Question

In Exercises 10 and 11, we used 16 bits to represent decimal numbers, allocating 10 bits for the mantissa and 6 bits for the exponent. What would be the impact on our representation if we still used 16 bits for each number but instead allocated 12 bits for the mantissa and 4 bits for the exponent?

Short answer

Increasing mantissa bits improves precision but reduces the range of representable numbers.

Step-by-step solution

Understanding the current allocation

Currently, we use 16 bits to represent numbers, with 10 bits allocated to the mantissa and 6 bits to the exponent. This means we have a precision given by the 10 bits (mantissa) and a range determined by the 6 bits (exponent).

Assessing the new allocation

With the new allocation, 12 bits would go to the mantissa and 4 bits to the exponent. This means the precision would increase because the mantissa has more bits, allowing it to represent more decimal places accurately.

Calculating the impact on range

Previously, the 6-bit exponent allowed us to have a range of possible exponents, specifically from -31 to 31 (using a biased or two's complement system). With just 4 bits for the exponent, the range reduces to -7 to 7, significantly decreasing the range of representable numbers.

Evaluating overall impact

The trade-off here is between precision and range. By increasing the mantissa bits, we gain higher precision in representing numbers but lose the wider range of numbers that could be represented with the previous exponent bits allocation.

Key concepts

Mantissa

The mantissa is an essential part of floating-point numbers. It holds the significant digits of a number and determines its precision. The higher the number of bits allocated to the mantissa, the more precise the number representation can be. This is because those bits allow us to represent a number's significant figures in finer detail. For example, shifting from 10 bits to 12 bits in a mantissa increases how accurately we can depict the number's exactness, enabling more decimal places to be captured accurately. However, while increasing the mantissa bit count improves precision, it doesn't affect the scale of numbers that can be represented. That's where the exponent comes in handy.

Exponent

The exponent part of a floating-point number dictates the scale or range of the number. It effectively tells us how many times the base (usually 2) is raised to expand or compress the number represented by the mantissa. When more bits are allocated to the exponent, a broader range of values can be expressed. For instance, with 6 bits, one can represent a range of exponents from -31 to 31, allowing extremely large or small numbers to be captured. Switching to a 4-bit exponent narrows this range to -7 to 7, limiting the scale of numbers we can handle. While we gain more precision by allocating more bits to the mantissa, the limitation in exponent bits means some very large or very small numbers may not be representable.

Bit Allocation

Bit allocation is a core aspect of designing a floating-point representation system. It involves deciding how many of the available bits will be designated for the mantissa and how many for the exponent. This decision shapes the balance between the precision and range of representable numbers. In a given 16-bit system, choosing to allocate more bits to the mantissa increases precision, enabling us to represent numbers with greater detail. Conversely, reducing the number of exponent bits limits the range of possible values. Optimal bit allocation must consider the typical use case — whether higher precision is more crucial than a wider range, or vice versa.

Precision vs Range Trade-Off

When designing a floating-point representation, a trade-off often exists between precision and range. Increasing precision by allocating more bits to the mantissa allows for more accurate representation of numbers but narrows the range. This means we might not be able to represent extremely large or small numbers. On the other hand, having more bits for the exponent offers a wider range, accommodating larger scales but at the cost of precision. Understanding the needs of a specific application is critical:

• For applications needing highly accurate calculations, more bits should be reserved for the mantissa.
• If representing wide-ranging data is crucial, then allocating more bits to the exponent is necessary.

Achieving the right balance depends on assessing the precision and range requirements for your specific use case.