Eternal 2014 — PCG.SE New Year’s Puzzle 2015

Python 2, 166 bytes

n=input()
d=m=y=1
M=([3]+[3,2]*3)*2
while n>1721426:
 n-=1;d+=1;M[2]=y%400<1or y%4<1<y%100
 if d>M[m%12]+28:m+=1;d=1
 if m>12<2014>y:y+=1;m=1
print'%02d/'*2%(d,m)+`y`

This cycles through every day from January 1, 1 (1721426) to the given date, incrementing the current day, month and year as it goes. The last test case takes about a second on my computer.

Output is printed in the second format:

01/01/1
31/12/999
23/15059/2014

Ostrich 0.5.0, 197 bytes

G~:J1401+4J*274227+146097/F3*4/F+38~+:f4*3+:e1461:p%4/F:g5*2+:h153:s%5/F1+:D;hs/F2+12:n%1+:M;ep/F4716-n2+M-n/F+:Y;{2014:Y;D365+:D;{M1-12%[31:x28x 30:yxyxxyxyx]=:dD<.{Dd-:D;M1+:M;}*}(}Y2014-*D"/M"/Y

Ungolfed (ha):

G~:J;
4716:y;1401:j;2:m;12:n;4:r;1461:p;3:v;5:u;153:s;2:w;274277:B;38~:C;
Jj+4J*B+146097/F3*4/F+C+:f;
rf*v+:e;
ep%r/F:g;
ug*w+:h;
hs%u/F1+:D;
hs/Fm+n%1+:M;
ep/Fy-nm+M-n/F+:Y;
{
2014:Y;
D365+:D;
{
M1-12%[31 28 31 30 31 30 31 31 30 31 30 31]=:d
D<.{Dd-:D;M1+:M;}*
}(
}Y2014-*
D"/M"/Y

I'm... really just exhausted from all the New Year's happiness and what not. That's why I haven't golfed this very much. I may or may not come back to make it better later.

Algorithm from https://en.wikipedia.org/wiki/Julian_day#Gregorian_calendar_from_Julian_day_number