Group Project: Computing π

Now that we’ve implemented the easy way to compute π, I’ll let you work in groups to compute it faster, using a different approximation series:

$$\pi = 3 + \sum_{i=0}^\infty (-1)^i \frac{4}{(2i + 2) (2i+3) (2i+4)} = 3 + \frac{4}{2 \times 3 \times 4} - \frac{4}{4 \times 5 \times 6} + \cdots$$

This series converges much faster than the traditional one: while the traditional series takes nearly 500,00 iterations to get just five digits of precision, this series gets there in about 12. Here’s a skeleton program for you:

section .data

zero:   dq      0.0
one:    dq      1.0
two:    dq      2.0
four:   dq      4.0
negone: dq      -1.0
limit:  dq      0.000001

format: db      "%f", 10, 0

section .text

extern printf

global main
main:

    push rbp
    mov rbp, rsp

    ;; Compute pi 
    movsd xmm0, qword [limit]   
    call compute_pi
    ; Return value in xmm0  

    ;; Print result
    mov rdi, format
    mov al, 1
    call printf

    mov rax, 0
    pop rbp
    ret

compute_pi:
    push rbp
    mov rbp, rsp

    ; xmm0 = Approximation limit
    ; Return result in xmm0

    ; YOUR CODE HERE

    pop rbp
    ret

You can find a copy of this file on the server in /usr/local/class/src/pi.s.

Note that I’ve changed all the constants to qwords (i.e., doubles), so don’t forget to use the double-precision versions of the various operations.

It should be possible to change the limit in the .text section to get more (or less) precise approximations.

Submit this in the directory ~/cs241/group1.