Tamura-Kanada Circuit Method

Algorithm

The following algorithm can be used to calculate $\pi$ (pi):

$(1): \quad$ Set $A = X = 1$, set $B = \dfrac 1 {\sqrt 2}$, and set $C = \dfrac 1 4$

$(2): \quad$ Set $Y = A$.

$(3): \quad$ Set $A = \dfrac {A + B} 2$

$(4): \quad$ Set $B = \sqrt {B Y}$.

$(5): \quad$ Set $C = C - X \paren {A - Y}^2$

$(6): \quad$ Set $X = 2 X$

$(7): \quad$ Output $\dfrac {\paren {A + B}^2} {4 C}$ as an approximation to $\pi$.

$(8): \quad$ For a better approximation to $\pi$, set $Y$ equal to the output, return to step $(2)$ and continue.

Example

Starting with $A = X = 1$, $B = \dfrac 1 {\sqrt 2}$, $C = \dfrac 1 4$, the successive values of $\dfrac {\paren {A + B}^2} {4 C}$ on the first $3$ loops are:

\(\text {(1)}: \quad\)

\(\ds \)

\(\)

\(\ds 2 \cdotp 91421 \, 35\)

\(\text {(2)}: \quad\)

\(\ds \)

\(\)

\(\ds 3 \cdotp 14057 \, 97\)

\(\text {(3)}: \quad\)

\(\ds \)

\(\)

\(\ds 3 \cdotp 14159 \, 28\)

and it is seen that the value for $\pi$ is already correct to $6$ decimal places.

Proof

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Source of Name

This entry was named for Yoshiaki Tamura and Yasumasa Kanada.

Historical Note

This algorithm was used by Yasumasa Kanada, Sayaka Yoshino and Yoshiaki Tamura to calculate $\pi$ (pi) to $16 \, 777 \, 216$ digits in $1983$.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41971 \ldots$