Pi

In diagram:

Point ${\displaystyle P}$ has coordinates ${\displaystyle (1,t).}$

Point ${\displaystyle R}$ has coordinates ${\displaystyle ({\sqrt {1+t^{2}}},0).}$

Mid point of ${\displaystyle PR,\ M}$ has coordinates ${\displaystyle ({\frac {1+{\sqrt {1+t^{2}}}}{2}},{\frac {t}{2}}).}$

${\displaystyle \tan {\frac {A}{2}}={\frac {t}{2}}/{\frac {1+{\sqrt {1+t^{2}}}}{2}}={\frac {t}{1+{\sqrt {1+t^{2}}}}}}$ ${\displaystyle ={\frac {t}{1+{\sqrt {1+t^{2}}}}}\cdot {\frac {1-{\sqrt {1+t^{2}}}}{1-{\sqrt {1+t^{2}}}}}}$ ${\displaystyle ={\frac {t(1-{\sqrt {1+t^{2}}})}{1-(1+t^{2})}}}$ ${\displaystyle ={\frac {t(1-{\sqrt {1+t^{2}}})}{-t^{2}}}}$ ${\displaystyle ={\frac {-1+{\sqrt {1+t^{2}}}}{t}}}$

• ${\displaystyle \tan {\frac {A}{2}}={\frac {\tan(A)}{1+{\sqrt {1+\tan ^{2}(A)}}}}={\frac {-1+{\sqrt {1+\tan ^{2}(A)}}}{\tan(A)}}}$

• ${\displaystyle \tan(2A)={\frac {2\tan(A)}{1-\tan ^{2}(A)}}}$

This section calculates five values of ${\displaystyle \pi }$ using the following known values of ${\displaystyle \tan(A):}$

Angle ${\displaystyle A}$	${\displaystyle \tan(A)}$
${\displaystyle 45^{\circ }}$	${\displaystyle 1}$
${\displaystyle 36^{\circ }}$	${\displaystyle {\sqrt {5-2{\sqrt {5}}}}}$
${\displaystyle 30^{\circ }}$	${\displaystyle {\frac {\sqrt {3}}{3}}}$
${\displaystyle 27^{\circ }}$	${\displaystyle {\sqrt {11-4{\sqrt {5}}+({\sqrt {5}}-3){\sqrt {10-2{\sqrt {5}}}}}}}$
${\displaystyle 24^{\circ }}$	${\displaystyle {\frac {(3{\sqrt {5}}+7){\sqrt {5-2{\sqrt {5}}}}-({\sqrt {5}}+3){\sqrt {3}}}{2}}}$

Values of ${\displaystyle x}$ in table below are derived from the above values by using identity ${\displaystyle \tan({\frac {A}{2}})={\frac {-1+{\sqrt {1+\tan ^{2}(A)}}}{\tan(A)}}}$:

Angle ${\displaystyle \theta }$	${\displaystyle x=\tan(\theta )}$
${\displaystyle {\frac {45^{\circ }}{2^{33}}}}$	0.00000_00000_91432_37995_4197.....089_03901_63759_3912
${\displaystyle {\frac {36^{\circ }}{2^{33}}}}$	0.00000_00000_73145_90396_3357.....211_97500_56173_0713
${\displaystyle {\frac {30^{\circ }}{2^{33}}}}$	0.00000_00000_60954_91996_9464.....024_32806_94580_0689
${\displaystyle {\frac {27^{\circ }}{2^{33}}}}$	0.00000_00000_54859_42797_2518.....791_30634_03540_9738
${\displaystyle {\frac {24^{\circ }}{2^{32}}}}$	0.00000_00000_97527_87195_1143.....736_60376_04724_6778

# python code

desired_precision = 1001
number_of_leading_zeroes = 10 # See below.

import decimal
dD = decimal.Decimal # Decimal object is like float with (almost) infinite precision.
dgt = decimal.getcontext()

Precision = dgt.prec = desired_precision + 3  # Adjust as necessary.
Tolerance = dD("1e-" + str(Precision-2))      # Adjust as necessary.
adjustment_to_precision = number_of_leading_zeroes * 2 + 3

def tan_halfA(tan_A) :
    dgt.prec += adjustment_to_precision
    top = -1 + (1+tan_A**2).sqrt()
    dgt.prec -= adjustment_to_precision
    tan_A_2 = top/tan_A
    return tan_A_2
    
def tan_2A (tanA) :
    '''
          2 * tanA
tan(2A) = -----------
          1 - tanA**2
    '''
    if tanA in (1,-1) : return '1/0'
    dgt.prec += adjustment_to_precision
    bottom = (1 - tanA**2)
    output = 2*tanA/bottom
    dgt.prec -= adjustment_to_precision
    return output+0

def θ_tanθ_from_A_tanA (angleA, tanA) :
    '''
if input == 45,1
output is:
"dD(45) / (2 ** (33))", "0.00000_00000_91432_37995_....._63759_3912"
                           ^^^^^^^^^^^
number_of_leading_zeroes refers to these zeroes.
θ,tanθ = θ_tanθ_from_A_tanA (angleA, tanA)
    '''
    θ, tanθ = angleA, tanA
    for p in range (1,100) :
        θ /= 2
        tanθ = tan_halfA(tanθ)
        if tanθ >= dD('1e-' + str(number_of_leading_zeroes)) : continue
        str1 = str(tanθ)
        # str1 = "n.nnnnnnnnnnnnn ..... nnnnnnnnnnnnE-11"
        str1a = str1[0] + str1[2:-4]
        list1 = [ str1a[q:q+5] for q in range (0, len(str1a), 5) ]
        str2 = '0.00000_00000_' + ('_'.join(list1))
        dD2 = dD(str2)
        (dD2 == tanθ) or ({}[2])
        ((θ * (2**p)) == angleA ) or ({}[3])
        str3 = 'dD({}) / (2 ** ({}))'.format(angleA,p)
        (θ == eval(str3)) or ({}[4])
        return str3, str2
    ({}[5])


r3 = dD(3).sqrt()
r5 = dD(5).sqrt()
tan36 = (5 - 2*r5).sqrt()
tan45 = dD(1)
tan30 = r3/3

v1 = 3*r5+7
v2 = (5 - 2*r5).sqrt()
v3 = (r5+3)*r3
tan24 = ( v1*v2 - v3 )/2

v1 = r5 - 3 ; v2 = (10 - 2*r5).sqrt()
tan27 = ( 11 - 4*r5 + v1*v2 ).sqrt()


values_of_A_tanA = (
    (dD(45), tan45),
    (dD(36), tan36),
    (dD(30), tan30),
    (dD(27), tan27),
    (dD(24), tan24),
)

values_of_θ_tanθ = []
for (A, tanA) in values_of_A_tanA :
    θ, tanθ = θ_tanθ_from_A_tanA (A, tanA)
    print()
    sx = 'θ' ; print (sx, '=', eval(sx))
#    sx = 'tanθ' ; print (sx, '=', eval(sx))
    print ('tanθ =', '{}.....{}'.format(tanθ[:30], tanθ[-20:]))
    values_of_θ_tanθ += [ (θ, tanθ) ]

# Check
for (v1,v2),(v3,v4) in zip (values_of_A_tanA, values_of_θ_tanθ) :
    A, tanA = v1,v2
    θ = eval(v3)
    tanθ = dD(v4)
    status = 0
    for p in range (1,100) :
        θ *= 2
        tanθ = tan_2A (tanθ) 
        if θ == A :
            dgt.prec = desired_precision
            (+tanθ == +tanA) or ({}[10])
            dgt.prec = Precision            
            status = 1
            break
    status or ({}[11])

θ = dD(45) / (2 ** (33))
tanθ = 0.00000_00000_91432_37995_4197.....089_03901_63759_3912

θ = dD(36) / (2 ** (33))
tanθ = 0.00000_00000_73145_90396_3357.....211_97500_56173_0713

θ = dD(30) / (2 ** (33))
tanθ = 0.00000_00000_60954_91996_9464.....024_32806_94580_0689

θ = dD(27) / (2 ** (33))
tanθ = 0.00000_00000_54859_42797_2518.....791_30634_03540_9738

θ = dD(24) / (2 ** (32))
tanθ = 0.00000_00000_97527_87195_1143.....736_60376_04724_6778

# python code

def calculate_π (angleθ, tanθ) :
    '''
angleθ may be: "dD(27) / (2 ** (33))"
tanθ may be: "0.00000_00000_54859_42797_ ..... _03540_9738"
π = calculate_π (angleθ, tanθ)
    '''
    thisName = 'calculate_π (angleθ, tanθ) :'

    if isinstance(angleθ, dD) : pass
    elif isinstance(angleθ, str) : angleθ = eval(angleθ)
    else : ({}[21])
    if isinstance(tanθ, dD) : pass
    elif isinstance(tanθ, str) : tanθ = dD(tanθ)
    else : ({}[22])
    
    sum = X = x = tanθ ; xx = x*x ; multiplier = -1 ; count = status = 0

    #         x**3   x**5   x**7   x**9
    # y = x - ---- + ---- - ---- + ---- - ......
    #           3      5      7      9
    # 
    # Each term in the sequence is roughly the previous term multiplied by x**2.
    # Each value of x contains 10 leading zeroes after decimal point.
    # Therefore, each term in the sequence is roughly the previous term with 20 more leading zeroes.
    # Each pass through main loop adds about 20 digits to current value of sum
    # and θ is calculated to precision of 1004 digits with about 50 passes through main loop.
    # 
    for p in range (3,200,2) :
        # This is main loop.
        count += 1
        X *= xx
        addendum = (multiplier * X) / p
        sum += addendum
        if abs(addendum) < Tolerance :
            status = 1; break
        multiplier = -multiplier
    status or ({}[23])
    print(thisName, 'count =',count)
    π = sum * 180 / angleθ
    dgt.prec = desired_precision
    π += 0 # This forces π to adopt precision of desired_precision.
    dgt.prec = Precision
    return π

# Calculate five values of π:
values_of_π = []
for θ,tanθ in values_of_θ_tanθ :
    π = calculate_π (θ,tanθ)
    values_of_π += [ π ]

Each calculation of π required about 50 passes through main loop:

calculate_π (angleθ, tanθ) : count = 50
calculate_π (angleθ, tanθ) : count = 49
calculate_π (angleθ, tanθ) : count = 49
calculate_π (angleθ, tanθ) : count = 49
calculate_π (angleθ, tanθ) : count = 50

Check that all 5 values of π are equal:

# python code

set_π = set(values_of_π)
π, = set_π # Note the syntax. If length of set_π is not 1, this statement fails.

format_string = '''
Length of list values_of_π = {}
Length of set set_π = {}
set_π = {}.....{}'''

set_π_as_string = str(set_π)

print ( format_string.format(
    len(values_of_π),
    len(set_π),
    set_π_as_string[:42], set_π_as_string[-41:], 
) )

Length of list values_of_π = 5
Length of set set_π = 1
set_π = {Decimal('3.141592653589793238462643383279.....12268066130019278766111959092164201989')}

Because all five calculated values of π are equal, there is very high probability that this value of π is accurate.

Print value of π as python command formatted:

# python code

newLine = '''
'''[-1:]

def print_π (π) :
    '''
Input π is : Decimal('3.141592653589793238 ..... 66111959092164201989')
This function prints:
π = ( "3.14159_26535_89793_23846_26433_83279_50288_41971_69399_37510"
     +  "58209_74944_59230_78164_06286_20899_86280_34825_34211_70679"
     .....
     +  "18577_80532_17122_68066_13001_92787_66111_95909_21642_01989" )
    '''
    πstr = str(π)
    (len(πstr) == (desired_precision + 1)) or ({}[31])
    (πstr[:2] == '3.') or ({}[32])

    twenty_rows = []
    for p in range (2, len(πstr), 50) :
        str1a = πstr[p:p+50]
        list1a = [ str1a[q:q+5] for q in range(0, len(str1a), 5) ]
        str1b = '_'.join(list1a)
        twenty_rows += [str1b]
    twenty_rows[0] = '3.' + twenty_rows[0]
    joiner = '"{}     +  "'.format(newLine)
    str3 = '( "{}" )'.format(joiner.join(twenty_rows))
    
    str4 = eval(str3)
    (dD(str4) == π) or ({}[33])

    lines = str3.split(newLine)
    paragraphs = [ newLine.join(lines[p:p+4]) for p in range(0,len(lines),4) ]
    str5 = (newLine*2).join(paragraphs)

    str6 = eval(str5)
    (dD(str6) == π) or ({}[34])
    print ('π =', str5)
    
    return str5

π1 = print_π (π)

π = ( "3.14159_26535_89793_23846_26433_83279_50288_41971_69399_37510"
     +  "58209_74944_59230_78164_06286_20899_86280_34825_34211_70679"
     +  "82148_08651_32823_06647_09384_46095_50582_23172_53594_08128"
     +  "48111_74502_84102_70193_85211_05559_64462_29489_54930_38196"

     +  "44288_10975_66593_34461_28475_64823_37867_83165_27120_19091"
     +  "45648_56692_34603_48610_45432_66482_13393_60726_02491_41273"
     +  "72458_70066_06315_58817_48815_20920_96282_92540_91715_36436"
     +  "78925_90360_01133_05305_48820_46652_13841_46951_94151_16094"

     +  "33057_27036_57595_91953_09218_61173_81932_61179_31051_18548"
     +  "07446_23799_62749_56735_18857_52724_89122_79381_83011_94912"
     +  "98336_73362_44065_66430_86021_39494_63952_24737_19070_21798"
     +  "60943_70277_05392_17176_29317_67523_84674_81846_76694_05132"

     +  "00056_81271_45263_56082_77857_71342_75778_96091_73637_17872"
     +  "14684_40901_22495_34301_46549_58537_10507_92279_68925_89235"
     +  "42019_95611_21290_21960_86403_44181_59813_62977_47713_09960"
     +  "51870_72113_49999_99837_29780_49951_05973_17328_16096_31859"

     +  "50244_59455_34690_83026_42522_30825_33446_85035_26193_11881"
     +  "71010_00313_78387_52886_58753_32083_81420_61717_76691_47303"
     +  "59825_34904_28755_46873_11595_62863_88235_37875_93751_95778"
     +  "18577_80532_17122_68066_13001_92787_66111_95909_21642_01989" )

If you highlight the above expression for ${\displaystyle \pi }$ as shown in diagram, you can copy and paste it into your python source file as valid python code.