Ordinal date

An ordinal date is a calendar date typically consisting of a year and a day of year ranging between 1 and 366 (starting on January 1), though year may sometimes be omitted. The two numbers can be formatted as YYYY-DDD to comply with the ISO 8601 ordinal date format.

Computation of the ordinal date within a year is part of calculating the ordinal date throughout the years from a reference date, such as the Julian date. It is also part of calculating the day of the week, though for this purpose modulo-7 simplifications can be made.

For these purposes it is convenient to count January and February as month 13 and 14 of the previous year, for two reasons: the shortness of February and its variable length. In that case the date counted from 1 March is given by

which can also be written

with m the month number and d the date.

The formula reflects the fact that any five consecutive months in the range March–January have a total length of 153 days, due to a fixed pattern 31–30–31–30–31 repeating itself some more than twice.

"Doomsday" properties:

For m = 2n and d=m we get

giving consecutive differences of 63 (9 weeks) for n = 2, 3, 4, 5, and 6, i.e., between 4/4, 6/6, 8/8, 10/10, and 12/12.

For m = 2n + 1 and d=m + 4 we get

and with m and d interchanged

giving a difference of 119 (17 weeks) for n = 2 (difference between 5/9 and 9/5), and also for n = 3 (difference between 7/11 and 11/7).

The ordinal date from 1 January is:

or equivalently, the ordinal date from 1 March of the previous year (for which the formula above can be used) minus 306.

Again counting January and February as month 13 and 14 of the previous year, the date counted from 1 March is modulo 7 equal to

with m the month number and d the date.

Calculation can be done starting with January 1 mathematically without if statements if we take advantage of min and max algebraic logic
MAX is 1/2*(a+b+|a-b|)
MIN is 1/2*(a+b-|a-b|)

provided the month(m) day(d) and year(y)
(MAX(0,MIN(1,m-1))*31)+ //if Jan is a full month
(MAX(0,MIN(1,m-2))*28)+ //if Feb is a full month
(MAX(0,MIN(1,m-3))*31)+ //if Mar is a full month
(MAX(0,MIN(1,m-4))*30)+ //if Apr is a full month
(MAX(0,MIN(1,m-5))*31)+ //if May is a full month
(MAX(0,MIN(1,m-6))*30)+ //if June is a full month
(MAX(0,MIN(1,m-7))*31)+ //if July is a full month
(MAX(0,MIN(1,m-8))*31)+ //if Aug is a full month
(MAX(0,MIN(1,m-9))*30)+ //if Sept is a full month
(MAX(0,MIN(1,m-10))*31)+ //if Oct is a full month
(MAX(0,MIN(1,m-11))*30)+ //if Nov is a full month
d+ //days of current month
(((INT((y)/4)-INT((y)/100)+INT((y)/400)) //leap year logic
-(INT((y-1)/4)-INT((y-1)/100)+INT((y-1)/400)))*MAX(0,MIN(1,m-2))) //only count a leap year if date is >=3rd month //leap year logic

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Wikipedia