log(2) in NASM x86-64

    extern  printf
    section .data
fmt db      "log(2) = %.10lf",10,0
    section .text
    global  main
main:
    ; log2 = 2*(1/3 + 1/3 * 1/3^3 + 1/5 * 1/3^5 +..)
    ; xmm0 = log2/2, xmm1 = 1/3^(2k-1), xmm2 = 1/9, xmm3 = 2k-1, xmm4 = xmm1 / xmm3
    push    rbp
    mov     rbp, rsp
    sub     rsp, 0x10               ; make space for 2 doubles

    mov     rax, 0x0
    mov     qword [rbp - 0x8], rax
    movsd   xmm0, qword [rbp - 0x8]
    mov     rax, 0x3fd5555555555555  ; hex form of 1./3
    mov     qword [rbp - 0x8], rax
    movsd   xmm1, qword [rbp - 0x8]
    mov     rax, 0x3fbc71c71c71c71c ; hex for 1./9
    mov     qword [rbp - 0x8], rax
    movsd   xmm2, qword [rbp - 0x8]
    mov     rax, 0x1

    mov     ecx, 15
    movsd   xmm4, xmm1
series_sum:
    addsd   xmm0, xmm4
    mulsd   xmm1, xmm2
    movsd   xmm4, xmm1
    add     rax, 2
cvtsi2sd    xmm3, rax
    divsd   xmm4, xmm3
    loop    series_sum

mult_by_two:
    addsd   xmm0, xmm0

print_log2:
    mov     rdi, fmt ; string print format
    mov     rax, 0x1 ; 1 floating point number, xmm0
    call    printf

end_program:
    mov     rsp, rbp
    pop     rbp
    ret