[Plouffe's Inverter]
A few approximations of Pi
 
by Simon Plouffe
 
Here is a BIG table of approximations of Pi, but first read these notes about the measure of a good approximation.
 
For each approximation of X (here X= Pi), 2 values are given, R1 and R2 which gives a measure of the approximation of a number.
 
R1 is defined as the distance to X in absolute value.
 
R1 = log(1/(| X - a |))/log(10).
 
R2 = R1/log(max(ai))*log(10) and ai is the element of highest size in the expression of a.
 
In other words it is the maximum of length(op(a)) in Maple. If a=355/113, it gives back : 355.
 
So with Pi = 355/113 we have R1 = 6.57 and a = 355 -> R2 = 2.57
 
In more practical terms, R1 gives the maximum of exact digits of the approximation and R2 gives the value of an approximation. If R2 is big, better is the approximation. In our example, R2 = 2.57 which means that the relative size of 355 in regards to R1 is good. If R2 is small (near 1), then the approximation is bad.
If R2 >> 2, it is an excellent approximation.
 
2nd example : The Ramanujan number : exp(Pi*sqrt(163)) = 262537412640768743.9999999999992507... gives us a good approximation of Pi which is log(262537412640768744)/sqrt(163) = Pi to 30 digits and R1 = 30.65 , but R2 is only 1.759.
The number of digits is good but if we compare to the relative size of a then this approximation is an average one.

We should expect R2 to be near 2 for most approximations and find an R2 > 2 in some good examples. In general terms as with the continued fraction expansion of X, if we truncate the expression at any point we should expect a value of R2 near 2.

 

Approximation of Pi

Expression

R1

R2

Comment - (author)
3.14159265358979323846264338327972661934754988088

log(262537412640768744)/sqrt(163)

30.65

1.759

(S. Ramanujan)

3.14159292035398230088495575221238938053097345132

355/113

6.573

2.577

A very good rational approximation

3.14159265358979323232482478168718522102495836130

3+1/8+1/61+1/5020+1/128541455

17.11

1.324

From the egyptian expansion

3.14285714285714285714285714285714285714285714286

22/7

2.898

0.937

One of the 2 values given by Archimedes

3.14159265297229778439562243903476832945058472332

log(5280)/sqrt(67/9)

9.209

2.474

(S. Plouffe 1988)

3.14159434945008183015994893408428386595324120871

log(2198)/sqrt(6)

5.770

1.726

(S. Plouffe 1988)

3.14153985278295351258699144235404432987724178693

43^(7/23)

4.277

1.137

(S. Plouffe 1988)

3.14159265350877192982456140350877192982456140351

3+1/8+1/(8*8)+ 1/(8*8*17)+ 1/(8*8*17*19)+ 1/(8*8*17*19*300)

10.091

0.723

Expansion into egyptien product (S. Plouffe 1988 )

3.14159265359494408765142414297178409903697305215

log(60318/13387)*48/23

11.288

2.361

(S. Plouffe 1988)

3.14159267809890117154750857021071215173157249665

(13/4)^(1181/1216)

7.610

2.467

(S. Plouffe 1988)

3.14159265358677810789019357753871093140928997070

(228+16/1329)^(1/41) + 2

11.520

2.101

(S. Plouffe 1988)

3.14159265358979323846264920145525604153046519371

( 276694819753963/226588)^(1/158) + 2

23.235

1.608

(S. Plouffe 1988)

3.14159265349255372811271549779505742798109245613

(63023/30510)**(1/3)+1/4+1/2*(sqrt(5)+1)

10.012

2.086

(S. Plouffe 1988)

3.14163154625920525451599912710309835110117456654

log(20+Pi)

4.410

3.389

This approximation comes from exp(Pi)-Pi=20 (author=?)

3.14159259508835562119429079295031986122206920605

689/396/ln(689/396)

7.232

2.548

(S. Plouffe 1988)

3.14159265258264612520603717964402237155787798317

(2143/22)^(1/4)

8.996

2.7009

(S. Ramanujan)

3.14159265359120552148305839712274875168668714821

log(28102/1277)*125/123

11.850

2.664

(S. Plouffe 1988)