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Index > Projects and Ideas > Hex digit extraction of PI:

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bitRAKE



Joined: 21 Jul 2003
Posts: 2796
Location: dank orb
bitRAKE
I just coded and verified (10 000 digits, with two other sources) this algorithm. It isn't good for anything, but maybe someone else might like to play with it.

It is a direct conversion of the BBP algorithm. Here are some single digit timings on my Pentium M:
Code:
      digit   time
----------------------------
        10^4      16 ms
     10^5     219 ms
     10^6    2860 ms
     10^7   33938 ms
     10^8  407828 ms    
An SSE2 version might be interesting to code in the future...
Code:
; 16^EBP MOD ECX=8k+z
; Special cases:
;       EBP=0, return 1
;    ECX=1, return 0
;
; EDX is return value
mpow:      xor edx,edx
 cmp ecx,1
   je .x
       bsr ebx,ebp
 mov edx,1
   jne .1
.x:       retn

.0:     mul eax
     div ecx
.1:      bt ebp,ebx
  mov eax,16
  jnc .2
      mul edx
     div ecx
.2:      dec ebx
     mov eax, edx
        jns .0
      retn

series: xor edi,edi     ; fraction
.1:   call mpow       ; 16^EBP mod ECX
    xor eax,eax
 div ecx         ; EDX:0 / 8k+m
  add ecx,8       ; 8k+m, k[0,EBP]
    add edi,eax
 dec ebp
     jns .1

  mov ebp,10000000h
.2:    mov eax,ebp
 xor edx,edx
 div ecx
     add ecx,8
   shr ebp,4
   cmp ecx,ebp
 lea edi,[edi+eax]
   jna .2
      retn

PiHexDigit:
 pushad
      mov ebp,[esp+36]
    mov ecx,1
   call series
 lea esi,[edi*4]

 mov ebp,[esp+36]
    mov ecx,4
   call series
 sub esi,edi
 sub esi,edi

     mov ebp,[esp+36]
    mov ecx,5
   call series
 sub esi,edi

     mov ebp,[esp+36]
    mov ecx,6
   call series
 sub esi,edi
 mov [esp+28],esi
    popad
       retn 4

; use it like this...
 invoke PiHexDigit, 1000000
; top nibble of EAX is hex digit    
[3.]243F6 A8885 A308D 31319 8A2E0 37073 44A40 93822 299F3 1D008 2EFA9 8EC4E 6C894 52821 E638D 01377 ...

Typically, each call returns at least five valid hex digits, but I haven't tested the validity of that claim rigorously.
Post 11 Nov 2007, 06:32
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edfed



Joined: 20 Feb 2006
Posts: 4224
Location: 2018
edfed
good job!
Post 13 Nov 2007, 14:20
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bitRAKE



Joined: 21 Jul 2003
Posts: 2796
Location: dank orb
bitRAKE
Next step is to remove the divisions with Montgomery exponentiation. Then I'll re-examine what is required for an SSE2 implementation. Rounding is free with scaled integers, so I doubt moving to floating point would help.
Post 13 Nov 2007, 16:52
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