Why define PI = 4*ATAN(1.d0)

This style ensures that the maximum precision available on ANY architecture is used when assigning a value to PI.

Because Fortran does not have a built-in constant for PI. But rather than typing in the number manually and potentially making a mistake or not getting the maximum possible precision on the given implementation, letting the library calculate the result for you guarantees that neither of those downsides happen.

These are equivalent and you'll sometimes see them too:

PI=DACOS(-1.D0)
PI=2.D0*DASIN(1.D0)

I believe it's because this is the shortest series on pi. That also means it's the MOST ACCURATE.

The Gregory-Leibniz series (4/1 - 4/3 + 4/5 - 4/7...) equals pi.

atan(x) = x^1/1 - x^3/3 + x^5/5 - x^7/7...

So, atan(1) = 1/1 - 1/3 + 1/5 - 1/7 + 1/9... 4 * atan(1) = 4/1 - 4/3 + 4/5 - 4/7 + 4/9...

That equals the Gregory-Leibniz series, and therefore equals pi, approximately 3.1415926535 8979323846 2643383279 5028841971 69399373510.

Another way to use atan and find pi is:

pi = 16*atan(1/5) - 4*atan(1/239), but I think that's more complicated.

I hope this helps!

(To be honest, I think the Gregory-Leibniz series was based on atan, not 4*atan(1) based on the Gregory-Leibniz series. In other words, the REAL proof is:

sin^2 x + cos^2 x = 1 [Theorem] If x = pi/4 radians, sin^2 x = cos^2 x, or sin^2 x = cos^2 x = 1/2.

Then, sin x = cos x = 1/(root 2). tan x (sin x / cos x) = 1, atan x (1 / tan x) = 1.

So if atan(x) = 1, x = pi/4, and atan(1) = pi/4. Finally, 4*atan(1) = pi.)

Please don't load me with comments-I'm still a pre-teen.

It's because this is an exact way to compute pi to arbitrary precision. You can simply continue executing the function to get greater and greater precision and stop at any point to have an approximation.

By contrast, specifying pi as a constant provides you with exactly as much precision as was originally given, which may not be appropriate for highly scientific or mathematical applications (as Fortran is frequently used with).

That sounds an awful lot like a work-around for a compiler bug. Or it could be that this particular program depends on that identity being exact, and so the programmer made it guaranteed.