 1025/1024 | 1   | 1025/512    | 2.001953125       |
| 512                                   | 0x6000 | 1         | 9   | 512         | 512.0             |
| Largest denormalized                  | 0x03FF | 1023/1024 | -14 | 1023/(1^24) | 0.000060975551605 |
| −∞                                    | 0xFC00 | --        | --  | --          | --                |
| Number with hex representation `3BB0` | --     | 123/64    | -1  | 123/128     | 0.9609375         |

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