Find arctan formulae

A Go tool find_arctan_relations that searches for Machin‑style arctan formulas for π by selecting arctan(1/x) terms according to prime‑factor norms and applying the PSLQ integer‑relation algorithm.

This reposistory also contains score_arctan.py which is used to process the log files generated by find_arctan_relations.

There is an accompanying paper which explains how this program works which can be found on arXiv at https://arxiv.org/abs/2508.08307

The full set of results found with this program are available at https://doi.org/10.5281/zenodo.16758216

The top results are available in the results-by4-top.txt and results-by4-top.csv files in this repository.

Here is an example of formulae found this this program which are the lowest known Lehmer measure relations with 5 and 6 terms.

$$\begin{alignat*}{2} \tfrac{\pi}{4} &= 29\,\arctan\left(\tfrac{1}{68}\right) &+& 42\,\arctan\left(\tfrac{1}{117}\right) \\\ & -15\,\arctan\left(\tfrac{1}{2675143}\right) &-& 5 \,\arctan\left(\tfrac{1}{2976163}\right) \\\ & -5\,\arctan\left(\tfrac{1}{302342643}\right) \\\ \lambda &= 1.4572 \\\ \tfrac{\pi}{4} &= 83\,\arctan\left(\tfrac{1}{107}\right) &+& 17\,\arctan\left(\tfrac{1}{1710}\right) \\\ & -44\,\arctan\left(\tfrac{1}{225443}\right) &-& 68\,\arctan\left(\tfrac{1}{2513489}\right) \\\ & +22\,\arctan\left(\tfrac{1}{42483057}\right) &+& 34\,\arctan\left(\tfrac{1}{7939642926390344818}\right) \\\ \lambda &= 1.3291 \end{alignat*}$$

Features

• Groups primes (excluding 2) into combinations of configurable size.
• Computes norms of potential arctan terms y/x and filters by size.
• Runs PSLQ to find integer relations combining π and selected arctan terms.
• Supports evolving prime‑groups to discover new formulas.
• Caches intermediate values and runs in parallel.

Requirements

find_arctan_relations

• Go 1.23 or newer
• Dependencies (fetched via go get):
  • github.com/ncw/pslq
  • github.com/hashicorp/golang-lru/v2

score_arctan.py

• Python 3.12 (tested with this version, likely works with others)
• Optionally sympy if the prime groups need to be displayed.

Installation

Download the latest release or build from source:

git clone https://github.com/ncw/find_arctan_formulae.git
cd find_arctan_formulae
go build
./find_arctan_formulae -help
python score_arctan.py --help

Testing

go build
go test -v
./regression.sh

Usage for find_arctan_relations

./find_arctan_formulae [flags]

Usage of ./find_arctan_formulae:
  -algorithm int
    	PSLQ: which algorithm to use. 1: orig, 2: pslqm2 (default 2)
  -alpha uint
    	 Don't output arctan terms bigger than 1/alpha (default 1)
  -arctan-cache-size int
    	Cache this many arctan values (default 10000)
  -candidate-cache-size int
    	Cache this many candidate norm lists (default 64)
  -cpuprofile string
    	Write cpu profile to file
  -evolve int
    	If set, use this many groups to evolve new relations from
  -group string
    	A comma or space separated list of primes to use as the prime group
  -group-cache-size int
    	Cache this many group values (default 10000)
  -group-extra-primes value
    	Number of extra primes to add to a group iteration
  -groups-file string
    	a file of comma or spare separated groups, one per line
  -iteration-type string
    	Iteration type normal|random|weighted (default "normal")
  -iterations int
    	PSLQ: number of iterations to use max (default 1000000)
  -log-max-coeff uint
    	PSLQ: log2(max coefficient size) (default 64)
  -max-fails int
    	Stop PSLQ with this many failures in a row (default 1000)
  -max-group-length int
    	Maximum number of members for prime groups (default 10)
  -max-items int
    	Maximum number of entries to run PSLQ on (default 20)
  -max-lehmer-measure float
    	Don't print formulae with a Lehmer measure greater than this
  -max-no-new int
    	Stop PSLQ if no new results for this many iterations (default 10000)
  -max-numerator uint
    	arctan terms should have numerator <= this (default 1)
  -max-prime value
    	Maximum prime factor to consider for groups (default 5000)
  -min-group-length int
    	Minimum number of members for prime groups (default 5)
  -min-items int
    	Minimum number of items required to run PSLQ on (default 2)
  -norm-cache-size int
    	Cache this many norms lists (default 100000)
  -prec uint
    	PSLQ: precision to use (bits) (default 1024)
  -pslq-verbose
    	Cause the PSLQ algorithm to print lots of info
  -skip-relations
    	if set skip PSLQ on any relations which contain previously found relations
  -skip-zeros
    	if set skip PSLQ on any relations which contain previously found zero relations
  -sort-max-x
    	If set sort by max X rather than Lehmer measure when evolving
  -target-precision float
    	PSLQ: target precision of the result as fraction of prec (default 0.75)
  -trim-start
    	If set, trim PSLQ list from start if too big
  -verbose
    	Print more info while running
  -workers int
    	Use this many threads for the search (default 16)

Example

Find the best known 5-term relation from the prime group given

./find_arctan_formulae -max-fails 100 -min-items 1 -max-items 20 -alpha 50 -max-prime 100_000 -max-fails 1000 -max-lehmer-measure 1.86 -group "5,13,37,24113,76369"

Results are printed as Lehmer measure, prime list and the formula line:

# 1.4571904502 Primes [5 13 37 24113 76369] from group [5 13 37 24113 76369]
+1 * pi = +116 * arctan(1/68) +168 * arctan(1/117) -60 * arctan(1/2675143) -20 * arctan(1/2976163) -20 * arctan(1/302342643) 

Find record breaking formulae

./find_arctan_formulae -evolve 1000 -skip-relations -trim-start -max-items 30 -max-no-new 100_000 -max-fails 1000 -alpha 10 -max-prime 2_000_000 -max-lehmer-measure 1.86 -max-numerator 2 | tee results.log

After that has run for a few days use this to process the results

python -u score_arctan.py --group-by-terms --max-lehmer-measure 1.86 --purify | tee results.txt

Add --alternate 1000 to generate alternate completions using numbers up to 1000 bits.

Usage for score_arctan.py

This program is used for finding and organising the relations found in the output of the find_arctan_formulae program.

It can also "purify" relations with arctan(a/b) where a != 1 to arctan(1/b) terms with the --purify flag.

It can also extend relations using the --alternate flag.

python -u score_arctan.py [flags] file.log file2.log

usage: score_arctan.py [-h] [--max-lehmer-measure MAX_LEHMER_MEASURE] [--show] [--groups] [--pure] [--purify] [--verify] [--group-by-terms] [--top TOP]
                       [--save-groups SAVE_GROUPS] [--find-double-checks] [--alpha ALPHA] [--by4] [--alternate ALTERNATE]
                       [inputs ...]

Score lines containing multiple arctan(a/b) terms

positional arguments:
  inputs                Input files (reads stdin if none provided)

options:
  -h, --help            show this help message and exit
  --max-lehmer-measure MAX_LEHMER_MEASURE, -m MAX_LEHMER_MEASURE
                        Maximum Lehmer measure; skip lines above this
  --show, -s            Show source files for each relation
  --groups, -g          Show prime groups for each relation
  --pure, -u            Only include relations where numerator a == 1
  --purify, -P          Try to make numerator a == 1 if possible
  --verify, -x          Check to see the relations are correct
  --group-by-terms, -G  Group the output by number of terms, lowest first
  --top TOP, -n TOP     Only show this many results
  --save-groups SAVE_GROUPS, -S SAVE_GROUPS
                        Save prime groups found to this file, one per line
  --find-double-checks  If set find the best relation and double check pairs
  --alpha ALPHA, -a ALPHA
                        Only show relations with terms <= 1/alpha
  --by4, -4             If set output pi/4 = versions
  --alternate ALTERNATE, -A ALTERNATE
                        Calculate alternate endings of numbers up to this many bits

Example

Find pure relations only with Lehmer measure < 1.60 and group them by terms, verifying them in the process

python -u score_arctan.py --verify --pure --group-by-terms --groups --max-lehmer-measure 1.60 *.log  | tee results.txt

Find and purify all relations with Lehmer measure < 1.86

python -u score_arctan.py --group-by-terms --max-lehmer-measure 1.86 --purify *.log | results-1.86.txt

Extend all relations found up to terms up to 1000 bits

python -u score_arctan.py --group-by-terms --max-lehmer-measure 1.86 --purify --alternate 1000 *.log | results-1.86-extended.txt

License

MIT License.