Do you want us to convert decimals to the corresponding non-decimal expression and count accordingly? 1.423 is 1423/10^3 for 7 digits, for example.

It's hard for a non-mathematician like myself to beat the efficiency of the examples already posted, but here are a few more fairly simple approximations, making the most of our irrational nth root oracle. I used a pocket calculator for that, and show the answer to the first wrong digit.

If you grade the efficiency of each, that will really clarify your rules.

2 + 2^(.2) = 3.148...

3*2^(1/15) = 3.1418...

7^(1/6) + 52^(1/7) = 3.141591...

Use decimal quirks to boost efficiency:
4 + 4^.2 - 7^.4 = 3.1416...

Remarkably simple but accurate one:
3^(3/4) + 6^(1/2) - 4^(1/3) = 3.1415927...

Number raised to its own reciprocal:
2 + 24^(1/24) = 3.14158...

A lot of correct digits:
13^(7/78) + 55^(3/19) = 3.14159264...

No, I did not accomplish awesome Ramanujan-like feats of math to find those; I just wrote a little script to do a beam search across a likely solution space.

Another remarkable thing is that "fast" algorithms for calcularing pi were used to detect hardware defects in the prototypes of the Cray-2 supercomputer (super in 1986 :-))

Another remarkable thing is that "fast" algorithms for calcularing pi were used to detect hardware defects in the prototypes of the Cray-2 supercomputer (super in 1986 :-))>

That's interesting. I can't remember any calculations in math or science classes that required more than 3.14 as the precision of the other measurements was usually no more than that.

In 1897 the Indiana House (GOP led probably) passed a bill to make the ratio of the diameter and circumference as 5/4 to 4, so that pi would be 3.2

But I found some of the formulae highly interesting.

2 + 24^(1/24), the square root of 4 plus 4 factorial raised to its own reciprocal. Can anyone give some kind of geometric or number theoretic story on why it's close to pi?

3^(3/4) + 6^(1/2) - 4^(1/3) is pretty close to pi with only a handful of small integers. Again, can we find any deeper connections here?

<beatgiant> I would never ever do that. It's just that I rank it as #2 after global warming...:-).

(1) The sole standard by which "effectiveness" is measured is the *difference* (not the ratio) between the number of digits employed in the approximation and the number of leading digits in the decimal expansion of pi that it correctly reproduces.

(2) Only parentheses, decimal points, and the usual symbols for elementary arithmetical operations (+, −, /, × or *) and exponentiation (^ or **) can be used. They aren't included in the count.

But if someone wants to suggest a different set of rules and provides a convincing rationale for it, I won't object too strenuously.

<beatgiant: The first wrong digit I find is as early as the 9th decimal place (the estimate is 3.14159265<2>... where it should be 3.14159265<3>). Can you post what you got with your 12 decimal point precision?> I find the same thing.

The statement in Weisstein is as follows:

<Some approximations due to Ramanujan include [...] = (97 + 9/22)^(1/4) [...] which are accurate to 3, 4, 4, 8, 8, 9, 14, [...] digits respectively.>

(2) Only parentheses, decimal points, and the usual symbols for elementary arithmetical operations (+, −, /, × or *) and exponentiation (^ or **) can be used. They aren't included in the count. ...>

(97 + 9/22)^(1/4) = 3.141592652...
efficiency = 9-7 = 2

355/113 = 3.1415929...
efficiency = 7-6 = 1

2+2^.2 = 3.148...
efficiency = 3-3 = 0

1.423^3.245 = 3.1415928...
efficiency = 7-8 = -1

2 + 24^(1/24) = 3.14158...
efficiency = 5-6 = -1

Saving parentheses, my friend?