Question

Suppose a computer company is developing a new floating point system for use with their machines. They need your help in answering a few questions regarding their system. Following the terminology of Section \(2.2\), the company's floating point system is specified by \((\beta, t, L, U) .\) Assume the following: \- All floating point values are normalized (except the floating point representation of zero). \- All digits in the mantissa (i.e., fraction) of a floating point value are explicitly stored. \- The number 0 is represented by a float with a mantissa and an exponent of zeros. (Don't worry about special bit patterns for \(\pm \infty\) and NaN.) Here is your part: (a) How many different nonnegative floating point values can be represented by this floating point system? (b) Same question for the actual choice \((\beta, t, L, U)=(8,5,-100,100)\) (in decimal) which the company is contemplating in particular. (c) What is the approximate value (in decimal) of the largest and smallest positive numbers that can be represented by this floating point system? (d) What is the rounding unit?

Short answer

(a) The total number of different nonnegative floating point values can be represented by the formula: $$\sum_{e=0}^{U}{(\beta-1)^{t}} - 1$$ (b) For the given system with \((\beta, t, L, U)=(8,5,-100,100)\), the total number of different nonnegative floating point values is 516,813. (c) The smallest positive floating point value is approximately \(1.27 \times 10^{-302}\), and the largest is approximately \(7 \times 10^{89}\). (d) The rounding unit for the given system is approximately \(1.27 \times 10^{-302}\).

Step-by-step solution

(a) Nonnegative floating point values

A nonnegative floating point value can be represented as \(\beta^{e}\) multiplied by the mantissa, which has \(t\) digits that can take values from \(0\) to \(\beta-1\). The exponent can take values from \(0\) to \(U\) (inclusive). However, zero can also be represented by an all-zero mantissa with an exponent of \(0\). Therefore, the total number of different nonnegative floating point values is given by: $$\sum_{e=0}^{U}{(\beta-1)^{t}} - 1$$

(b) Nonnegative floating point values for \((\beta, t, L, U)=(8,5,-100,100)\)

For the given values, the total number of different nonnegative floating point values can be calculated by plugging the values into the formula: $$\sum_{e=0}^{100}{(8-1)^{5}} - 1 = \sum_{e=0}^{100}{(7^{5})} - 1 = 101 \cdot 7^5 - 1 = 516813$$

(c) Largest and smallest positive floating point values

The smallest positive floating point value is the one with the smallest mantissa and exponent (\(t=1\) and \(e=L\)). It can be calculated as: $$\beta^{L}(1)\Rightarrow \beta^{-100}$$ The largest positive floating point value is the one with the largest mantissa and exponent (\(t=5\) and \(e=U\)). It can be calculated as: $$(\beta-1) \cdot \beta^{U} \Rightarrow (8-1) \cdot 8^{100}$$ In decimal: $$\beta^{-100} \approx 1.27 \times 10^{-302} \text{ (smallest)}$$ $$(7) \cdot 8^{100} \approx 7 \times 10^{89}\text{ (largest)}$$

(d) Rounding unit

The rounding unit for any floating point system can be determined by calculating the smallest value that can be represented, which is the minimum distance between adjacent representable numbers. The rounding unit can be represented as: $$\beta^{e_{min}} = \beta^{L} $$ For the given values, this can be calculated as \(8^{-100}\), which is approximately \(1.27 \times 10^{-302}\) in decimal.

Key concepts

Normalized Floating Point Numbers

In the world of computing, floating point numbers are often used to represent real numbers that require a certain level of precision. A key concept in this representation is the idea of a "normalized" floating point number.

A floating point number is considered normalized when its mantissa starts with a non-zero digit. This ensures that the number is stored in a form that maximizes the use of the available precision.

In a binary system, for example, normalized numbers have a leading digit of 1 in the mantissa. This normalization helps reduce redundancy and makes computations more efficient.

Except for the representation of zero, all non-zero floating point values in our new system are normalized, which maximizes precision and efficiency while avoiding unnecessary repetition of numbers with leading zeroes.

Mantissa Representation

The mantissa is a vital part of a floating point number as it represents the significant digits of the number.

In our system, the mantissa is explicitly stored, meaning each digit is saved accurately. This stored value, along with the base, determines the exact value of the floating point number.

For instance, if our floating point system uses a base (\(\beta\)) of 8, and our mantissa has 5 digits, those digits will range from 0 to 7. The explicit storage of these digits allows for precise mathematical operations and numerical stability.

Each digit in the mantissa can contribute significantly to the overall system's range and accuracy when performing calculations, especially when processing a large dataset or when precision is paramount in applications.

Floating Point Rounding Unit

The floating point rounding unit, also known as the round-off error, is a crucial aspect of floating point arithmetic.

It defines the smallest difference that can be represented between two distinct floating point numbers. In many systems, this is influenced by the least significant bit in the mantissa.

In our context, the rounding unit is derived from the smallest exponent used in the floating point system, specifically \\(\beta^{L}\). For a base of 8 and the smallest exponent \(L = -100\), the rounding unit would be \(8^{-100}\), translating to approximately \(1.27 \times 10^{-302}\).

This rounding unit is crucial for ensuring that small variations are minimized during computations, maintaining numerical stability, and ensuring minimal loss of precision.

Floating Point Range Calculation

Calculating the range of representable values in a floating point system involves understanding both the largest and smallest values possible.

The smallest positive number in the system is determined by the smallest mantissa and the smallest exponent. This can be calculated using \(\beta^{L}\) where \(L\) is the lowest exponent. For our system with \(L = -100\) and \(\beta = 8\), this gives a smallest value of \(8^{-100}\), or approximately \(1.27 \times 10^{-302}\).

On the other hand, the largest floating point number is achieved with the largest mantissa and exponent. Here, the largest number is calculated as \((\beta - 1) \times \beta^{U}\), where \(U\) represents the maximum exponent, which in our example is 100. Evaluating this gives us \(7 \times 8^{100}\), or approximately \(7 \times 10^{89}\).

These calculations define the upper and lower limits on the scale of numbers that can be accurately represented within the system, outlining the system's capabilities to handle very large or very small numbers without loss of precision.