Machin-like formulas   are useful for efficiently computing numerical approximations for 



π


{\displaystyle \pi }





Task

Verify the following Machin-like formulas are correct by calculating the value of tan   (right hand side) for each equation using exact arithmetic and showing they equal 1:

 





π
4


=
arctan
⁡


1
2


+
arctan
⁡


1
3




{\displaystyle {\pi \over 4}=\arctan {1 \over 2}+\arctan {1 \over 3}}

 
 





π
4


=
2
arctan
⁡


1
3


+
arctan
⁡


1
7




{\displaystyle {\pi \over 4}=2\arctan {1 \over 3}+\arctan {1 \over 7}}


 





π
4


=
4
arctan
⁡


1
5


−
arctan
⁡


1
239




{\displaystyle {\pi \over 4}=4\arctan {1 \over 5}-\arctan {1 \over 239}}


 





π
4


=
5
arctan
⁡


1
7


+
2
arctan
⁡


3
79




{\displaystyle {\pi \over 4}=5\arctan {1 \over 7}+2\arctan {3 \over 79}}


 





π
4


=
5
arctan
⁡


29
278


+
7
arctan
⁡


3
79




{\displaystyle {\pi \over 4}=5\arctan {29 \over 278}+7\arctan {3 \over 79}}


 





π
4


=
arctan
⁡


1
2


+
arctan
⁡


1
5


+
arctan
⁡


1
8




{\displaystyle {\pi \over 4}=\arctan {1 \over 2}+\arctan {1 \over 5}+\arctan {1 \over 8}}

 
 





π
4


=
4
arctan
⁡


1
5


−
arctan
⁡


1
70


+
arctan
⁡


1
99




{\displaystyle {\pi \over 4}=4\arctan {1 \over 5}-\arctan {1 \over 70}+\arctan {1 \over 99}}

 
 





π
4


=
5
arctan
⁡


1
7


+
4
arctan
⁡


1
53


+
2
arctan
⁡


1
4443




{\displaystyle {\pi \over 4}=5\arctan {1 \over 7}+4\arctan {1 \over 53}+2\arctan {1 \over 4443}}


 





π
4


=
6
arctan
⁡


1
8


+
2
arctan
⁡


1
57


+
arctan
⁡


1
239




{\displaystyle {\pi \over 4}=6\arctan {1 \over 8}+2\arctan {1 \over 57}+\arctan {1 \over 239}}


 





π
4


=
8
arctan
⁡


1
10


−
arctan
⁡


1
239


−
4
arctan
⁡


1
515




{\displaystyle {\pi \over 4}=8\arctan {1 \over 10}-\arctan {1 \over 239}-4\arctan {1 \over 515}}


 





π
4


=
12
arctan
⁡


1
18


+
8
arctan
⁡


1
57


−
5
arctan
⁡


1
239




{\displaystyle {\pi \over 4}=12\arctan {1 \over 18}+8\arctan {1 \over 57}-5\arctan {1 \over 239}}


 





π
4


=
16
arctan
⁡


1
21


+
3
arctan
⁡


1
239


+
4
arctan
⁡


3
1042




{\displaystyle {\pi \over 4}=16\arctan {1 \over 21}+3\arctan {1 \over 239}+4\arctan {3 \over 1042}}


 





π
4


=
22
arctan
⁡


1
28


+
2
arctan
⁡


1
443


−
5
arctan
⁡


1
1393


−
10
arctan
⁡


1
11018




{\displaystyle {\pi \over 4}=22\arctan {1 \over 28}+2\arctan {1 \over 443}-5\arctan {1 \over 1393}-10\arctan {1 \over 11018}}


 





π
4


=
22
arctan
⁡


1
38


+
17
arctan
⁡


7
601


+
10
arctan
⁡


7
8149




{\displaystyle {\pi \over 4}=22\arctan {1 \over 38}+17\arctan {7 \over 601}+10\arctan {7 \over 8149}}


 





π
4


=
44
arctan
⁡


1
57


+
7
arctan
⁡


1
239


−
12
arctan
⁡


1
682


+
24
arctan
⁡


1
12943




{\displaystyle {\pi \over 4}=44\arctan {1 \over 57}+7\arctan {1 \over 239}-12\arctan {1 \over 682}+24\arctan {1 \over 12943}}


 





π
4


=
88
arctan
⁡


1
172


+
51
arctan
⁡


1
239


+
32
arctan
⁡


1
682


+
44
arctan
⁡


1
5357


+
68
arctan
⁡


1
12943




{\displaystyle {\pi \over 4}=88\arctan {1 \over 172}+51\arctan {1 \over 239}+32\arctan {1 \over 682}+44\arctan {1 \over 5357}+68\arctan {1 \over 12943}}


and confirm that the following formula is incorrect by showing   tan   (right hand side)   is not   1:

 





π
4


=
88
arctan
⁡


1
172


+
51
arctan
⁡


1
239


+
32
arctan
⁡


1
682


+
44
arctan
⁡


1
5357


+
68
arctan
⁡


1
12944




{\displaystyle {\pi \over 4}=88\arctan {1 \over 172}+51\arctan {1 \over 239}+32\arctan {1 \over 682}+44\arctan {1 \over 5357}+68\arctan {1 \over 12944}}


These identities are useful in calculating the values:

 



tan
⁡
(
a
+
b
)
=



tan
⁡
(
a
)
+
tan
⁡
(
b
)


1
−
tan
⁡
(
a
)
tan
⁡
(
b
)





{\displaystyle \tan(a+b)={\tan(a)+\tan(b) \over 1-\tan(a)\tan(b)}}


 



tan
⁡

(
arctan
⁡


a
b


)

=


a
b




{\displaystyle \tan \left(\arctan {a \over b}\right)={a \over b}}


 



tan
⁡
(
−
a
)
=
−
tan
⁡
(
a
)


{\displaystyle \tan(-a)=-\tan(a)}




You can store the equations in any convenient data structure, but for extra credit parse them from human-readable text input.

Note: to formally prove the formula correct, it would have to be shown that 






−
3
p
i

4




{\displaystyle {-3pi \over 4}}

 < right hand side < 






5
p
i

4




{\displaystyle {5pi \over 4}}

 due to 



tan
⁡
(
)


{\displaystyle \tan()}

 periodicity.



package require Tcl 8.5
 
# Compute tan(atan(p)+atan(q)) using rationals
proc tadd {p q} {
    lassign $p pp pq
    lassign $q qp qq
    set topp [expr {$pp*$qq + $qp*$pq}]
    set topq [expr {$pq*$qq}]
    set prodp [expr {$pp*$qp}]
    set prodq [expr {$pq*$qq}]
    set lowp [expr {$prodq - $prodp}]
    set resultp [set gcd1 [expr {$topp * $prodq}]]
    set resultq [set gcd2 [expr {$topq * $lowp}]]
    # Critical! Normalize using the GCD
    while {$gcd2 != 0} {
	lassign [list $gcd2 [expr {$gcd1 % $gcd2}]] gcd1 gcd2
    }
    list [expr {$resultp / abs($gcd1)}] [expr {$resultq / abs($gcd1)}]
}
proc termTan {n a b} {
    if {$n < 0} {
	set n [expr {-$n}]
	set a [expr {-$a}]
    }
    if {$n == 1} {
	return [list $a $b]
    }
    set k [expr {$n - [set m [expr {$n / 2}]]*2}]
    set t2 [termTan $m $a $b]
    set m2 [tadd $t2 $t2]
    if {$k == 0} {
	return $m2
    }
    return [tadd [termTan $k $a $b] $m2]
}
proc machinTan {terms} {
    set sum {0 1}
    foreach term $terms {
	set sum [tadd $sum [termTan {*}$term]]
    }
    return $sum
}
 
# Assumes that the formula is in the very specific form below!
proc parseFormula {formula} {
    set RE {(-?\s*\d*\s*\*?)\s*arctan\s*\(\s*(-?\s*\d+)\s*/\s*(-?\s*\d+)\s*\)}
    set nospace {" " "" "*" ""}
    foreach {all n a b} [regexp -inline -all $RE $formula] {
	if {![regexp {\d} $n]} {append n 1}
	lappend result [list [string map $nospace $n] [string map $nospace $a] [string map $nospace $b]]
    }
    return $result
}
 
foreach formula {
    "pi/4 = arctan(1/2) + arctan(1/3)"
    "pi/4 = 2*arctan(1/3) + arctan(1/7)"
    "pi/4 = 4*arctan(1/5) - arctan(1/239)"
    "pi/4 = 5*arctan(1/7) + 2*arctan(3/79)"
    "pi/4 = 5*arctan(29/278) + 7*arctan(3/79)"
    "pi/4 = arctan(1/2) + arctan(1/5) + arctan(1/8)"
    "pi/4 = 4*arctan(1/5) - arctan(1/70) + arctan(1/99)"
    "pi/4 = 5*arctan(1/7) + 4*arctan(1/53) + 2*arctan(1/4443)"
    "pi/4 = 6*arctan(1/8) + 2*arctan(1/57) + arctan(1/239)"
    "pi/4 = 8*arctan(1/10) - arctan(1/239) - 4*arctan(1/515)"
    "pi/4 = 12*arctan(1/18) + 8*arctan(1/57) - 5*arctan(1/239)"
    "pi/4 = 16*arctan(1/21) + 3*arctan(1/239) + 4*arctan(3/1042)"
    "pi/4 = 22*arctan(1/28) + 2*arctan(1/443) - 5*arctan(1/1393) - 10*arctan(1/11018)"
    "pi/4 = 22*arctan(1/38) + 17*arctan(7/601) + 10*arctan(7/8149)"
    "pi/4 = 44*arctan(1/57) + 7*arctan(1/239) - 12*arctan(1/682) + 24*arctan(1/12943)"
    "pi/4 = 88*arctan(1/172) + 51*arctan(1/239) + 32*arctan(1/682) + 44*arctan(1/5357) + 68*arctan(1/12943)"
    "pi/4 = 88*arctan(1/172) + 51*arctan(1/239) + 32*arctan(1/682) + 44*arctan(1/5357) + 68*arctan(1/12944)"
} {
    if {[tcl::mathop::== {*}[machinTan [parseFormula $formula]]]} {
	puts "Yes! '$formula' is true"
    } else {
	puts "No! '$formula' not true"
    }
}