A competition to find the best approximations for pi
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       This is an open-ended competition to find the best possible approximations to pi (about 3.14159 26535 89793 23846 26433). A good approximation would be an expression which matches pi to more significant digits than the number of digits contained in that expression. For example, the most common approximation to pi is 22/7 which is about 3.14286. This contains 3 digits, and matches the first 3 digits of pi, so it is a fair approximation. By comparison, the expression

√√√√√√√√√√√√√√√√√√√√√√√√√√√√8 + √√√√√√√√√√√√14 + √√√√√68

contains 5 digits, but matches pi to 9 decimal places, so it is considered an outstanding approximation.
       We will consider 4 types of approximations. The first type uses only the mathematical operations of addition, subtraction, multiplication, division, and square root. It may not use other operations such as decimal fractions, exponentiation, logarithms, or any trig functions. For example, 355/113 (approximately pi+0.00000 0266) would be a valid expression, but 2 arcsin(1) would not. These approximations can be constructed with ruler and compass.
       The second type adds exponentiation and higher roots to the list of allowed operators. For example 710/17 would be a valid expression containing 5 digits, and evaluating to about pi-0.000241.
       The third type approximates pi as the root of a polynomial. So far only polynomial equations with integer coefficients have been considered.
       The fourth type approximates pi as the root of a more general equation.
       The 4 tables below show the best approximations that I have found for each of the 4 types.
       Here √19 is the square root of 19, and 3√52 is the cube root of 52. In counting digits, there are two digits in √19 and 3 digits in 3√52.
       In the equations, I count only the digits in the coefficients, and not the digits in the exponents, which are standard. A coefficient of 1 or -1 is counted as 1 digit, even though the 1 is not normally written. This avoids "sparse" polynomials with many terms, most of them having +1 or -1 as the coefficient.
       Anyone who wishes is invited to contribute to this effort, and all worthy results will be posted, with credit. The best approximations, where the number of digits of accuracy exceed the number of digits in the expression by at least 3, have been highlighted in yellow.
       Approximations using only small numbers will be considered superior to those using larger numbers. Please try to avoid any number with more than 5 digits.


Type 1 - Fractions and Square Roots
Digits Expression Value Contributor
1 3 pi - .142 Ancient
2 √10 pi + .0206 Ancient
2 √(7+√8) pi - .00656 Jean Jacquelin
3 22/7 pi + .00126 Ancient
3 √(2+√62) pi + .000701 Frank Rubin
3 √(6+√15) pi + .000538 Frank Rubin
3 √(8+√7/2) pi + .000195 Frank Rubin
3 √51 - 4 pi - .000164 ?
3 √(7+√(6+√5)) pi + .00003 99 Jean Jacquelin
4 √63 - √23 pi - .000170 Frank Rubin
4 √14 - 3/5 pi + .00006 47 Frank Rubin
4 9/5 + √9/5 pi + .00004 81 Brian Astle
4 √(7+√(9-√(2-√2))) pi + .00000 0103 Jean Jacquelin
5 √(227/23) pi - .00000 624 Frank Rubin
6 355/113 pi + .00000 0266 Tsu Ch'ung Chi 450AD
7 √527 - √354 - 1 pi - .00000 0190 Frank Rubin
7 √73 - √29 + 8/43 pi + .00000 00839 Frank Rubin
7 43/(√425-√48) pi + .00000 00479 Frank Rubin
8 √61/16 + √41/29 pi + .00000 00859 Frank Rubin
8 22/7 - 6/4745 pi + .00000 00003 32 Brian Astle
9 (√179-√97)/(√59-√43) pi + .00000 00001 08 Frank Rubin
10 √238+√668-√1454 pi + .00000 000372 Frank Rubin
10 √2471-√380-√733 pi + .00000 000198 Frank Rubin
10 (√2925-√73)/(√29+√83) pi - .00000 00009 02 Frank Rubin
10 (√1781+√17)/(√23+√99) pi + .00000 00005 33 Frank Rubin
11 √(525665/53261) pi - .00000 00000 192 N.V. Subramanyan


Type 2 - Higher Roots
Digits Expression Value Contributor
2 √√97 pi - .00330 Jean Jacquelin
2 2 + √√√√8 pi - .00280 Frank Rubin
3 3√31 pi - .000212 ?
3 √√√√√√√√2 + √√√√√√√√√√√√√√5 + √√√√8 pi + .00000 549 Frank Rubin
3 √√√7 [√√3+√√√3] pi - .00000 421 Jean Jacquelin
3 √ [√√√√2 / (√√√√5 - √√√√√√√√√√√√√√√√√8)] pi - .00000 143 Jean Jacquelin
4 5√306 pi - .00004 04 Frank Rubin
4 8 [√√√(7^3)-√√8] pi + .00000 0727 Jean Jacquelin
4 √√√√2 + √√√√√7 + √√√√√√√78 pi + .00000 0452 Frank Rubin
4 √√√√√√√√√√√√2 + √√√√√√√√√√√√7 +
√√√√√68 		pi - .00000 00884 Frank Rubin
4 (√√√√√√√√√√2 + √√√√√√√√√3) /
(√√8 - √√√√2) 		pi + .00000 00056 Jean Jacquelin
4 √√√√√√√√√√2 + √√√√√5 +
√√√√√(2+√√√√√√√7) 		pi - .00000 000409 Frank Rubin
4 √√√√√√√√√√√√√√√√√√3 +
√√√√√√√7 + √√√√(8-√√√8) 		pi + .00000 00006 70 Frank Rubin
5 2 + √√√√√√4794 pi - .00000 0680 N.V. Subramanyan
5 √√(767 / √62) pi + .00000 00513 Jean Jacquelin
5 √√√√√√√√√√√√√√6 +
√√√√√√√√√√√√√√√√√√45 + √√√√√69 		pi + .00000 000453 Frank Rubin
5 √√√√√√√√√√√√√986 + √√√√5 + √√√√√3 pi - .00000 000452 Joel Li
5 √√√√√√√√√√√√√√√√436 + √√√√8 +
√√√√√√√√2 		pi - .00000 000202 Joel Li
5 √√√√√√√√√√√√√√√√√√√√√√√√√√√√8 +
√√√√√√√√√√√√14 + √√√√√68 		pi - .00000 00002 13 Frank Rubin
5 √√√√√√√√√√√√√√√√5 +
√√√√√√√√√(3+√√√√√√√√√√2) +
√√√√(7+√√√√√√√√7) 		pi - .00000 00000 213 Frank Rubin
6 (7^7+√√√5) / (4^9-√3) pi + .00000 000130 Jean Jacquelin
6 √√2143/22 pi + .00000 000107 Frank Rubin
6 √√√√√√√√√√√√√√√√√√√√√√96 + √√√√√67 + √√√√√√√√√√√11 pi + .00000 00004 30 Joel Li
6 √√√√√√√√√√√√√√√34 + √√√√√69 +
√√√√√√√√√√√√√√√√√√71 		pi - .00000 00004 01 Frank Rubin
6 √√√√√√√√√√√√14 + √√√√√65 +
√√√√√√√√√√√√√√√√√√√√√√√√√√√√√72 		pi + .00000 00000 0593 Frank Rubin
6 √√√√√√√√√√(4+√√√√5) +
√√√√√√√√√√(3+√√√√√√√√6) +
√√√√(9-√√√√√√√8) 		pi - .00000 00000 00775 Frank Rubin
6 √√√√√√√√√√(6+√√√√√√√8) + √√√√(7√√√√7) +
√√√√√√√√√√(7-√√√√√√5) 		pi - .00000 00000 00219 Frank Rubin
7 3√(31+3/478) pi + .00000 00179 Frank Rubin and N.V. Subramanyan
7 √√√√√√√√√√√√8 +
√√√√√√√√√√√√√√√√7777 + √√√√√68 		pi + .00000 00000 796 Joel Li
9 √√√√√√√√√√√√√√√√√21777 + √√√√√60 + √√√√√√√√√13 pi + 0.00000 00000 0119 Joel Li
9 3√(31+25/3983) pi + .00000 00001 50 Frank Rubin and N.V. Subramanyan
10 9√√888582403 pi - .00000 00000 140 Rory Kulz
10 5 - 25√5352083 pi - .00000 00000 111 N.V. Subramanyan
11 √√(5√(8769956796)) pi - .00000 00000 0148 Nick McGrath
13 √√(6√(854273519914)) pi + .00000 00000 000172 Nick McGrath
14 √√√(199002961/20973) pi - .00000 00000 000952 N.V. Subramanyan
16 √√√√√8105800789910710 pi + .00000 00000 00000 005 Sudipta Das
5 43 ^ 7/23 pi - .00005 28 Frank Rubin
9 97 ^ 272/1087 pi + .00000 00006 78 Frank Rubin


Type 3 - Roots of Polynomials
Digits Polynomial Value Contributor
3 x3-7x-9 pi+.000669 Frank Rubin
4 x5-99x+5 pi+.00000 519 Frank Rubin
4 x9-31x6-6 pi+.00000 121 Frank Rubin
5 7x3-15x2-69 pi-.00000 114 Frank Rubin
5 10x4-99x2+3 pi+.00000 0121 Frank Rubin
5 9x5-877x+1 pi+.00000 0116 Frank Rubin
5 x12-99x8+5x7-6 pi+.00000 00658 Frank Rubin
6 34x2-4x-323 pi+.00000 0854 Frank Rubin
6 9x3-9x2-44x-52 pi-.00000 0158 Frank Rubin
7 x8-x6-x-8524 pi-.00000 00106 N.V. Subramanyan
7 8x3+54x2+7x-803 pi-.00000 00005 55 Frank Rubin
8 x11-x5-293898 pi+.00000 000166 N.V. Subramanyan
8 x8+x5-x4-x-9694 pi-.00000 00005 90 N.V. Subramanyan
8 8x4-37x2+77x-656 pi-.00000 00002 89 Frank Rubin
8 9x5+7x4-15x3-946x+1 pi-.00000 00000 505 Frank Rubin
8 2x9-4x3+9x2-59583 pi-.00000 00000 0128 Frank Rubin
9 85x3-69x2-597x-79 pi-.00000 00000 288 Frank Rubin
9 x14-x12-8197902 pi-.00000 00000 0622 N.V. Subramanyan
9 4x5+77x3+81x2-4411 pi+.00000 00000 0263 Frank Rubin
9 2x8-55x3+2x-17278 pi+.00000 00000 006817 Frank Rubin
10 3x12-73x9-x7-593723 pi+.00000 00000 00178 Frank Rubin
10 33x5+x4-74x2-5x-9450 pi-.00000 00000 000172 Frank Rubin
10 6x6-4x5+5x4+2x3-2x2+3x-5083 pi-.00000 00000 00003 16 Jean Jacquelin
11 18x4+39x3-67x2-744x+36 pi+.00000 00000 000254 Frank Rubin
11 6x5-8x4-41x2+85x2-53x-458 pi+.00000 00000 00240 Frank Rubin
11 4x11-22x10-90x6+969966 pi+.00000 00000 000158 Frank Rubin
12 76x4-11x3+51x2+68x-7779 pi+.00000 00000 000580 Frank Rubin
12 14x5+8x4+21x3-7x2-96x-5344 pi+.00000 00000 00001 26 Frank Rubin
13 40x5+x4-4x3+25x2-70x-12241 pi+.00000 00000 00000 497 Frank Rubin
14 66x5-6x4-29x3+57x2-44x-19138 pi+.00000 00000 00000 293 Frank Rubin
15 79x5+70x4-56x3+87x2-60x-29928 pi+.00000 00000 00000 046 Frank Rubin
16 87x5+15x4+22x3+13x2-560x-27136 pi-.00000 00000 00000 079 Frank Rubin
The 13 to 16 digit approximations are pushing the limit of extended precision
arithmetic on my PC, so may not be fully accurate.

Type 4 - Roots of General Equations
Digits Equation Value Contributor
2 √(4e-1) pi+.000561 Alexander R. Povolotsky
3 ex-x=20 pi+.00004 06 Frank Rubin
7 x5+x4=e6 pi+.00000 0029 Jean Jacquelin, based on an approximation by Simon Plouffe


Pandigital: uses all digits once each
Digits Equation Value Contributor
9 3.14 + (7-.9^-6+2/8)5 pi-.00000 00000 93 Joel Karnofsky
9 3 + (2^((((5+8)/4)^-6)*9))/7.1 pi-.00000 00000 281 Sergey Ioffe
9 2^5^.4 - .6 - ((.3^9)/7)^.8^.1 pi+.00000 00000 00660 Richard Hess
10 3 + 4/28 - 1/(790 + 5/6) pi+.00000 00003 32 Brian Astle
10 3 + (5^-8^-((10+2-6-9)/4))/7 pi-.00000 00000 193 Sergey Ioffe


Simon Plouffe uses a measure R2 for the effectiveness of a pi approximation which is the number of digits of accuracy divided by the number of digits in the largest number in the expression. Values over 2 are considered best. Here are some approximations that score well on this scale.
Digits R2 Equation Value Contributor
6 4.574 8/3+3/5-1/8 pi+.00007 40 Frank Rubin
9 5.502 5/8+5/9+8/7+9/11 pi+.00000 186 Frank Rubin
11 4.592 42/25+49/33-1/43 pi+.00000 00173 Frank Rubin
12 4.690 73/39+92/73+1/105 pi+.00000 00003 32 Frank Rubin
12 3.796 427/596+405/167 pi+.00000 00000 291 Frank Rubin
47 10** -1/3-4/5-4/7-9/11+9/13+13/17+16/19+16/23
+14/29+26/31+19/37+14/41+2/43+21/47 		** (error < 10-18) Simon Plouffe
** Beyond the accuracy of my computer


LINKS
Robert Chin's Pi Approximation webpage.
Simon Plouffe's Pi Approximation webpage.
Carlos Rivera's Pi Approximation webpage.
Sage Weil's Pi Approximation webpage.
Eric Weisstein's Pi Approximation webpage.


David Blatner's The Joy of Pi webpage.
Ramon Lloren's Pi Webpage.
Martin Rebas's Pi Approximation Day webpage.


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