# Calculating π using the Gregory Leibniz series until n terms

based off my previous challenge, this wikipedia article, and a Scratch project

Your task: given i, calculate $$\\pi\$$ till i terms of the Gregory-Leibniz series.

The series:

$$\pi=\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}-...$$

Here, 4/1 is the first term, -4/3 is the second, 4/5 is the second and so on.

Note that for the nth term,

• $$\text S_n = \frac{4 \times (-1)^{n+1}}{2n-1}$$
• $$\pi_n = \text S_1 + \text S_2 + ... + \text S_n,$$ where $$\\pi_n\$$ is $$\\pi\$$ approximated to $$\n\$$ terms.

Test cases:

In - Out
1 - 4
2 - 2.66666667
3 - 3.46666667
4 - 2.8952381


Floating point issues are OK.

You may not calculate infinite terms of pi using this as we are calculating a number rather than terms of a series here.

This is , so shortest answer wins!

EDIT: It's strange that this question got some new... activity.

• You may not calculate infinite terms of pi [...] -> Does that mean that we can't use standard sequence I/O? If so, I don't understand why. Sep 19, 2022 at 13:20
• @Arnauld I think what he meant to say is that the challenge is to output the terms of the Gregory Leibniz sequence, and not the (potentially infinite) digits of pi itself. Sep 19, 2022 at 13:22
• Nope. Here pi is being calculated by adding up the terms of the series. So you add up the terms, and then print the approx. of pi when summing up Sep 19, 2022 at 13:24
• I don't understand why we cannot output $\pi_n$ with standard sequence I/O (this would mean also allowing output of approximations of $\pi$ up to $n$ terms or outputting more approximations without end). Sep 20, 2022 at 5:26
• Normally, I would have let you as you'd be printing terms of a series, but here we are printing approximations. Why should we print all approximations to a certain term? When calculating pi, we aim for the most accurate, not keeping a log of inaccurate periods. Sep 20, 2022 at 7:01

# Python 3, 36 35 bytes

f=lambda n,k=1:n and 4/k-f(n-1,k+2)


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-1 thanks to Mukundan314.

• 35 bytes Sep 19, 2022 at 14:07
• FWIW, we can accelerate the convergence: (not golfed) sagecell.sagemath.org/… Sep 23, 2022 at 12:19

# 05AB1E, 14 9 bytes

L·<z··R®β


-5 bytes porting @JonathanAllan's Jelly answer

0λè®N>m4*N·</+


Explanation:

L          # Push a list in the range [1, (implicit) input]
·         # Double each value: [2,4,6,...,2n]
<        # Decrease each by 1: [1,3,5,...,2n-1]
z       # Calculate 1/z for each: [1,1/3,1/5,...,1/(2n-1)]
4*     # Multiply by 4: [4,4/3,4/5,...,4/(2n-1)]
R    # Reverse the list: [4/(2n-1),...,4/5,4/3,4]
®β  # Convert it from a base-(-1) list to a base-10 integer:
#  (-1)^(n-1)*4/(2n-1) + ... + 1*4/5 + -1*4/3 + 1*4/1
# (after which the result is output implicitly)


The 14-byter is pretty similar as my 05AB1E answer of the previous challenge.

 λ         # Start a recursive environment,
è        # to calculate a(input)
# (which is output implicitly afterwards)
# Where every following a(n) is calculated by:
#  (implicitly push the previous term a(n-1))
®N>m    #  Push (-1) to the power (n+1)
4*  #  Multiply it by 4
N·<     #  Push 2n-1
/       #  Divide the earlier 4*(-1)**(n+1) by this (2n-1)
+      #  Add it to the previous term a(n-1)


# R, 31 28 bytes

Edit: -3 bytes thanks to @Dominic van Essen.

\(n,m=2*1:n-1)sum(4i/1i^m/m)


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Uses $$(-1)^{n+1} = -1 \times (-1)^{n} = i^2\times i^{-2n} = i^{-2n+2} = i \times i^{-(2n-1)}$$ And then $$\text S_n = \frac{4 \times (-1)^{n+1}}{2n-1} = \frac{\frac{4i}{i^{2n-1}} }{2n-1}$$

• Alternative 28 byte version: \(n)sum(4/(1:n*2-1)*c(1,-1)) (generates a warning for odd n, but still computes the correct answer)
– JDL
Sep 20, 2022 at 10:38
• Nevermind, I can't make it work for n=1 :(
– JDL
Sep 20, 2022 at 10:49
• 28 bytes (and works for n=1)... Sep 26, 2022 at 11:43

# Jelly, 8 bytes

RḤ’4÷Ṛḅ-


A monadic Link that accepts a positive integer, $$\n\$$, and yields a floating point number, $$\\pi_n\$$.

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### How?

RḤ’4÷Ṛḅ- - Link: positive integer, n
R        - range n           -> [1,2,3,...,n]
Ḥ       - double            -> [2,4,6,...,2n]
’      - decrement         -> [1,3,5,...,2n-1]
4     - 4
÷    - divide            -> [4/1,4/3,4/5,...,4/(2n-1)]
Ṛ   - reverse           -> [4/(2n-1),...,4/5,4/3,4/1]
- - -1
ḅ  - convert from base -> (-1)^(n-1)*4/(2n-1)+...+1*4/5+-1*4/3+1*4/1 = pi_n


# C (gcc), 63 60 bytes

float f(n){float m=4,p=1,d=1;for(;n--;d+=2,m=-m)p+=m/d;--p;}


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Saved 3 bytes thanks to jdt!!!

• float f(n){float m=4,p=1,d=1;for(;n--;d+=2,m=-m)p+=m/d;--p;}
– jdt
Sep 19, 2022 at 14:23
• @jdt Nice one - thanks! :D Sep 19, 2022 at 14:40

# Excel, 51 48 bytes

Saved 3 bytes thanks to some clever math from JvdV

=LET(s,SEQUENCE(A1),SUM((8*ISODD(s)-4)/(2*s-1)))


Input is in the cell A1. Output is wherever the formula is.

• LET(s,SEQUENCE(A1) defines s to be an array of numbers from 1 to the value in A1. This is what lets us calculate the first n terms.
• (8*ISODD(s)-4) creates an array of the same sized filled with 4, -4, 4, -4, .... The only trick is that ISODD() returns a boolean and Excel turns TRUE into 1 and FALSE into 0 when you force it to do math. That means it becomes an array of 8-4, 0-4, 8-4, 0-4, ... which then simplifies as shown above.
• (2*s-1) creates an array of the same size filled with 1, 3, 5, 7, ...
• SUM((~)/(~)) divides the first array by the second (giving us each term in the sequence) and then adds them up. • Nice. If you'd rely on math you can scratch the IF() saving two bytes using =LET(s,SEQUENCE(A1),SUM((-4+8*ISODD(s))/(2*s-1)))
– JvdV
Oct 1, 2022 at 10:33

# Python 3, 41 39 bytes

f=lambda x:x and(x%2*4-2)/(x-.5)+f(x-1)


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# Desmos, 30 bytes

f(n)=∑_{k=1}^n4(-1)^k/(1-2k)


Pretty self-explanatory.

Try It On Desmos!

Try It On Desmos! - Prettified

• Congrats - This is my first time seeing a desmos answer... Sep 28, 2022 at 3:39
• @py3programmer If you enjoy them, I got many more! Sep 28, 2022 at 3:43
• It is true... but I suppose Desmos only works on math-related questions... Sep 28, 2022 at 3:50
• @py3programmer Yeah true, Desmos works really well on math-related decision problems especially. Sep 28, 2022 at 3:53
• True, because, after all, Desmos is a graphing calculator, but I'm surprised it can be used as a code-golfing language! Sep 28, 2022 at 3:54

# Vyxalḋ, 10 bytes

4ÞNÞ∞d‹/¦i


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4          # 4
ÞN        # [4, -4, 4, -4, 4, -4...
Þ∞      # [1, 2, 3, 4, 5, 6...
d     # [2, 4, 6, 8, 10, 12...
‹    # [1, 3, 5, 7, 9, 11...
/   # [4/1, -4/3, 4/5, -4/7...
¦  # Take the cumulative sums of that
i # Index the input into that

• "You may not calculate infinite terms of pi using this as we are calculating a number rather than terms of a series here." Sep 19, 2022 at 21:08

f n=foldr((-).(2/))0[1/2..n]


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# Japt-mx, 11 bytes

4*JpU /(UÑÄ


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## Without Flags, 11 bytes

Ç±Z4/°ZÃÔìJ


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4*JpU /(UÑÄ     :Implicit map of each U in the range [0,input)
4*              :Multiply 4 by
J             :-1
pU           :Raised to the power of U
/(        :Divide by
UÑ      :U times 2
Ä     :Plus 1
:Implicit output of sum of resulting array

Ç±Z4/°ZÃÔìJ     :Implicit input of integer U
Ç               :Map each Z in the range [0,U)
±Z             :  Increment Z by itself
4/           :  Divide 4 by
°Z         :  Z, prefix incremented
Ã        :End map
Ô       :Reverse
ì      :Convert from digit array in base
J     :  -1


# Vyxalḋ, 13 bytes

Flag ḋ prints out rationals in their decimal form

ƛ4nd‹/un›e*;∑


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Explanation:

ƛ4nd‹/un›e*;∑
ƛ          ;    Map trough range of input
4nd‹/          Divide four by 2n-1
un›e*     Raise -1 to the n+1 and multiply
∑   Sum the fractions


Another 13 byter: ƛ4nd‹/k-ni*;∑

# Factor + koszul math.unicode, 51 bytes

[ [1,b] [ dup 1 + -1^ 4 * swap 2 * 1 - / ] map Σ ]


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• [1,b] [ ... ] map Σ Map each integer in 1..input using [ ... ] and add them together.
• dup 1 + -1^ 4 * swap 2 * 1 - / Just a straight translation of $$\text S_n = \frac{4 \times (-1)^{n+1}}{2n-1}$$ Here's how the data stack looks arriving at the second term to add together:
       ! 2
dup    ! 2 2
1      ! 2 2 1
+      ! 2 3
-1^    ! 2 -1
4      ! 2 -1 4
*      ! 2 -4
swap   ! -4 2
2      ! -4 2 2
*      ! -4 4
1      ! -4 4 1
-      ! -4 3
/      ! -1-1/3


# Fortran (GFortran), 60 bytes

function a(j)
do i=1,j
a=a+4.0*(-1)**(i+1)/(2*i-1)
enddo
end


Try it online!

Mostly just exploiting implicit typing so no need to declare variable types. Also saves 1 byte by removing the space between end and do.

• Welcome to Code Golf, and nice answer! Sep 20, 2022 at 14:53

# Husk, 10 9 bytes

-1 thanks to a clever rearrangement by Dominic van Essen

Ḟ-↑M/İ14


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### Explanation

Husk's reversed operand order for subtraction really hurts here, but at least Dominic found a nice workaround for the reversed division.

Ḟ-↑M/İ14
İ1   Infinite list of positive odd numbers
4  Integer 4
M      For each element in the first argument,
/     divide the second argument by it
↑       Take the first N elements, where N is the program's argument
Ḟ          Right-fold the resulting list
-        on subtraction

• ...but you can avoid one of the reversed operands here by using M instead of m for 9 bytes... Aug 23 at 15:51
• (I had to golf this just to let it beat Fig... Aug 23 at 15:52

# C (gcc), 47 bytes

float f(n){return--n?(4.-n%2*8)/(n-~n)+f(n):4;}


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# APL (Dyalog Unicode), 18 bytes SBCS

(4×¯1*⍳)+.÷(1+2×⍳)


Try it on APLgolf!

A tacit function.

# Charcoal, 14 bytes

ＩΣＥＮ×Ｘ±¹ι∕⁴⊕⊗ι


Try it online! Link is to verbose version of code. Explanation:

   Ｎ            Input n as a number
Ｅ             Map over implicit range
¹        Literal integer 1
±         Negated
Ｘ          Raised to power
ι       Current value
×           Multiplied by
⁴     Literal integer 4
∕      Divided by
ι  Current value
⊗   Doubled
⊕    Incremented
Σ              Take the sum
Ｉ               Cast to string
Implicitly print


f i=g[1,3..i*2]
g(h:t)=4/h-g t
g[]=0


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I think it's quite self descriptive due to Haskell feature of describing something.

I try to visualize what g does, pretty simple : taking a list of i denominators but all positive...

[1,      3,      5,      7]
4/7
4/5 -  (4/7)
4 - ( 4/3 - ( 4/5 -  (4/7)))

hence the signs result alternating
4 -   4/3 +   4/5 -   4/7


# Wolfram Language (Mathematica), 26 bytes

Tr[4I^#/I/#&[2Range@#-1]]&


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Unnamed function that takes a positive integer argument and returns an exact fraction (the TIO footer shows both the output fraction and its decimal equivalent). Managed to beat the super straightforward Sum[-4(-1)^j/(2j-1),{j,#}]& by one byte by using powers of the complex unit I instead of powers of -1.

# x86 32-bit machine code, 23 bytes

D9 EE D9 E8 D9 E8 D8 C0 DC E9 D8 F1 DE E2 E2 F4 DD D8 D8 C0 D9 E1 C3


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Following the fastcall calling convention, this takes a number n in ECX and returns the sum of the first n terms on the FPU register stack.

This code creates −2/1, −2/3, −2/5, ..., and combines them using the reverse-subtract instruction. This is equivalent to negating the running total before adding each term; this handles the alternating signs, but leaves the result with the wrong sign for odd n, which is corrected for by taking the absolute value at the end (because all the correct results are positive). The value is also doubled at the end.

In assembly:

                        # Example execution for i=2
# FPU register stack (left is bottom):
f:  fldz                # 0
fld1                # 0   1
r:  fld1                # 0   1   1
fadd st(0), st(0)   # 0   1   2
fsub st(1), st(0)   # 0  -1   2
fdiv st(0), st(1)   # 0  -1  -2/1
fsubrp st(2), st(0) # -2/1  -1
loop r              #[The loop executes again:
# -2/1  -1   1
# -2/1  -1   2
# -2/1  -3   2
# -2/1  -3  -2/3
# -2/3+2/1  -3            ]
fstp st(0)          # -2/3+2/1
fabs                # -4/3+4/1   (but for odd n, this fixes the sign)
ret


# BQN, 9 bytes

-´4÷1+2×↕


Try it at BQN online!

### Explanation

-´4÷1+2×↕
↕  Range [0, n)
2×   Double each number
1+     Add 1 to each number
4÷       Divide 4 by each number
-´         Right fold on subtraction


# MATLAB, 34 bytes

f=@(n)sum(4./[1:4:2*n,-(3:4:2*n)])


An anonymous function that makes an array of [ 1 5 9 ... -3 -7 -11 ...], member-wise divides 4 by the array and then sums the result.

Test:

arrayfun(f, 1:4)
ans = 4.0000    2.6667    3.4667    2.8952


# Rust, 62 bytes

fn f(n:f32,k:f32)->f32{if n==0.{return 4./k}4./k-f(n-1.,k+2.)}


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## Math++, 49 bytes

?>x
1>a
1>b
k+b*4/a>k
a+2>a
-b>b
x-1>x
4+5*!x>$k  # Raku, 26 bytes [\+] (4,-*...*)Z/(1,3...*)  Try it online! This is an expression for the infinite sequence of partial sums. • (4, -* ... *) is the sequence 4, -4, 4, -4, .... • (1, 3 ... *) is the sequence 1, 3, 5, .... • Z/ zips those two sequences together using division. • [\+] produces the sequence of partial sums. # ><>, 33 bytes 0$\~n;
(?\:2%2*1-4*$:}2*1-,+$1-:1


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# TI-Basic, 22 bytes

sum(4/seq((1-2I)(-1)^I,I,1,Ans


Takes input in Ans. The first - is the subtraction sign, and the second is the negative sign. Only works for inputs less than or equal to 999, or 99 on a TI-83. For larger inputs, use this:

### 29 bytes

Input N
For(I,1,N
Ans+4(-1)^I/(1-2I
End
Ans


The first - is the negative sign, and the second is the subtraction sign. Assumes being run on fresh/RAM cleared calculator.

# APL (Dyalog Unicode), 10 bytes

4×-/÷1+2×⍳


Try it on TryAPL.org!

The environmental variable ⎕IO←0.

A tacit function.

# Pyth, 11 bytes

*4smc^_1dhy


Test suite

##### Explanation:
*4smc^_1dhy   | Full code
*4smc^_1dhydQ | with implicit variables
--------------+
m        Q | For each d from 0 to input:
c^_1dhyd  |  (-1)^d/(2d+1)
s           | Sum the results
*4            | Print sum * 4
`