created 05/22/2004

#
Chapter 32 Programming Exercises

**NOTE:**
If you have read the chapters on
subroutine linkage,
write each exercise as a
main program calling a subroutine.
Use either the stack-based linkage or the
frame-based linkage convention.

In the *Settings* menu of SPIM set
Bare Machine **OFF**, Allow Pseudo Instructions **ON**,
Load Trap File **ON**, Delayed Branches **OFF**,
Delayed Loads **OFF**, Mapped IO **ON**, Quiet **OFF**.

Run the programs
by clicking __S__imulator__G__o and then OK in the pop-up panel.

## *Exercise 1 — exp(x)

Write a program that computes
exp(x) by using a Taylor series:

exp(x) = 1 + x + x^{2}/2 + x^{3}/3! + x^{4}/4! + ...

The user enters x.
exp(x) is e^{x}, where e is the base of the natural
logarithms, 2.718281828...
You don't need to worry about the math behind this.
Just compute a sum of terms.
Each term in the sum looks like:

x^{n}/n!

Designate a register to hold the current term.
Initialize it to 1.0 (the zeroeth term).
Term one is calculated by multiplying:

term*x/1

Term one is the value x.
Term two is calculated by multiplying:

term*x/2

Term two is the value x^{2}/2.
Term three is calculated by multiplying:

term*x/3

Term three is the value x^{3}/3!.
Term four is calculated by multiplying:

term*x/4

... and so on.
In general, after you have added term (n-1)to the sum,
calculate term n by
multiplying:

term *x/n

Keep doing this in a loop
until the term becomes very small.
Here are some example runs (with SPIM using single precision).
As usual, only the first seven digits printed have any
significance.

It might be helpful to first write this program in
Java or C to verify your design,
and then to proceed with the version in
assembly language.

Click here to go back to the main menu.

## *Exercise 2 — pi

Various infinite series for pi have been discovered.
The first such series was

pi = 4( 1 - 1/3 + 1/5 - 1/7 + 1/9 - . . .)

This series is of little practical value because
enormous numbers of terms are required to achieve
good approximations (Acton, *Calculus*, 1999).
However, it does provide good programming practice.

Write a SPIM program that writes out the sum
of the first 1000, 2000, 3000, ... , 10000 terms
of the series.

As with the previous program,
write a version of this program in C or Java
first to check that your design is correct.

Click here
to go back to the main menu.

**End of Exercises**