How is this code calculating pi and what principle is it based on?

Question

I've stumbled upon this snippet in the signature of a mail:

int f[9814],b,c=9814,g,i;long a=1e4,d,e,h;main(){for(;b=c,c-=14;i=printf(
"%04d",e+d/a),e=d%a)while(g=--b*2)d=h*b+a*(i?f[b]:a/5),h=d/--g,f[b]=d%g;}

This calculates the output of pi (π):

3141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420198938095257201065485863278865936153381827968230301952035301852968995773622599413891249721775283479131515574857242454150695950829533116861727855889075098381754637464939319255060400927701671139009848824012858361603563707660104710181942955596198946767837449448255379774726847104047534646208046684259069491293313677028989152104752162056966024058038150193511253382430035587640247496473263914199272604269922796782354781636009341721641219924586315030286182974555706749838505494588586926995690927210797509302955321165344987202755960236480665499119881834797753566369807426542527862551818417574672890977772793800081647060016145249192173217214772350141441973568548161361157352552133475741849468438523323907394143334547762416862518983569485562099219222184272550254256887671790494601653466804988627232791786085784383827967976681454100953883786360950680064225125205117392984896084128488626945604241965285022210661186306744278622039194945047123713786960956364371917287467764657573962413890865832645995813390478027590099465764078951269468398352595709825822620522489407726719478268482601476990902640136394437455305068203496252451749399651431429809190659250937221696461515709858387410597885959772975498930161753928468138268683868942774155991855925245953959431049972524680845987273644695848653836736222626099124608051243884390451244136549762780797715691435997700129616089441694868555848406353422072225828488648158456028506016842739452267467678895252138522549954666727823986456596116354886230577456498035593634568174324112515076069479451096596094025228879710893145669136867228748940560101503308617928680920874760917824938589009714909675985261365549781893129784821682998948722658804857564014270477555132379641451523746234364542858444795265867821051141354735739523113427166102135969536231442952484937187110145765403590279934

Some questions I have regarding this:

• Which mathematical principle it is based upon?
• What sort of iterative approximation is this (which series)?
• Why is the output 2800 characters long?

Answer 1

This appears to be based off of Dik T. Winter's 160-byte C program to compute the first 800 digits of pi, extended for higher precision.

His original code:

int a=10000,b,c=2800,d,e,f[2801],g;main(){for(;b-c;)f[b++]=a/5;
for(;d=0,g=c*2;c-=14,printf("%.4d",e+d/a),e=d%a)for(b=c;d+=f[b]*a,
f[b]=d%--g,d/=g--,--b;d*=b);}

can be rewritten more readably thus:

#include <stdio.h>

int main() {
    int r[2800 + 1];
    int i, k;
    int b, d;
    int c = 0;

    for (i = 0; i < 2800; i++) {
        r[i] = 2000;
    }

    for (k = 2800; k > 0; k -= 14) {
        d = 0;

        i = k;
        for (;;) {
            d += r[i] * 10000;
            b = 2 * i - 1;

            r[i] = d % b;
            d /= b;
            i--;
            if (i == 0) break;
            d *= i;
        }
        printf("%.4d", c + d / 10000);
        c = d % 10000;
    }

    return 0;
}

During the k loop, this is computing the digits of:

Which by Beeler et al. (1972) is an approximation of pi:

See here for further details:

• https://crypto.stanford.edu/pbc/notes/pi/code.html

Answer 2

I'll answer the last question: "Why is the output 2800 characters long?"

In the initialization line of the code, we see

c=9814

Then in the loop condition expression of the for loop we see

c-=14

So the loop subtracts 14 from c on each loop, starting from 9814, and stops when c reaches zero. But we see that 9814 / 14 is 701. Since that termination condition is executed before the start of the first loop, the loop executes 700 times.

The last part of the for loop, which usually increments the loop variable, instead executes

printf("%04d",e+d/a)

So 4 characters of an integer are printed in each loop. Those 4 characters, printed 700 times, result in 2800 characters printed total.