NVIDIA has a “half” format, which is similar to IEEE 754 except that it is only 16 bits wide. The leftmost bit is still the sign bit, the exponent is 5 bits wide and stored in excess-56 format, and the mantissa is 10 bits long. A hidden 1 is assumed. Write down the bit pattern assuming a modified version of this format, which uses an excess-16 format to store the exponent. Comment on how the range and accuracy of this 16-bit floating point format compares to the single precision IEEE 754 standard.

In the IEEE 754 floating point standard the exponent is stored in "bias" (also known as "Excess-N") format. This approach was selected because we want an all-zero pattern to be as close to zero as possible. Because of the use of a hidden 1, if we were to represent the exponent in two's complement format an all-zero pattern would actually be the number 1! (Remember, anything raised to the zeroth power is 1, so 1.0⁰ = 1.) There are many other aspects of the IEEE 754 standard that exist in order to help hardware floating point units work more quickly. However, in many older machines floating point calculations were handled in software, and therefore other formats were used. The following table shows decimal numbers.

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a. b. -1.5625 x 10-1¹ 9.356875 x 10²