Write a declaration for each of the following: a) Integer count that should be maintained in a register. Initialize count to 0. b) Double-precision, floating-point variable lastVal that is to retain its value between calls to the function in which it’s defined.

In C++ programming, when optimizing for speed, you might come across a situation where you need a variable to be accessed as quickly as possible. For such cases, the register keyword was traditionally used as a hint to the compiler to store the variable in a register of the CPU. Registers are the fastest memory locations available, and accessing a register is quicker than accessing RAM.

For example, declaring an integer to be kept in a register would look like this: register int count = 0;. This declaration also initializes count to 0. However, it's important to note that modern compilers are very sophisticated and often ignore the register keyword, as they employ their algorithms to optimize register usage. Therefore, while it's semantically significant to denote the intent of having a variable in a register, it doesn't guarantee that the compiler will comply.

This keyword is less relevant in modern C++ coding and often considered deprecated. Compilers nowadays do an excellent job of optimizing code, so the use of this keyword might not lead to the expected performance improvements.

If you've ever needed a variable to maintain its value across multiple invocations of a function, you've likely encountered the static storage class. In C++, declaring a variable as static within a function means that the variable retains its value even after the function exits. It's essentially persisted across the entire run-time of the program.

For instance, a double-precision floating-point variable can be declared with static storage duration like so: static double lastVal;. The first time the function containing this declaration is called, lastVal is initialized, and its value will persist through subsequent calls to the function.

This behavior is particularly useful for implementing stateful functions or for counting occurrences without having to use global variables. It's a powerful feature in C++ that allows encapsulated variables to have a persistent state.

When dealing with numerical calculations that require a high degree of accuracy in C++, you might need to use double-precision floating-point numbers. A double-precision floating-point variable can store a much larger range of values as well as more precise decimal points compared to a single-precision floating-point variable. It is called 'double-precision' because it uses double the number of bits to store the number, typically 64 bits.

These extra bits provide a significant increase in precision which is essential for certain mathematical computations, such as scientific calculations or when dealing with very large or very small numbers. The declaration of a double-precision floating-point in C++ would be simply: double myVariable;. If you want a variable like this to maintain its value across function calls, you would combine it with the static keyword discussed previously: static double myVariable;.

Keep in mind that while double-precision offers more accuracy, it also consumes more memory and can be slower to process than single-precision, so it's important to use them thoughtfully depending on the requirements of your program.