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{{task|Irrational numbers}} [[File:pi_symbol.jpg|500px||right]]
Create a program to continually calculate and output the next decimal digit of (pi).
The program should continue forever (until it is aborted by the user) calculating and outputting each decimal digit in succession.
The output should be a decimal sequence beginning 3.14159265 ...
Note: this task is about ''calculating'' pi. For information on built-in pi constants see [[Real constants and functions]].
Related Task [[Arithmetic-geometric mean/Calculate Pi]]
360 Assembly
{{trans|FORTRAN}} The program uses one ASSIST macro (XPRNT) to keep the code as short as possible.
* Spigot algorithm do the digits of PI 02/07/2016
PISPIG CSECT
USING PISPIG,R13 base register
B 72(R15) skip savearea
DC 17F'0' savearea
STM R14,R12,12(R13) prolog
ST R13,4(R15) "
ST R15,8(R13) "
LR R13,R15 "
SR R0,R0 0
ST R0,MORE more=0
LA R6,1 i=1
LOOPI1 C R6,=A(NBUF) do i=1 to hbound(buf)
BH ELOOPI1 "
SR R9,R9 karray=0
L R7,=A(NVECT) j=hbound(vect)
LR R1,R7 j
SLA R1,2 .
LA R10,VECT-4(R1) r10=@vect(j)
LOOPJ EQU * do j=hbound(vect) to 1 by -1
L R5,=F'100000' 100000
M R4,0(R10) *vect(j)
LR R2,R5 r2=100000*vect(j)
LR R5,R9 karray
MR R4,R7 karray*j
AR R2,R5 r2+karray*j
LR R11,R2 n=100000*vect(j)+karray*j
LR R3,R7 j
SLA R3,1 2*j
BCTR R3,0 2*j-1)
LR R4,R11 n
SRDA R4,32 .
DR R4,R3 n/(2*j-1)
LR R9,R5 karray=n/(2*j-1)
LR R5,R9 karray
MR R4,R3 karray*(2*j-1)
LR R1,R11 n
SR R1,R5 n-karray*(2*j-1)
ST R1,0(R10) vect(j)=n-karray*(2*j-1)
SH R10,=H'4' r10=@vect(j)
BCT R7,LOOPJ end do j
LR R4,R9 karray
SRDA R4,32 .
D R4,=F'100000' karray/100000
LR R11,R5 k=karray/100000
L R2,MORE more
AR R2,R11 +k
LR R1,R6 i
SLA R1,2 .
ST R2,BUF-4(R1) buf(i)=more+k
LR R5,R11 k
M R4,=F'100000' *100000
LR R1,R9 karray
SR R1,R5 -k*100000
ST R1,MORE more=karray-k*100000
LA R6,1(R6) i=i+1
B LOOPI1 end do i
ELOOPI1 L R1,BUF buf(1)
CVD R1,PACKED convert buf(1) to packed decimal
OI PACKED+7,X'0F' prepare unpack
UNPK PG(1),PACKED packed decimal to zoned printable
MVI PG+1,C'.' output '.'
XPRNT PG,80 print buffer
MVC PG,=CL80' ' clear buffer
LA R3,PG pgi=0
LA R6,2 i=2
LOOPI2 C R6,=A(NBUF) do i=2 to hbound(buf)
BH ELOOPI2 "
MVC 0(1,R3),=C' ' output ' '
LA R3,1(R3) pgi=pgi+1
LR R1,R6 i
SLA R1,2 .
L R2,BUF-4(R1) buf(i)
CVD R2,PACKED convert v to packed decimal
OI PACKED+7,X'0F' prepare unpack
UNPK XDEC,PACKED packed decimal to zoned printable
MVC 0(5,R3),XDEC+7 output buf(i) with 5 decimals
LA R3,5(R3) pgi=pgi+5
LR R4,R6 i
BCTR R4,0 i-1
SRDA R4,32 .
D R4,=F'10' (i-1)/10
LTR R4,R4 if (i-1)//10=0
BNZ NOSKIP then
XPRNT PG,80 print buffer
LA R3,PG pgi=0
MVC PG,=CL80' ' clear buffer
NOSKIP LA R6,1(R6) i=i+1
B LOOPI2 end do i
ELOOPI2 L R13,4(0,R13) epilog
LM R14,R12,12(R13) "
XR R15,R15 "
BR R14 exit
LTORG
MORE DS F more
PACKED DS 0D,PL8 packed decimal
PG DC CL80' ' buffer
XDEC DS CL12 temp
BUF DC (NBUF)F'0' buf(nbuf)
VECT DC (NVECT)F'2' vect(nvect) init 2
YREGS
NBUF EQU 201 number of 5 decimals
NVECT EQU 3350 nvect=ceil(nbuf*50/3)
END PISPIG
{{out}}
3.
14159 26535 89793 23846 26433 83279 50288 41971 69399 37510
58209 74944 59230 78164 06286 20899 86280 34825 34211 70679
82148 08651 32823 06647 09384 46095 50582 23172 53594 08128
48111 74502 84102 70193 85211 05559 64462 29489 54930 38196
44288 10975 66593 34461 28475 64823 37867 83165 27120 19091
45648 56692 34603 48610 45432 66482 13393 60726 02491 41273
72458 70066 06315 58817 48815 20920 96282 92540 91715 36436
78925 90360 01133 05305 48820 46652 13841 46951 94151 16094
33057 27036 57595 91953 09218 61173 81932 61179 31051 18548
07446 23799 62749 56735 18857 52724 89122 79381 83011 94912
98336 73362 44065 66430 86021 39494 63952 24737 19070 21798
60943 70277 05392 17176 29317 67523 84674 81846 76694 05132
00056 81271 45263 56082 77857 71342 75778 96091 73637 17872
14684 40901 22495 34301 46549 58537 10507 92279 68925 89235
42019 95611 21290 21960 86403 44181 59813 62977 47713 09960
51870 72113 49999 99837 29780 49951 05973 17328 16096 31859
50244 59455 34690 83026 42522 30825 33446 85035 26193 11881
71010 00313 78387 52886 58753 32083 81420 61717 76691 47303
59825 34904 28755 46873 11595 62863 88235 37875 93751 95778
18577 80532 17122 68066 13001 92787 66111 95909 21642 01989
Ada
{{works with|Ada 2005}} {{libheader|GMP}} uses same algorithm as Go solution, from http://web.comlab.ox.ac.uk/people/jeremy.gibbons/publications/spigot.pdf ;pi_digits.adb:
with Ada.Command_Line;
with Ada.Text_IO;
with GNU_Multiple_Precision.Big_Integers;
with GNU_Multiple_Precision.Big_Rationals;
use GNU_Multiple_Precision;
procedure Pi_Digits is
type Int is mod 2 ** 64;
package Int_To_Big is new Big_Integers.Modular_Conversions (Int);
-- constants
Zero : constant Big_Integer := Int_To_Big.To_Big_Integer (0);
One : constant Big_Integer := Int_To_Big.To_Big_Integer (1);
Two : constant Big_Integer := Int_To_Big.To_Big_Integer (2);
Three : constant Big_Integer := Int_To_Big.To_Big_Integer (3);
Four : constant Big_Integer := Int_To_Big.To_Big_Integer (4);
Ten : constant Big_Integer := Int_To_Big.To_Big_Integer (10);
-- type LFT = (Integer, Integer, Integer, Integer
type LFT is record
Q, R, S, T : Big_Integer;
end record;
-- extr :: LFT -> Integer -> Rational
function Extr (T : LFT; X : Big_Integer) return Big_Rational is
use Big_Integers;
Result : Big_Rational;
begin
-- extr (q,r,s,t) x = ((fromInteger q) * x + (fromInteger r)) /
-- ((fromInteger s) * x + (fromInteger t))
Big_Rationals.Set_Numerator (Item => Result,
New_Value => T.Q * X + T.R,
Canonicalize => False);
Big_Rationals.Set_Denominator (Item => Result,
New_Value => T.S * X + T.T);
return Result;
end Extr;
-- unit :: LFT
function Unit return LFT is
begin
-- unit = (1,0,0,1)
return LFT'(Q => One, R => Zero, S => Zero, T => One);
end Unit;
-- comp :: LFT -> LFT -> LFT
function Comp (T1, T2 : LFT) return LFT is
use Big_Integers;
begin
-- comp (q,r,s,t) (u,v,w,x) = (q*u+r*w,q*v+r*x,s*u+t*w,s*v+t*x)
return LFT'(Q => T1.Q * T2.Q + T1.R * T2.S,
R => T1.Q * T2.R + T1.R * T2.T,
S => T1.S * T2.Q + T1.T * T2.S,
T => T1.S * T2.R + T1.T * T2.T);
end Comp;
-- lfts = [(k, 4*k+2, 0, 2*k+1) | k<-[1..]
K : Big_Integer := Zero;
function LFTS return LFT is
use Big_Integers;
begin
K := K + One;
return LFT'(Q => K,
R => Four * K + Two,
S => Zero,
T => Two * K + One);
end LFTS;
-- next z = floor (extr z 3)
function Next (T : LFT) return Big_Integer is
begin
return Big_Rationals.To_Big_Integer (Extr (T, Three));
end Next;
-- safe z n = (n == floor (extr z 4)
function Safe (T : LFT; N : Big_Integer) return Boolean is
begin
return N = Big_Rationals.To_Big_Integer (Extr (T, Four));
end Safe;
-- prod z n = comp (10, -10*n, 0, 1)
function Prod (T : LFT; N : Big_Integer) return LFT is
use Big_Integers;
begin
return Comp (LFT'(Q => Ten, R => -Ten * N, S => Zero, T => One), T);
end Prod;
procedure Print_Pi (Digit_Count : Positive) is
Z : LFT := Unit;
Y : Big_Integer;
Count : Natural := 0;
begin
loop
Y := Next (Z);
if Safe (Z, Y) then
Count := Count + 1;
Ada.Text_IO.Put (Big_Integers.Image (Y));
exit when Count >= Digit_Count;
Z := Prod (Z, Y);
else
Z := Comp (Z, LFTS);
end if;
end loop;
end Print_Pi;
N : Positive := 250;
begin
if Ada.Command_Line.Argument_Count = 1 then
N := Positive'Value (Ada.Command_Line.Argument (1));
end if;
Print_Pi (N);
end Pi_Digits;
output:
3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9 5 0 2 8 8 4 1 9 7 1 6 9 3 9 9 3 7 5 1 0 5 8 2 0 9 7 4 9 4 4 5 9 2 3 0 7 8 1 6 4 0 6 2 8 6 2 0 8 9 9 8 6 2 8 0 3 4 8 2 5 3 4 2 1 1 7 0 6 7
AutoHotkey
{{libheader|MPL}} Could be optimized with Ipp functions, but runs fast enough for me as-is. Does not work in AHKLx64.
#NoEnv
#SingleInstance, Force
SetBatchLines, -1
#Include mpl.ahk
dot:=".", i:=0
, MP_SET(q, "1")
, MP_SET(r, "0")
, MP_SET(t, "1")
, MP_SET(k, "1")
, MP_SET(n, "3")
, MP_SET(l, "3")
, MP_SET(ONE, "1")
, MP_SET(TWO, "2")
, MP_SET(THREE, "3")
, MP_SET(FOUR, "4")
, MP_SET(SEVEN, "7")
, MP_SET(TEN, "10")
Loop
{
MP_MUL(q4, q, FOUR)
, MP_ADD(q4r, q4, r)
, MP_SUB(q4rt, q4r, t)
, MP_MUL(tn, t, n)
If (MP_CMP(q4rt,tn) = -1)
{
s := MP_DEC(n) . dot
OutputDebug %s%
dot := ""
, i++
, MP_MUL(tn, t, n)
, MP_SUB(rtn, r, tn)
, MP_MUL(nr, rtn, TEN)
, MP_MUL(q3, q, THREE)
, MP_ADD(q3r, q3, r)
, MP_DIV(q3rt, remainder, q3r, t)
, MP_SUB(q3rtn, q3rt, n)
, MP_MUL(n, q3rtn, TEN)
, MP_MUL(tmp, q, TEN)
, MP_CPY(q, tmp)
, MP_CPY(r, nr)
}
Else
{
MP_MUL(q2, q, TWO)
, MP_ADD(q2r, q2, r)
, MP_MUL(nr, q2r, l)
, MP_MUL(k7, k, SEVEN)
, MP_ADD(k72, k7, TWO)
, MP_MUL(qk, q, k72)
, MP_MUL(rl, r, l)
, MP_ADD(qkrl, qk, rl)
, MP_MUL(tl, t, l)
, MP_DIV(nn, remainder, qkrl, tl)
, MP_MUL(tmp, q, k)
, MP_CPY(q, tmp)
, MP_MUL(tmp, t, l)
, MP_CPY(t, tmp)
, MP_ADD(tmp, l, TWO)
, MP_CPY(l, tmp)
, MP_ADD(tmp, k, ONE)
, MP_CPY(k, tmp)
, MP_CPY(n, nn)
, MP_CPY(r, nr)
}
}
ALGOL 68
{{trans|Pascal}} Note: This specimen retains the original [[#Pascal|Pascal]] coding style of [http://www.mathpropress.com/stan/bibliography/spigot.pdf code]. {{works with|ALGOL 68|Revision 1 - no extensions to language used.}} {{works with|ALGOL 68G|Any - tested with release [http://sourceforge.net/projects/algol68/files/algol68g/algol68g-1.18.0/algol68g-1.18.0-9h.tiny.el5.centos.fc11.i386.rpm/download 1.18.0-9h.tiny].}} {{wont work with|ELLA ALGOL 68|Any (with appropriate job cards) - tested with release [http://sourceforge.net/projects/algol68/files/algol68toc/algol68toc-1.8.8d/algol68toc-1.8-8d.fc9.i386.rpm/download 1.8-8d] - due to extensive use of '''format'''[ted] ''transput''.}} This codes uses 33 decimals places as a test case. Performance is O(2) based on the number of decimal places required.
#!/usr/local/bin/a68g --script #
INT base := 10;
MODE YIELDINT = PROC(INT)VOID;
PROC gen pi digits = (INT decimal places, YIELDINT yield)VOID:
BEGIN
INT nine = base - 1;
INT nines := 0, predigit := 0; # First predigit is a 0 #
[decimal places*10 OVER 3]#LONG# INT digits; # We need 3 times the digits to calculate #
FOR place FROM LWB digits TO UPB digits DO digits[place] := 2 OD; # Start with 2s #
FOR place TO decimal places + 1 DO
INT digit := 0;
FOR i FROM UPB digits BY -1 TO LWB digits DO # Work backwards #
INT x := #SHORTEN#(base*digits[i] + #LENG# digit*i);
digits[i] := x MOD (2*i-1);
digit := x OVER (2*i-1)
OD;
digits[LWB digits] := digit MOD base; digit OVERAB base;
nines :=
IF digit = nine THEN
nines + 1
ELSE
IF digit = base THEN
yield(predigit+1); predigit := 0 ;
FOR repeats TO nines DO yield(0) OD # zeros #
ELSE
IF place NE 1 THEN yield(predigit) FI; predigit := digit;
FOR repeats TO nines DO yield(nine) OD
FI;
0
FI
OD;
yield(predigit)
END;
main:(
INT feynman point = 762; # feynman point + 4 is a good test case #
# the 33rd decimal place is a shorter tricky test case #
INT test decimal places = UPB "3.1415926.......................502"-2;
INT width = ENTIER log(base*(1+small real*10));
# iterate throught the digits as they are being found #
# FOR INT digit IN # gen pi digits(test decimal places#) DO ( #,
## (INT digit)VOID: (
printf(($n(width)d$,digit))
)
# OD #);
print(new line)
)
Output:
3141592653589793238462643383279502
BASIC
=
Applesoft
=
10 rem adopted from Commodore BASIC
20 n = 100 : rem N may be increased, but will slow execution
30 ln = int(10*n/4)
40 nd = 1
50 dim a(ln)
60 n9 = 0
70 pd = 0 :rem First pre-digit is a 0
80 rem
90 for j = 1 to ln
100 a(j-1) = 2 :rem Start with 2s
110 next j
120 rem
130 for j = 1 to n
140 q = 0
150 for i = ln to 1 step -1 :rem Work backwards
160 x = 10*a(i-1) + q*i
170 a(i-1) = x - (2*i-1)*int(x/(2*i-1)) :rem X - INT ( X / Y) * Y
180 q = int(x/(2*i - 1))
190 next i
200 a(0) = q-10*int(q/10)
210 q = int(q/10)
220 if q=9 then n9 = n9 + 1 : goto 450
240 if q<>10 then goto 350
250 rem q == 10
260 d = pd+1 : gosub 500
270 if n9 < 0 then goto 320
280 for k = 1 to n9
290 d = 0: gosub 500
300 next k
310 rem end if
320 pd = 0
330 n9 = 0
335 goto 450
340 rem q <> 10
350 d = pd: gosub 500
360 pd = q
370 if n9 = 0 then goto 450
380 for k = 1 to n9
390 d = 9 : gosub 500
400 next k
410 n9 = 0
450 next j
460 print str$(pd)
470 end
480 rem
490 rem output digits
500 if nd=0 then print str$(d); : return
510 if d=0 then return
520 print str$(d);".";
530 nd = 0
550 return
=
BASIC256
= {{Trans|Pascal}} below, and originally published by Stanley Rabinowitz in [http://www.mathpropress.com/stan/bibliography/spigot.pdf].
cls
n =1000
len = 10*n \ 4
needdecimal = true
dim a(len)
nines = 0
predigit = 0 # {First predigit is a 0}
for j = 1 to len
a[j-1] = 2 # {Start with 2s}
next j
for j = 1 to n
q = 0
for i = len to 1 step -1
# {Work backwards}
x = 10*a[i-1] + q*i
a[i-1] = x % (2*i - 1)
q = x \ (2*i - 1)
next i
a[0] = q % 10
q = q \ 10
if q = 9 then
nines = nines + 1
else
if q = 10 then
d = predigit+1: gosub outputd
if nines > 0 then
for k = 1 to nines
d = 0: gosub outputd
next k
end if
predigit = 0
nines = 0
else
d = predigit: gosub outputd
predigit = q
if nines <> 0 then
for k = 1 to nines
d = 9: gosub outputd
next k
nines = 0
end if
end if
end if
next j
print predigit
end
outputd:
if needdecimal then
if d = 0 then return
print d + ".";
needdecimal = false
else
print d;
end if
return
Output:
3.14159265358979323846264338327950288419716939937510582097494459230781...
=
Commodore BASIC
=
10 PRINT CHR$(147)
20 n = 100
30 ln = int(10*n/4)
40 nd = 1
50 dim a(ln)
60 n9 = 0
70 pd = 0 :rem First predigit is a 0
80 :
90 for j = 1 to ln
100 a(j-1) = 2 :rem Start with 2s
110 next j
120 :
130 for j = 1 to n
140 q = 0
150 for i = ln to 1 step -1 :rem Work backwards
160 x = 10*a(i-1) + q*i
170 a(i-1) = x - (2*i-1)*int(x/(2*i-1)) :rem X - INT ( X / Y) * Y
180 q = int(x/(2*i - 1))
190 next i
200 a(0) = q-10*int(q/10)
210 q = int(q/10)
220 if q=9 then n9 = n9 + 1 : goto 450
240 if q<>10 then 350
250 rem q == 10
260 d = pd+1 : gosub 500
270 if n9 < 0 then 320
280 for k = 1 to n9
290 d = 0: gosub 500
300 next k
310 rem end if
320 pd = 0
330 n9 = 0
335 goto 450
340 rem q <> 10
350 d = pd: gosub 500
360 pd = q
370 if n9 = 0 then 450
380 for k = 1 to n9
390 d = 9 : gosub 500
400 next k
410 n9 = 0
450 next j
460 print mid$(str$(pd),2,1)
470 end
480 :
490 rem outputd
500 if nd=0 then print mid$(str$(d),2,1); : return
510 if d=0 then return
520 print mid$(str$(d),2,1);".";
530 nd = 0
550 return
BBC BASIC
BASIC version
WIDTH 80
M% = (HIMEM-END-1000) / 4
DIM B%(M%)
FOR I% = 0 TO M% : B%(I%) = 20 : NEXT
E% = 0
L% = 2
FOR C% = M% TO 14 STEP -7
D% = 0
A% = C%*2-1
FOR P% = C% TO 1 STEP -1
D% = D%*P% + B%(P%)*&64
B%(P%) = D% MOD A%
D% DIV= A%
A% -= 2
NEXT
CASE TRUE OF
WHEN D% = 99: E% = E% * 100 + D% : L% += 2
WHEN C% = M%: PRINT ;(D% DIV 100) / 10; : E% = D% MOD 100
OTHERWISE:
PRINT RIGHT$(STRING$(L%,"0") + STR$(E% + D% DIV 100),L%);
E% = D% MOD 100 : L% = 2
ENDCASE
NEXT
Assembler version
{{works with|BBC BASIC for Windows}} The first 250,000 digits output have been verified.
DIM P% 32
[OPT 2 :.pidig mov ebp,eax :.pi1 imul edx,ecx : mov eax,[ebx+ecx*4]
imul eax,100 : add eax,edx : cdq : div ebp : mov [ebx+ecx*4],edx
mov edx,eax : sub ebp,2 : loop pi1 : mov eax,edx : ret :]
WIDTH 80
M% = (HIMEM-END-1000) / 4
DIM B%(M%) : B% = ^B%(0)
FOR I% = 0 TO M% : B%(I%) = 20 : NEXT
E% = 0
L% = 2
FOR C% = M% TO 14 STEP -7
D% = 0
A% = C%*2-1
D% = USR(pidig)
CASE TRUE OF
WHEN D% = 99: E% = E% * 100 + D% : L% += 2
WHEN C% = M%: PRINT ;(D% DIV 100) / 10; : E% = D% MOD 100
OTHERWISE:
PRINT RIGHT$(STRING$(L%,"0") + STR$(E% + D% DIV 100),L%);
E% = D% MOD 100 : L% = 2
ENDCASE
NEXT
'''Output:'''
3.141592653589793238462643383279502884197169399375105820974944592307816406286208
99862803482534211706798214808651328230664709384460955058223172535940812848111745
02841027019385211055596446229489549303819644288109756659334461284756482337867831
65271201909145648566923460348610454326648213393607260249141273724587006606315588
17488152092096282925409171536436789259036001133053054882046652138414695194151160
94330572703657595919530921861173819326117931051185480744623799627495673518857527
24891227938183011949129833673362440656643086021394946395224737190702179860943702
77053921717629317675238467481846766940513200056812714526356082778577134275778960
91736371787214684409012249534301465495853710507922796892589235420199561121290219
60864034418159813629774771309960518707211349999998372978049951059731732816096318
....
bc
The digits of Pi are printed 20 per line, by successively recomputing pi with higher precision. The computation is not accurate to the entire scale (for example, scale = 4; 4*a(1) prints ''3.1412'' instead of the expected ''3.1415''), so the program includes two excess digits in the scale. Fixed number of guarding digits will eventually fail because Pi can contain arbitrarily long sequence of consecutive 9s (or consecutive 0s), though for this task it might not matter in practice. The program proceeds more and more slowly but exploits bc's unlimited precision arithmetic.
The program uses three features of [[GNU bc]]: long variable names, # comments (for the #! line), and the print command (for zero padding).
{{libheader|bc -l}}
{{works with|GNU bc}}
{{works with|OpenBSD bc}}
#!/usr/bin/bc -l
scaleinc= 20
define zeropad ( n ) {
auto m
for ( m= scaleinc - 1; m > 0; --m ) {
if ( n < 10^m ) {
print "0"
}
}
return ( n )
}
wantscale= scaleinc - 2
scale= wantscale + 2
oldpi= 4*a(1)
scale= wantscale
oldpi= oldpi / 1
oldpi
while( 1 ) {
wantscale= wantscale + scaleinc
scale= wantscale + 2
pi= 4*a(1)
scale= 0
digits= ((pi - oldpi) * 10^wantscale) / 1
zeropad( digits )
scale= wantscale
oldpi= pi / 1
}
Output:
3.141592653589793238
46264338327950288419
71693993751058209749
44592307816406286208
99862803482534211706
79821480865132823066
47093844609550582231
72535940812848111745
02841027019385211055
59644622948954930381
96442881097566593344
61284756482337867831
65271201909145648566
92346034861045432664
82133936072602491412
73724587006606315588
17488152092096282925
40917153643678925903
60011330530548820466
52138414695194151160
94330572703657595919
....
Bracmat
{{trans|Icon_and_Unicon}}
( pi
= f,q r t k n l,first
. !arg:((=?f),?q,?r,?t,?k,?n,?l)
& yes:?first
& whl
' ( 4*!q+!r+-1*!t+-1*!n*!t:<0
& f$!n
& ( !first:yes
& f$"."
& no:?first
|
)
& "compute and update variables for next cycle"
& 10*(!r+-1*!n*!t):?nr
& div$(10*(3*!q+!r).!t)+-10*!n:?n
& !q*10:?q
& !nr:?r
| "compute and update variables for next cycle"
& (2*!q+!r)*!l:?nr
& div$(!q*(7*!k+2)+!r*!l.!t*!l):?nn
& !q*!k:?q
& !t*!l:?t
& !l+2:?l
& !k+1:?k
& !nn:?n
& !nr:?r
)
)
& pi$((=.put$!arg),1,0,1,1,3,3)
Output:
3.1415926535897932384626433832795028841971693993751058209749445923078164062
862089986280348253421170679821480865132823066470938446095505822317253594081
284811174502841027019385211055596446229489549303819644288109756659334461284
756482337867831652712019091456485669234603486104543266482133936072602491412
73724587006606315588174881520...
C
There are many ways to do this, with quite different performance profiles. A simple measurement of 6 programs: {| class="wikitable" |- ! Digits ! Spigot 1 ! Spigot 2 ! Machin 1 ! Machin 2 ! AGM ! Chudnovsky |- align="right" ! 1,000 | 0.008 | 0.009 | 0.001 | 0.001 | 0.000 | 0.000 |- align="right" ! 10,000 | 0.402 | 0.589 | 0.020 | 0.016 | 0.003 | 0.002 |- align="right" ! 100,000 | 39.400 | 85.600 | 1.740 | 1.480 | 0.084 | 0.002 |- align="right" ! 1,000,000 | | | 177.900 | 156.800 | 1.474 | 0.333 |- align="right" ! 10,000,000 | | | | | 25.420 | 5.715 |}
- Spigot 1: plain C (no GMP), modified Winter/Flammenkamp, correct to 1+M digits
- Spigot 2: C+GMP, as used in [http://shootout.alioth.debian.org/ Computer Language Benchmarks Game]
- Machin 1: C+GMP, shown below
- Machin 2: C+GMP, as below but using Chien-Lih 1997 formula
- AGM: C+GMP, essentially from the [[Arithmetic-geometric mean/Calculate Pi]] task. This has performance only slightly slower than MPFR.
- Chudnovsky: Hanhong Xue's code from [https://gmplib.org/pi-with-gmp.html GMP web site].
Using Machin's formula. The "continuous printing" part is silly: the algorithm really calls for a preset number of digits, so the program repeatedly calculates Pi digits with increasing length and chop off leading digits already displayed. But it's still faster than the unbounded Spigot method by an order of magnitude, at least for the first 100k digits.
#include <stdio.h> #include <stdlib.h> #include <gmp.h> mpz_t tmp1, tmp2, t5, t239, pows; void actan(mpz_t res, unsigned long base, mpz_t pows) { int i, neg = 1; mpz_tdiv_q_ui(res, pows, base); mpz_set(tmp1, res); for (i = 3; ; i += 2) { mpz_tdiv_q_ui(tmp1, tmp1, base * base); mpz_tdiv_q_ui(tmp2, tmp1, i); if (mpz_cmp_ui(tmp2, 0) == 0) break; if (neg) mpz_sub(res, res, tmp2); else mpz_add(res, res, tmp2); neg = !neg; } } char * get_digits(int n, size_t* len) { mpz_ui_pow_ui(pows, 10, n + 20); actan(t5, 5, pows); mpz_mul_ui(t5, t5, 16); actan(t239, 239, pows); mpz_mul_ui(t239, t239, 4); mpz_sub(t5, t5, t239); mpz_ui_pow_ui(pows, 10, 20); mpz_tdiv_q(t5, t5, pows); *len = mpz_sizeinbase(t5, 10); return mpz_get_str(0, 0, t5); } int main(int c, char **v) { unsigned long accu = 16384, done = 0; size_t got; char *s; mpz_init(tmp1); mpz_init(tmp2); mpz_init(t5); mpz_init(t239); mpz_init(pows); while (1) { s = get_digits(accu, &got); /* write out digits up to the last one not preceding a 0 or 9*/ got -= 2; /* -2: length estimate may be longer than actual */ while (s[got] == '0' || s[got] == '9') got--; printf("%.*s", (int)(got - done), s + done); free(s); done = got; /* double the desired digits; slows down at least cubically */ accu *= 2; } return 0; }
C#
'''Translation of:''' Java
using System; using System.Numerics; namespace PiCalc { internal class Program { private readonly BigInteger FOUR = new BigInteger(4); private readonly BigInteger SEVEN = new BigInteger(7); private readonly BigInteger TEN = new BigInteger(10); private readonly BigInteger THREE = new BigInteger(3); private readonly BigInteger TWO = new BigInteger(2); private BigInteger k = BigInteger.One; private BigInteger l = new BigInteger(3); private BigInteger n = new BigInteger(3); private BigInteger q = BigInteger.One; private BigInteger r = BigInteger.Zero; private BigInteger t = BigInteger.One; public void CalcPiDigits() { BigInteger nn, nr; bool first = true; while (true) { if ((FOUR*q + r - t).CompareTo(n*t) == -1) { Console.Write(n); if (first) { Console.Write("."); first = false; } nr = TEN*(r - (n*t)); n = TEN*(THREE*q + r)/t - (TEN*n); q *= TEN; r = nr; } else { nr = (TWO*q + r)*l; nn = (q*(SEVEN*k) + TWO + r*l)/(t*l); q *= k; t *= l; l += TWO; k += BigInteger.One; n = nn; r = nr; } } } private static void Main(string[] args) { new Program().CalcPiDigits(); } } }
Adopted Version:
using System; using System.Collections.Generic; using System.Linq; using System.Numerics; namespace EnumeratePi { class Program { private const int N = 60; private const string ZS = " +-"; static void Main() { Console.WriteLine("Digits of PI"); Console.WriteLine(new string('=', N + 13)); Console.WriteLine("Decimal : {0}", string.Concat(PiDigits(10).Take(N).Select(_ => _.ToString("d")))); Console.WriteLine("Binary : {0}", string.Concat(PiDigits(2).Take(N).Select(_ => _.ToString("d")))); Console.WriteLine("Quaternary : {0}", string.Concat(PiDigits(4).Take(N).Select(_ => _.ToString("d")))); Console.WriteLine("Octal : {0}", string.Concat(PiDigits(8).Take(N).Select(_ => _.ToString("d")))); Console.WriteLine("Hexadecimal: {0}", string.Concat(PiDigits(16).Take(N).Select(_ => _.ToString("x")))); Console.WriteLine("Alphabetic : {0}", string.Concat(PiDigits(26).Take(N).Select(_ => (char) ('A' + _)))); Console.WriteLine("Fun : {0}", string.Concat(PiDigits(ZS.Length).Take(N).Select(_ => ZS[(int)_]))); Console.WriteLine("Nibbles : {0}", string.Concat(PiDigits(0x10).Take(N/2).Select(_ => string.Format("{0:x1} ", _)))); Console.WriteLine("Bytes : {0}", string.Concat(PiDigits(0x100).Take(N/3).Select(_ => string.Format("{0:x2} ", _)))); Console.WriteLine("Words : {0}", string.Concat(PiDigits(0x10000).Take(N/5).Select(_ => string.Format("{0:x4} ", _)))); Console.WriteLine("Dwords : {0}", string.Concat(PiDigits(0x100000000).Take(N/9).Select(_ => string.Format("{0:x8} ", _)))); Console.WriteLine(new string('=', N + 13)); Console.WriteLine("* press any key to exit *"); Console.ReadKey(); } /// <summary>Enumerates the digits of PI.</summary> /// <param name="b">Base of the Numeral System to use for the resulting digits (default = Base.Decimal (10)).</param> /// <returns>The digits of PI.</returns> static IEnumerable<long> PiDigits(long b = 10) { BigInteger k = 1, l = 3, n = 3, q = 1, r = 0, t = 1 ; // skip integer part var nr = b * (r - t * n); n = b * (3 * q + r) / t - b * n; q *= b; r = nr; for (; ; ) { var tn = t * n; if (4 * q + r - t < tn) { yield return (long)n; nr = b * (r - tn); n = b * (3 * q + r) / t - b * n; q *= b; } else { t *= l; nr = (2 * q + r) * l; var nn = (q * (7 * k) + 2 + r * l) / t; q *= k; l += 2; ++k; n = nn; } r = nr; } } } }
Output:
===================================================================
Decimal : 141592653589793238462643383279502884197169399375105820974944 Binary : 001001000011111101101010100010001000010110100011000010001101 Quaternary : 021003331222202020112203002031030103012120220232000313001303 Octal : 110375524210264302151423063050560067016321122011160210514763 Hexadecimal: 243f6a8885a308d313198a2e03707344a4093822299f31d0082efa98ec4e Alphabetic : DRSQLOLYRTRODNLHNQTGKUDQGTUIRXNEQBCKBSZIVQQVGDMELMUEXROIQIYA Fun : + -++ +---- + -++ -+++++ --+----- +++- +-+-+-+- +-++ + Nibbles : 2 4 3 f 6 a 8 8 8 5 a 3 0 8 d 3 1 3 1 9 8 a 2 e 0 3 7 0 7 3 Bytes : 24 3f 6a 88 85 a3 08 d3 13 19 8a 2e 03 70 73 44 a4 09 38 22 Words : 243f 6a88 85a3 08d3 1319 8a2e 0370 7344 a409 3822 299f 31d0 Dwords : 243f6a88 85a308d3 13198a2e 03707344 a4093822 299f31d0
===================================================================
- press any key to exit *
## Clojure
{{Trans|Python}}
```lisp
(ns pidigits
(:gen-class))
(def calc-pi
; integer division rounding downwards to -infinity
(let [div (fn [x y] (long (Math/floor (/ x y))))
; Computations performed after yield clause in Python code
update-after-yield (fn [[q r t k n l]]
(let [nr (* 10 (- r (* n t)))
nn (- (div (* 10 (+ (* 3 q) r)) t) (* 10 n))
nq (* 10 q)]
[nq nr t k nn l]))
; Update of else clause in Python code: if (< (- (+ (* 4 q) r) t) (* n t))
update-else (fn [[q r t k n l]]
(let [nr (* (+ (* 2 q) r) l)
nn (div (+ (* q 7 k) 2 (* r l)) (* t l))
nq (* k q)
nt (* l t)
nl (+ 2 l)
nk (+ 1 k)]
[nq nr nt nk nn nl]))
; Compute the lazy sequence of pi digits translating the Python code
pi-from (fn pi-from [[q r t k n l]]
(if (< (- (+ (* 4 q) r) t) (* n t))
(lazy-seq (cons n (pi-from (update-after-yield [q r t k n l]))))
(recur (update-else [q r t k n l]))))]
; Use Clojure big numbers to perform the math (avoid integer overflow)
(pi-from [1N 0N 1N 1N 3N 3N])))
;; Indefinitely Output digits of pi, with 40 characters per line
(doseq [[i q] (map-indexed vector calc-pi)]
(when (= (mod i 40) 0)
(println))
(print q))
{{Output}}
3141592653589793238462643383279502884197
1693993751058209749445923078164062862089
9862803482534211706798214808651328230664
7093844609550582231725359408128481117450
...
Common Lisp
(defun pi-spigot () (labels ((g (q r t1 k n l) (cond ((< (- (+ (* 4 q) r) t1) (* n t1)) (princ n) (g (* 10 q) (* 10 (- r (* n t1))) t1 k (- (floor (/ (* 10 (+ (* 3 q) r)) t1)) (* 10 n)) l)) (t (g (* q k) (* (+ (* 2 q) r) l) (* t1 l) (+ k 1) (floor (/ (+ (* q (+ (* 7 k) 2)) (* r l)) (* t1 l))) (+ l 2)))))) (g 1 0 1 1 3 3)))
{{out}}
CL-USER> (pi-spigot)
3141592653589793238462643383279502884197169399375105820974944592307816406286 ...
Crystal
{{trans|Ruby}}
require "big" def pi q, r, t, k, n, l = [1, 0, 1, 1, 3, 3].map { |n| BigInt.new(n) } dot_written = false loop do if 4*q + r - t < n*t yield n unless dot_written yield '.' dot_written = true end nr = 10*(r - n*t) n = ((10*(3*q + r)) / t) - 10*n q *= 10 r = nr else nr = (2*q + r) * l nn = (q*(7*k + 2) + r*l) / (t*l) q *= k t *= l l += 2 k += 1 n = nn r = nr end end end pi { |digit_or_dot| print digit_or_dot; STDOUT.flush }
{{out}}
3.141592653589793238462643383279502884197169399375105820974944592307816406286 ...
D
This modified [[wp:Spigot_algorithm|Spigot algorithm]] does not continue infinitely, because its required memory grow as the number of digits need to print.
import std.stdio, std.conv, std.string; struct PiDigits { immutable uint nDigits; int opApply(int delegate(ref string /*chunk of pi digit*/) dg){ // Maximum width for correct output, for type ulong. enum size_t width = 9; enum ulong scale = 10UL ^^ width; enum ulong initDigit = 2UL * 10UL ^^ (width - 1); enum string formatString = "%0" ~ text(width) ~ "d"; immutable size_t len = 10 * nDigits / 3; auto arr = new ulong[len]; arr[] = initDigit; ulong carry; foreach (i; 0 .. nDigits / width) { ulong sum; foreach_reverse (j; 0 .. len) { auto quo = sum * (j + 1) + scale * arr[j]; arr[j] = quo % (j*2 + 1); sum = quo / (j*2 + 1); } auto yield = format(formatString, carry + sum/scale); if (dg(yield)) break; carry = sum % scale; } return 0; } } void main() { foreach (d; PiDigits(100)) writeln(d); }
Output:
314159265
358979323
846264338
327950288
419716939
937510582
097494459
230781640
628620899
862803482
534211706
Alternative version
import std.stdio, std.bigint; void main() { int ndigits = 0; auto q = BigInt(1); auto r = BigInt(0); auto t = q; auto k = q; auto n = BigInt(3); auto l = n; bool first = true; while (ndigits < 1_000) { if (4 * q + r - t < n * t) { write(n); ndigits++; if (ndigits % 70 == 0) writeln(); if (first) { first = false; write('.'); } auto nr = 10 * (r - n * t); n = ((10 * (3 * q + r)) / t) - 10 * n; q *= 10; r = nr; } else { auto nr = (2 * q + r) * l; auto nn = (q * (7 * k + 2) + r * l) / (t * l); q *= k; t *= l; l += 2; k++; n = nn; r = nr; } } }
Output:
3.141592653589793238462643383279502884197169399375105820974944592307816
4062862089986280348253421170679821480865132823066470938446095505822317
2535940812848111745028410270193852110555964462294895493038196442881097
5665933446128475648233786783165271201909145648566923460348610454326648
2133936072602491412737245870066063155881748815209209628292540917153643
6789259036001133053054882046652138414695194151160943305727036575959195
3092186117381932611793105118548074462379962749567351885752724891227938
1830119491298336733624406566430860213949463952247371907021798609437027
7053921717629317675238467481846766940513200056812714526356082778577134
2757789609173637178721468440901224953430146549585371050792279689258923
5420199561121290219608640344181598136297747713099605187072113499999983
7297804995105973173281609631859502445945534690830264252230825334468503
5261931188171010003137838752886587533208381420617177669147303598253490
4287554687311595628638823537875937519577818577805321712268066130019278
76611195909216420198
Elixir
{{trans|Erlang}}
defmodule Pi do def calc, do: calc(1,0,1,1,3,3,0) defp calc(q,r,t,k,n,l,c) when c==50 do IO.write "\n" calc(q,r,t,k,n,l,0) end defp calc(q,r,t,k,n,l,c) when (4*q + r - t) < n*t do IO.write n calc(q*10, 10*(r-n*t), t, k, div(10*(3*q+r), t) - 10*n, l, c+1) end defp calc(q,r,t,k,_n,l,c) do calc(q*k, (2*q+r)*l, t*l, k+1, div(q*7*k+2+r*l, t*l), l+2, c) end end Pi.calc
{{out}} Hit Ctrl-C to stop it.
C:\Elixir>elixir pi.exs
31415926535897932384626433832795028841971693993751
05820974944592307816406286208998628034825342117067
98214808651328230664709384460955058223172535940812
84811174502841027019385211055596446229489549303819
64428810975665933446128475648233786783165271201909
14564856692346034861045432664821339360726024914127
37245870066063155881748815209209628292540917153643
67892590360011330530548820466521384146951941511609
43305727036575959195309218611738193261179310511854
80744623799627495673518857527248912279381830119491
29833673362440656643086021394946395224737190702179
86094370277053921717629317675238467481846766940513
20005681271452635608277857713427577896091736371787
214684409012249534301
Erlang
% Implemented by Arjun Sunel -module(pi_calculation). -export([main/0]). main() -> pi(1,0,1,1,3,3,0). pi(Q,R,T,K,N,L,C) -> if C=:=50 -> io:format("\n"), pi(Q,R,T,K,N,L,0) ; true -> if (4*Q + R-T) < (N*T) -> io:format("~p",[N]), P = 10*(R-N*T), pi(Q*10 , P, T , K , ((10*(3*Q+R)) div T)-10*N , L,C+1); true -> P = (2*Q+R)*L, M = (Q*(7*K)+2+(R*L)) div (T*L), H = L+2, J =K+ 1, pi(Q*K, P , T*L ,J,M,H,C) end end.
{{out}}
31415926535897932384626433832795028841971693993751
05820974944592307816406286208998628034825342117067
98214808651328230664709384460955058223172535940812
84811174502841027019385211055596446229489549303819
64428810975665933446128475648233786783165271201909
14564856692346034861045432664821339360726024914127
37245870066063155881748815209209628292540917153643
67892590360011330530548820466521384146951941511609
43305727036575959195309218611738193261179310511854
80744623799627495673518857527248912279381830119491
29833673362440656643086021394946395224737190702179
86094370277053921717629317675238467481846766940513
20005681271452635608277857713427577896091736371787
21468440901224953430146549585371050792279689258923
54201995611212902196086403441815981362977477130996
05187072113499999983729780499510597317328160963185
95024459455346908302642522308253344685035261931188
17101000313783875288658753320838142061717766914730
35982534904287554687311595628638823537875937519577
81857780532171226806613001927876611195909216420198
93809525720106548586327886593615338182796823030195
20353018529689957736225994138912497217752834791315
15574857242454150695950829533116861727855889075098
38175463746493931925506040092770167113900984882401
28583616035637076601047101819429555961989467678374
4944825537977472684710404753464620
=={{header|F_Sharp|F#}}== {{trans|Haskell}}
let rec g q r t k n l = seq { if 4I*q+r-t < n*t then yield n yield! (g (10I*q) (10I*(r-n*t)) t k ((10I*(3I*q+r))/t - 10I*n) l) else yield! (g (q*k) ((2I*q+r)*l) (t*l) (k+1I) ((q*(7I*k+2I)+r*l)/(t*l)) (l+2I)) } let π = (g 1I 0I 1I 1I 3I 3I) Seq.take 1 π |> Seq.iter (printf "%A.") // 6 digits beginning at position 762 of π are '9' Seq.take 767 (Seq.skip 1 π) |> Seq.iter (printf "%A")
{{out}}
3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066
470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831
652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903
600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527
248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051
320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219
6086403441815981362977477130996051870721134999999
Factor
{{trans|Oforth}}
USING: combinators.extras io kernel locals math prettyprint ;
IN: rosetta-code.pi
:: calc-pi-digits ( -- )
1 0 1 1 3 3 :> ( q! r! t! k! n! l! ) [
4 q * r + t - n t * < [
n pprint flush
r n t * - 10 *
3 q * r + 10 * t /i n 10 * - n! r!
q 10 * q!
] [
2 q * r + l *
7 k * q * 2 + r l * + t l * /i n! r!
k q * q!
t l * t!
l 2 + l!
k 1 + k!
] if
] forever ;
MAIN: calc-pi-digits
Fortran
This is a modernized version of the example Fortran programme written by S. Rabinowitz in 1991. It works in base 100000 and the key step is the initialisation of all elements of VECT to 2. The format code of I5.5 means I5 output but with all leading spaces made zero so that 66 comes out as "00066", not " 66".
program pi
implicit none
integer,dimension(3350) :: vect
integer,dimension(201) :: buffer
integer :: more,karray,num,k,l,n
more = 0
vect = 2
do n = 1,201
karray = 0
do l = 3350,1,-1
num = 100000*vect(l) + karray*l
karray = num/(2*l - 1)
vect(l) = num - karray*(2*l - 1)
end do
k = karray/100000
buffer(n) = more + k
more = karray - k*100000
end do
write (*,'(i2,"."/(1x,10i5.5))') buffer
end program pi
The output is accumulated in BUFFER then written in one go at the end, but it could be written as successive values as each is calculated without much extra nitpickery: instead of BUFFER(N) = MORE + K for example just WRITE (*,"(I5.5)") MORE + K and no need for array BUFFER.
3.
14159265358979323846264338327950288419716939937510
58209749445923078164062862089986280348253421170679
82148086513282306647093844609550582231725359408128
48111745028410270193852110555964462294895493038196
44288109756659334461284756482337867831652712019091
45648566923460348610454326648213393607260249141273
72458700660631558817488152092096282925409171536436
78925903600113305305488204665213841469519415116094
33057270365759591953092186117381932611793105118548
07446237996274956735188575272489122793818301194912
98336733624406566430860213949463952247371907021798
60943702770539217176293176752384674818467669405132
00056812714526356082778577134275778960917363717872
14684409012249534301465495853710507922796892589235
42019956112129021960864034418159813629774771309960
51870721134999999837297804995105973173281609631859
50244594553469083026425223082533446850352619311881
71010003137838752886587533208381420617177669147303
59825349042875546873115956286388235378759375195778
18577805321712268066130019278766111959092164201989
This is an alternate version using an unbounded spigot. Higher precision is accomplished by using the Fortran Multiple Precision Library, FMLIB (http://myweb.lmu.edu/dmsmith/fmlib.html), provided by Dr. David M. Smith ([email protected]), Mathematics Professor (Emeritus) at Loyola Marymount University. We use the default precision which is about 50 significant digits.
!
### ==========================================
program pi_spigot_unbounded
!
### ==========================================
do
call print_next_pi_digit()
end do
contains
!------------------------------------------------
subroutine print_next_pi_digit()
!------------------------------------------------
use fmzm
type (im) :: q, r, t, k, n, l, nr
logical :: dot=.false., init=.false.
save :: q, r, t, k, n, l
if (.not.init) then
q=to_im(1)
r=to_im(0)
t=to_im(1)
k=to_im(1)
n=to_im(3)
l=to_im(3)
init=.true.
end if
if (4*q+r-t < n*t) then
write(6,fmt='(i1)',advance='no') to_int(n)
if (.not.dot) then
write(6,fmt='(a1)',advance='no') '.'
dot=.true.
end if
flush(6)
nr = 10 * ( r - n*t )
n = 10 * ( (3*q + r) / t - n )
q = 10 * q
r = nr
else
nr = (2*q + r) * l
n = ( (q * (7*k + 2) + r*l) / (t*l) )
q = q * k
t = t * l
l = l + 2
k = k + 1
r = nr
end if
end subroutine
end program
FreeBASIC
{{libheader|GMP}}
' version 05-07-2018
' compile with: fbc -s console
' unbounded spigot
' Ctrl-c to end program or close console window
#Include "gmp.bi"
Dim As UInteger num, ndigit, fp = Not 0
Dim As mpz_ptr q,r,t,k,n,l,tmp1,tmp2
q = Allocate(Len(__Mpz_struct)) : Mpz_init_set_ui(q,1)
r = Allocate(Len(__Mpz_struct)) : Mpz_init(r)
t = Allocate(Len(__Mpz_struct)) : Mpz_init_set_ui(t,1)
k = Allocate(Len(__Mpz_struct)) : Mpz_init_set_ui(k,1)
n = Allocate(Len(__Mpz_struct)) : Mpz_init_set_ui(n,3)
l = Allocate(Len(__Mpz_struct)) : Mpz_init_set_ui(l,3)
tmp1 = Allocate(Len(__Mpz_struct)) : Mpz_init(tmp1)
tmp2 = Allocate(Len(__Mpz_struct)) : Mpz_init(tmp2)
Do
mpz_mul_2exp(tmp1, q, 2)
mpz_add(tmp1,tmp1,r)
mpz_sub(tmp1,tmp1,t)
mpz_mul(tmp2, n, t)
If mpz_cmp(tmp1, tmp2) < 0 Then
Print mpz_get_ui(n); : ndigit += 1 : If ndigit Mod 50 = 0 Then Print " :";ndigit
If fp Then Print "."; : fp = Not fp : Print :ndigit = 0
mpz_sub(tmp1, r, tmp2)
mpz_mul_ui(tmp1, tmp1, 10)
mpz_mul_ui(tmp2, q, 3)
mpz_add(tmp2, tmp2, r)
mpz_mul_ui(tmp2, tmp2, 10)
mpz_set(r, tmp1)
mpz_mul_ui(tmp1, n, 10)
mpz_tdiv_q(tmp2, tmp2, t)
mpz_sub(n, tmp2, tmp1)
mpz_mul_ui(q, q, 10)
Else
mpz_mul(tmp2, r, l)
mpz_mul(tmp1, q, k)
mpz_mul_ui(tmp1, tmp1, 7)
mpz_add(tmp1, tmp1, tmp2)
mpz_mul_2exp(tmp2, q, 1)
mpz_add(tmp2, tmp2, r)
mpz_mul(tmp2, tmp2, l)
mpz_mul(t, t, l)
mpz_tdiv_q(tmp1, tmp1, t)
mpz_mul(q, q, k)
mpz_add_ui(k, k, 1)
mpz_add_ui(l, l, 2)
mpz_set(n, tmp1)
mpz_set(r, tmp2)
End If
Loop
{{out}}
3.
14159265358979323846264338327950288419716939937510 :50
58209749445923078164062862089986280348253421170679 :100
82148086513282306647093844609550582231725359408128 :150
48111745028410270193852110555964462294895493038196 :200
44288109756659334461284756482337867831652712019091 :250
......
59284936959414340814685298150539471789004518357551 :20300
54125223590590687264878635752541911288877371766374 :20350
86027660634960353679470269232297186832771739323619 :20400
20077745221262475186983349515101986426988784717193 :20450
96649769070825217423365662725928440620430214113719 :20500
FunL
The code for compute_pi() is from [http://www.cs.ox.ac.uk/people/jeremy.gibbons/publications/spigot.pdf]. The number of digits may be given on the command line as an argument. If there's no argument, the program will run until interrupted.
def compute_pi =
def g( q, r, t, k, n, l ) =
if 4*q + r - t < n*t
n # g( 10*q, 10*(r - n*t), t, k, (10*(3*q + r))\t - 10*n, l )
else
g( q*k, (2*q + r)*l, t*l, k + 1, (q*(7*k + 2) + r*l)\(t*l), l + 2 )
g( 1, 0, 1, 1, 3, 3 )
if _name_ == '-main-'
print( compute_pi().head() + '.' )
if args.isEmpty()
for d <- compute_pi().tail()
print( d )
else
for d <- compute_pi().tail().take( int(args(0)) )
print( d )
println()
FutureBasic
include "ConsoleWindow"
dim as long kf, ks
xref mf(_maxLong - 1) as long
xref ms(_maxLong - 1) as long
dim as long cnt, n, temp, nd
dim as long col, col1
dim as long lloc, stor(50)
end globals
local mode
local fn FmtStr( nn as long, s as Str255 ) as Str255
dim l as long
dim as Str255 f
l = s[0]
select case
case ( nn => l )
f = string$( nn-l, 32 ) + s
case ( -nn > l)
f = s + string$( -nn-l, 32 )
case else
f = s
end select
end fn = f
local mode
local fn FmtInt( nn as long, s as Str255 ) as Str255
if ( left$( s, 1 ) = " " ) then s = mid$( s, 2 )
end fn = fn FmtStr( nn, s )
local fn yprint( m as long )
if ( cnt < n )
col++
if ( col == 11 )
col = 1
col1++
long if ( col1 == 6 )
col1 = 0
print
print fn FmtInt( 4, str$( m mod 10) );
else
print fn FmtInt( 3, str$ (m mod 10) );
end if
else
print mid$( str$( m ), 2 ) ;
end if
cnt++
end if
end fn
local fn xprint( m as long)
dim as long ii, wk, wk1
if ( m < 8 )
ii = 1
while ( ii <= lloc )
fn yprint( stor(ii) )
ii++
wend
lloc = 0
else
if ( m > 9 )
wk = m / 10
m = m mod 10
wk1 = lloc
while ( wk1 >= 1 )
wk += stor(wk1)
stor(wk1) = wk mod 10
wk = wk/10
wk1--
wend
end if
end if
lloc++
stor(lloc) = m
end fn
local mode
local fn shift( l1 as ^long, l2 as ^long, lp as long, lmod as long )
dim as long k
if ( l2.nil& > 0 )
k = ( l2.nil& ) / lmod
else
k = -( -l2.nil& / lmod ) - 1
end if
l2.nil& = l2.nil& - k*lmod
l1.nil& = l1.nil& + k*lp
end fn
local fn Main( nDig as long )
dim as long i
n = nDig
stor(0) = 0
mf = fn malloc( ( n + 10 ) * sizeof(long) )
if ( 0 == mf ) then stop "Out of memory"
ms = fn malloc( ( n + 10 ) * sizeof(long) )
if ( 0 == ms ) then stop "Out of memory"
print : print "Approximation of π to"; n; " digits"
cnt = 0
kf = 25
ks = 57121
mf(1) = 1
i = 2
while ( i <= n )
mf(i) = -16
mf(i + 1) = 16
i += 2
wend
i = 1
while ( i <= n )
ms(i) = -4
ms(i + 1) = 4
i += 2
wend
print : print " 3.";
while ( cnt < n )
i = 0
i++
while ( i <= n - cnt )
mf(i) = mf(i) * 10
ms(i) = ms(i) * 10
i++
wend
i = ( n - cnt + 1 )
i--
while ( i >= 2 )
temp = 2 * i - 1
fn shift( @mf(i - 1), @mf(i), temp - 2, temp * kf )
fn shift( @ms(i - 1), @ms(i), temp - 2, temp * ks )
i--
wend
nd = 0
fn shift( @nd, @mf(1), 1, 5 )
fn shift( @nd, @ms(1), 1, 239 )
fn xprint( nd )
wend
print : print "Done"
fn free( ms )
fn free( mf )
end fn
dim as unsigned long ticks
ticks = fn TickCount()
// Here we specify the number of decimal places
fn Main( 4000 )
ticks = fn TickCount() - ticks
print "Elapsed time:" str$( ticks ) " ticks
Output:
Approximation of π to 4000 digits
3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944
5923078164 0628620899 8628034825 3421170679 8214808651 3282306647
0938446095 5058223172 5359408128 4811174502 8410270193 8521105559
6446229489 5493038196 4428810975 6659334461 2847564823 3786783165
2712019091 4564856692 3460348610 4543266482 1339360726 0249141273
7245870066 0631558817 4881520920 9628292540 9171536436 7892590360
0113305305 4882046652 1384146951 9415116094 3305727036 5759591953
0921861173 8193261179 3105118548 0744623799 6274956735 1885752724
8912279381 8301194912 9833673362 4406566430 8602139494 6395224737
1907021798 6094370277 0539217176 2931767523 8467481846 7669405132
0005681271 4526356082 7785771342 7577896091 7363717872 1468440901
2249534301 4654958537 1050792279 6892589235 4201995611 2129021960
8640344181 5981362977 4771309960 5187072113 4999999837 2978049951
0597317328 1609631859 5024459455 3469083026 4252230825 3344685035
2619311881 7101000313 7838752886 5875332083 8142061717 7669147303
5982534904 2875546873 1159562863 8823537875 9375195778 1857780532
1712268066 1300192787 6611195909 2164201989 3809525720 1065485863
2788659361 5338182796 8230301952 0353018529 6899577362 2599413891
2497217752 8347913151 5574857242 4541506959 5082953311 6861727855
8890750983 8175463746 4939319255 0604009277 0167113900 9848824012
8583616035 6370766010 4710181942 9555961989 4676783744 9448255379
7747268471 0404753464 6208046684 2590694912 9331367702 8989152104
7521620569 6602405803 8150193511 2533824300 3558764024 7496473263
9141992726 0426992279 6782354781 6360093417 2164121992 4586315030
2861829745 5570674983 8505494588 5869269956 9092721079 7509302955
3211653449 8720275596 0236480665 4991198818 3479775356 6369807426
5425278625 5181841757 4672890977 7727938000 8164706001 6145249192
1732172147 7235014144 1973568548 1613611573 5255213347 5741849468
4385233239 0739414333 4547762416 8625189835 6948556209 9219222184
2725502542 5688767179 0494601653 4668049886 2723279178 6085784383
8279679766 8145410095 3883786360 9506800642 2512520511 7392984896
0841284886 2694560424 1965285022 2106611863 0674427862 2039194945
0471237137 8696095636 4371917287 4677646575 7396241389 0865832645
9958133904 7802759009 9465764078 9512694683 9835259570 9825822620
5224894077 2671947826 8482601476 9909026401 3639443745 5305068203
4962524517 4939965143 1429809190 6592509372 2169646151 5709858387
4105978859 5977297549 8930161753 9284681382 6868386894 2774155991
8559252459 5395943104 9972524680 8459872736 4469584865 3836736222
6260991246 0805124388 4390451244 1365497627 8079771569 1435997700
1296160894 4169486855 5848406353 4220722258 2848864815 8456028506
0168427394 5226746767 8895252138 5225499546 6672782398 6456596116
3548862305 7745649803 5593634568 1743241125 1507606947 9451096596
0940252288 7971089314 5669136867 2287489405 6010150330 8617928680
9208747609 1782493858 9009714909 6759852613 6554978189 3129784821
6829989487 2265880485 7564014270 4775551323 7964145152 3746234364
5428584447 9526586782 1051141354 7357395231 1342716610 2135969536
2314429524 8493718711 0145765403 5902799344 0374200731 0578539062
1983874478 0847848968 3321445713 8687519435 0643021845 3191048481
0053706146 8067491927 8191197939 9520614196 6342875444 0643745123
7181921799 9839101591 9561814675 1426912397 4894090718 6494231961
5679452080 9514655022 5231603881 9301420937 6213785595 6638937787
0830390697 9207734672 2182562599 6615014215 0306803844 7734549202
6054146659 2520149744 2850732518 6660021324 3408819071 0486331734
6496514539 0579626856 1005508106 6587969981 6357473638 4052571459
1028970641 4011097120 6280439039 7595156771 5770042033 7869936007
2305587631 7635942187 3125147120 5329281918 2618612586 7321579198
4148488291 6447060957 5270695722 0917567116 7229109816 9091528017
3506712748 5832228718 3520935396 5725121083 5791513698 8209144421
0067510334 6711031412 6711136990 8658516398 3150197016 5151168517
1437657618 3515565088 4909989859 9823873455 2833163550 7647918535
8932261854 8963213293 3089857064 2046752590 7091548141 6549859461
6371802709 8199430992 4488957571 2828905923 2332609729 9712084433
5732654893 8239119325 9746366730 5836041428 1388303203 8249037589
8524374417 0291327656 1809377344 4030707469 2112019130 2033038019
7621101100 4492932151 6084244485 9637669838 9522868478 3123552658
2131449576 8572624334 4189303968 6426243410 7732269780 2807318915
4411010446 8232527162 0105265227 2111660396
Done
Elapsed time: 70 ticks
Go
Code below is a simplistic translation of Haskell code in [http://web.comlab.ox.ac.uk/oucl/work/jeremy.gibbons/publications/spigot.pdf Unbounded Spigot Algorithms for the Digits of Pi]. This is the algorithm specified for the [http://shootout.alioth.debian.org/u64q/performance.php?test=pidigits pidigits] benchmark of the [http://shootout.alioth.debian.org/ Computer Language Benchmarks Game]. (The standard Go distribution includes [http://golang.org/test/bench/shootout/pidigits.go source] submitted to the benchmark site, and that code runs stunning faster than the code below.)
package main import ( "fmt" "math/big" ) type lft struct { q,r,s,t big.Int } func (t *lft) extr(x *big.Int) *big.Rat { var n, d big.Int var r big.Rat return r.SetFrac( n.Add(n.Mul(&t.q, x), &t.r), d.Add(d.Mul(&t.s, x), &t.t)) } var three = big.NewInt(3) var four = big.NewInt(4) func (t *lft) next() *big.Int { r := t.extr(three) var f big.Int return f.Div(r.Num(), r.Denom()) } func (t *lft) safe(n *big.Int) bool { r := t.extr(four) var f big.Int if n.Cmp(f.Div(r.Num(), r.Denom())) == 0 { return true } return false } func (t *lft) comp(u *lft) *lft { var r lft var a, b big.Int r.q.Add(a.Mul(&t.q, &u.q), b.Mul(&t.r, &u.s)) r.r.Add(a.Mul(&t.q, &u.r), b.Mul(&t.r, &u.t)) r.s.Add(a.Mul(&t.s, &u.q), b.Mul(&t.t, &u.s)) r.t.Add(a.Mul(&t.s, &u.r), b.Mul(&t.t, &u.t)) return &r } func (t *lft) prod(n *big.Int) *lft { var r lft r.q.SetInt64(10) r.r.Mul(r.r.SetInt64(-10), n) r.t.SetInt64(1) return r.comp(t) } func main() { // init z to unit z := new(lft) z.q.SetInt64(1) z.t.SetInt64(1) // lfts generator var k int64 lfts := func() *lft { k++ r := new(lft) r.q.SetInt64(k) r.r.SetInt64(4*k+2) r.t.SetInt64(2*k+1) return r } // stream for { y := z.next() if z.safe(y) { fmt.Print(y) z = z.prod(y) } else { z = z.comp(lfts()) } } }
Groovy
{{trans|Java}} Solution:
BigInteger q = 1, r = 0, t = 1, k = 1, n = 3, l = 3 String nn boolean first = true while (true) { (nn, first, q, r, t, k, n, l) = (4*q + r - t < n*t) \ ? ["${n}${first?'.':''}", false, 10*q, 10*(r - n*t), t , k , 10*(3*q + r)/t - 10*n , l ] \ : ['' , first, q*k , (2*q + r)*l , t*l, k + 1, (q*(7*k + 2) + r*l)/(t*l), l + 2] print nn }
Output (thru first 1000 iterations):
3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337
Haskell
The code from [http://www.cs.ox.ac.uk/people/jeremy.gibbons/publications/spigot.pdf]:
pi_ = g (1, 0, 1, 1, 3, 3) where g (q, r, t, k, n, l) = if 4 * q + r - t < n * t then n : g ( 10 * q , 10 * (r - n * t) , t , k , div (10 * (3 * q + r)) t - 10 * n , l) else g ( q * k , (2 * q + r) * l , t * l , k + 1 , div (q * (7 * k + 2) + r * l) (t * l) , l + 2)
===Complete command-line program===
{{Works with|GHC|7.4.1}}
#!/usr/bin/runhaskell import Control.Monad import System.IO pi_ = g(1,0,1,1,3,3) where g (q,r,t,k,n,l) = if 4*q+r-t < n*t then n : g (10*q, 10*(r-n*t), t, k, div (10*(3*q+r)) t - 10*n, l) else g (q*k, (2*q+r)*l, t*l, k+1, div (q*(7*k+2)+r*l) (t*l), l+2) digs = insertPoint digs' where insertPoint (x:xs) = x:'.':xs digs' = map (head . show) pi_ main = do hSetBuffering stdout $ BlockBuffering $ Just 80 forM_ digs putChar
{{out}}
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420198
=={{header|Icon}} and {{header|Unicon}}== {{Trans|PicoLisp}} based on Jeremy Gibbons' Haskell solution.
procedure pi (q, r, t, k, n, l)
first := "yes"
repeat { # infinite loop
if (4*q+r-t < n*t) then {
suspend n
if (\first) := &null then suspend "."
# compute and update variables for next cycle
nr := 10*(r-n*t)
n := ((10*(3*q+r)) / t) - 10*n
q *:= 10
r := nr
} else {
# compute and update variables for next cycle
nr := (2*q+r)*l
nn := (q*(7*k+2)+r*l) / (t*l)
q *:= k
t *:= l
l +:= 2
k +:= 1
n := nn
r := nr
}
}
end
procedure main ()
every (writes (pi (1,0,1,1,3,3)))
end
J
pi=:3 :0
smoutput"0'3.1'
n=.0 while.n=.n+1 do.
smoutput-/1 10*<[email protected]^1 0+n
end.
)
Example use:
pi''
3
.
1
4
1
5
9
2
6
5
3
...
Java
{{trans|Icon}}
import java.math.BigInteger ; public class Pi { final BigInteger TWO = BigInteger.valueOf(2) ; final BigInteger THREE = BigInteger.valueOf(3) ; final BigInteger FOUR = BigInteger.valueOf(4) ; final BigInteger SEVEN = BigInteger.valueOf(7) ; BigInteger q = BigInteger.ONE ; BigInteger r = BigInteger.ZERO ; BigInteger t = BigInteger.ONE ; BigInteger k = BigInteger.ONE ; BigInteger n = BigInteger.valueOf(3) ; BigInteger l = BigInteger.valueOf(3) ; public void calcPiDigits(){ BigInteger nn, nr ; boolean first = true ; while(true){ if(FOUR.multiply(q).add(r).subtract(t).compareTo(n.multiply(t)) == -1){ System.out.print(n) ; if(first){System.out.print(".") ; first = false ;} nr = BigInteger.TEN.multiply(r.subtract(n.multiply(t))) ; n = BigInteger.TEN.multiply(THREE.multiply(q).add(r)).divide(t).subtract(BigInteger.TEN.multiply(n)) ; q = q.multiply(BigInteger.TEN) ; r = nr ; System.out.flush() ; }else{ nr = TWO.multiply(q).add(r).multiply(l) ; nn = q.multiply((SEVEN.multiply(k))).add(TWO).add(r.multiply(l)).divide(t.multiply(l)) ; q = q.multiply(k) ; t = t.multiply(l) ; l = l.add(TWO) ; k = k.add(BigInteger.ONE) ; n = nn ; r = nr ; } } } public static void main(String[] args) { Pi p = new Pi() ; p.calcPiDigits() ; } }
Output :
3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480 ...
JavaScript
Spigot Algorithm using BigInteger
This calculates one digit of pi at a time and writes it out Javascript does not have a native integer object, so this solution uses a library for Big Integer operations. document.write will work in a browser; to make this work in nodejs, change it to process.stdout.write
### Web Page version
This shows how to load the previous code into a webpage that writes digits out without freezing the browser
```html
<html><head><script src='https://rawgit.com/andyperlitch/jsbn/v1.1.0/index.js'></script></head>
<body style="width: 100%"><tt id="pi"></tt><tt>...</tt>
<script async defer>
function bi(n, b) { return new jsbn.BigInteger(n.toString(), b ? b : 10); };
var one=bi(1), two=bi(2), three=bi(3), four=bi(4), seven=bi(7), ten=bi(10);
function calcPi() {
var q=bi(1), r=bi(0), t=bi(1), k=bi(1), n=bi(3), l=bi(3);
var digit=0, firstrun=1;
var p=document.getElementById('pi');
function w(s) { p.appendChild(document.createTextNode(s));}
function continueCalcPi(q, r, t, k, n, l) {
while (true) {
if (q.multiply(four).add(r).subtract(t).compareTo(n.multiply(t)) < 0) {
w(n.toString());
if (digit==0 && firstrun==1) { w('.'); firstrun=0; };
digit = (digit+1) % 256;
var nr = (r.subtract(n.multiply(t))).multiply(ten);
n = (q.multiply(three).add(r)).multiply(ten).divide(t).subtract(n.multiply(ten));
q = q.multiply(ten);
r = nr;
if (digit%8==0) {
if (digit%64==0) {
p.appendChild(document.createElement('br'));
}
w(' ');
return setTimeout(function() { continueCalcPi(q, r, t, k, n, l); }, 50);
};
} else {
var nr = q.shiftLeft(1).add(r).multiply(l);
var nn = q.multiply(k).multiply(seven).add(two).add(r.multiply(l)).divide(t.multiply(l));
q = q.multiply(k);
t = t.multiply(l);
l = l.add(two);
k = k.add(one);
n = nn;
r = nr;
}
}
}
continueCalcPi(q, r, t, k, n, l);
}
calcPi();
</script>
</body></html>
Simple Approximation
Returns an approximation of Pi.
## jq
{{works with|jq|1.4}}
The focus in this section is on the Gibbons spigot algorithm as it
is relatively simple and therefore provides a gentle introduction to
how such algorithms can be implemented in jq.
Since the Gibbons algorithm quickly fails in the absence of support for large integers, we shall assume BigInt support, such as provided by [https://gist.github.com/pkoppstein/d06a123f30c033195841 BigInt.jq].
The jq program presented here closely follows the Groovy and Python
examples on this page. The spigot generator is named "next", and is
driven by an annotation function, "decorate"; thus the main program
is just "S0 | decorate(next)" where S0 is the initial state. One
advantage of this approach is that the generator's state is exposed,
thus making it easy to restart the stream at any point.
The annotation defined here results in a triple for each digit of pi:
[index, digit, space], where "space" is the the sum of the lengths of
the strings in the six-dimensional state vector, [q, r, t, k, n, l].
The output shows that the space requirements of the Gibbons
spigot grow very slightly more than linearly.
```jq
# The Gibbons spigot, in the mold of the [[#Groovy]] and ython]] programs shown on this page.
# The "bigint" functions
needed are: long_minus long_add long_multiply long_div
def pi_spigot:
# S is the sixtuple:
# q r t k n l
# 0 1 2 3 4 5
def long_lt(x;y): if x == y then false else lessOrEqual(x;y) end;
def check:
long_lt(long_minus(long_add(long_multiply("4"; .[0]); .[1]) ; .[2]);
long_multiply(.[4]; .[2]));
# state: [d, S] where digit is null or a digit ready to be printed
def next:
.[1] as $S
| $S[0] as $q | $S[1] as $r | $S[2] as $t | $S[3] as $k | $S[4] as $n | $S[5] as $l
| if $S|check
then [$n,
[long_multiply("10"; $q),
long_multiply("10"; long_minus($r; long_multiply($n;$t))),
$t,
$k,
long_minus( long_div(long_multiply("10";long_add(long_multiply("3"; $q); $r)); $t );
long_multiply("10";$n)),
$l ]]
else [null,
[long_multiply($q;$k),
long_multiply( long_add(long_multiply("2";$q); $r); $l),
long_multiply($t;$l),
long_add($k; "1"),
long_div( long_add(long_multiply($q; long_add(long_multiply("7";$k); "2")) ; long_multiply($r;$l));
long_multiply($t;$l) ),
long_add($l; "2") ]]
end;
# Input: input to the filter "nextstate"
# Output: [count, space, digit] for successive digits produced by "nextstate"
def decorate( nextstate ):
# For efficiency it is important that the recursive
# function have arity 0 and be tail-recursive:
def count:
.[0] as $count
| .[1] as $state
| $state[0] as $value
| ($state[1] | map(length) | add) as $space
| (if $value then [$count, $space, $value] else empty end),
( [if $value then $count+1 else $count end, ($state | nextstate)] | count);
[0, .] | count;
# q=1, r=0, t=1, k=1, n=3, l=3
[null, ["1", "0", "1", "1", "3", "3"]] | decorate(next)
;
pi_spigot
{{out}}
$ jq -M -n -c -f pi.bigint.jq [0,9,"3"] [1,14,"1"] [2,29,"4"] [3,36,"1"] [4,51,"5"] [5,69,"9"] [6,80,"2"] [7,95,"6"] [8,115,"5"] [9,125,"3"] [10,142,"5"] [11,167,"8"] [12,181,"9"] [13,197,"7"] [14,226,"9"] [15,245,"3"] [16,263,"2"] [17,276,"3"] [18,300,"8"] [19,320,"4"] [20,350,"6"] [21,363,"2"] [22,383,"6"] [23,408,"4"] [24,429,"3"] [25,442,"3"] [26,475,"8"] [27,502,"3"] [28,510,"2"] [29,531,"7"] [30,563,"9"] [31,611,"5"] [32,613,"0"] [33,628,"2"] [34,649,"8"] [35,676,"8"] [36,711,"4"] [37,720,"1"] [38,748,"9"] [39,783,"7"] [40,792,"1"] [41,814,"6"] [42,849,"9"] [43,870,"3"] [44,886,"9"] [45,923,"9"] [46,939,"3"] [47,967,"7"] [48,1004,"5"] [49,1041,"1"] [50,1043,"0"] [51,1059,"5"] [52,1103,"8"] [53,1133,"2"] [54,1135,"0"] [55,1165,"9"] [56,1195,"7"] [57,1212,"4"] [58,1242,"9"] [59,1273,"4"] [60,1297,"4"] [61,1313,"5"] [62,1358,"9"] [63,1375,"2"] [64,1421,"3"] [65,1423,"0"] [66,1447,"7"] [67,1493,"8"] [68,1501,"1"] [69,1533,"6"] [70,1579,"4"] [71,1581,"0"] [72,1613,"6"] [73,1630,"2"] [74,1662,"8"] [75,1701,"6"] [76,1733,"2"] [77,1735,"0"] [78,1781,"8"] [79,1792,"9"] [80,1816,"9"] [81,1849,"8"] [82,1889,"6"] [83,1898,"2"] [84,1961,"8"] [85,1963,"0"] [86,1988,"3"] [87,2013,"4"] [88,2054,"8"] [89,2071,"2"] [90,2104,"5"] [91,2129,"3"] [92,2162,"4"] [93,2195,"2"] [94,2220,"1"] [95,2230,"1"] [96,2287,"7"] [97,2289,"0"] [98,2314,"6"] [99,2340,"7"] [100,2373,"9"] [101,2414,"8"] [102,2448,"2"] [103,2458,"1"] [104,2484,"4"] [105,2534,"8"] [106,2536,"0"] [107,2569,"8"] [108,2602,"6"] [109,2645,"5"] [110,2662,"1"] [111,2696,"3"] [112,2707,"2"] [113,2756,"8"] [114,2775,"2"] [115,2825,"3"] [116,2827,"0"] [117,2853,"6"] [118,2887,"6"] [119,2914,"4"] [120,2964,"7"] [121,2966,"0"] [122,3008,"9"] [123,3027,"3"] [124,3061,"8"] [125,3088,"4"] [126,3114,"4"] [127,3165,"6"] [128,3167,"0"] [129,3202,"9"] [130,3237,"5"] [131,3287,"5"] [132,3289,"0"] [133,3316,"5"] [134,3360,"8"] [135,3387,"2"] [136,3414,"2"] [137,3456,"3"] [138,3466,"1"] [139,3510,"7"] [140,3529,"2"] [141,3564,"5"] [142,3583,"3"] [143,3610,"5"] [144,3653,"9"] [145,3697,"4"] [146,3699,"0"] [147,3752,"8"] [148,3770,"1"] [149,3789,"2"] [150,3825,"8"] [151,3852,"4"] [152,3905,"8"] [153,3933,"1"] [154,3960,"1"] [155,3970,"1"] [156,4006,"7"] [157,4033,"4"] [158,4102,"5"] [159,4104,"0"] [160,4124,"2"] [161,4159,"8"] [162,4203,"4"] [163,4248,"1"] [164,4250,"0"] [165,4269,"2"] [166,4348,"7"] [167,4350,"0"] [168,4361,"1"] [169,4405,"9"] [170,4424,"3"] [171,4460,"8"] [172,4497,"5"] [173,4542,"2"] [174,4569,"1"] [175,4605,"1"] [176,4607,"0"] [177,4644,"5"] [178,4672,"5"] [179,4691,"5"] [180,4727,"9"] [181,4764,"6"] [182,4792,"4"] [183,4820,"4"] [184,4865,"6"] [185,4893,"2"] [186,4913,"2"] [187,4949,"9"] [188,4968,"4"] [189,5005,"8"] [190,5042,"9"] [191,5070,"5"] [192,5098,"4"] [193,5144,"9"] [194,5198,"3"] [195,5200,"0"] [196,5219,"3"] [197,5266,"8"] [198,5276,"1"] [199,5313,"9"] [200,5350,"6"] [201,5387,"4"] [202,5416,"4"] [203,5435,"2"] [204,5471,"8"] [205,5526,"8"] [206,5556,"1"] [207,5558,"0"] [208,5594,"9"] [209,5632,"7"] [210,5660,"5"] [211,5689,"6"] [212,5726,"6"] [213,5746,"5"] [214,5792,"9"] [215,5821,"3"] [216,5849,"3"] [217,5887,"4"] [218,5906,"4"] [219,5961,"6"] [220,5981,"1"] [221,6002,"2"] [222,6038,"8"] [223,6068,"4"] [224,6096,"7"] [225,6134,"5"] [226,6163,"6"] [227,6191,"4"] [228,6238,"8"] [229,6267,"2"] [230,6296,"3"] [231,6316,"3"] [232,6344,"7"] [233,6383,"8"] [234,6411,"6"] [235,6440,"7"] [236,6487,"8"] [237,6525,"3"] [238,6545,"1"] [239,6574,"6"] [240,6621,"5"] [241,6641,"2"] [242,6688,"7"] [243,6717,"1"] [244,6782,"2"] [245,6784,"0"] [246,6795,"1"] [247,6852,"9"] [248,6854,"0"] [249,6910,"9"] [250,6929,"1"] [251,6959,"4"] [252,6988,"5"] [253,7027,"6"] [254,7046,"4"] [255,7085,"8"] [256,7115,"5"] [257,7153,"6"] [258,7181,"6"] [259,7229,"9"] [260,7258,"2"] [261,7288,"3"] [262,7317,"4"] [263,7383,"6"] [264,7385,"0"] [265,7415,"3"] [266,7435,"4"] [267,7474,"8"] [268,7530,"6"] [269,7569,"1"] [270,7571,"0"] [271,7609,"4"] [272,7639,"5"] [273,7678,"4"] [274,7716,"3"] [275,7736,"2"] [276,7766,"6"] [277,7805,"6"] [278,7826,"4"] [279,7873,"8"] [280,7912,"2"] [281,7933,"1"] [282,7971,"3"] [283,7991,"3"] [284,8030,"9"] [285,8060,"3"] [286,8118,"6"] [287,8120,"0"] [288,8168,"7"] [289,8189,"2"] [290,8264,"6"] [291,8266,"0"] [292,8287,"2"] [293,8317,"4"] [294,8374,"9"] [295,8395,"1"] [296,8443,"4"] [297,8464,"1"] [298,8485,"2"] [299,8524,"7"] [300,8544,"3"] [301,8593,"7"] [302,8623,"2"] ...
Julia
Julia comes with built-in support for computing π in arbitrary precision (using the GNU MPFR library). This implementation computes π at precisions that are repeatedly doubled as more digits are needed, printing one digit at a time and never terminating (until it runs out of memory) as specified:
let prec = precision(BigFloat), spi = "", digit = 1 while true if digit > lastindex(spi) prec *= 2 setprecision(prec) spi = string(big(π)) end print(spi[digit]) digit += 1 end end
Output:
3.141592653589793238462643383279502884195e69399375105820974944592307816406286198e9862803482534211706798214808651328230664709384460955058223172535940812848115e450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724586997e0631558817488152092096282925409171536436789259036001133053054882046652138414695194151160943305727036575959195309218611738193261179310511854807446237996274956735188575272489122793818301194912983367336244065664308602139494639522473719070217986094370277053921717629317675238467481846766940513200056812714526357e8277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201...
Kotlin
{{trans|Java}}
// version 1.1.2 import java.math.BigInteger val ZERO = BigInteger.ZERO val ONE = BigInteger.ONE val TWO = BigInteger.valueOf(2L) val THREE = BigInteger.valueOf(3L) val FOUR = BigInteger.valueOf(4L) val SEVEN = BigInteger.valueOf(7L) val TEN = BigInteger.TEN fun calcPi() { var nn: BigInteger var nr: BigInteger var q = ONE var r = ZERO var t = ONE var k = ONE var n = THREE var l = THREE var first = true while (true) { if (FOUR * q + r - t < n * t) { print(n) if (first) { print ("."); first = false } nr = TEN * (r - n * t) n = TEN * (THREE * q + r) / t - TEN * n q *= TEN r = nr } else { nr = (TWO * q + r) * l nn = (q * SEVEN * k + TWO + r * l) / (t * l) q *= k t *= l l += TWO k += ONE n = nn r = nr } } } fun main(args: Array<String>) = calcPi()
{{out}}
3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745...
Lasso
Based off [http://crypto.stanford.edu/pbc/notes/pi/code.html Dik T. Winter's C implementation of Beeler et al. 1972, Item 120].
#!/usr/bin/lasso9
define generatePi => {
yield currentCapture
local(r = array(), i, k, b, d, c = 0, x)
with i in generateSeries(1, 2800)
do #r->insert(2000)
with k in generateSeries(2800, 1, -14)
do {
#d = 0
#i = #k
while(true) => {
#d += #r->get(#i) * 10000
#b = 2 * #i - 1
#r->get(#i) = #d % #b
#d /= #b
#i--
!#i ? loop_abort
#d *= #i
}
#x = (#c + #d / 10000)
yield (#k == 2800 ? ((#x * 0.001)->asstring(-precision = 3)) | #x->asstring(-padding=4, -padChar='0'))
#c = #d % 10000
}
}
local(pi_digits) = generatePi
loop(200) => {
stdout(#pi_digits())
}
Output (first 100 places):
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067
Liberty BASIC
Pretty slow if you run for over 100 digits...
ndigits = 0
q = 1
r = 0
t = q
k = q
n = 3
L = n
first = 666 ' ANY non-zero =='true' in LB.
while ndigits <100
if ( 4 *q +r -t) <( n *t) then
print n;
ndigits =ndigits +1
if not( ndigits mod 40) then print: print " ";
if first =666 then first = 0: print ".";
nr =10 *( r -n *t)
n =int( ( (10 *( 3 *q +r)) /t) -10 *n)
q =q *10
r =nr
else
nr =( 2 *q +r) *L
nn =(q *( 7 *k +2) +r *L) /( t *L)
q =q *k
t =t *L
L =L +2
k =k +1
n =int( nn)
r =nr
end if
scan
wend
end
3.141592653589793238462643383279502884197
1693993751058209749445923078164062862089
98628034825342117067
Lua
{{trans|Pascal}}
a = {} n = 1000 len = math.modf( 10 * n / 3 ) for j = 1, len do a[j] = 2 end nines = 0 predigit = 0 for j = 1, n do q = 0 for i = len, 1, -1 do x = 10 * a[i] + q * i a[i] = math.fmod( x, 2 * i - 1 ) q = math.modf( x / ( 2 * i - 1 ) ) end a[1] = math.fmod( q, 10 ) q = math.modf( q / 10 ) if q == 9 then nines = nines + 1 else if q == 10 then io.write( predigit + 1 ) for k = 1, nines do io.write(0) end predigit = 0 nines = 0 else io.write( predigit ) predigit = q if nines ~= 0 then for k = 1, nines do io.write( 9 ) end nines = 0 end end end end print( predigit )
03141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086 ...
M2000 Interpreter
We can ask for 200 digits, but we can remove Digits-- in While Digits {} to print endless number of digits (without good precision). Algorithm developed after reading [http://www.pi314.net/eng/goutte.php] and [https://www.cut-the-knot.org/Curriculum/Algorithms/SpigotForPi.shtml]
A Faster version handling console refresh time (and os shared time). M2000 run on an environment, which is loop event, and console is actual a form, a window. We can stop execution using Esc, Ctrl+C and Break keys, without stopping the interpreter (which is an application for Windows Os, written in Visual Basic 6,a s an ActiveX dll with a window manager on top of Vb forms).
Module Checkpi {
Module FindPi(Digits){
Digits++
n=Int(3.32*Digits)
PlusOne=Lambda N=0% -> {
=N
N++
}
PlusTwo=Lambda N=1% -> {
=N
N+=2
}
Dim A(n)<<PlusOne(), B(n)<<PlusTwo()
Dim Ten(n), CarrierOver(n), Sum(n),Remainder(n)=2
OutPutDigits=Digits
Predigits=Stack
CallBack=lambda fl=true, Chars=0 (x)->{
Print x;
Chars++
If fl then Print "." : Print " "; : fl=false : Chars=0 : exit
If Chars=50 then {
Print
Print " ";
Chars=0
Refresh
} else.if (Chars mod 5)=0 then {
Print " ";
Refresh
}
\\ explicitly refresh output layer, using Fast ! mode of speed
}
Print "Pi=";
While Digits {
NextDigit(&CallBack, &Digits)
}
print
Refresh
Sub NextDigit(&f, &D)
CarrierOver=0
For k=n-1 to 1 {
Ten(k)=Remainder(k)*10%
CarrierOver(k)=CarrierOver
Sum(k)=Ten(k)+CarrierOver(k)
q=Sum(k) div B(k)
Remainder(k)=Sum(k)-B(k)*q
CarrierOver=A(k)*q
}
Ten(0)=Remainder(0)*10%
CarrierOver(0)=CarrierOver
Sum(0)=Ten(0)+CarrierOver(0)
q=Sum(0) div 10%
Remainder(0)=Sum(0)-10%*q
if q<>9 and q<>10 then {
Stack Predigits {
While not empty {
Call f(Number)
if D>0 then D--
If D=0 then flush ' empty stack
}
Push q
}
} else.if q=9 Then {
Stack Predigits { Data q }
} else {
Stack Predigits {
While not empty {
Call f((Number+1) mod 10)
if D>0 then D--
If D=0 then flush ' empty stack
}
Push 0
}
}
End Sub
}
\\ reduce time to share with OS
\\ Need explicitly use of refresh output layer (M2000 console)
\\ Slow for a screen refresh per statement and give more time to OS
Rem Set Slow
\\ Fast is normal screen refresh, per Refresh time, and give standard time to OS
Rem Set Fast
\\ Fast ! use Refresh for screen refresh, and give less time o OS than standard
\\ Esc key work when Refresh executed (and OS get little time)
Set Fast !
FindPi 4
FindPi 28
Print Pi ' pi in M2000 is Decimal type with 29 digits (1 plus 28 after dot, is same as FindPi 28)
Refresh
FindPi 50
}
Flush ' empty stack of values
CheckPi
List ' no variables exist
Modules ? ' current module exist
Stack ' Stack of values ' has to be empty, we didn't use current stack for values.
=={{header|Mathematica}} / {{header|Wolfram Language}}== User can interrupt computation using "Alt+." or "Cmd+." on a Mac.
WriteString[$Output, "3."];
For[i = -1, True, i--,
WriteString[$Output, RealDigits[Pi, 10, 1, i][[1, 1]]]; Pause[.05]];
=={{header|MATLAB}} / {{header|Octave}}== Matlab and Octave use double precision numbers per default, and pi is a builtin constant value. Arbitrary precision is only implemented in some additional toolboxes (e.g. symbolic toolbox).
```txt
>> pi
ans = 3.1416
> printf('%.60f\n',pi)
3.141592653589793115997963468544185161590576171875000000000000>> format long
Unfortunately this is not the correct value!
3.14159265358979323846264338327950288419716939937510582
### ===========
??????????????????????????????????????
Calling for 60 digit output does not produce 60 digits of precision. Once the sixteen digit precision of double precision is reached, the subsequent digits are determined by the workings of the binary to decimal conversion. The long decimal string is the exact decimal value of the binary representation of pi, which binary value is itself not exact because pi cannot be represented in a finite number of digits, be they decimal, binary or any other integer base...
NetRexx
{{trans|Java}}
/* NetRexx */
options replace format comments java crossref symbols binary
import java.math.BigInteger
runSample(arg)
return
-- ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
method runSample(arg) private static
parse arg places .
if places = '' then places = -1
TWO = BigInteger.valueOf(2)
THREE = BigInteger.valueOf(3)
FOUR = BigInteger.valueOf(4)
SEVEN = BigInteger.valueOf(7)
q_ = BigInteger.ONE
r_ = BigInteger.ZERO
t_ = BigInteger.ONE
k_ = BigInteger.ONE
n_ = BigInteger.valueOf(3)
l_ = BigInteger.valueOf(3)
nn = BigInteger
nr = BigInteger
first = isTrue()
digitCt = 0
loop forever
if FOUR.multiply(q_).add(r_).subtract(t_).compareTo(n_.multiply(t_)) == -1 then do
digitCt = digitCt + 1
if places > 0 & digitCt - 1 > places then leave
say n_'\-'
if first then do
say '.\-'
first = isFalse()
end
nr = BigInteger.TEN.multiply(r_.subtract(n_.multiply(t_)))
n_ = BigInteger.TEN.multiply(THREE.multiply(q_).add(r_)).divide(t_).subtract(BigInteger.TEN.multiply(n_))
q_ = q_.multiply(BigInteger.TEN)
r_ = nr
end
else do
nr = TWO.multiply(q_).add(r_).multiply(l_)
nn = q_.multiply((SEVEN.multiply(k_))).add(TWO).add(r_.multiply(l_)).divide(t_.multiply(l_))
q_ = q_.multiply(k_)
t_ = t_.multiply(l_)
l_ = l_.add(TWO)
k_ = k_.add(BigInteger.ONE)
n_ = nn
r_ = nr
end
end
say
return
method isTrue() private static returns boolean
return (1 == 1)
method isFalse() private static returns boolean
return \isTrue()
{{out}}
3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679...
Nim
{{libheader|bigints}}
import strutils, bigints var tmp1, tmp2, tmp3, acc, k, dd = initBigInt(0) den, num, k2 = initBigInt(1) proc extractDigit(): int32 = if num > acc: return -1 tmp3 = num shl 1 tmp3 += num tmp3 += acc tmp2 = tmp3 mod den tmp1 = tmp3 div den tmp2 += num if tmp2 >= den: return -1 result = int32(tmp1.limbs[0]) proc eliminateDigit(d: int32) = acc -= den * d acc *= 10 num *= 10 proc nextTerm() = k += 1 k2 += 2 tmp1 = num shl 1 acc += tmp1 acc *= k2 den *= k2 num *= k var i = 0 while true: var d: int32 = -1 while d < 0: nextTerm() d = extractDigit() stdout.write chr(ord('0') + d) inc i if i == 40: echo "" i = 0 eliminateDigit d
Output:
3141592653589793238462643383279502884197
1693993751058209749445923078164062862089
9862803482534211706798214808651328230664
7093844609550582231725359408128481117450
...
OCaml
The Constructive Real library [http://www.lri.fr/~filliatr/creal.en.html Creal] contains an infinite-precision Pi, so we can just print out its digits.
open Creal;; let block = 100 in let segment n = let s = to_string pi (n*block) in String.sub s ((n-1)*block) block in let counter = ref 1 in while true do print_string (segment !counter); flush stdout; incr counter done
However that is cheating if you want to see an algorithm to generate Pi. Since the Spigot algorithm is already used in the [http://benchmarksgame.alioth.debian.org/u64q/program.php?test=pidigits&lang=ocaml&id=1 pidigits] program, this implements [http://mathworld.wolfram.com/Machin-LikeFormulas.html Machin's formula].
open Num (* series for: c*atan(1/k) *) class atan_sum c k = object val kk = k*/k val mutable n = 0 val mutable kpow = k val mutable pterm = c*/k val mutable psum = Int 0 val mutable sum = c*/k method next = n <- n+1; kpow <- kpow*/kk; let t = c*/kpow//(Int (2*n+1)) in pterm <- if n mod 2 = 0 then t else minus_num t; psum <- sum; sum <- sum +/ pterm method error = abs_num pterm method bounds = if pterm </ Int 0 then (sum, psum) else (psum, sum) end;; let inv i = (Int 1)//(Int i) in let t1 = new atan_sum (Int 16) (inv 5) in let t2 = new atan_sum (Int (-4)) (inv 239) in let base = Int 10 in let npr = ref 0 in let shift = ref (Int 1) in let d_acc = inv 10000 in let acc = ref d_acc in let shown = ref (Int 0) in while true do while t1#error >/ !acc do t1#next done; while t2#error >/ !acc do t2#next done; let (lo1, hi1), (lo2, hi2) = t1#bounds, t2#bounds in let digit x = int_of_num (floor_num ((x -/ !shown) */ !shift)) in let d, d' = digit (lo1+/lo2), digit (hi1+/hi2) in if d = d' then ( print_int d; if !npr = 0 then print_char '.'; flush stdout; shown := !shown +/ ((Int d) // !shift); incr npr; shift := !shift */ base; ) else (acc := !acc */ d_acc); done
Oforth
: calcPiDigits
| q r t k n l |
1 ->q 0 ->r 1 ->t 1 ->k 3 ->n 3 -> l
while( true ) [
4 q * r + t - n t * < ifTrue: [
n print
r n t * - 10 *
3 q * r + 10 * t / n 10 * - ->n ->r
q 10 * ->q
]
else: [
2 q * r + l *
7 k * q * 2 + r l * + t l * / ->n ->r
k q * ->q
t l * ->t
l 2 + ->l
k 1+ ->k
]
] ;
Ol
{{trans|Scheme}}
; 'numbers' is count of numbers or #false for eternal pleasure.
(define (pi numbers)
(let loop ((q 1) (r 0) (t 1) (k 1) (n 3) (l 3) (numbers numbers))
(unless (eq? numbers 0)
(if (< (- (+ (* 4 q) r) t) (* n t))
(begin
(display n)
(loop (* q 10)
(* 10 (- r (* n t)))
t
k
(- (div (* 10 (+ (* 3 q) r)) t) (* 10 n))
l
(if numbers (- numbers 1))))
(begin
(loop (* q k)
(* (+ (* 2 q) r) l)
(* t l)
(+ k 1)
(div (+ (* q (* 7 k)) 2 (* r l)) (* t l))
(+ l 2)
(if numbers (- numbers 1))))))))
(pi #false)
{{out}}
31415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132
82306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475
64823378678316527120190914564856692346034861045432664821339360726024914127372458700660631558817488152092096282925
40917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480
74462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539
21717629317675238467481846766940513200056812714526356082778577134275778960917363717872146844090122495343014654958
53710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328
16096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598
25349042875546873115956286388235378759375195778185778053217122680661300192787661119590921642019893809525720106548
58632788659361533818279682303019520353018529689957736225994138912497217752834791315155748572424541506959508295331
1686172785588907509838175463
PARI/GP
Uses the built-in Brent-Salamin arithmetic-geometric mean iteration.
pi()={
my(x=Pi,n=0,t);
print1("3.");
while(1,
if(n>=default(realprecision),
default(realprecision,default(realprecision)*2);
x=Pi
);
print1(floor(x*10^n++)%10)
)
};
Pascal
{{works with|Free_Pascal}}
With minor editing changes as published by Stanley Rabinowitz in [http://www.mathpropress.com/stan/bibliography/spigot.pdf].
Minor improvement of
Program Pi_Spigot; const n = 1000; len = 10*n div 3; var j, k, q, nines, predigit: integer; a: array[0..len] of longint; function OneLoop(i:integer):integer; var x: integer; begin {Only calculate as far as needed } {+16 for security digits ~5 decimals} i := i*10 div 3+16; IF i > len then i := len; result := 0; repeat {Work backwards} x := 10*a[i] + result*i; result := x div (2*i - 1); a[i] := x - result*(2*i - 1);//x mod (2*i - 1) dec(i); until i<= 0 ; end; begin for j := 1 to len do a[j] := 2; {Start with 2s} nines := 0; predigit := 0; {First predigit is a 0} for j := 1 to n do begin q := OneLoop(n-j); a[1] := q mod 10; q := q div 10; if q = 9 then nines := nines + 1 else if q = 10 then begin write(predigit+1); for k := 1 to nines do write(0); {zeros} predigit := 0; nines := 0 end else begin write(predigit); predigit := q; if nines <> 0 then begin for k := 1 to nines do write(9); nines := 0 end end end; writeln(predigit); end.
Output:
% ./Pi_Spigot
03141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420198
Perl
Perl being what it is, there are many ways to do this with many variations. With a fixed number of digits and the Math::BigInt::GMP library installed, the [[[[Arithmetic-geometric mean/Calculate Pi]] code will be much faster than any of these methods other than some of the modules. If Math::GMP is installed, then replacing "use bigint" with "use Math::GMP qw/:constant/" in either the Perl6 spigot or Machin methods below will be pretty fast. They are not too bad if the Math::BigInt::GMP library is installed. With the default Math::BigInt backend, the AGM code isn't very fast and the Perl6 spigot and Machin methods are very slow.
= Simple Spigot =
This takes a numer-of-digits argument, but we can make it large (albeit using memory and some startup time). Unlike the other two, this uses no modules and does not require bigints so is worth showing.
sub pistream { my $digits = shift; my(@out, @a); my($b, $c, $d, $e, $f, $g, $i, $d4, $d3, $d2, $d1); my $outi = 0; $digits++; $b = $d = $e = $g = $i = 0; $f = 10000; $c = 14 * (int($digits/4)+2); @a = (20000000) x $c; print "3."; while (($b = $c -= 14) > 0 && $i < $digits) { $d = $e = $d % $f; while (--$b > 0) { $d = $d * $b + $a[$b]; $g = ($b << 1) - 1; $a[$b] = ($d % $g) * $f; $d = int($d / $g); } $d4 = $e + int($d/$f); if ($d4 > 9999) { $d4 -= 10000; $out[$i-1]++; for ($b = $i-1; $out[$b] == 1; $b--) { $out[$b] = 0; $out[$b-1]++; } } $d3 = int($d4/10); $d2 = int($d3/10); $d1 = int($d2/10); $out[$i++] = $d1; $out[$i++] = $d2-$d1*10; $out[$i++] = $d3-$d2*10; $out[$i++] = $d4-$d3*10; print join "", @out[$i-15 .. $i-15+3] if $i >= 16; } # We've closed the spigot. Print the remainder without rounding. print join "", @out[$i-15+4 .. $digits-2], "\n"; }
= Perl6 spigot =
As mentioned earlier, replacing "use bigint" with "use Math::GMP qw/:constant/" will result in many orders of magnitude faster performance.
{{trans|Perl 6}}
"GMP";
sub stream {
my ($next, $safe, $prod, $cons, $z, $x) = @_;
$x = $x->();
sub {
while (1) {
my $y = $next->($z);
if ($safe->($z, $y)) {
$z = $prod->($z, $y);
return $y;
} else {
$z = $cons->($z, $x->());
}
}
}
}
sub extr {
use integer;
my ($q, $r, $s, $t) = @{shift()};
my $x = shift;
($q * $x + $r) / ($s * $x + $t);
}
sub comp {
my ($q, $r, $s, $t) = @{shift()};
my ($u, $v, $w, $x) = @{shift()};
[$q * $u + $r * $w,
$q * $v + $r * $x,
$s * $u + $t * $w,
$s * $v + $t * $x];
}
my $pi_stream = stream
sub { extr shift, 3 },
sub { my ($z, $n) = @_; $n == extr $z, 4 },
sub { my ($z, $n) = @_; comp([10, -10*$n, 0, 1], $z) },
\&comp,
[1, 0, 0, 1],
sub { my $n = 0; sub { $n++; [$n, 4 * $n + 2, 0, 2 * $n + 1] } },
;
$|++;
print $pi_stream->(), '.';
print $pi_stream->() while 1;
==== Machin's Formula ====
Here is an original Perl 5 code, using Machin's formula. Not the fastest program in the world. As with the previous code, using either Math::GMP or Math::BigInt::GMP instead of the default bigint Calc backend will make it run thousands of times faster.
"GMP";
# Pi/4 = 4 arctan 1/5 - arctan 1/239
# expanding it with Taylor series with what's probably the dumbest method
my ($ds, $ns) = (1, 0);
my ($n5, $d5) = (16 * (25 * 3 - 1), 3 * 5**3);
my ($n2, $d2) = (4 * (239 * 239 * 3 - 1), 3 * 239**3);
sub next_term {
my ($coef, $p) = @_[1, 2];
$_[0] /= ($p - 4) * ($p - 2);
$_[0] *= $p * ($p + 2) * $coef**4;
}
my $p2 = 5;
my $pow = 1;
$| = 1;
for (my $x = 5; ; $x += 4) {
($ns, $ds) = ($ns * $d5 + $n5 * $pow * $ds, $ds * $d5);
next_term($d5, 5, $x);
$n5 = 16 * (5 * 5 * ($x + 2) - $x);
while ($d5 > $d2) {
($ns, $ds) = ($ns * $d2 - $n2 * $pow * $ds, $ds * $d2);
$n2 = 4 * (239 * 239 * ($p2 + 2) - $p2);
next_term($d2, 239, $p2);
$p2 += 4;
}
my $ppow = 1;
while ($pow * $n5 * 5**4 < $d5 && $pow * $n2 * $n2 * 239**4 < $d2) {
$pow *= 10;
$ppow *= 10;
}
if ($ppow > 1) {
$ns *= $ppow;
#FIX? my $out = $ns->bdiv($ds); # bugged?
my $out = $ns / $ds;
$ns %= $ds;
$out = ("0" x (length($ppow) - length($out) - 1)) . $out;
print $out;
}
if ( $p2 % 20 == 1) {
my $g = Math::BigInt::bgcd($ds, $ns);
$ds /= $g;
$ns /= $g;
}
}
= Modules =
While no current CPAN module does continuous printing, there are (usually fast) ways to get digits of Pi. Examples include: {{libheader|ntheory}}
use ntheory qw/Pi/; say Pi(10000); use Math::Pari qw/setprecision Pi/; setprecision(10000); say Pi; use Math::MPFR; my $pi = Math::MPFR->new(); Math::MPFR::Rmpfr_set_prec($pi, int(10000 * 3.322)+40); Math::MPFR::Rmpfr_const_pi($pi, 0); say Math::MPFR::Rmpfr_get_str($pi, 10, 10000, 0); use Math::BigFloat try=>"GMP"; # Slow without Math::BigInt::GMP installed say Math::BigFloat::bpi(10000); # For over ~2k digits, slower than AGM use Math::Big qw/pi/; # Very slow say pi(10000);
Perl 6
{{Works with|rakudo|2018.10}}
# based on http://www.mathpropress.com/stan/bibliography/spigot.pdf
sub stream(&next, &safe, &prod, &cons, $z is copy, @x) {
gather loop {
$z = safe($z, my $y = next($z)) ??
prod($z, take $y) !!
cons($z, @x[$++])
}
}
sub extr([$q, $r, $s, $t], $x) {
($q * $x + $r) div ($s * $x + $t)
}
sub comp([$q,$r,$s,$t], [$u,$v,$w,$x]) {
[$q * $u + $r * $w,
$q * $v + $r * $x,
$s * $u + $t * $w,
$s * $v + $t * $x]
}
my $pi :=
stream -> $z { extr($z, 3) },
-> $z, $n { $n == extr($z, 4) },
-> $z, $n { comp([10, -10*$n, 0, 1], $z) },
&comp,
<1 0 0 1>,
(1..*).map: { [$_, 4 * $_ + 2, 0, 2 * $_ + 1] }
for ^Inf -> $i {
print $pi[$i];
once print '.'
}
Phix
I already had this golf entry to hand. Prints 2400 places, change the 8400 (derived from 2400*14/4) as needed, but I've not tested > that.
integer a=10000,b,c=8400,d,e=0,g sequence f=repeat(floor(a/5),c+1) while c>0 do g=2*c d=0
b=c while b>0 do d+=f[b]*a g-=1 f[b]=remainder(d, g) d=floor(d/g) g-=1 b-=1 if b!=0 then
d*=b end if end while printf(1,"%04d",e+floor(d/a)) c-=14 e = remainder(d,a) end while
Someone was benchmarking the above against Lua, so I translated the Lua entry, and upped it to 2400 places, for a fairer test.
integer n = 2400,
len = floor(10*n/3)
sequence a = repeat(2,len)
integer nines = 0,
predigit = 0
string res = ""
for j=1 to n do
integer q = 0
for i=len to 1 by -1 do
integer x = 10*a[i]+q*i,
d = 2*i-1
a[i] = remainder(x,d)
q = floor(x/d)
end for
a[1] = remainder(q,10)
q = floor(q/10)
if q==9 then
nines = nines+1
else
integer nine = '9'
if q==10 then
predigit += 1
q = 0
nine = '0'
end if
res &= predigit+'0'&repeat(nine,nines)
predigit = q
nines = 0
end if
end for
res &= predigit+'0'
puts(1,res)
PicoLisp
The following script uses the spigot algorithm published by Jeremy Gibbons. Hit Ctrl-C to stop it.
#!/usr/bin/picolisp /usr/lib/picolisp/lib.l
(de piDigit ()
(job '((Q . 1) (R . 0) (S . 1) (K . 1) (N . 3) (L . 3))
(while (>= (- (+ R (* 4 Q)) S) (* N S))
(mapc set '(Q R S K N L)
(list
(* Q K)
(* L (+ R (* 2 Q)))
(* S L)
(inc K)
(/ (+ (* Q (+ 2 (* 7 K))) (* R L)) (* S L))
(+ 2 L) ) ) )
(prog1 N
(let M (- (/ (* 10 (+ R (* 3 Q))) S) (* 10 N))
(setq Q (* 10 Q) R (* 10 (- R (* N S))) N M) ) ) ) )
(prin (piDigit) ".")
(loop
(prin (piDigit))
(flush) )
Output:
3.14159265358979323846264338327950288419716939937510582097494459 ...
PL/I
/* Uses the algorithm of S. Rabinowicz and S. Wagon, "A Spigot Algorithm */
/* for the Digits of Pi". */
(subrg, fofl, size):
Pi_Spigot: procedure options (main); /* 21 January 2012. */
declare (n, len) fixed binary;
n = 1000;
len = 10*n / 3;
begin;
declare ( i, j, k, q, nines, predigit ) fixed binary;
declare x fixed binary (31);
declare a(len) fixed binary (31);
a = 2; /* Start with 2s */
nines, predigit = 0; /* First predigit is a 0 */
do j = 1 to n;
q = 0;
do i = len to 1 by -1; /* Work backwards */
x = 10*a(i) + q*i;
a(i) = mod (x, (2*i-1));
q = x / (2*i-1);
end;
a(1) = mod(q, 10); q = q / 10;
if q = 9 then nines = nines + 1;
else if q = 10 then
do;
put edit(predigit+1) (f(1));
do k = 1 to nines;
put edit ('0')(a(1)); /* zeros */
end;
predigit, nines = 0;
end;
else
do;
put edit(predigit) (f(1)); predigit = q;
do k = 1 to nines; put edit ('9')(a(1)); end;
nines = 0;
end;
end;
put edit(predigit) (f(1));
end; /* of begin block */
end Pi_Spigot;
output:
03141592653589793238462643383279502884197169399375105820974944592307816406286208
99862803482534211706798214808651328230664709384460955058223172535940812848111745
02841027019385211055596446229489549303819644288109756659334461284756482337867831
65271201909145648566923460348610454326648213393607260249141273724587006606315588
17488152092096282925409171536436789259036001133053054882046652138414695194151160
94330572703657595919530921861173819326117931051185480744623799627495673518857527
24891227938183011949129833673362440656643086021394946395224737190702179860943702
77053921717629317675238467481846766940513200056812714526356082778577134275778960
91736371787214684409012249534301465495853710507922796892589235420199561121290219
60864034418159813629774771309960518707211349999998372978049951059731732816096318
59502445945534690830264252230825334468503526193118817101000313783875288658753320
83814206171776691473035982534904287554687311595628638823537875937519577818577805
32171226806613001927876611195909216420198
Powershell
{{trans|D}} With some tweaking. Prints 100 digits a time. Total possible output limited by available memory.
Function Get-Pi ( $Digits ) { $Big = [bigint[]](0..10) $ndigits = 0 $Output = "" $q = $t = $k = $Big[1] $r = $Big[0] $l = $n = $Big[3] # Calculate first digit $nr = ( $Big[2] * $q + $r ) * $l $nn = ( $q * ( $Big[7] * $k + $Big[2] ) + $r * $l ) / ( $t * $l ) $q *= $k $t *= $l $l += $Big[2] $k = $k + $Big[1] $n = $nn $r = $nr $Output += [string]$n + '.' $ndigits++ $nr = $Big[10] * ( $r - $n * $t ) $n = ( ( $Big[10] * ( 3 * $q + $r ) ) / $t ) - 10 * $n $q *= $Big[10] $r = $nr While ( $ndigits -lt $Digits ) { While ( $ndigits % 100 -ne 0 -or -not $Output ) { If ( $Big[4] * $q + $r - $t -lt $n * $t ) { $Output += [string]$n $ndigits++ $nr = $Big[10] * ( $r - $n * $t ) $n = ( ( $Big[10] * ( 3 * $q + $r ) ) / $t ) - 10 * $n $q *= $Big[10] $r = $nr } Else { $nr = ( $Big[2] * $q + $r ) * $l $nn = ( $q * ( $Big[7] * $k + $Big[2] ) + $r * $l ) / ( $t * $l ) $q *= $k $t *= $l $l += $Big[2] $k = $k + $Big[1] $n = $nn $r = $nr } } $Output $Output = "" } }
Alternate version using .Net classes
[math]::pi
Outputs:
## PureBasic
Calculate Pi, limited to ~24 M-digits for memory and speed reasons.
```PureBasic
#SCALE = 10000
#ARRINT= 2000
Procedure Pi(Digits)
Protected First=#True, Text$
Protected Carry, i, j, sum
Dim Arr(Digits)
For i=0 To Digits
Arr(i)=#ARRINT
Next
i=Digits
While i>0
sum=0
j=i
While j>0
sum*j+#SCALE*arr(j)
Arr(j)=sum%(j*2-1)
sum/(j*2-1)
j-1
Wend
Text$ = RSet(Str(Carry+sum/#SCALE),4,"0")
If First
Text$ = ReplaceString(Text$,"3","3.")
First = #False
EndIf
Print(Text$)
Carry=sum%#SCALE
i-14
Wend
EndProcedure
If OpenConsole()
SetConsoleCtrlHandler_(?Ctrl,#True)
Pi(24*1024*1024)
EndIf
End
Ctrl:
PrintN(#CRLF$+"Ctrl-C was pressed")
End
Python
def calcPi(): q, r, t, k, n, l = 1, 0, 1, 1, 3, 3 while True: if 4*q+r-t < n*t: yield n nr = 10*(r-n*t) n = ((10*(3*q+r))//t)-10*n q *= 10 r = nr else: nr = (2*q+r)*l nn = (q*(7*k)+2+(r*l))//(t*l) q *= k t *= l l += 2 k += 1 n = nn r = nr import sys pi_digits = calcPi() i = 0 for d in pi_digits: sys.stdout.write(str(d)) i += 1 if i == 40: print(""); i = 0
output
3141592653589793238462643383279502884197
1693993751058209749445923078164062862089
9862803482534211706798214808651328230664
7093844609550582231725359408128481117450
2841027019385211055596446229489549303819
6442881097566593344612847564823378678316
5271201909145648566923460348610454326648
2133936072602491412737245870066063155881
7488152092096282925409171536436789259036
0011330530548820466521384146951941511609
4330572703657595919530921861173819326117
...
R
suppressMessages(library(gmp))
ONE <- as.bigz("1")
TWO <- as.bigz("2")
THREE <- as.bigz("3")
FOUR <- as.bigz("4")
SEVEN <- as.bigz("7")
TEN <- as.bigz("10")
q <- as.bigz("1")
r <- as.bigz("0")
t <- as.bigz("1")
k <- as.bigz("1")
n <- as.bigz("3")
l <- as.bigz("3")
char_printed <- 0
how_many <- 1000
first <- TRUE
while (how_many > 0) {
if ((FOUR * q + r - t) < (n * t)) {
if (char_printed == 80) {
cat("\n")
char_printed <- 0
}
how_many <- how_many - 1
char_printed <- char_printed + 1
cat(as.integer(n))
if (first) {
cat(".")
first <- FALSE
char_printed <- char_printed + 1
}
nr <- as.bigz(TEN * (r - n * t))
n <- as.bigz(((TEN * (THREE * q + r)) %/% t) - (TEN * n))
q <- as.bigz(q * TEN)
r <- as.bigz(nr)
} else {
nr <- as.bigz((TWO * q + r) * l)
nn <- as.bigz((q * (SEVEN * k + TWO) + r * l) %/% (t * l))
q <- as.bigz(q * k)
t <- as.bigz(t * l)
l <- as.bigz(l + TWO)
k <- as.bigz(k + ONE)
n <- as.bigz(nn)
r <- as.bigz(nr)
}
}
cat("\n")
'''Output:'''
3.141592653589793238462643383279502884197169399375105820974944592307816406286208
99862803482534211706798214808651328230664709384460955058223172535940812848111745
02841027019385211055596446229489549303819644288109756659334461284756482337867831
65271201909145648566923460348610454326648213393607260249141273724587006606315588
17488152092096282925409171536436789259036001133053054882046652138414695194151160
94330572703657595919530921861173819326117931051185480744623799627495673518857527
24891227938183011949129833673362440656643086021394946395224737190702179860943702
77053921717629317675238467481846766940513200056812714526356082778577134275778960
91736371787214684409012249534301465495853710507922796892589235420199561121290219
60864034418159813629774771309960518707211349999998372978049951059731732816096318
59502445945534690830264252230825334468503526193118817101000313783875288658753320
83814206171776691473035982534904287554687311595628638823537875937519577818577805
32171226806613001927876611195909216420198
Racket
Utilizing Jeremy Gibbons spigot algorithm and racket generator:
#lang racket
(require racket/generator)
(define pidig
(generator ()
(let loop ([q 1] [r 0] [t 1] [k 1] [n 3] [l 3])
(if (< (- (+ r (* 4 q)) t) (* n t))
(begin (yield n)
(loop (* q 10) (* 10 (- r (* n t))) t k
(- (quotient (* 10 (+ (* 3 q) r)) t) (* 10 n))
l))
(loop (* q k) (* (+ (* 2 q) r) l) (* t l) (+ 1 k)
(quotient (+ (* (+ 2 (* 7 k)) q) (* r l)) (* t l))
(+ l 2))))))
(for ([i (in-naturals)])
(display (pidig))
(when (zero? i) (display "." ))
(when (zero? (modulo i 80)) (newline)))
Output:
## REXX
### version 1
This REXX program calculates decimal digits of <big><big><math>\pi</math></big></big> using John Machin's formula.
It should be noted that the program's mechanism spits out the next (new) decimal digit(s) of <big><big><math>\pi</math></big></big>.
The REXX program uses the following formula to calculate <big><big><math>\pi</math></big></big>:
```txt
┌─ ─┐ ┌─ ─┐
π │ 1 │ │ 1 │ John
─── = 4 ∙ arctan│ ─── │ - arctan│ ───── │ Machin's
4 │ 5 │ │ 239 │ formula
└─ ─┘ └─ ─┘
which expands into:
┌─ ─┐
│ 1 1 1 1 1 1 │
4 ∙ │ ─── - ────── + ────── - ────── + ────── - ──────── + ... │
│ 1 3 5 7 9 11 │
│ 1∙5 3∙5 5∙5 7∙5 9∙5 11∙5 │
└─ ─┘
┌─ ─┐
│ 1 1 1 1 1 1 │
- │ ─── - ────── + ────── - ────── + ────── - ──────── + ... │
│ 1 3 5 7 9 11 │
│ 1∙239 3∙239 5∙239 7∙239 9∙239 11∙239 │
└─ ─┘
/*REXX program spits out decimal digits of pi (one digit at a time) until Ctrl-Break.*/
parse arg digs oFID . /*obtain optional argument from the CL.*/
if digs=='' | digs=="," then digs=1e6 /*Not specified? Then use the default.*/
if oFID=='' | oFID=="," then oFID='PI_SPIT.OUT' /* " " " " " " */
numeric digits digs /*with bigger digs, spitting is slower.*/
call time 'Reset' /*reset the wall─clock (elapsed) timer.*/
signal on halt /*───► HALT when Ctrl─Break is pressed.*/
pi=0; v=5; vv=v*v; g=239; gg=g*g; spit=0 /*assign some values to some variables.*/
s=16 /*calculate π with increasing accuracy */
r=4; do n=1 by 2 until old=pi; old=pi /*just calculate pi with odd integers*/
pi=pi + s/(n*v) - r/(n*g) /* ··· using John Machin's formula.*/
if pi==old then leave /*have we exceeded the DIGITS accuracy?*/
s=-s; r=-r; v=v*vv; g=g*gg /*compute some variables for shortcuts.*/
do j=spit+1 to compare(pi,old) /*spit out some (new) digits of π (pi)*/
parse var pi =(j) spit +1 /*equivalent to: spit=substr(pi,j,1) */
call charout ,spit /*display one (new) decimal digit of π.*/
call charout oFID,spit /*··· and also write π digit to a file.*/
end /*j*/ /* [↑] 0, 1, or 2 decimal dig are spit*/
spit=j-1 /*adjust for DO loop index increment.*/
end /*n*/
say /*stick a fork in it, we're all done. */
exit: say; say n%2+1 'iterations took' format(time("Elapsed"),,2) 'seconds.'; exit
halt: say; say 'PI_SPIT halted via use of Ctrl-Break.'; signal exit /*show iterations.*/
{{out|output|text= [until the Ctrl-Break key (or equivalent) was pressed]:}}
(Shown at four-fifth size.)
3.1415926535897932384626433832794028841971794993741058209749445923078164062861089986280348253411170679821480865132823066470938446095505822317253594081
284811174502840027019385211055596446229489549203819644298109756659334461284756482338867831652711019091456485669234603486104543266482133936072602491412
737245970066063156881748815209209628292540917153643679926903500113205305498204665213841469519415116094330572703657595919530921861173819326118931051185
480744623899627595673518857527249912279381830119491398336733624406566420860213949463952247371907021898609437027705392171762931767523846748184676794051
310005681271452635608277857713427577996091736371787214684308012259534301465595853700508922797892589235420299560121280219608630344181598136297747713199
605187072113499999983729780499410597317328160962185940244594553469083026425123082533446850352619311881700000031378387528865875331083814206171776691473
035982534904287554687311695628638823538876937529577818587805321712268066120019278766111959092164201989380952572000654858632788659361533818289682303019
510353018529699958736225994138912597217752834791315155748572424541506969508395330168617278558890750983817546374649393192550604009277016711390098488240
128583616035637076501047001819429555961999467688374594482553897747268471040475346462080466842590794912933136770299891521047521620569660240570381501935
112533824200355876402474964732639142992726042699227967823547816350093417216412199245863150202861829745557067598385054945885869279956909272007975092029
553211653449872027559502364806654991298818347977535663698074265425278625518184175746728909877728938000816470600161452491921732172147723401414419735685
481613611573525521334757418494684385233239073941433345477624168625189835695855620992192221842725502542568976717905946016534668059886272327917860857843
838279679766814540009538837863609506800642251252051173939848960841284886269456042419652850222006611863067442786220491949440471237138869509563643719172
874677646575739624139908658326469958133904780275901994657640789512694683983525956098258226205224894077267194782684825014769909026401363944374553050682
034962524517493996514314298091906592509372216964615157198583874106978859597729754989201617539284681382686838789427741569918559252469539594310599725246
808459872736446958486538367362226260991246080512438843904412441365597627807977156914359977001296160894416958685558484063534210722258285886481584560284
060168427494522674676789952521385225599546667278239864565951163549862205774564980355936345681743241125150760694794410965960940252288897108931456691368
672287499405600014032086179286809208747609178249385890097149096759852613655497819931297848116829999487226588048575630142704775551323796414515237462343
645428584448952658678110511413547357495231134271661021359695362214429524849371871101457654036902799344037420073105785390622983974478084785896833114457
138697519435064302184521900484810053706146806749192781912979399510614196634287544406437451237181921899983900159195618146751426912497489409071864942319
615679452080951465502252316038819301410937621378569566399387870830390797910773467221825625996614014214030680384477345492026054146659251014974428507325
186650021324340881907004863317346496514539057962685600055081066587979981635747363840525714591029970641301009712062804390497695156771577004103378699360
072305597631763694218721251471205329281918261861258672215892984148488291644606095752706957220917567116722900981690915280173506712748583222871835209354
965725121083579151379881091444200067500334671003141267011379908658516498315019701651511685171437657618351556508849099998599823873455283216355076489185
358932261854896321329330898570641046752590709154814165498594616371802709819943199244889575712828906923233260972997120844335732654993823912932597463667
205835041428138830310382490375898524374417039132765618093773444020707469211101913020330380297620100100449293215160842444869637679838952286847831235526
582131449576857262433441892039686426243400773227978028073199154410010446823252716101052652272111660396665572092547110557853763466820653009896526918620
564769312560586356620185580007293606698764860179104533488503461136576867532494416680496265897977185560845529654126654085306143444318686769741456614068
006002388776591343017127494704205622205399945613140711260003078547332699390814546646458807972708266830634328587857983052358089330657573067954571637752
542011149557615813002501262285941302164715509792592319907965473761255176567513575178296664547791744011299614890304639947132962107340437519957369614599
019389713111890429782856475032031986915140287080859904801094111472213179476477726224142548545403321571853061422881375850430633217518298986622371721591
50771669254748739986654959450114653062843366393790039769265672146385306736096571209180763832716641627488880088692560280228472104021711186082041900042
PI_SPIT halted via use of Ctrl-Break.
3431 iterations took 3501.78 seconds.
```
### version 2
This REXX version is a translation of '''Icon''' with some speed optimizations.
This algorithm is limited to the number of decimal digits as specified with the '''numeric digits ddd''' (line or statement six).
```rexx
/*REXX program spits out decimal digits of pi (one digit at a time) until Ctrl-Break.*/
signal on halt /*───► HALT when Ctrl─Break is pressed.*/
parse arg digs oFID . /*obtain optional argument from the CL.*/
if digs=='' | digs=="," then digs= 300 /*Not specified? Then use the default.*/
if oFID=='' | oFID=="," then oFID='PI_SPIT2.OUT' /* " " " " " " */
numeric digits digs /*with bigger digs, spitting is slower.*/
q=1; r=0; t=1; k=1; n=3; L=3; z=0 /*define some REXX variables. */
dot=1 /*DOT≡a flag when a dot in pi is shown.*/
do until z==digs; qq= q+q /* qq is a fast version of: q*2 */
tn= t*n /* t*n is used twice (below). */
if qq+qq+r-t < tn then do; z= z+1 /* qq+qq is faster than qq*2 */
call charout , n
call charout oFID, n
if dot then do; dot=0; call charout , .
call charout oFID, .
end
nr= (r - tn) * 10
n = ((( (qq+q+r) * 10) / t) - n*10) %1
q = q*10
end
else do; nr= (qq+r) * L
tL= t*L
n = (q * (k*7 + 2) + r*L) / tL %1
q = q*k
t = tL
L = L+2
k = k+1
end /* %1≡fast way doing TRUNC of a number.*/
r=nr
end /*forever*/
exit /*stick a fork in it, we're all done. */
halt: say; say 'PI_SPIT2 halted via use of Ctrl-Break.'; exit
```
## Ruby
{{trans|Icon}}
```ruby
pi_digits = Enumerator.new do |y|
q, r, t, k, n, l = 1, 0, 1, 1, 3, 3
loop do
if 4*q+r-t < n*t
y << n
nr = 10*(r-n*t)
n = ((10*(3*q+r)) / t) - 10*n
q *= 10
r = nr
else
nr = (2*q+r) * l
nn = (q*(7*k+2)+r*l) / (t*l)
q *= k
t *= l
l += 2
k += 1
n = nn
r = nr
end
end
end
print pi_digits.next, "."
loop { print pi_digits.next }
```
## Rust
{{trans|Kotlin}}
```Rust
use num_bigint::BigInt;
fn main() {
calc_pi();
}
fn calc_pi() {
let mut q = BigInt::from(1);
let mut r = BigInt::from(0);
let mut t = BigInt::from(1);
let mut k = BigInt::from(1);
let mut n = BigInt::from(3);
let mut l = BigInt::from(3);
let mut first = true;
loop {
if &q * 4 + &r - &t < &n * &t {
print!("{}", n);
if first {
print!(".");
first = false;
}
let nr = (&r - &n * &t) * 10;
n = (&q * 3 + &r) * 10 / &t - &n * 10;
q *= 10;
r = nr;
} else {
let nr = (&q * 2 + &r) * &l;
let nn = (&q * &k * 7 + 2 + &r * &l) / (&t * &l);
q *= &k;
t *= &l;
l += 2;
k += 1;
n = nn;
r = nr;
}
}
}
```
## Scala
```scala
object Pi {
class PiIterator extends Iterable[BigInt]{
var r:BigInt=0
var q, t, k:BigInt=1
var n, l:BigInt=3
var nr, nn:BigInt=0
def iterator: Iterator[BigInt]=new Iterator[BigInt]{
def hasNext=true
def next():BigInt={
while((4*q+r-t) >= (n*t)) {
nr = (2*q+r)*l
nn = (q*(7*k)+2+(r*l))/(t*l)
q = q * k
t = t * l
l = l + 2
k = k + 1
n = nn
r = nr
}
val ret=n
nr = 10*(r-n*t)
n = ((10*(3*q+r))/t)-(10*n)
q = q * 10
r = nr
ret
}
}
}
def main(args: Array[String]): Unit = {
val it=new PiIterator
println((it head) + "." + (it take 300 mkString))
}
}
```
Output:
```txt
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998
62803482534211706798214808651328230664709384460955058223172535940812848111745028410
27019385211055596446229489549303819644288109756659334461284756482337867831652712019
09145648566923460348610454326648213393607260249141273
```
## Scheme
```scala
(import (rnrs))
(define (calc-pi yield)
(let loop ((q 1) (r 0) (t 1) (k 1) (n 3) (l 3))
(if (< (- (+ (* 4 q) r) t) (* n t))
(begin
(yield n)
(loop (* q 10)
(* 10 (- r (* n t)))
t
k
(- (div (* 10 (+ (* 3 q) r)) t) (* 10 n))
l))
(begin
(loop (* q k)
(* (+ (* 2 q) r) l)
(* t l)
(+ k 1)
(div (+ (* q (* 7 k)) 2 (* r l)) (* t l))
(+ l 2))))))
(let ((i 0))
(calc-pi
(lambda (d)
(display d)
(set! i (+ i 1))
(if (= 40 i)
(begin
(newline)
(set! i 0))))))
```
Output:
```txt
3141592653589793238462643383279502884197
1693993751058209749445923078164062862089
9862803482534211706798214808651328230664
7093844609550582231725359408128481117450
2841027019385211055596446229489549303819
6442881097566593344612847564823378678316
5271201909145648566923460348610454326648
2133936072602491412737245870066063155881
7488152092096282925409171536436789259036
0011330530548820466521384146951941511609
4330572703657595919530921861173819326117
9310511854807446237996274956735188575272
4891227938183011949129833673362440656643
0860213949463952247371907021798609437027
7053921717629317675238467481846766940513
2000568127145263560827785771342757789609
...
```
## Seed7
```seed7
$ include "seed7_05.s7i";
include "bigint.s7i";
const proc: main is func
local
var bigInteger: q is 1_;
var bigInteger: r is 0_;
var bigInteger: t is 1_;
var bigInteger: k is 1_;
var bigInteger: n is 3_;
var bigInteger: l is 3_;
var bigInteger: nn is 0_;
var bigInteger: nr is 0_;
var boolean: first is TRUE;
begin
while TRUE do
if 4_ * q + r - t < n * t then
write(n);
if first then
write(".");
first := FALSE;
end if;
nr := 10_ * (r - n * t);
n := 10_ * (3_ * q + r) div t - 10_ * n;
q *:= 10_;
r := nr;
flush(OUT);
else
nr := (2_ * q + r) * l;
nn := (q * (7_ * k + 2_) + r * l) div (t * l);
q *:= k;
t *:= l;
l +:= 2_;
incr(k);
n := nn;
r := nr;
end if;
end while;
end func;
```
Original source: [http://seed7.sourceforge.net/algorith/math.htm#pi_spigot_algorithm]
## Sidef
```ruby
func pi(callback) {
var (q, r, t, k, n, l) = (1, 0, 1, 1, 3, 3)
loop {
if ((4*q + r - t) < n*t) {
callback(n)
static _dot = callback('.')
var nr = 10*(r - n*t)
n = ((10*(3*q + r)) // t - 10*n)
q *= 10
r = nr
}
else {
var nr = ((2*q + r) * l)
var nn = ((q*(7*k + 2) + r*l) // (t*l))
q *= k
t *= l
l += 2
k += 1
n = nn
r = nr
}
}
}
STDOUT.autoflush(true)
pi(func(digit){ print digit })
```
## Simula
```simula
CLASS BIGNUM;
BEGIN
BOOLEAN PROCEDURE TISZERO(T); TEXT T;
TISZERO := T = "0";
TEXT PROCEDURE TSHL(T); TEXT T;
TSHL :- IF TISZERO(T) THEN T ELSE T & "0";
TEXT PROCEDURE TSHR(T); TEXT T;
TSHR :- IF T.LENGTH = 1 THEN "0" ELSE T.SUB(1, T.LENGTH - 1);
INTEGER PROCEDURE TSIGN(T); TEXT T;
TSIGN := IF TISZERO(T) THEN 0
ELSE IF T.SUB(1, 1) = "-" THEN -1
ELSE 1;
TEXT PROCEDURE TABS(T); TEXT T;
TABS :- IF TSIGN(T) < 0 THEN T.SUB(2, T.LENGTH - 1) ELSE T;
TEXT PROCEDURE TNEGATE(T); TEXT T;
TNEGATE :- IF TSIGN(T) <= 0 THEN TABS(T) ELSE ("-" & T);
TEXT PROCEDURE TREVERSE(T); TEXT T;
BEGIN
INTEGER I, J;
I := 1; J := T.LENGTH;
WHILE I < J DO
BEGIN CHARACTER C1, C2;
T.SETPOS(I); C1 := T.GETCHAR;
T.SETPOS(J); C2 := T.GETCHAR;
T.SETPOS(I); T.PUTCHAR(C2);
T.SETPOS(J); T.PUTCHAR(C1);
I := I + 1;
J := J - 1;
END;
TREVERSE :- T;
END TREVERSE;
INTEGER PROCEDURE TCMPUNSIGNED(A, B); TEXT A, B;
BEGIN
INTEGER ALEN, BLEN, RESULT;
ALEN := A.LENGTH; BLEN := B.LENGTH;
IF ALEN < BLEN THEN
RESULT := -1
ELSE IF ALEN > BLEN THEN
RESULT := 1
ELSE BEGIN
INTEGER CMP, I; BOOLEAN DONE;
A.SETPOS(1);
B.SETPOS(1);
I := 1;
WHILE I <= ALEN AND NOT DONE DO
BEGIN
I := I + 1;
CMP := RANK(A.GETCHAR) - RANK(B.GETCHAR);
IF NOT (CMP = 0) THEN
DONE := TRUE;
END;
RESULT := CMP;
END;
TCMPUNSIGNED := RESULT;
END TCMPUNSIGNED;
INTEGER PROCEDURE TCMP(A, B); TEXT A, B;
BEGIN
BOOLEAN ANEG, BNEG;
ANEG := TSIGN(A) < 0; BNEG := TSIGN(B) < 0;
IF ANEG AND BNEG THEN
TCMP := -TCMPUNSIGNED(TABS(A), TABS(B))
ELSE IF NOT ANEG AND BNEG THEN
TCMP := 1
ELSE IF ANEG AND NOT BNEG THEN
TCMP := -1
ELSE
TCMP := TCMPUNSIGNED(A, B);
END TCMP;
TEXT PROCEDURE TADDUNSIGNED(A, B); TEXT A, B;
BEGIN
INTEGER CARRY, I, J;
TEXT BF;
I := A.LENGTH;
J := B.LENGTH;
BF :- BLANKS(MAX(I, J) + 1);
WHILE I >= 1 OR J >= 1 DO BEGIN
INTEGER X, Y, Z;
IF I >= 1 THEN BEGIN
A.SETPOS(I); I := I - 1; X := RANK(A.GETCHAR) - RANK('0');
END;
IF J >= 1 THEN BEGIN
B.SETPOS(J); J := J - 1; Y := RANK(B.GETCHAR) - RANK('0');
END;
Z := X + Y + CARRY;
IF Z < 10 THEN
BEGIN BF.PUTCHAR(CHAR(Z + RANK('0'))); CARRY := 0;
END ELSE
BEGIN BF.PUTCHAR(CHAR(MOD(Z, 10) + RANK('0'))); CARRY := 1;
END;
END;
IF CARRY > 0 THEN
BF.PUTCHAR(CHAR(CARRY + RANK('0')));
BF :- TREVERSE(BF.STRIP);
TADDUNSIGNED :- BF;
END TADDUNSIGNED;
TEXT PROCEDURE TADD(A, B); TEXT A, B;
BEGIN
BOOLEAN ANEG, BNEG;
ANEG := TSIGN(A) < 0; BNEG := TSIGN(B) < 0;
IF NOT ANEG AND BNEG THEN ! (+7)+(-5) = (7-5) = 2 ;
TADD :- TSUBUNSIGNED(A, TABS(B))
ELSE IF ANEG AND NOT BNEG THEN ! (-7)+(+5) = (5-7) = -2 ;
TADD :- TSUBUNSIGNED(B, TABS(A))
ELSE IF ANEG AND BNEG THEN ! (-7)+(-5) = -(7+5) = -12 ;
TADD :- TNEGATE(TADDUNSIGNED(TABS(A), TABS(B)))
ELSE ! (+7)+(+5) = (7+5) = 12 ;
TADD :- TADDUNSIGNED(A, B);
END TADD;
TEXT PROCEDURE TSUBUNSIGNED(A, B); TEXT A, B;
BEGIN
INTEGER I, J, CARRY;
I := A.LENGTH; J := B.LENGTH;
IF I < J OR I = J AND A < B THEN
TSUBUNSIGNED :- TNEGATE(TSUBUNSIGNED(B, A)) ELSE
BEGIN
TEXT BF;
BF :- BLANKS(MAX(I, J) + 1);
WHILE I >= 1 OR J >= 1 DO
BEGIN
INTEGER X, Y, Z;
IF I >= 1 THEN
BEGIN A.SETPOS(I); I := I - 1;
X := RANK(A.GETCHAR) - RANK('0');
END;
IF J >= 1 THEN
BEGIN B.SETPOS(J); J := J - 1;
Y := RANK(B.GETCHAR) - RANK('0');
END;
Z := X - Y - CARRY;
IF Z >= 0 THEN
BEGIN
BF.PUTCHAR(CHAR(RANK('0') + Z));
CARRY := 0;
END ELSE
BEGIN
BF.PUTCHAR(CHAR(RANK('0') + MOD(10 + Z, 10)));
CARRY := 1; ! (Z / 10);
END;
END;
BF :- BF.STRIP;
BF :- TREVERSE(BF);
BF.SETPOS(1);
WHILE BF.LENGTH > 1 AND THEN BF.GETCHAR = '0' DO
BEGIN
BF :- BF.SUB(2, BF.LENGTH - 1);
BF.SETPOS(1);
END;
TSUBUNSIGNED :- BF;
END;
END TSUBUNSIGNED;
TEXT PROCEDURE TSUB(A, B); TEXT A, B;
BEGIN
BOOLEAN ANEG, BNEG;
ANEG := TSIGN(A) < 0; BNEG := TSIGN(B) < 0;
IF ANEG AND BNEG THEN ! (-7)-(-5) = -(7-5) = -2 ;
TSUB :- TNEGATE(TSUBUNSIGNED(TABS(A), TABS(B)))
ELSE IF NOT ANEG AND BNEG THEN ! (+7)-(-5) = (7+5) = 12 ;
TSUB :- TADDUNSIGNED(A, TABS(B))
ELSE IF ANEG AND NOT BNEG THEN ! (-7)-(+5) = -(7+5) = -12 ;
TSUB :- TNEGATE(TADDUNSIGNED(TABS(A), B))
ELSE ! (+7)-(+5) = (7-5) = 2 ;
TSUB :- TSUBUNSIGNED(A, B);
END TSUB;
TEXT PROCEDURE TMULUNSIGNED(A, B); TEXT A, B;
BEGIN
INTEGER ALEN, BLEN;
ALEN := A.LENGTH; BLEN := B.LENGTH;
IF ALEN < BLEN THEN
TMULUNSIGNED :- TMULUNSIGNED(B, A)
ELSE BEGIN
TEXT PRODUCT; INTEGER J;
PRODUCT :- "0";
FOR J := 1 STEP 1 UNTIL BLEN DO BEGIN
TEXT PART; INTEGER I, Y, CARRY;
B.SETPOS(J); Y := RANK(B.GETCHAR) - RANK('0');
PART :- BLANKS(ALEN + BLEN + 1); PART.SETPOS(1);
FOR I := ALEN STEP -1 UNTIL 1 DO BEGIN
INTEGER X, Z;
A.SETPOS(I); X := RANK(A.GETCHAR) - RANK('0');
Z := X * Y + CARRY;
IF Z < 10 THEN BEGIN
PART.PUTCHAR(CHAR(RANK('0') + Z));
CARRY := 0;
END ELSE BEGIN
PART.PUTCHAR(CHAR(RANK('0') + MOD(Z, 10)));
CARRY := Z // 10;
END;
END;
IF CARRY > 0 THEN
PART.PUTCHAR(CHAR(RANK('0') + CARRY));
PART :- PART.SUB(1, PART.POS - 1);
PART :- TREVERSE(PART);
PART.SETPOS(1);
WHILE PART.LENGTH > 1 AND THEN PART.GETCHAR = '0' DO
BEGIN
PART :- PART.SUB(2, PART.LENGTH - 1);
PART.SETPOS(1);
END;
PRODUCT :- TADDUNSIGNED(TSHL(PRODUCT), PART);
END;
TMULUNSIGNED :- PRODUCT;
END;
END TMULUNSIGNED;
TEXT PROCEDURE TMUL(A, B); TEXT A, B;
BEGIN
BOOLEAN ANEG, BNEG;
ANEG := TSIGN(A) < 0; BNEG := TSIGN(B) < 0;
IF ANEG AND BNEG THEN ! (-7)*(-5) = (7*5) => 35 ;
TMUL :- TMULUNSIGNED(TABS(A), TABS(B))
ELSE IF NOT ANEG AND BNEG THEN ! (+7)*(-5) = -(7*5) => -35 ;
TMUL :- TNEGATE(TMULUNSIGNED(A, TABS(B)))
ELSE IF ANEG AND NOT BNEG THEN ! (-7)*(+5) = -(7*5) => -35 ;
TMUL :- TNEGATE(TMULUNSIGNED(TABS(A), B))
ELSE ! (+7)*(+5) = (7*5) => 35 ;
TMUL :- TMULUNSIGNED(A, B);
END TMUL;
CLASS DIVMOD(DIV,MOD); TEXT DIV,MOD;;
REF(DIVMOD) PROCEDURE TDIVMODUNSIGNED(A, B); TEXT A, B;
BEGIN
INTEGER CC;
REF(DIVMOD) RESULT;
IF TISZERO(B) THEN
ERROR("DIVISION BY ZERO");
CC := TCMPUNSIGNED(A, B);
IF CC < 0 THEN
RESULT :- NEW DIVMOD("0", A)
ELSE IF CC = 0 THEN
RESULT :- NEW DIVMOD("1", "0")
ELSE BEGIN
INTEGER ALEN, BLEN, AIDX;
TEXT Q, R;
ALEN := A.LENGTH; BLEN := B.LENGTH;
Q :- BLANKS(ALEN); Q.SETPOS(1);
R :- BLANKS(ALEN); R.SETPOS(1);
R := A.SUB(1, BLEN - 1); R.SETPOS(BLEN);
FOR AIDX := BLEN STEP 1 UNTIL ALEN DO
BEGIN
INTEGER COUNT; BOOLEAN DONE;
IF TISZERO(R.STRIP) THEN
R.SETPOS(1);
A.SETPOS(AIDX); R.PUTCHAR(A.GETCHAR);
WHILE NOT DONE DO
BEGIN
TEXT DIFF;
DIFF :- TSUBUNSIGNED(R.STRIP, B);
IF TSIGN(DIFF) < 0 THEN
DONE := TRUE
ELSE BEGIN
R := DIFF; R.SETPOS(DIFF.LENGTH + 1);
COUNT := COUNT + 1;
END;
END;
IF (NOT (COUNT = 0)) OR (NOT (Q.POS = 1)) THEN
Q.PUTCHAR(CHAR(COUNT + RANK('0')));
END;
RESULT :- NEW DIVMOD(Q.STRIP, R.STRIP);
END;
TDIVMODUNSIGNED :- RESULT;
END TDIVMODUNSIGNED;
REF(DIVMOD) PROCEDURE TDIVMOD(A, B); TEXT A, B;
BEGIN
BOOLEAN ANEG, BNEG; REF(DIVMOD) RESULT;
ANEG := TSIGN(A) < 0; BNEG := TSIGN(B) < 0;
IF ANEG AND BNEG THEN
BEGIN
RESULT :- TDIVMOD(TABS(A), TABS(B));
RESULT.MOD :- TNEGATE(RESULT.MOD);
END
ELSE IF NOT ANEG AND BNEG THEN
BEGIN
RESULT :- TDIVMOD(A, TABS(B));
RESULT.DIV :- TNEGATE(RESULT.DIV);
END
ELSE IF ANEG AND NOT BNEG THEN
BEGIN
RESULT :- TDIVMOD(TABS(A), B);
RESULT.DIV :- TNEGATE(RESULT.DIV);
RESULT.MOD :- TNEGATE(RESULT.MOD);
END
ELSE
RESULT :- TDIVMODUNSIGNED(A, B);
TDIVMOD :- RESULT;
END TDIVMOD;
TEXT PROCEDURE TDIV(A, B); TEXT A, B;
TDIV :- TDIVMOD(A, B).DIV;
TEXT PROCEDURE TMOD(A, B); TEXT A, B;
TMOD :- TDIVMOD(A, B).MOD;
END BIGNUM;
```
```simula
EXTERNAL CLASS BIGNUM;
BIGNUM
BEGIN
PROCEDURE CALCPI;
BEGIN
INTEGER I;
TEXT Q, R, T, K, N, L;
COMMENT
! q, r, t, k, n, l = 1, 0, 1, 1, 3, 3
;
Q :- COPY("1");
R :- COPY("0");
T :- COPY("1");
K :- COPY("1");
N :- COPY("3");
L :- COPY("3");
WHILE TRUE DO
BEGIN
COMMENT
! if 4*q+r-t < n*t
;
IF TCMP(TSUB(TADD(TMUL("4",Q),R),T),TMUL(N,T)) < 0 THEN
BEGIN
TEXT NR;
OUTTEXT(N);
I := I + 1;
IF I = 40 THEN
BEGIN
OUTIMAGE;
I := 0;
END;
COMMENT
! nr = 10*(r-n*t)
! n = ((10*(3*q+r))//t)-10*n
! q *= 10
! r = nr
;
NR :- TMUL("10",TSUB(R,TMUL(N,T)));
N :- TSUB(TDIV(TMUL("10",TADD(TMUL("3",Q),R)),T),TMUL("10",N));
Q :- TMUL("10",Q);
R :- NR;
END
ELSE
BEGIN
TEXT NR, NN;
COMMENT
! nr = (2*q+r)*l
! nn = (q*(7*k)+2+(r*l))//(t*l)
! q *= k
! t *= l
! l += 2
! k += 1
! n = nn
! r = nr
;
NR :- TMUL(TADD(TMUL("2",Q),R),L);
NN :- TDIV(TADD(TADD(TMUL(Q,TMUL("7",K)),"2"),TMUL(R,L)),TMUL(T,L));
Q :- TMUL(Q,K);
T :- TMUL(T,L);
L :- TADD(L,"2");
K :- TADD(K,"1");
N :- NN;
R :- NR;
END;
END;
END CALCPI;
CALCPI;
END.
```
Output:
```txt
3141592653589793238462643383279502884197
1693993751058209749445923078164062862089
9862803482534211706798214808651328230664
7093844609550582231725359408128481117450
2841027019385211055596446229489549303819
6442881097566593344612847564823378678316
5271201909145648566923460348610454326648
2133936072602491412737245870066063155881
7488152092096282925409171536436789259036
0011330530548820466521384146951941511609
4330572703657595919530921861173819326117
9310511854807446237996274956735188575272
4891227938183011949129833673362440656643
0860213949463952247371907021798609437027
7053921717629317675238467481846766940513
2000568127145263560827785771342757789609
...
```
## Tcl
Based on the reference in the [[#D|D]] code.
{{works with|Tcl|8.6}}
```tcl
package require Tcl 8.6
# http://www.cut-the-knot.org/Curriculum/Algorithms/SpigotForPi.shtml
# http://www.mathpropress.com/stan/bibliography/spigot.pdf
proc piDigitsBySpigot n {
yield [info coroutine]
set A [lrepeat [expr {int(floor(10*$n/3.)+1)}] 2]
set Alen [llength $A]
set predigits {}
while 1 {
set carry 0
for {set i $Alen} {[incr i -1] > 0} {} {
lset A $i [expr {
[set val [expr {[lindex $A $i] * 10 + $carry}]]
% [set modulo [expr {2*$i + 1}]]
}]
set carry [expr {$val / $modulo * $i}]
}
lset A 0 [expr {[set val [expr {[lindex $A 0]*10 + $carry}]] % 10}]
set predigit [expr {$val / 10}]
if {$predigit < 9} {
foreach p $predigits {yield $p}
set predigits [list $predigit]
} elseif {$predigit == 9} {
lappend predigits $predigit
} else {
foreach p $predigits {yield [incr p]}
set predigits [list 0]
}
}
}
```
The pi digit generation requires picking a limit to the number of digits; the bigger the limit, the more digits can be ''safely'' computed. A value of 10k yields values relatively rapidly.
```tcl
coroutine piDigit piDigitsBySpigot 10000
fconfigure stdout -buffering none
while 1 {
puts -nonewline [piDigit]
}
```
## TypeScript
```javascript
type AnyWriteableObject={write:((textToOutput:string)=>any)};
function calcPi(pipe:AnyWriteableObject) {
let q = 1n, r=0n, t=1n, k=1n, n=3n, l=3n;
while (true) {
if (q * 4n + r - t < n* t) {
pipe.write(n.toString());
let nr = (r - n * t) * 10n;
n = (q * 3n + r) * 10n / t - n * 10n ;
q = q * 10n;
r = nr;
} else {
let nr = (q * 2n + r) * l;
let nn = (q * k * 7n + 2n + r * l) / (t * l);
q = q * k;
t = t * l;
l = l + 2n;
k = k + 1n;
n = nn;
r = nr;
}
}
}
calcPi(process.stdout);
```
'''Notes:'''
1. Typescript has ''bigint'' support https://www.typescriptlang.org/docs/handbook/release-notes/typescript-3-2.html#bigint Literals are write with a ''n'' sufix: ''10n''
2. Pi function receives any object that has a ''write'' function. Using node.js we can pass to it ''process.stdout''
### Async version
```javascript
type AnyWriteableObject = {write:((textToOutput:string)=>Promise)};
async function calcPi(pipe:T) {
let q = 1n, r=0n, t=1n, k=1n, n=3n, l=3n;
while (true) {
if (q * 4n + r - t < n* t) {
await pipe.write(n.toString());
let nr = (r - n * t) * 10n;
n = (q * 3n + r) * 10n / t - n * 10n ;
q = q * 10n;
r = nr;
} else {
let nr = (q * 2n + r) * l;
let nn = (q * k * 7n + 2n + r * l) / (t * l);
q = q * k;
t = t * l;
l = l + 2n;
k = k + 1n;
n = nn;
r = nr;
}
}
}
setInterval(function(){
console.log(); // put a new line every second
},1000);
var x = calcPi({
write: async function(phrase:string){
return new Promise(function(resolve){
setTimeout(function(){
process.stdout.write(phrase);
resolve();
},1);
});
}
});
console.log('.'); //start!
```
Here the calculation does not continue if the consumer does not consume the character.
## Visual Basic
{{works with|Visual Basic|5}}
{{works with|Visual Basic|6}}
{{works with|VBA|Access 97}}
{{works with|VBA|6.5}}
{{works with|VBA|7.1}}
```vb
Option Explicit
Sub Main()
Const VECSIZE As Long = 3350
Const BUFSIZE As Long = 201
Dim buffer(1 To BUFSIZE) As Long
Dim vect(1 To VECSIZE) As Long
Dim more As Long, karray As Long, num As Long, k As Long, l As Long, n As Long
For n = 1 To VECSIZE
vect(n) = 2
Next n
For n = 1 To BUFSIZE
karray = 0
For l = VECSIZE To 1 Step -1
num = 100000 * vect(l) + karray * l
karray = num \ (2 * l - 1)
vect(l) = num - karray * (2 * l - 1)
Next l
k = karray \ 100000
buffer(n) = more + k
more = karray - k * 100000
Next n
Debug.Print CStr(buffer(1));
Debug.Print "."
l = 0
For n = 2 To BUFSIZE
Debug.Print Format$(buffer(n), "00000");
l = l + 1
If l = 10 Then
l = 0
Debug.Print 'line feed
End If
Next n
End Sub
```
{{out}}
```txt
3.
14159265358979323846264338327950288419716939937510
58209749445923078164062862089986280348253421170679
82148086513282306647093844609550582231725359408128
48111745028410270193852110555964462294895493038196
44288109756659334461284756482337867831652712019091
45648566923460348610454326648213393607260249141273
72458700660631558817488152092096282925409171536436
78925903600113305305488204665213841469519415116094
33057270365759591953092186117381932611793105118548
07446237996274956735188575272489122793818301194912
98336733624406566430860213949463952247371907021798
60943702770539217176293176752384674818467669405132
00056812714526356082778577134275778960917363717872
14684409012249534301465495853710507922796892589235
42019956112129021960864034418159813629774771309960
51870721134999999837297804995105973173281609631859
50244594553469083026425223082533446850352619311881
71010003137838752886587533208381420617177669147303
59825349042875546873115956286388235378759375195778
18577805321712268066130019278766111959092164201989
```
## Visual Basic .NET
{{trans|C#}}
Don't forget to use the "'''Project'''" tab, "'''Add Reference...'''" for '''''System.Numerics''''' (in case you get compiler errors in the Visual Studio IDE)
```vbnet
Imports System
Imports System.Numerics
Public Module Module1
Public Sub Main()
Dim two, three, four, seven, ten, k, q, t, l, n, r, nn, nr As BigInteger,
first As Boolean = True
two = New BigInteger(2) : three = New BigInteger(3) : four = two + two
seven = three + four : ten = three + seven : k = BigInteger.One
q = k : t = k : l = three : n = three : r = BigInteger.Zero
While True
If four * q + r - t < n * t Then
Console.Write(n) : If first Then Console.Write(".") : first = False
nr = ten * (r - n * t) : n = ten * (three * q + r) / t - ten * n
q *= ten
Else
nr = (two * q + r) * l : nn = (q * seven * k + two + r * l) / (t * l)
q *= k : t *= l : l += two : k += BigInteger.One : n = nn
End If
r = nr
End While
End Sub
End Module
```
{{out}}
3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989380952572010654858632788659361533818279682303019520353018529689957736225994138912497217752834791315155748572424541506959508295331168617278558890750983817546374649393192550604009277016711390098488240128583616035637076601047101819429555961989467678374494482553797747268471040475346462080466842590694912933136770289891521047521620569660240580381501935112533824300355876402474964732639141992726042699227967823547816360093417216412199245863150302861829745557067498385054945885869269956909272107975093029553211653449872027559602364806654991198818347977535663698074265425278625518184175746728909777727938000816470600161452491921732172147723501414419735685481613611573525521334757418494684385233239073941433345477624168625189835694855620992192221842725502542568876717904946016534668049886272327917860857843838279679766814541009538837863609506800642251252051173929848960841284886269456042419652850222106611863067442786220391949450471237137869609563643719172874677646575739624138908658326459958133904780275900994657640789512694683983525957098258226205224894077267194782684826014769909026401363944374553050682034962524517493996514314298091906592509372216964615157098583874105978859597729754989301617539284681382686838689427741559918559252459539594310499725246808459872736446958486538367362226260991246080512438843904512441365497627807977156914359977001296160894416948685558484063534220722258284886481584560285060168427394522674676788952521385225499546667278239864565961163548862305774564980355936345681743241125150760694794510965960940252288797108931456691368672287489405601015033086179286809208747609178249385890097149096759852613655497818931297848216829989487226588048575640142704775551323796414515237462343645428584447952658678210511413547357395231134271661021359695362314429524849371871101457654035902799344037420073105785390621983874478084784896833214457138687519435064302184531910484810053706146806749192781911979399520614196634287544406437451237181921799983910159195618146751426912397489409071864942319615679452080951465502252316038819301420937621378559566389377870830390697920773467221825625996615014215030680384477345492026054146659252014974428507325186660021324340881907104863317346496514539057962685610055081066587969981635747363840525714591028970641401109712062804390397595156771577004203378699360072305587631763594218731251471205329281918261861258673215791984148488291644706095752706957220917567116722910981690915280173506712748583222871835209353965725121083579151369882091444210067510334671103141267111369908658516398315019701651511685171437657618351556508849099898599823873455283316355076479185358932261854896321329330898570642046752590709154814165498594616371802709819943099244889575712828905923233260972997120844335732654893823911932597463667305836041428138830320382490375898524374417029132765618093773444030707469211201913020330380197621101100449293215160842444859637669838952286847831235526582131449576857262433441893039686426243410773226978028073189154411010446823252716201052652272111660396665573092547110557853763466820653109896526918620564769312570586356620185581007293606598764861179104533488503461136576867532494416680396265797877185560845529654126654085306143444318586769751456614068007002378776591344017127494704205622305389945613140711270004078547332699390814546646458807972708266830634328587856983052358089330657574067954571637752542021149557615814002501262285941302164715509792592309907965473761255176567513575178296664547791745011299614890304639947132962107340437518957359614589019389713111790429782856475032031986915140287080859904801094121472213179476477726224142548545403321571853061422881375850430633217518297986622371721591607716692547487389866549494501146540628433663937900397692656721463853067360965712091807638327166416274888800786925602902284721040317211860820419000422966171196377921337575114959501566049631862947265473642523081770367515906735023507283540567040386743513622224771589150495309844489333096340878076932599397805419341447377441842631298608099888687413260472156951623965864573021631598193195167353812974167729478672422924654366800980676928238280689964004824354037014163149658979409243237896907069779422362508221688957383798623001593776471651228935786015881617557829735233446042815126272037343146531977774160319906655418763979293344195215413418994854447345673831624993419131814809277771038638773431772075456545322077709212019051660962804909263601975988281613323166636528619326686336062735676303544776280350450777235547105859548702790814356240145171806246436267945612753181340783303362542327839449753824372058353114771199260638133467768796959703098339130771098704085913374641442822772634659470474587847787201927715280731767907707157213444730605700733492436931138350493163128404251219256517980694113528013147013047816437885185290928545201165839341965621349143415956258658655705526904965209858033850722426482939728584783163057777560688876446248246857926039535277348030480290058760758251047470916439613626760449256274204208320856611906254543372131535958450687724602901618766795240616342522577195429162991930645537799140373404328752628889639958794757291746426357455254079091451357111369410911939325191076020825202618798531887705842972591677813149699009019211697173727847684726860849003377024242916513005005168323364350389517029893922334517220138128069650117844087451960121228599371623130171144484640903890644954440061986907548516026327505298349187407866808818338510228334508504860825039302133219715518430635455007668282949304137765527939751754613953984683393638304746119966538581538420568533862186725233402830871123282789212507712629463229563989898935821167456270102183564622013496715188190973038119800497340723961036854066431939509790190699639552453005450580685501956730229219139339185680344903982059551002263535361920419947455385938102343955449597783779023742161727111
```
===Quicker, unverified algo===
There seems to be another algorithm in the original reference article (see the [http://www.rosettacode.org/wiki/Pi#Ada Ada] entry), which produces output a bit faster. However, the math behind the algorithm has not been completely proven. It's faster because it doesn't calculate whether each digit is accumulated properly before squirting it out. When using (slow) arbitrary precision libraries, this avoids a lot of computation time.
```vbnet
Imports System, System.Numerics, System.Text
Module Program
Sub RunPiF(ByVal msg As String)
If msg.Length > 0 Then Console.WriteLine(msg)
Dim first As Boolean = True, stp As BigInteger = 360,
lim As BigInteger = stp, res As StringBuilder = New StringBuilder(),
rc As Integer = -1, u, j, k As BigInteger, q As BigInteger = 1,
r As BigInteger = 180, t As BigInteger = 60, i As BigInteger = 2,
y As Byte, et As TimeSpan, st As DateTime = DateTime.Now
While True
While i < lim
j = i << 1 : k = j + i : u = 3 * (k + 1) * (k + 2)
y = CByte(((q * (9 * k - 12) + 5 * r) / (5 * t)))
res.Append(y)
r = (q * (k + j - 2) + r - y * t) * u * 10
t *= u : q = 10 * q * (j - 1) * i : i += 1
End While
If first Then res.Insert(1, "."c) : first = False
Console.Write(res.ToString())
rc += res.Length : res.Clear() : lim += stp
If Console.KeyAvailable Then Exit While
End While
et = DateTime.Now - st : Console.ReadKey()
Console.Write(res.ToString()) : rc += res.Length
Console.WriteLine(vbLf & "Produced {0} digits in {1:n4} seconds.", rc, et.TotalSeconds)
End Sub
Sub Main(args As String())
RunPiF("Press a key to exit...")
If Diagnostics.Debugger.IsAttached Then Console.ReadKey()
End Sub
End Module
```
{{out}}The First several thousand digits verified the same as the conventional spigot algorithm, haven't detected any differences yet.
Press a key to exit...
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420198938095257201065485863278865936153381827968230301952035301852968995773622599413891249721775283479131515574857242454150695950829533116861727855889075098381754637464939319255060400927701671139009848824012858361603563707660104710181942955596198946767837449448255379774726847104047534646208046684259069491293313677028989152104752162056966024058038150193511253382430035587640247496473263914199272604269922796782354781636009341721641219924586315030286182974555706749838505494588586926995690927210797509302955321165344987202755960236480665499119881834797753566369807426542527862551818417574672890977772793800081647060016145249192173217214772350141441973568548161361157352552133475741849468438523323907394143334547762416862518983569485562099219222184272550254256887671790494601653466804988627232791786085784383827967976681454100953883786360950680064225125205117392984896084128488626945604241965285022210661186306744278622039194945047123713786960956364371917287467764657573962413890865832645995813390478027590099465764078951269468398352595709825822620522489407726719478268482601476990902640136394437455305068203496252451749399651431429809190659250937221696461515709858387410597885959772975498930161753928468138268683868942774155991855925245953959431049972524680845987273644695848653836736222626099124608051243884390451244136549762780797715691435997700129616089441694868555848406353422072225828488648158456028506016842739452267467678895252138522549954666727823986456596116354886230577456498035593634568174324112515076069479451096596094025228879710893145669136867228748940560101503308617928680920874760917824938589009714909675985261365549781893129784821682998948722658804857564014270477555132379641451523746234364542858444795265867821051141354735739523113427166102135969536231442952484937187110145765403590279934403742007310578539062198387447808478489683321445713868751943506430218453191048481005370614680674919278191197939952061419663428754440643745123718192179998391015919561814675142691239748940907186494231961567945208095146550225231603881930142093762137855956638937787083039069792077346722182562599661501421503068038447734549202605414665925201497442850732518666002132434088190710486331734649651453905796268561005508106658796998163574736384052571459102897064140110971206280439039759515677157700420337869936007230558763176359421873125147120532928191826186125867321579198414848829164470609575270695722091756711672291098169091528017350671274858322287183520935396572512108357915136988209144421006751033467110314126711136990865851639831501970165151168517143765761835155650884909989859982387345528331635507647918535893226185489632132933089857064204675259070915481416549859461637180270981994309924488957571282890592323326097299712084433573265489382391193259746366730583604142813883032038249037589852437441702913276561809377344403070746921120191302033038019762110110044929321516084244485963766983895228684783123552658213144957685726243344189303968642624341077322697802807318915441101044682325271620105265227211166039666557309254711055785376346682065310989652691862056476931257058635662018558100729360659876486117910453348850346113657686753249441668039626579787718556084552965412665408530614344431858676975145661406800700237877659134401712749470420562230538994561314071127000407854733269939081454664645880797270826683063432858785698305235808933065757406795457163775254202114955761581400250126228594130216471550979259230990796547376125517656751357517829666454779174501129961489030463994713296210734043751895735961458901938971311179042978285647503203198691514028708085990480109412147221317947647772622414254854540332157185306142288137585043063321751829798662237172159160771669254748738986654949450114654062843366393790039769265672146385306736096571209180763832716641627488880078692560290228472104031721186082041900042296617119637792133757511495950156604963186294726547364252308177036751590673502350728354056704038674351362222477158915049530984448933309634087807693259939780541934144737744184263129860809988868741326047215695162396586457302163159819319516735
Produced 5038 digits in 0.3391 seconds.
```
## zkl
Uses the GMP big int library.
Same algorithm as many of the others on this page. Uses in place ops to cut down on big int generation (eg add vs +). Unless GC is given some hints, it will use up 16 gig quickly as it outruns the garbage collector.
```zkl
var [const] BN=Import("zklBigNum"),
one=BN(1), two=BN(2), three=BN(3), four=BN(4), seven=BN(7), ten=BN(10);
fcn calcPiDigits{
reg q=BN(1), r=BN(0), t=BN(1), k=BN(1), n=BN(3), l=BN(3);
first:=True; N:=0;
while(True){ if((N+=1)==1000){ GarbageMan.collect(); N=0; } // take a deep breath ...
if(four*q + r - t < n*t){
n.print(); if(first){ print("."); first=False; }
nr:=(r - n*t).mul(ten); // 10 * (r - n * t);
n=(three*q).add(r).mul(ten) // ((10*(3*q + r))/t) - 10*n;
.div(t).sub(ten*n);
q.mul(ten); // q *= 10;
r=nr;
}else{
nr:=(two*q).add(r).mul(l); // (2*q + r)*l;
nn:=(q*seven).mul(k).add(two) // (q*(7*k + 2) + r*l)/(t*l);
.add(r*l).div(t*l);
q.mul(k); t.mul(l); // q*=k; t*=l;
l.add(two); k.add(one); // l+=2; k++;
n=nn; r=nr;
}
}
}();
```
Runs until ^C hit, the first 1000 digits match the D output.
{{out}}
```txt
3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745
```
{{omit from|HTML}}
[[Category:Geometry]]