addition,
subtraction,
negation,
multiply and divide by powers of 2,

Now you want to add other arithmetic functions, especially multiplication and division.

We will look at 4 problems:

P_m = multiplication of two numbers (a * b)
P_d = division of two numbers (a / b)
P_r = reciprocal of a number (1 / a)
P_s = square of a number (a²)

We will be writing stuff like P₁ ≤ P₂ to mean a reduction of P₁ to P₂, similar to the book's ≤_p. However, here it's not polynomial-time, but rather an efficient reduction comparable to the other operations on these numbers. Intuitively, this means: if you can compute P₂ efficiently, then you can compute P₁ efficiently.

1. If 1000 binary bits are used for the significant part, roughly how many decimal digits of accuracy would we have?

2. Two of the reductions are trivial. Which are they? (Think about this carefully.)

3. Use the formula below to come up with another reduction:
   

4. Use the formula below to come up with yet another reduction:
   

5. Use the following iteration formula from Newton's method (which we studied) to find another reduction.
   

   In this case you should explain in some detail how you intend to handle the initial guess for Newton's method, which should not be too far from the correct answer. [Hint: here is the page that covers this material.]

6. Argue that if you can do both P_m and P_r efficiently, then you can also do P_d efficiently.

7. Argue that if we can do P_s efficiently, then we can also do the other three efficiently: P_m, P_r, and P_d. (This might be a quick and dirty way to do a half-way efficient implementation.) [Note: Not all the reductions above are needed to show this part vii.]