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{{task}}
[[wp:Machin-like_formula|Machin-like formulas]] are useful for efficiently computing numerical approximations for
;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''':
: : : : : : : : : : : : : : : :
and confirm that the following formula is incorrect by showing '''tan''' (''right hand side)'' is ''not'' '''1''':
:
These identities are useful in calculating the values: :
:
:
You can store the equations in any convenient data structure, but for extra credit parse them from human-readable [[Check_Machin-like_formulas/text_equations|text input]].
Note: to formally prove the formula correct, it would have to be shown that '' < right hand side < '' due to '''' periodicity.
Clojure
Clojure automatically handles ratio of numbers as fractions {{trans|Go}}
(ns tanevaulator
(:gen-class))
;; Notation: [a b c] -> a x arctan(a/b)
(def test-cases [
[[1, 1, 2], [1, 1, 3]],
[[2, 1, 3], [1, 1, 7]],
[[4, 1, 5], [-1, 1, 239]],
[[5, 1, 7], [2, 3, 79]],
[[1, 1, 2], [1, 1, 5], [1, 1, 8]],
[[4, 1, 5], [-1, 1, 70], [1, 1, 99]],
[[5, 1, 7], [4, 1, 53], [2, 1, 4443]],
[[6, 1, 8], [2, 1, 57], [1, 1, 239]],
[[8, 1, 10], [-1, 1, 239], [-4, 1, 515]],
[[12, 1, 18], [8, 1, 57], [-5, 1, 239]],
[[16, 1, 21], [3, 1, 239], [4, 3, 1042]],
[[22, 1, 28], [2, 1, 443], [-5, 1, 1393], [-10, 1, 11018]],
[[22, 1, 38], [17, 7, 601], [10, 7, 8149]],
[[44, 1, 57], [7, 1, 239], [-12, 1, 682], [24, 1, 12943]],
[[88, 1, 172], [51, 1, 239], [32, 1, 682], [44, 1, 5357], [68, 1, 12943]],
[[88, 1, 172], [51, 1, 239], [32, 1, 682], [44, 1, 5357], [68, 1, 12944]]
])
(defn tan-sum [a b]
" tan (a + b) "
(/ (+ a b) (- 1 (* a b))))
(defn tan-eval [m]
" Evaluates tan of a triplet (e.g. [1, 1, 2])"
(let [coef (first m)
rat (/ (nth m 1) (nth m 2))]
(cond
(= 1 coef) rat
(neg? coef) (tan-eval [(- (nth m 0)) (- (nth m 1)) (nth m 2)])
:else (let [
ca (quot coef 2)
cb (- coef ca)
a (tan-eval [ca (nth m 1) (nth m 2)])
b (tan-eval [cb (nth m 1) (nth m 2)])]
(tan-sum a b)))))
(defn tans [m]
" Evaluates tan of set of triplets (e.g. [[1, 1, 2], [1, 1, 3]])"
(if (= 1 (count m))
(tan-eval (nth m 0))
(let [a (tan-eval (first m))
b (tans (rest m))]
(tan-sum a b))))
(doseq [q test-cases]
" Display results "
(println "tan " q " = "(tans q)))
{{Output}}
tan [[1 1 2] [1 1 3]] = 1N
tan [[2 1 3] [1 1 7]] = 1N
tan [[4 1 5] [-1 1 239]] = 1N
tan [[5 1 7] [2 3 79]] = 1N
tan [[1 1 2] [1 1 5] [1 1 8]] = 1N
tan [[4 1 5] [-1 1 70] [1 1 99]] = 1N
tan [[5 1 7] [4 1 53] [2 1 4443]] = 1N
tan [[6 1 8] [2 1 57] [1 1 239]] = 1N
tan [[8 1 10] [-1 1 239] [-4 1 515]] = 1N
tan [[12 1 18] [8 1 57] [-5 1 239]] = 1N
tan [[16 1 21] [3 1 239] [4 3 1042]] = 1N
tan [[22 1 28] [2 1 443] [-5 1 1393] [-10 1 11018]] = 1N
tan [[22 1 38] [17 7 601] [10 7 8149]] = 1N
tan [[44 1 57] [7 1 239] [-12 1 682] [24 1 12943]] = 1N
tan [[88 1 172] [51 1 239] [32 1 682] [44 1 5357] [68 1 12943]] = 1N
tan [[88 1 172] [51 1 239] [32 1 682] [44 1 5357] [68 1 12944]] = 1009288018000944050967896710431587186456256928584351786643498522649995492271475761189348270710224618853590682465929080006511691833816436374107451368838065354726517908250456341991684635768915704374493675498637876700129004484434187627909285979251682006538817341793224963346197503893270875008524149334251672855130857035205217929335932890740051319216343365800342290782260673215928499123722781078448297609548233999010983373327601187505623621602789012550584784738082074783523787011976757247516095289966708782862528690942242793667539020699840402353522108223 /
1009288837315638583415701528780402795721935641614456853534313491853293025565940011104051964874275710024625850092154664245109626053906509780125743180758231049920425664246286578958307532545458843067352531217230461290763258378749459637420702619029075083089762088232401888676895047947363883809724322868121990870409574061477638203859217672620508200713073485398199091153535700094640095900731630771349477187594074169815106104524371099618096164871416282464532355211521113449237814080332335526420331468258917484010722587072087349909684004660371264507984339711
(equals 0.9999991882257445)
D
This uses the module of the Arithmetic Rational Task. {{trans|Python}}
import std.stdio, std.regex, std.conv, std.string, std.range,
arithmetic_rational;
struct Pair { int x; Rational r; }
Pair[][] parseEquations(in string text) /*pure nothrow*/ {
auto r = regex(r"\s*(?P<sign>[+-])?\s*(?:(?P<mul>\d+)\s*\*)?\s*" ~
r"arctan\((?P<num>\d+)/(?P<denom>\d+)\)");
Pair[][] machins;
foreach (const line; text.splitLines) {
Pair[] formula;
foreach (part; line.split("=")[1].matchAll(r)) {
immutable mul = part["mul"],
num = part["num"],
denom = part["denom"];
formula ~= Pair((part["sign"] == "-" ? -1 : 1) *
(mul.empty ? 1 : mul.to!int),
Rational(num.to!int,
denom.empty ? 1 : denom.to!int));
}
machins ~= formula;
}
return machins;
}
Rational tans(in Pair[] xs) pure nothrow {
static Rational tanEval(in int coef, in Rational f)
pure nothrow {
if (coef == 1)
return f;
if (coef < 0)
return -tanEval(-coef, f);
immutable a = tanEval(coef / 2, f),
b = tanEval(coef - coef / 2, f);
return (a + b) / (1 - a * b);
}
if (xs.length == 1)
return tanEval(xs[0].tupleof);
immutable a = xs[0 .. $ / 2].tans,
b = xs[$ / 2 .. $].tans;
return (a + b) / (1 - a * b);
}
void main() {
immutable equationText =
"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)";
const machins = equationText.parseEquations;
foreach (const machin, const eqn; machins.zip(equationText.splitLines)) {
immutable ans = machin.tans;
writefln("%5s: %s", ans == 1 ? "OK" : "ERROR", eqn);
}
}
{{out}}
OK: pi/4 = arctan(1/2) + arctan(1/3)
OK: pi/4 = 2*arctan(1/3) + arctan(1/7)
OK: pi/4 = 4*arctan(1/5) - arctan(1/239)
OK: pi/4 = 5*arctan(1/7) + 2*arctan(3/79)
OK: pi/4 = 5*arctan(29/278) + 7*arctan(3/79)
OK: pi/4 = arctan(1/2) + arctan(1/5) + arctan(1/8)
OK: pi/4 = 4*arctan(1/5) - arctan(1/70) + arctan(1/99)
OK: pi/4 = 5*arctan(1/7) + 4*arctan(1/53) + 2*arctan(1/4443)
OK: pi/4 = 6*arctan(1/8) + 2*arctan(1/57) + arctan(1/239)
OK: pi/4 = 8*arctan(1/10) - arctan(1/239) - 4*arctan(1/515)
OK: pi/4 = 12*arctan(1/18) + 8*arctan(1/57) - 5*arctan(1/239)
OK: pi/4 = 16*arctan(1/21) + 3*arctan(1/239) + 4*arctan(3/1042)
OK: pi/4 = 22*arctan(1/28) + 2*arctan(1/443) - 5*arctan(1/1393) - 10*arctan(1/11018)
OK: pi/4 = 22*arctan(1/38) + 17*arctan(7/601) + 10*arctan(7/8149)
OK: pi/4 = 44*arctan(1/57) + 7*arctan(1/239) - 12*arctan(1/682) + 24*arctan(1/12943)
OK: pi/4 = 88*arctan(1/172) + 51*arctan(1/239) + 32*arctan(1/682) + 44*arctan(1/5357) + 68*arctan(1/12943)
ERROR: pi/4 = 88*arctan(1/172) + 51*arctan(1/239) + 32*arctan(1/682) + 44*arctan(1/5357) + 68*arctan(1/12944)
EchoLisp
(lib 'math)
(lib 'match)
(math-precision 1.e-10)
;; formally derive (tan ..) expressions
;; copied from Racket
;; adapted and improved for performance
(define (reduce e)
;; (set! rcount (1+ rcount)) ;; # of calls
(match e
[(? number? a) a]
[('+ (? number? a) (? number? b)) (+ a b)]
[('- (? number? a) (? number? b)) (- a b)]
[('- (? number? a)) (- a)]
[('* (? number? a) (? number? b)) (* a b)]
[('/ (? number? a) (? number? b)) (/ a b)] ; patch
[( '+ a b) (reduce `(+ ,(reduce a) ,(reduce b)))]
[( '- a b) (reduce `(- ,(reduce a) ,(reduce b)))]
[( '- a) (reduce `(- ,(reduce a)))]
[( '* a b) (reduce `(* ,(reduce a) ,(reduce b)))]
[( '/ a b) (reduce `(/ ,(reduce a) ,(reduce b)))]
[( 'tan ('arctan a)) (reduce a)]
[( 'tan ( '- a)) (reduce `(- (tan ,a)))]
;; x 100 # calls reduction : derive (tan ,a) only once
[( 'tan ( '+ a b))
(let ((alpha (reduce `(tan ,a))) (beta (reduce `(tan ,b))))
(reduce `(/ (+ ,alpha ,beta) (- 1 (* ,alpha ,beta)))))]
[( 'tan ( '+ a b c ...)) (reduce `(tan (+ ,a (+ ,b ,@c))))]
[( 'tan ( '- a b))
(let ((alpha (reduce `(tan ,a))) (beta (reduce `(tan ,b))))
(reduce `(/ (- ,alpha ,beta) (+ 1 (* ,alpha ,beta)))))]
;; add formula for (tan 2 (arctan a)) = 2 a / (1 - a^2))
[( 'tan ( '* 2 ('arctan a))) (reduce `(/ (* 2 ,a) (- 1 (* ,a ,a))))]
[( 'tan ( '* 1 ('arctan a))) (reduce a)] ; added
[( 'tan ( '* (? number? n) a))
(cond [(< n 0) (reduce `(- (tan (* ,(- n) ,a))))]
[(= n 0) 0]
[(= n 1) (reduce `(tan ,a))]
[(even? n)
(let ((alpha (reduce `(tan (* ,(/ n 2) ,a))))) ;; # calls reduction
(reduce `(/ (* 2 ,alpha) (- 1 (* ,alpha ,alpha)))))]
[else (reduce `(tan (+ ,a (* ,(- n 1) ,a))))])]
))
(define (task)
(for ((f machins))
(if (~= 1 (reduce f))
(writeln 'π f 'βΎ 1 )
(writeln 'β f 'β½ (reduce f) ))))
{{out}}
(define machins
'((tan (+ (arctan 1/2) (arctan 1/3)))
(tan (+ (* 2 (arctan 1/3)) (arctan 1/7)))
(tan (- (* 4 (arctan 1/5)) (arctan 1/239)))
(tan (+ (* 5 (arctan 1/7)) (* 2 (arctan 3/79))))
(tan (+ (* 5 (arctan 29/278)) (* 7 (arctan 3/79))))
(tan (+ (arctan 1/2) (arctan 1/5) (arctan 1/8)))
(tan (+ (* 4 (arctan 1/5)) (* -1 (arctan 1/70)) (arctan 1/99)))
(tan (+ (* 5 (arctan 1/7)) (* 4 (arctan 1/53)) (* 2 (arctan 1/4443))))
(tan (+ (* 6 (arctan 1/8)) (* 2 (arctan 1/57)) (arctan 1/239)))
(tan (+ (* 8 (arctan 1/10)) (* -1 (arctan 1/239)) (* -4 (arctan 1/515))))
(tan (+ (* 12 (arctan 1/18)) (* 8 (arctan 1/57)) (* -5 (arctan 1/239))))
(tan (+ (* 16 (arctan 1/21)) (* 3 (arctan 1/239)) (* 4 (arctan 3/1042))))
(tan (+ (* 22 (arctan 1/28)) (* 2 (arctan 1/443)) (* -5 (arctan 1/1393)) (* -10 (arctan 1/11018))))
(tan (+ (* 22 (arctan 1/38)) (* 17 (arctan 7/601)) (* 10 (arctan 7/8149))))
(tan (+ (* 44 (arctan 1/57)) (* 7 (arctan 1/239)) (* -12 (arctan 1/682)) (* 24 (arctan 1/12943))))
(tan (+ (* 88 (arctan 1/172)) (* 51 (arctan 1/239)) (* 32 (arctan 1/682))
(* 44 (arctan 1/5357)) (* 68 (arctan 1/12943))))
(tan (+ (* 88 (arctan 1/172)) (* 51 (arctan 1/239)) (* 32 (arctan 1/682))
(* 44 (arctan 1/5357)) (* 68 (arctan 1/12944))))))
(task)
π (tan (+ (arctan 1/2) (arctan 1/3))) βΎ 1
π (tan (+ (* 2 (arctan 1/3)) (arctan 1/7))) βΎ 1
π (tan (- (* 4 (arctan 1/5)) (arctan 1/239))) βΎ 1
π (tan (+ (* 5 (arctan 1/7)) (* 2 (arctan 3/79)))) βΎ 1
π (tan (+ (* 5 (arctan 29/278)) (* 7 (arctan 3/79)))) βΎ 1
π (tan (+ (arctan 1/2) (arctan 1/5) (arctan 1/8))) βΎ 1
π (tan (+ (* 4 (arctan 1/5)) (* -1 (arctan 1/70)) (arctan 1/99))) βΎ 1
π (tan (+ (* 5 (arctan 1/7)) (* 4 (arctan 1/53)) (* 2 (arctan 1/4443)))) βΎ 1
π (tan (+ (* 6 (arctan 1/8)) (* 2 (arctan 1/57)) (arctan 1/239))) βΎ 1
π (tan (+ (* 8 (arctan 1/10)) (* -1 (arctan 1/239)) (* -4 (arctan 1/515)))) βΎ 1
π (tan (+ (* 12 (arctan 1/18)) (* 8 (arctan 1/57)) (* -5 (arctan 1/239)))) βΎ 1
π (tan (+ (* 16 (arctan 1/21)) (* 3 (arctan 1/239)) (* 4 (arctan 3/1042)))) βΎ 1
π (tan (+ (* 22 (arctan 1/28)) (* 2 (arctan 1/443)) (* -5 (arctan 1/1393)) (* -10 (arctan 1/11018)))) βΎ 1
π (tan (+ (* 22 (arctan 1/38)) (* 17 (arctan 7/601)) (* 10 (arctan 7/8149)))) βΎ 1
π (tan (+ (* 44 (arctan 1/57)) (* 7 (arctan 1/239)) (* -12 (arctan 1/682)) (* 24 (arctan 1/12943)))) βΎ 1
π (tan (+ (* 88 (arctan 1/172)) (* 51 (arctan 1/239)) (* 32 (arctan 1/682))
(* 44 (arctan 1/5357)) (* 68 (arctan 1/12943)))) βΎ 1
β (tan (+ (* 88 (arctan 1/172)) (* 51 (arctan 1/239)) (* 32 (arctan 1/682))
(* 44 (arctan 1/5357)) (* 68 (arctan 1/12944)))) β½ 0.9999991882257442
Factor
USING: combinators formatting kernel locals math sequences ;
IN: rosetta-code.machin
: tan+ ( x y -- z ) [ + ] [ * 1 swap - / ] 2bi ;
:: tan-eval ( coef frac -- x )
{
{ [ coef zero? ] [ 0 ] }
{ [ coef neg? ] [ coef neg frac tan-eval neg ] }
{ [ coef odd? ] [ frac coef 1 - frac tan-eval tan+ ] }
[ coef 2/ frac tan-eval dup tan+ ]
} cond ;
: tans ( seq -- x ) [ first2 tan-eval ] [ tan+ ] map-reduce ;
: machin ( -- )
{
{ { 1 1/2 } { 1 1/3 } }
{ { 2 1/3 } { 1 1/7 } }
{ { 4 1/5 } { -1 1/239 } }
{ { 5 1/7 } { 2 3/79 } }
{ { 5 29/278 } { 7 3/79 } }
{ { 1 1/2 } { 1 1/5 } { 1 1/8 } }
{ { 5 1/7 } { 4 1/53 } { 2 1/4443 } }
{ { 6 1/8 } { 2 1/57 } { 1 1/239 } }
{ { 8 1/10 } { -1 1/239 } { -4 1/515 } }
{ { 12 1/18 } { 8 1/57 } { -5 1/239 } }
{ { 16 1/21 } { 3 1/239 } { 4 3/1042 } }
{ { 22 1/28 } { 2 1/443 }
{ -5 1/1393 } { -10 1/11018 } }
{ { 22 1/38 } { 17 7/601 } { 10 7/8149 } }
{ { 44 1/57 } { 7 1/239 } { -12 1/682 } { 24 1/12943 } }
{ { 88 1/172 } { 51 1/239 } { 32 1/682 }
{ 44 1/5357 } { 68 1/12943 } }
{ { 88 1/172 } { 51 1/239 } { 32 1/682 }
{ 44 1/5357 } { 68 1/12944 } }
} [ dup tans "tan %u = %u\n" printf ] each ;
MAIN: machin
{{out}}
tan { { 1 1/2 } { 1 1/3 } } = 1
tan { { 2 1/3 } { 1 1/7 } } = 1
tan { { 4 1/5 } { -1 1/239 } } = 1
tan { { 5 1/7 } { 2 3/79 } } = 1
tan { { 5 29/278 } { 7 3/79 } } = 1
tan { { 1 1/2 } { 1 1/5 } { 1 1/8 } } = 1
tan { { 5 1/7 } { 4 1/53 } { 2 1/4443 } } = 1
tan { { 6 1/8 } { 2 1/57 } { 1 1/239 } } = 1
tan { { 8 1/10 } { -1 1/239 } { -4 1/515 } } = 1
tan { { 12 1/18 } { 8 1/57 } { -5 1/239 } } = 1
tan { { 16 1/21 } { 3 1/239 } { 4 3/1042 } } = 1
tan { { 22 1/28 } { 2 1/443 } { -5 1/1393 } { -10 1/11018 } } = 1
tan { { 22 1/38 } { 17 7/601 } { 10 7/8149 } } = 1
tan { { 44 1/57 } { 7 1/239 } { -12 1/682 } { 24 1/12943 } } = 1
tan {
{ 88 1/172 }
{ 51 1/239 }
{ 32 1/682 }
{ 44 1/5357 }
{ 68 1/12943 }
} = 1
tan {
{ 88 1/172 }
{ 51 1/239 }
{ 32 1/682 }
{ 44 1/5357 }
{ 68 1/12944 }
} = 10092...08223/10092...39711
FreeBASIC
{{libheader|GMP}}
' version 07-04-2018
' compile with: fbc -s console
#Include "gmp.bi"
#Define _a(Q) (@(Q)->_mp_num) 'a
#Define _b(Q) (@(Q)->_mp_den) 'b
Data "[1, 1, 2] [1, 1, 3]"
Data "[2, 1, 3] [1, 1, 7]"
Data "[4, 1, 5] [-1, 1, 239]"
Data "[5, 1, 7] [2, 3, 79]"
Data "[1, 1, 2] [1, 1, 5] [1, 1, 8]"
Data "[4, 1, 5] [-1, 1, 70] [1, 1, 99]"
Data "[5, 1, 7] [4, 1, 53] [2, 1, 4443]"
Data "[6, 1, 8] [2, 1, 57] [1, 1, 239]"
Data "[8, 1, 10] [-1, 1, 239] [-4, 1, 515]"
Data "[12, 1, 18] [8, 1, 57] [-5, 1, 239]"
Data "[16, 1, 21] [3, 1, 239] [4, 3, 1042]"
Data "[22, 1, 28] [2, 1, 443] [-5, 1, 1393] [-10, 1, 11018]"
Data "[22, 1, 38] [17, 7, 601] [10, 7, 8149]"
Data "[44, 1, 57] [7, 1, 239] [-12, 1, 682] [24, 1, 12943]"
Data "[88, 1, 172] [51, 1, 239] [32, 1, 682] [44, 1, 5357] [68, 1, 12943]"
Data "[88, 1, 172] [51, 1, 239] [32, 1, 682] [44, 1, 5357] [68, 1, 12944]"
Data ""
Sub work2do (ByRef a As LongInt, f1 As mpq_ptr)
Dim As LongInt flag = -1
Dim As Mpq_ptr x, y, z
x = Allocate(Len(__mpq_struct)) : Mpq_init(x)
y = Allocate(Len(__mpq_struct)) : Mpq_init(y)
z = Allocate(Len(__mpq_struct)) : Mpq_init(z)
Dim As Mpz_ptr temp1, temp2
temp1 = Allocate(Len(__Mpz_struct)) : Mpz_init(temp1)
temp2 = Allocate(Len(__Mpz_struct)) : Mpz_init(temp2)
mpq_set(y, f1)
While a > 0
If (a And 1) = 1 Then
If flag = -1 Then
mpq_set(x, y)
flag = 0
Else
Mpz_mul(temp1, _a(x), _b(y))
Mpz_mul(temp2, _b(x), _a(y))
Mpz_add(_a(z), temp1, temp2)
Mpz_mul(temp1, _b(x), _b(y))
Mpz_mul(temp2, _a(x), _a(y))
Mpz_sub(_b(z), temp1, temp2)
mpq_canonicalize(z)
mpq_set(x, z)
End If
End If
Mpz_mul(temp1, _a(y), _b(y))
Mpz_mul(temp2, _b(y), _a(y))
Mpz_add(_a(z), temp1, temp2)
Mpz_mul(temp1, _b(y), _b(y))
Mpz_mul(temp2, _a(y), _a(y))
Mpz_sub(_b(z), temp1, temp2)
mpq_canonicalize(z)
mpq_set(y, z)
a = a Shr 1
Wend
mpq_set(f1, x)
End Sub
' ------=< MAIN >=------
Dim As Mpq_ptr f1, f2, f3
f1 = Allocate(Len(__mpq_struct)) : Mpq_init(f1)
f2 = Allocate(Len(__mpq_struct)) : Mpq_init(f2)
f3 = Allocate(Len(__mpq_struct)) : Mpq_init(f3)
Dim As Mpz_ptr temp1, temp2
temp1 = Allocate(Len(__Mpz_struct)) : Mpz_init(temp1)
temp2 = Allocate(Len(__Mpz_struct)) : Mpz_init(temp2)
Dim As mpf_ptr float
float = Allocate(Len(__mpf_struct)) : Mpf_init(float)
Dim As LongInt m1, a1, b1, flag, t1, t2, t3, t4
Dim As String s, s1, s2, s3, sign
Dim As ZString Ptr zstr
Do
Read s
If s = "" Then Exit Do
flag = -1
While s <> ""
t1 = InStr(s, "[") +1
t2 = InStr(t1, s, ",") +1
t3 = InStr(t2, s, ",") +1
t4 = InStr(t3, s, "]")
s1 = Trim(Mid(s, t1, t2 - t1 -1))
s2 = Trim(Mid(s, t2, t3 - t2 -1))
s3 = Trim(Mid(s, t3, t4 - t3))
m1 = Val(s1)
a1 = Val(s2)
b1 = Val(s3)
sign = IIf(m1 < 0, " - ", " + ")
If m1 < 0 Then a1 = -a1 : m1 = Abs(m1)
s = Mid(s, t4 +1)
Print IIf(flag = 0, sign, ""); IIf(m1 = 1, "", Str(m1));
Print "Atn("; s2; "/" ;s3; ")";
If flag = -1 Then
flag = 0
Mpz_set_si(_a(f1), a1)
Mpz_set_si(_b(f1), b1)
If m1 > 1 Then work2do(m1, f1)
Continue While
End If
Mpz_set_si(_a(f2), a1)
Mpz_set_si(_b(f2), b1)
If m1 > 1 Then work2do(m1, f2)
Mpz_mul(temp1, _a(f1), _b(f2))
Mpz_mul(temp2, _b(f1), _a(f2))
Mpz_add(_a(f3), temp1, temp2)
Mpz_mul(temp1, _b(f1), _b(f2))
Mpz_mul(temp2, _a(f1), _a(f2))
Mpz_sub(_b(f3), temp1, temp2)
mpq_canonicalize(f3)
mpq_set(f1, f3)
Wend
If Mpz_cmp_ui(_b(f1), 1) = 0 AndAlso Mpz_cmp(_a(f1), _b(f1)) = 0 Then
Print " = 1"
Else
Mpf_set_q(float, f1)
gmp_printf(!" = %.*Ff\n", 15, float)
End If
Loop
' empty keyboard buffer
While InKey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End
{{out}}
Atn(1/2) + Atn(1/3) = 1
2Atn(1/3) + Atn(1/7) = 1
4Atn(1/5) - Atn(1/239) = 1
5Atn(1/7) + 2Atn(3/79) = 1
Atn(1/2) + Atn(1/5) + Atn(1/8) = 1
4Atn(1/5) - Atn(1/70) + Atn(1/99) = 1
5Atn(1/7) + 4Atn(1/53) + 2Atn(1/4443) = 1
6Atn(1/8) + 2Atn(1/57) + Atn(1/239) = 1
8Atn(1/10) - Atn(1/239) - 4Atn(1/515) = 1
12Atn(1/18) + 8Atn(1/57) - 5Atn(1/239) = 1
16Atn(1/21) + 3Atn(1/239) + 4Atn(3/1042) = 1
22Atn(1/28) + 2Atn(1/443) - 5Atn(1/1393) - 10Atn(1/11018) = 1
22Atn(1/38) + 17Atn(7/601) + 10Atn(7/8149) = 1
44Atn(1/57) + 7Atn(1/239) - 12Atn(1/682) + 24Atn(1/12943) = 1
88Atn(1/172) + 51Atn(1/239) + 32Atn(1/682) + 44Atn(1/5357) + 68Atn(1/12943) = 1
88Atn(1/172) + 51Atn(1/239) + 32Atn(1/682) + 44Atn(1/5357) + 68Atn(1/12944) = 0.999999188225744
GAP
The formula is entered as a list of pairs [k, x], where each pair means k*atan(x), and all the terms in the list are summed. Like most other solutions, the program will only check that the tangent of the resulting sum is 1. For instance, Check([[5, 1/2], [5, 1/3]]); returns also true, though the result is 5pi/4.
TanPlus := function(a, b)
return (a + b) / (1 - a * b);
end;
TanTimes := function(n, a)
local x;
x := 0;
while n > 0 do
if IsOddInt(n) then
x := TanPlus(x, a);
fi;
a := TanPlus(a, a);
n := QuoInt(n, 2);
od;
return x;
end;
Check := function(a)
local x, p;
x := 0;
for p in a do
x := TanPlus(x, SignInt(p[1]) * TanTimes(AbsInt(p[1]), p[2]));
od;
return x = 1;
end;
ForAll([
[[1, 1/2], [1, 1/3]],
[[2, 1/3], [1, 1/7]],
[[4, 1/5], [-1, 1/239]],
[[5, 1/7], [2, 3/79]],
[[5, 29/278], [7, 3/79]],
[[1, 1/2], [1, 1/5], [1, 1/8]],
[[5, 1/7], [4, 1/53], [2, 1/4443]],
[[6, 1/8], [2, 1/57], [1, 1/239]],
[[8, 1/10], [-1, 1/239], [-4, 1/515]],
[[12, 1/18], [8, 1/57], [-5, 1/239]],
[[16, 1/21], [3, 1/239], [4, 3/1042]],
[[22, 1/28], [2, 1/443], [-5, 1/1393], [-10, 1/11018]],
[[22, 1/38], [17, 7/601], [10, 7/8149]],
[[44, 1/57], [7, 1/239], [-12, 1/682], [24, 1/12943]],
[[88, 1/172], [51, 1/239], [32, 1/682], [44, 1/5357], [68, 1/12943]]], Check);
Check([[88, 1/172], [51, 1/239], [32, 1/682], [44, 1/5357], [68, 1/12944]]);
Go
{{trans|Python}}
package main
import (
"fmt"
"math/big"
)
type mTerm struct {
a, n, d int64
}
var testCases = [][]mTerm{
{{1, 1, 2}, {1, 1, 3}},
{{2, 1, 3}, {1, 1, 7}},
{{4, 1, 5}, {-1, 1, 239}},
{{5, 1, 7}, {2, 3, 79}},
{{1, 1, 2}, {1, 1, 5}, {1, 1, 8}},
{{4, 1, 5}, {-1, 1, 70}, {1, 1, 99}},
{{5, 1, 7}, {4, 1, 53}, {2, 1, 4443}},
{{6, 1, 8}, {2, 1, 57}, {1, 1, 239}},
{{8, 1, 10}, {-1, 1, 239}, {-4, 1, 515}},
{{12, 1, 18}, {8, 1, 57}, {-5, 1, 239}},
{{16, 1, 21}, {3, 1, 239}, {4, 3, 1042}},
{{22, 1, 28}, {2, 1, 443}, {-5, 1, 1393}, {-10, 1, 11018}},
{{22, 1, 38}, {17, 7, 601}, {10, 7, 8149}},
{{44, 1, 57}, {7, 1, 239}, {-12, 1, 682}, {24, 1, 12943}},
{{88, 1, 172}, {51, 1, 239}, {32, 1, 682}, {44, 1, 5357}, {68, 1, 12943}},
{{88, 1, 172}, {51, 1, 239}, {32, 1, 682}, {44, 1, 5357}, {68, 1, 12944}},
}
func main() {
for _, m := range testCases {
fmt.Printf("tan %v = %v\n", m, tans(m))
}
}
var one = big.NewRat(1, 1)
func tans(m []mTerm) *big.Rat {
if len(m) == 1 {
return tanEval(m[0].a, big.NewRat(m[0].n, m[0].d))
}
half := len(m) / 2
a := tans(m[:half])
b := tans(m[half:])
r := new(big.Rat)
return r.Quo(new(big.Rat).Add(a, b), r.Sub(one, r.Mul(a, b)))
}
func tanEval(coef int64, f *big.Rat) *big.Rat {
if coef == 1 {
return f
}
if coef < 0 {
r := tanEval(-coef, f)
return r.Neg(r)
}
ca := coef / 2
cb := coef - ca
a := tanEval(ca, f)
b := tanEval(cb, f)
r := new(big.Rat)
return r.Quo(new(big.Rat).Add(a, b), r.Sub(one, r.Mul(a, b)))
}
{{out}} Last line edited to show only most significant digits of fraction which is near, but not exactly equal to 1.
tan [{1 1 2} {1 1 3}] = 1/1
tan [{2 1 3} {1 1 7}] = 1/1
tan [{4 1 5} {-1 1 239}] = 1/1
tan [{5 1 7} {2 3 79}] = 1/1
tan [{1 1 2} {1 1 5} {1 1 8}] = 1/1
tan [{4 1 5} {-1 1 70} {1 1 99}] = 1/1
tan [{5 1 7} {4 1 53} {2 1 4443}] = 1/1
tan [{6 1 8} {2 1 57} {1 1 239}] = 1/1
tan [{8 1 10} {-1 1 239} {-4 1 515}] = 1/1
tan [{12 1 18} {8 1 57} {-5 1 239}] = 1/1
tan [{16 1 21} {3 1 239} {4 3 1042}] = 1/1
tan [{22 1 28} {2 1 443} {-5 1 1393} {-10 1 11018}] = 1/1
tan [{22 1 38} {17 7 601} {10 7 8149}] = 1/1
tan [{44 1 57} {7 1 239} {-12 1 682} {24 1 12943}] = 1/1
tan [{88 1 172} {51 1 239} {32 1 682} {44 1 5357} {68 1 12943}] = 1/1
tan [{88 1 172} {51 1 239} {32 1 682} {44 1 5357} {68 1 12944}] =
100928801... /
100928883...
Haskell
import Data.Ratio
import Data.List (foldl')
tanPlus :: Fractional a => a -> a -> a
tanPlus a b = (a + b) / (1 - a * b)
tanEval :: (Integral a, Fractional b) => (a, b) -> b
tanEval (0,_) = 0
tanEval (coef,f)
| coef < 0 = -tanEval (-coef, f)
| odd coef = tanPlus f $ tanEval (coef - 1, f)
| otherwise = tanPlus a a
where a = tanEval (coef `div` 2, f)
tans :: (Integral a, Fractional b) => [(a, b)] -> b
tans = foldl' tanPlus 0 . map tanEval
machins = [
[(1, 1%2), (1, 1%3)],
[(2, 1%3), (1, 1%7)],
[(12, 1%18), (8, 1%57), (-5, 1%239)],
[(88, 1%172), (51, 1%239), (32 , 1%682), (44, 1%5357), (68, 1%12943)]]
not_machin = [(88, 1%172), (51, 1%239), (32 , 1%682), (44, 1%5357), (68, 1%12944)]
main = do
putStrLn "Machins:"
mapM_ (\x -> putStrLn $ show (tans x) ++ " <-- " ++ show x) machins
putStr "\nnot Machin: "; print not_machin
print (tans not_machin)
A crazier way to do the above, exploiting the built-in exponentiation algorithms:
import Data.Ratio
-- Private type. Do not use outside of the tans function
newtype Tan a = Tan a deriving (Eq, Show)
instance Fractional a => Num (Tan a) where
_ + _ = undefined
Tan a * Tan b = Tan $ (a + b) / (1 - a * b)
negate _ = undefined
abs _ = undefined
signum _ = undefined
fromInteger 1 = Tan 0 -- identity for the (*) above
fromInteger _ = undefined
instance Fractional a => Fractional (Tan a) where
fromRational _ = undefined
recip (Tan f) = Tan (-f) -- inverse for the (*) above
tans :: (Integral a, Fractional b) => [(a, b)] -> b
tans xs = x where
Tan x = product [Tan f ^^ coef | (coef,f) <- xs]
machins = [
[(1, 1%2), (1, 1%3)],
[(2, 1%3), (1, 1%7)],
[(12, 1%18), (8, 1%57), (-5, 1%239)],
[(88, 1%172), (51, 1%239), (32 , 1%682), (44, 1%5357), (68, 1%12943)]]
not_machin = [(88, 1%172), (51, 1%239), (32 , 1%682), (44, 1%5357), (68, 1%12944)]
main = do
putStrLn "Machins:"
mapM_ (\x -> putStrLn $ show (tans x) ++ " <-- " ++ show x) machins
putStr "\nnot Machin: "; print not_machin
print (tans not_machin)
J
'''Solution''':
machin =: 1r4p1 = [: +/ ({. * _3 o. %/@:}.)"1@:x:
'''Example''' (''test cases from task description''):
R =: <@:(0&".);._2 ];._2 noun define
Β 1 Β 1 Β Β 2
Β 1 Β 1 Β Β 3
------------
Β 2 Β 1 Β Β 3
Β 1 Β 1 Β Β 7
------------
Β 4 Β 1 Β Β 5
Β _1 Β 1 Β 239
------------
Β 5 Β 1 Β Β 7
Β 2 Β 3 Β Β 79
------------
Β 5 29 Β 278
Β 7 Β 3 Β Β 79
------------
Β 1 Β 1 Β Β 2
Β 1 Β 1 Β Β 5
Β 1 Β 1 Β Β 8
------------
Β 4 Β 1 Β Β 5
Β _1 Β 1 Β Β 70
Β 1 Β 1 Β Β 99
------------
Β 5 Β 1 Β Β 7
Β 4 Β 1 Β Β 53
Β 2 Β 1 Β 4443
------------
Β 6 Β 1 Β Β 8
Β 2 Β 1 Β Β 57
Β 1 Β 1 Β 239
------------
Β 8 Β 1 Β Β 10
Β _1 Β 1 Β 239
Β _4 Β 1 Β 515
------------
Β 12 Β 1 Β Β 18
Β 8 Β 1 Β Β 57
Β _5 Β 1 Β 239
------------
Β 16 Β 1 Β Β 21
Β 3 Β 1 Β 239
Β 4 Β 3 Β 1042
------------
Β 22 Β 1 Β Β 28
Β 2 Β 1 Β 443
Β _5 Β 1 Β 1393
_10 Β 1 11018
------------
Β 22 Β 1 Β Β 38
Β 17 Β 7 Β 601
Β 10 Β 7 Β 8149
------------
Β 44 Β 1 Β Β 57
Β 7 Β 1 Β 239
_12 Β 1 Β 682
Β 24 Β 1 12943
------------
Β 88 Β 1 Β 172
Β 51 Β 1 Β 239
Β 32 Β 1 Β 682
Β 44 Β 1 Β 5357
Β 68 Β 1 12943
------------
)
Β Β machin&> R
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
'''Example''' (''counterexample''):
Β Β counterExample=. 12944 (<_1;_1)} >{:R
Β Β counterExample Β NB. Same as final test case with 12943 incremented to 12944
88 1 Β 172
51 1 Β 239
32 1 Β 682
44 1 Β 5357
68 1 12944
Β Β machin counterExample
0
'''Notes''': The function machin compares the results of each formula to Ο/4 (expressed as 1r4p1 in J's numeric notation). The first example above shows the results of these comparisons for each formula (with 1 for true and 0 for false). In J, arctan is expressed as _3 o. ''values'' and the function x: coerces values to exact representation; thereafter J will maintain exactness throughout its calculations, as long as it can.
Julia
using AbstractAlgebra # implements arbitrary precision rationals
tanplus(x,y) = (x + y) / (1 - x * y)
function taneval(coef, frac)
if coef == 0
return 0
elseif coef < 0
return -taneval(-coef, frac)
elseif isodd(coef)
return tanplus(frac, taneval(coef - 1, frac))
else
x = taneval(div(coef, 2), frac)
return tanplus(x, x)
end
end
taneval(tup::Tuple) = taneval(tup[1], tup[2])
tans(v::Vector{Tuple{BigInt, Rational{BigInt}}}) = foldl(tanplus, map(taneval, v), init=0)
const testmats = Dict{Vector{Tuple{BigInt, Rational{BigInt}}}, Bool}([
([(1, 1//2), (1, 1//3)], true), ([(2, 1//3), (1, 1//7)], true),
([(12, 1//18), (8, 1//57), (-5, 1//239)], true),
([(88, 1//172), (51, 1//239), (32, 1//682), (44, 1//5357), (68, 1//12943)], true),
([(88, 1//172), (51, 1//239), (32, 1//682), (44, 1//5357), (68, 1//12944)], false)])
function runtestmats()
println("Testing matrices:")
for (k, m) in testmats
ans = tans(k)
println((ans == 1) == m ? "Verified as $m: " : "Not Verified as $m: ", "tan $k = $ans")
end
end
runtestmats()
{{output}}
Testing matrices:
Verified as true: tan Tuple{BigInt,Rational{BigInt}}[(1, 1//2), (1, 1//3)] = 1//1
Verified as true: tan Tuple{BigInt,Rational{BigInt}}[(2, 1//3), (1, 1//7)] = 1//1
Verified as true: tan Tuple{BigInt,Rational{BigInt}}[(88, 1//172), (51, 1//239), (32, 1//682), (44, 1//5357), (68, 1//12943)] = 1//1
Verified as true: tan Tuple{BigInt,Rational{BigInt}}[(12, 1//18), (8, 1//57), (-5, 1//239)] = 1//1
Verified as false: tan Tuple{BigInt,Rational{BigInt}}[(88, 1//172), (51, 1//239), (32, 1//682), (44, 1//5357), (68, 1//12944)] = 1009288018000944050967896710431587186456256928584351786643498522649995492271475761189348270710224618853590682465929080006511691833816436374107451368838065354726517908250456341991684635768915704374493675498637876700129004484434187627909285979251682006538817341793224963346197503893270875008524149334251672855130857035205217929335932890740051319216343365800342290782260673215928499123722781078448297609548233999010983373327601187505623621602789012550584784738082074783523787011976757247516095289966708782862528690942242793667539020699840402353522108223//1009288837315638583415701528780402795721935641614456853534313491853293025565940011104051964874275710024625850092154664245109626053906509780125743180758231049920425664246286578958307532545458843067352531217230461290763258378749459637420702619029075083089762088232401888676895047947363883809724322868121990870409574061477638203859217672620508200713073485398199091153535700094640095900731630771349477187594074169815106104524371099618096164871416282464532355211521113449237814080332335526420331468258917484010722587072087349909684004660371264507984339711
Kotlin
As the JVM and Kotlin standard libraries lack a BigRational class, I've written one which just provides sufficient functionality to complete this task:
// version 1.1.3
import java.math.BigInteger
val bigZero = BigInteger.ZERO
val bigOne = BigInteger.ONE
class BigRational : Comparable<BigRational> {
val num: BigInteger
val denom: BigInteger
constructor(n: BigInteger, d: BigInteger) {
require(d != bigZero)
var nn = n
var dd = d
if (nn == bigZero) {
dd = bigOne
}
else if (dd < bigZero) {
nn = -nn
dd = -dd
}
val g = nn.gcd(dd)
if (g > bigOne) {
nn /= g
dd /= g
}
num = nn
denom = dd
}
constructor(n: Long, d: Long) : this(BigInteger.valueOf(n), BigInteger.valueOf(d))
operator fun plus(other: BigRational) =
BigRational(num * other.denom + denom * other.num, other.denom * denom)
operator fun unaryMinus() = BigRational(-num, denom)
operator fun minus(other: BigRational) = this + (-other)
operator fun times(other: BigRational) = BigRational(this.num * other.num, this.denom * other.denom)
fun inverse(): BigRational {
require(num != bigZero)
return BigRational(denom, num)
}
operator fun div(other: BigRational) = this * other.inverse()
override fun compareTo(other: BigRational): Int {
val diff = this - other
return when {
diff.num < bigZero -> -1
diff.num > bigZero -> +1
else -> 0
}
}
override fun equals(other: Any?): Boolean {
if (other == null || other !is BigRational) return false
return this.compareTo(other) == 0
}
override fun toString() = if (denom == bigOne) "$num" else "$num/$denom"
companion object {
val ZERO = BigRational(bigZero, bigOne)
val ONE = BigRational(bigOne, bigOne)
}
}
/** represents a term of the form: c * atan(n / d) */
class Term(val c: Long, val n: Long, val d: Long) {
override fun toString() = when {
c == 1L -> " + "
c == -1L -> " - "
c < 0L -> " - ${-c}*"
else -> " + $c*"
} + "atan($n/$d)"
}
val one = BigRational.ONE
fun tanSum(terms: List<Term>): BigRational {
if (terms.size == 1) return tanEval(terms[0].c, BigRational(terms[0].n, terms[0].d))
val half = terms.size / 2
val a = tanSum(terms.take(half))
val b = tanSum(terms.drop(half))
return (a + b) / (one - (a * b))
}
fun tanEval(c: Long, f: BigRational): BigRational {
if (c == 1L) return f
if (c < 0L) return -tanEval(-c, f)
val ca = c / 2
val cb = c - ca
val a = tanEval(ca, f)
val b = tanEval(cb, f)
return (a + b) / (one - (a * b))
}
fun main(args: Array<String>) {
val termsList = listOf(
listOf(Term(1, 1, 2), Term(1, 1, 3)),
listOf(Term(2, 1, 3), Term(1, 1, 7)),
listOf(Term(4, 1, 5), Term(-1, 1, 239)),
listOf(Term(5, 1, 7), Term(2, 3, 79)),
listOf(Term(5, 29, 278), Term(7, 3, 79)),
listOf(Term(1, 1, 2), Term(1, 1, 5), Term(1, 1, 8)),
listOf(Term(4, 1, 5), Term(-1, 1, 70), Term(1, 1, 99)),
listOf(Term(5, 1, 7), Term(4, 1, 53), Term(2, 1, 4443)),
listOf(Term(6, 1, 8), Term(2, 1, 57), Term(1, 1, 239)),
listOf(Term(8, 1, 10), Term(-1, 1, 239), Term(-4, 1, 515)),
listOf(Term(12, 1, 18), Term(8, 1, 57), Term(-5, 1, 239)),
listOf(Term(16, 1, 21), Term(3, 1, 239), Term(4, 3, 1042)),
listOf(Term(22, 1, 28), Term(2, 1, 443), Term(-5, 1, 1393), Term(-10, 1, 11018)),
listOf(Term(22, 1, 38), Term(17, 7, 601), Term(10, 7, 8149)),
listOf(Term(44, 1, 57), Term(7, 1, 239), Term(-12, 1, 682), Term(24, 1, 12943)),
listOf(Term(88, 1, 172), Term(51, 1, 239), Term(32, 1, 682), Term(44, 1, 5357), Term(68, 1, 12943)),
listOf(Term(88, 1, 172), Term(51, 1, 239), Term(32, 1, 682), Term(44, 1, 5357), Term(68, 1, 12944))
)
for (terms in termsList) {
val f = String.format("%-5s << 1 == tan(", tanSum(terms) == one)
print(f)
print(terms[0].toString().drop(3))
for (i in 1 until terms.size) print(terms[i])
println(")")
}
}
{{out}}
true << 1 == tan(atan(1/2) + atan(1/3))
true << 1 == tan(2*atan(1/3) + atan(1/7))
true << 1 == tan(4*atan(1/5) - atan(1/239))
true << 1 == tan(5*atan(1/7) + 2*atan(3/79))
true << 1 == tan(5*atan(29/278) + 7*atan(3/79))
true << 1 == tan(atan(1/2) + atan(1/5) + atan(1/8))
true << 1 == tan(4*atan(1/5) - atan(1/70) + atan(1/99))
true << 1 == tan(5*atan(1/7) + 4*atan(1/53) + 2*atan(1/4443))
true << 1 == tan(6*atan(1/8) + 2*atan(1/57) + atan(1/239))
true << 1 == tan(8*atan(1/10) - atan(1/239) - 4*atan(1/515))
true << 1 == tan(12*atan(1/18) + 8*atan(1/57) - 5*atan(1/239))
true << 1 == tan(16*atan(1/21) + 3*atan(1/239) + 4*atan(3/1042))
true << 1 == tan(22*atan(1/28) + 2*atan(1/443) - 5*atan(1/1393) - 10*atan(1/11018))
true << 1 == tan(22*atan(1/38) + 17*atan(7/601) + 10*atan(7/8149))
true << 1 == tan(44*atan(1/57) + 7*atan(1/239) - 12*atan(1/682) + 24*atan(1/12943))
true << 1 == tan(88*atan(1/172) + 51*atan(1/239) + 32*atan(1/682) + 44*atan(1/5357) + 68*atan(1/12943))
false << 1 == tan(88*atan(1/172) + 51*atan(1/239) + 32*atan(1/682) + 44*atan(1/5357) + 68*atan(1/12944))
=={{header|Mathematica}} / {{header|Wolfram Language}}==
{{Out}}
```txt
True
True
True
True
True
True
True
True
True
True
True
True
True
True
True
True
False
Maxima
trigexpand:true$
is(tan(atan(1/2)+atan(1/3))=1);
is(tan(2*atan(1/3)+atan(1/7))=1);
is(tan(4*atan(1/5)-atan(1/239))=1);
is(tan(5*atan(1/7)+2*atan(3/79))=1);
is(tan(5*atan(29/278)+7*atan(3/79))=1);
is(tan(atan(1/2)+atan(1/5)+atan(1/8))=1);
is(tan(4*atan(1/5)-atan(1/70)+atan(1/99))=1);
is(tan(5*atan(1/7)+4*atan(1/53)+2*atan(1/4443))=1);
is(tan(6*atan(1/8)+2*atan(1/57)+atan(1/239))=1);
is(tan(8*atan(1/10)-atan(1/239)-4*atan(1/515))=1);
is(tan(12*atan(1/18)+8*atan(1/57)-5*atan(1/239))=1);
is(tan(16*atan(1/21)+3*atan(1/239)+4*atan(3/1042))=1);
is(tan(22*atan(1/28)+2*atan(1/443)-5*atan(1/1393)-10*atan(1/11018))=1);
is(tan(22*atan(1/38)+17*atan(7/601)+10*atan(7/8149))=1);
is(tan(44*atan(1/57)+7*atan(1/239)-12*atan(1/682)+24*atan(1/12943))=1);
is(tan(88*atan(1/172)+51*atan(1/239)+32*atan(1/682)+44*atan(1/5357)+68*atan(1/12943))=1);
is(tan(88*atan(1/172)+51*atan(1/239)+32*atan(1/682)+44*atan(1/5357)+68*atan(1/12944))=1);
{{out}}
(%i2)
(%o2) true
(%i3)
(%o3) true
(%i4)
(%o4) true
(%i5)
(%o5) true
(%i6)
(%o6) true
(%i7)
(%o7) true
(%i8)
(%o8) true
(%i9)
(%o9) true
(%i10)
(%o10) true
(%i11)
(%o11) true
(%i12)
(%o12) true
(%i13)
(%o13) true
(%i14)
(%o14) true
(%i15)
(%o15) true
(%i16)
(%o16) true
(%i17)
(%o17) true
(%i18)
(%o18) false
OCaml
open Num;; (* use exact rationals for results *)
let tadd p q = (p +/ q) // ((Int 1) -/ (p */ q)) in
(* tan(n*arctan(a/b)) *)
let rec tan_expr (n,a,b) =
if n = 1 then (Int a)//(Int b) else
if n = -1 then (Int (-a))//(Int b) else
let m = n/2 in
let tm = tan_expr (m,a,b) in
let m2 = tadd tm tm and k = n-m-m in
if k = 0 then m2 else tadd (tan_expr (k,a,b)) m2 in
let verify (k, tlist) =
Printf.printf "Testing: pi/%d = " k;
let t_str = List.map (fun (x,y,z) -> Printf.sprintf "%d*atan(%d/%d)" x y z) tlist in
print_endline (String.concat " + " t_str);
let ans_terms = List.map tan_expr tlist in
let answer = List.fold_left tadd (Int 0) ans_terms in
Printf.printf " tan(RHS) is %s\n" (if answer = (Int 1) then "one" else "not one") in
(* example: prog 4 5 29 278 7 3 79 represents pi/4 = 5*atan(29/278) + 7*atan(3/79) *)
let args = Sys.argv in
let nargs = Array.length args in
let v k = int_of_string args.(k) in
let rec triples n =
if n+2 > nargs-1 then []
else (v n, v (n+1), v (n+2)) :: triples (n+3) in
if nargs > 4 then
let dat = (v 1, triples 2) in
verify dat
else
List.iter verify [
(4,[(1,1,2);(1,1,3)]);
(4,[(2,1,3);(1,1,7)]);
(4,[(4,1,5);(-1,1,239)]);
(4,[(5,1,7);(2,3,79)]);
(4,[(5,29,278);(7,3,79)]);
(4,[(1,1,2);(1,1,5);(1,1,8)]);
(4,[(4,1,5);(-1,1,70);(1,1,99)]);
(4,[(5,1,7);(4,1,53);(2,1,4443)]);
(4,[(6,1,8);(2,1,57);(1,1,239)]);
(4,[(8,1,10);(-1,1,239);(-4,1,515)]);
(4,[(12,1,18);(8,1,57);(-5,1,239)]);
(4,[(16,1,21);(3,1,239);(4,3,1042)]);
(4,[(22,1,28);(2,1,443);(-5,1,1393);(-10,1,11018)]);
(4,[(22,1,38);(17,7,601);(10,7,8149)]);
(4,[(44,1,57);(7,1,239);(-12,1,682);(24,1,12943)]);
(4,[(88,1,172);(51,1,239);(32,1,682);(44,1,5357);(68,1,12943)]);
(4,[(88,1,172);(51,1,239);(32,1,682);(44,1,5357);(68,1,12944)])
]
Compile with ocamlopt -o verify_machin.opt nums.cmxa verify_machin.ml or run with ocaml nums.cma verify_machin.ml
{{out}}
Testing: pi/4 = 1*atan(1/2) + 1*atan(1/3)
tan(RHS) is one
Testing: pi/4 = 2*atan(1/3) + 1*atan(1/7)
tan(RHS) is one
Testing: pi/4 = 4*atan(1/5) + -1*atan(1/239)
tan(RHS) is one
Testing: pi/4 = 5*atan(1/7) + 2*atan(3/79)
tan(RHS) is one
Testing: pi/4 = 5*atan(29/278) + 7*atan(3/79)
tan(RHS) is one
Testing: pi/4 = 1*atan(1/2) + 1*atan(1/5) + 1*atan(1/8)
tan(RHS) is one
Testing: pi/4 = 4*atan(1/5) + -1*atan(1/70) + 1*atan(1/99)
tan(RHS) is one
Testing: pi/4 = 5*atan(1/7) + 4*atan(1/53) + 2*atan(1/4443)
tan(RHS) is one
Testing: pi/4 = 6*atan(1/8) + 2*atan(1/57) + 1*atan(1/239)
tan(RHS) is one
Testing: pi/4 = 8*atan(1/10) + -1*atan(1/239) + -4*atan(1/515)
tan(RHS) is one
Testing: pi/4 = 12*atan(1/18) + 8*atan(1/57) + -5*atan(1/239)
tan(RHS) is one
Testing: pi/4 = 16*atan(1/21) + 3*atan(1/239) + 4*atan(3/1042)
tan(RHS) is one
Testing: pi/4 = 22*atan(1/28) + 2*atan(1/443) + -5*atan(1/1393) + -10*atan(1/11018)
tan(RHS) is one
Testing: pi/4 = 22*atan(1/38) + 17*atan(7/601) + 10*atan(7/8149)
tan(RHS) is one
Testing: pi/4 = 44*atan(1/57) + 7*atan(1/239) + -12*atan(1/682) + 24*atan(1/12943)
tan(RHS) is one
Testing: pi/4 = 88*atan(1/172) + 51*atan(1/239) + 32*atan(1/682) + 44*atan(1/5357) + 68*atan(1/12943)
tan(RHS) is one
Testing: pi/4 = 88*atan(1/172) + 51*atan(1/239) + 32*atan(1/682) + 44*atan(1/5357) + 68*atan(1/12944)
tan(RHS) is not one
ooRexx
/*REXX ----------------------------------------------------------------
* 09.04.2014 Walter Pachl the REXX solution adapted for ooRexx
* which provides a function package rxMath
*--------------------------------------------------------------------*/
Numeric Digits 16
Numeric Fuzz 3; pi=rxCalcpi(); a.=''
a.1 = 'pi/4 = rxCalcarctan(1/2,16,'R') + rxCalcarctan(1/3,16,'R')'
a.2 = 'pi/4 = 2*rxCalcarctan(1/3,16,'R') + rxCalcarctan(1/7,16,'R')'
a.3 = 'pi/4 = 4*rxCalcarctan(1/5,16,'R') - rxCalcarctan(1/239,16,'R')'
a.4 = 'pi/4 = 5*rxCalcarctan(1/7,16,'R') + 2*rxCalcarctan(3/79,16,'R')'
a.5 = 'pi/4 = 5*rxCalcarctan(29/278,16,'R') + 7*rxCalcarctan(3/79,16,'R')'
a.6 = 'pi/4 = rxCalcarctan(1/2,16,'R') + rxCalcarctan(1/5,16,'R') + rxCalcarctan(1/8,16,'R')'
a.7 = 'pi/4 = 4*rxCalcarctan(1/5,16,'R') - rxCalcarctan(1/70,16,'R') + rxCalcarctan(1/99,16,'R')'
a.8 = 'pi/4 = 5*rxCalcarctan(1/7,16,'R') + 4*rxCalcarctan(1/53,16,'R') + 2*rxCalcarctan(1/4443,16,'R')'
a.9 = 'pi/4 = 6*rxCalcarctan(1/8,16,'R') + 2*rxCalcarctan(1/57,16,'R') + rxCalcarctan(1/239,16,'R')'
a.10 = 'pi/4 = 8*rxCalcarctan(1/10,16,'R') - rxCalcarctan(1/239,16,'R') - 4*rxCalcarctan(1/515,16,'R')'
a.11 = 'pi/4 = 12*rxCalcarctan(1/18,16,'R') + 8*rxCalcarctan(1/57,16,'R') - 5*rxCalcarctan(1/239,16,'R')'
a.12 = 'pi/4 = 16*rxCalcarctan(1/21,16,'R') + 3*rxCalcarctan(1/239,16,'R') + 4*rxCalcarctan(3/1042,16,'R')'
a.13 = 'pi/4 = 22*rxCalcarctan(1/28,16,'R') + 2*rxCalcarctan(1/443,16,'R') - 5*rxCalcarctan(1/1393,16,'R') - 10*rxCalcarctan(1/11018,16,'R')'
a.14 = 'pi/4 = 22*rxCalcarctan(1/38,16,'R') + 17*rxCalcarctan(7/601,16,'R') + 10*rxCalcarctan(7/8149,16,'R')'
a.15 = 'pi/4 = 44*rxCalcarctan(1/57,16,'R') + 7*rxCalcarctan(1/239,16,'R') - 12*rxCalcarctan(1/682,16,'R') + 24*rxCalcarctan(1/12943,16,'R')'
a.16 = 'pi/4 = 88*rxCalcarctan(1/172,16,'R') + 51*rxCalcarctan(1/239,16,'R') + 32*rxCalcarctan(1/682,16,'R') + 44*rxCalcarctan(1/5357,16,'R') + 68*rxCalcarctan(1/12943,16,'R')'
a.17 = 'pi/4 = 88*rxCalcarctan(1/172,16,'R') + 51*rxCalcarctan(1/239,16,'R') + 32*rxCalcarctan(1/682,16,'R') + 44*rxCalcarctan(1/5357,16,'R') + 68*rxCalcarctan(1/12944,16,'R')'
do j=1 while a.j\=='' /*evaluate each of the formulas. */
interpret 'answer=' "(" a.j ")" /*the heavy lifting.*/
say right(word('bad OK',answer+1),3)": " space(a.j,0)
end /*j*/ /* [?] show OK | bad, formula. */
::requires rxmath library
{{out}}
OK: pi/4=rxCalcarctan(1/2,16,R)+rxCalcarctan(1/3,16,R)
OK: pi/4=2*rxCalcarctan(1/3,16,R)+rxCalcarctan(1/7,16,R)
OK: pi/4=4*rxCalcarctan(1/5,16,R)-rxCalcarctan(1/239,16,R)
OK: pi/4=5*rxCalcarctan(1/7,16,R)+2*rxCalcarctan(3/79,16,R)
OK: pi/4=5*rxCalcarctan(29/278,16,R)+7*rxCalcarctan(3/79,16,R)
OK: pi/4=rxCalcarctan(1/2,16,R)+rxCalcarctan(1/5,16,R)+rxCalcarctan(1/8,16,R)
OK: pi/4=4*rxCalcarctan(1/5,16,R)-rxCalcarctan(1/70,16,R)+rxCalcarctan(1/99,16,R)
OK: pi/4=5*rxCalcarctan(1/7,16,R)+4*rxCalcarctan(1/53,16,R)+2*rxCalcarctan(1/4443,16,R)
OK: pi/4=6*rxCalcarctan(1/8,16,R)+2*rxCalcarctan(1/57,16,R)+rxCalcarctan(1/239,16,R)
OK: pi/4=8*rxCalcarctan(1/10,16,R)-rxCalcarctan(1/239,16,R)-4*rxCalcarctan(1/515,16,R)
OK: pi/4=12*rxCalcarctan(1/18,16,R)+8*rxCalcarctan(1/57,16,R)-5*rxCalcarctan(1/239,16,R)
OK: pi/4=16*rxCalcarctan(1/21,16,R)+3*rxCalcarctan(1/239,16,R)+4*rxCalcarctan(3/1042,16,R)
OK: pi/4=22*rxCalcarctan(1/28,16,R)+2*rxCalcarctan(1/443,16,R)-5*rxCalcarctan(1/1393,16,R)-10*rxCalcarctan(1/11018,16,R)
OK: pi/4=22*rxCalcarctan(1/38,16,R)+17*rxCalcarctan(7/601,16,R)+10*rxCalcarctan(7/8149,16,R)
OK: pi/4=44*rxCalcarctan(1/57,16,R)+7*rxCalcarctan(1/239,16,R)-12*rxCalcarctan(1/682,16,R)+24*rxCalcarctan(1/12943,16,R)
OK: pi/4=88*rxCalcarctan(1/172,16,R)+51*rxCalcarctan(1/239,16,R)+32*rxCalcarctan(1/682,16,R)+44*rxCalcarctan(1/5357,16,R)+68*rxCalcarctan(1/12943,16,R)
bad: pi/4=88*rxCalcarctan(1/172,16,R)+51*rxCalcarctan(1/239,16,R)+32*rxCalcarctan(1/682,16,R)+44*rxCalcarctan(1/5357,16,R)+68*rxCalcarctan(1/12944,16,R)
PARI/GP
tanEval(coef, f)={
if (coef <= 1, return(if(coef<1,-tanEval(-coef, f),f)));
my(a=tanEval(coef\2, f), b=tanEval(coef-coef\2, f));
(a + b)/(1 - a*b)
};
tans(xs)={
if (#xs == 1, return(tanEval(xs[1][1], xs[1][2])));
my(a=tans(xs[1..#xs\2]),b=tans(xs[#xs\2+1..#xs]));
(a + b)/(1 - a*b)
};
test(v)={
my(t=tans(v));
if(t==1,print("OK"),print("Error: "v))
};
test([[1,1/2],[1,1/3]]);
test([[2,1/3],[1,1/7]]);
test([[4,1/5],[-1,1/239]]);
test([[5,1/7],[2,3/79]]);
test([[5,29/278],[7,3/79]]);
test([[1,1/2],[1,1/5],[1,1/8]]);
test([[4,1/5],[-1,1/70],[1,1/99]]);
test([[5,1/7],[4,1/53],[2,1/4443]]);
test([[6,1/8],[2,1/57],[1,1/239]]);
test([[8,1/10],[-1,1/239],[-4,1/515]]);
test([[12,1/18],[8,1/57],[-5,1/239]]);
test([[16,1/21],[3,1/239],[4,3/1042]]);
test([[22,1/28],[2,1/443],[-5,1/1393],[-10,1/11018]]);
test([[22,1/38],[17,7/601],[10,7/8149]]);
test([[44,1/57],[7,1/239],[-12,1/682],[24,1/12943]]);
test([[88,1/172],[51,1/239],[32,1/682],[44,1/5357],[68,1/12943]]);
test([[88,1/172],[51,1/239],[32,1/682],[44,1/5357],[68,1/12944]]);
{{out}}
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
OK
Error: [[88, 1/172], [51, 1/239], [32, 1/682], [44, 1/5357], [68, 1/12944]]
Perl
"GMP";
sub taneval {
my($coef,$f) = @_;
$f = Math::BigRat->new($f) unless ref($f);
return 0 if $coef == 0;
return $f if $coef == 1;
return -taneval(-$coef, $f) if $coef < 0;
my($a,$b) = ( taneval($coef>>1, $f), taneval($coef-($coef>>1),$f) );
($a+$b)/(1-$a*$b);
}
sub tans {
my @xs=@_;
return taneval(@{$xs[0]}) if scalar(@xs)==1;
my($a,$b) = ( tans(@xs[0..($#xs>>1)]), tans(@xs[($#xs>>1)+1..$#xs]) );
($a+$b)/(1-$a*$b);
}
sub test {
printf "%5s (%s)\n", (tans(@_)==1)?"OK":"Error", join(" ",map{"[@$_]"} @_);
}
test([1,'1/2'], [1,'1/3']);
test([2,'1/3'], [1,'1/7']);
test([4,'1/5'], [-1,'1/239']);
test([5,'1/7'],[2,'3/79']);
test([5,'29/278'],[7,'3/79']);
test([1,'1/2'],[1,'1/5'],[1,'1/8']);
test([4,'1/5'],[-1,'1/70'],[1,'1/99']);
test([5,'1/7'],[4,'1/53'],[2,'1/4443']);
test([6,'1/8'],[2,'1/57'],[1,'1/239']);
test([8,'1/10'],[-1,'1/239'],[-4,'1/515']);
test([12,'1/18'],[8,'1/57'],[-5,'1/239']);
test([16,'1/21'],[3,'1/239'],[4,'3/1042']);
test([22,'1/28'],[2,'1/443'],[-5,'1/1393'],[-10,'1/11018']);
test([22,'1/38'],[17,'7/601'],[10,'7/8149']);
test([44,'1/57'],[7,'1/239'],[-12,'1/682'],[24,'1/12943']);
test([88,'1/172'],[51,'1/239'],[32,'1/682'],[44,'1/5357'],[68,'1/12943']);
test([88,'1/172'],[51,'1/239'],[32,'1/682'],[44,'1/5357'],[68,'1/12944']);
{{out}}
OK ([1 1/2] [1 1/3])
OK ([2 1/3] [1 1/7])
OK ([4 1/5] [-1 1/239])
OK ([5 1/7] [2 3/79])
OK ([5 29/278] [7 3/79])
OK ([1 1/2] [1 1/5] [1 1/8])
OK ([4 1/5] [-1 1/70] [1 1/99])
OK ([5 1/7] [4 1/53] [2 1/4443])
OK ([6 1/8] [2 1/57] [1 1/239])
OK ([8 1/10] [-1 1/239] [-4 1/515])
OK ([12 1/18] [8 1/57] [-5 1/239])
OK ([16 1/21] [3 1/239] [4 3/1042])
OK ([22 1/28] [2 1/443] [-5 1/1393] [-10 1/11018])
OK ([22 1/38] [17 7/601] [10 7/8149])
OK ([44 1/57] [7 1/239] [-12 1/682] [24 1/12943])
OK ([88 1/172] [51 1/239] [32 1/682] [44 1/5357] [68 1/12943])
Error ([88 1/172] [51 1/239] [32 1/682] [44 1/5357] [68 1/12944])
Perl 6
{{Works with|rakudo|2018.03}} The coercion to FatRat provides for exact computation for all input.
{{trans|Perl}}
sub taneval ($coef, $f) {
return 0 if $coef == 0;
return $f if $coef == 1;
return -taneval(-$coef, $f) if $coef < 0;
my $a = taneval($coef+>1, $f);
my $b = taneval($coef - $coef+>1, $f);
($a+$b)/(1-$a*$b);
}
sub tans (@xs) {
return taneval(@xs[0;0], @xs[0;1].FatRat) if @xs == 1;
my $a = tans(@xs[0 .. (-1+@xs+>1)]);
my $b = tans(@xs[(-1+@xs+>1)+1 .. -1+@xs]);
($a+$b)/(1-$a*$b);
}
sub verify (@eqn) {
printf "%5s (%s)\n", (tans(@eqn) == 1) ?? "OK" !! "Error",
(map { "[{.[0]} {.[1].nude.join('/')}]" }, @eqn).join(' ');
}
verify($_) for
([[1,1/2], [1,1/3]],
[[2,1/3], [1,1/7]],
[[4,1/5], [-1,1/239]],
[[5,1/7], [2,3/79]],
[[5,29/278], [7,3/79]],
[[1,1/2], [1,1/5], [1,1/8]],
[[4,1/5], [-1,1/70], [1,1/99]],
[[5,1/7], [4,1/53], [2,1/4443]],
[[6,1/8], [2,1/57], [1,1/239]],
[[8,1/10], [-1,1/239], [-4,1/515]],
[[12,1/18], [8,1/57], [-5,1/239]],
[[16,1/21], [3,1/239], [4,3/1042]],
[[22,1/28], [2,1/443], [-5,1/1393], [-10,1/11018]],
[[22,1/38], [17,7/601], [10,7/8149]],
[[44,1/57], [7,1/239], [-12,1/682], [24,1/12943]],
[[88,1/172], [51,1/239], [32,1/682], [44,1/5357], [68,1/12943]],
[[88,1/172], [51,1/239], [32,1/682], [44,1/5357], [68,1/21944]]
);
{{out}}
OK ([1 1/2] [1 1/3])
OK ([2 1/3] [1 1/7])
OK ([4 1/5] [-1 1/239])
OK ([5 1/7] [2 3/79])
OK ([5 29/278] [7 3/79])
OK ([1 1/2] [1 1/5] [1 1/8])
OK ([4 1/5] [-1 1/70] [1 1/99])
OK ([5 1/7] [4 1/53] [2 1/4443])
OK ([6 1/8] [2 1/57] [1 1/239])
OK ([8 1/10] [-1 1/239] [-4 1/515])
OK ([12 1/18] [8 1/57] [-5 1/239])
OK ([16 1/21] [3 1/239] [4 3/1042])
OK ([22 1/28] [2 1/443] [-5 1/1393] [-10 1/11018])
OK ([22 1/38] [17 7/601] [10 7/8149])
OK ([44 1/57] [7 1/239] [-12 1/682] [24 1/12943])
OK ([88 1/172] [51 1/239] [32 1/682] [44 1/5357] [68 1/12943])
Error ([88 1/172] [51 1/239] [32 1/682] [44 1/5357] [68 1/21944])
Phix
Naieve version
Hint: rather than test tan(a) for 1.0, test whether the sprint of it, which is rounded to 10 significant digits, is "1.0".
The failing test case, I believe, is only accurate to 6 (or perhaps 7, see fractions output) significant digits.
procedure test(atom a)
if -3*PI/4 >= a then ?9/0 end if
if 5*PI/4 <= a then ?9/0 end if
string s = sprint(tan(a))
?s -- or test for "1.0", but not 1.0
end procedure
test( arctan(1 / 2) + arctan(1 / 3))
test( 2*arctan(1 / 3) + arctan(1 / 7))
test( 4*arctan(1 / 5) - arctan(1 / 239))
test( 5*arctan(1 / 7) + 2*arctan(3 / 79))
test( 5*arctan(29/ 278) + 7*arctan(3 / 79))
test( arctan(1 / 2) + arctan(1 / 5) + arctan(1 / 8))
test( 4*arctan(1 / 5) - arctan(1 / 70) + arctan(1 / 99))
test( 5*arctan(1 / 7) + 4*arctan(1 / 53) + 2*arctan(1 / 4443))
test( 6*arctan(1 / 8) + 2*arctan(1 / 57) + arctan(1 / 239))
test( 8*arctan(1 / 10) - arctan(1 / 239) - 4*arctan(1 / 515))
test(12*arctan(1 / 18) + 8*arctan(1 / 57) - 5*arctan(1 / 239))
test(16*arctan(1 / 21) + 3*arctan(1 / 239) + 4*arctan(3 / 1042))
test(22*arctan(1 / 28) + 2*arctan(1 / 443) - 5*arctan(1 / 1393) - 10*arctan(1 / 11018))
test(22*arctan(1 / 38) + 17*arctan(7 / 601) +10*arctan(7 / 8149))
test(44*arctan(1 / 57) + 7*arctan(1 / 239) -12*arctan(1 / 682) + 24*arctan(1 / 12943))
test(88*arctan(1 / 172) + 51*arctan(1 / 239) +32*arctan(1 / 682) + 44*arctan(1 / 5357) + 68*arctan(1 / 12943))
?"==="
test(88*arctan(1 / 172) + 51*arctan(1 / 239) + 32*arctan(1 / 682) + 44*arctan(1 / 5357) + 68*arctan(1 / 12944))
{{out}}
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
"==="
0.9999991882
Using proper fractions
{{trans|D}} Routines adapted from [[Arithmetic/Rational#Phix]]
include builtins\pfrac.e -- (provisional/0.8.0+)
function tans(sequence x)
frac a,b
integer h
if length(x)=1 then
{integer m, frac f} = x[1]
if m=1 then
return f
elsif m<0 then
return frac_uminus(tans({{-m,f}}))
end if
h = floor(m/2)
a = tans({{h,f}})
b = tans({{m-h,f}})
else
h = floor(length(x)/2)
a = tans(x[1..h])
b = tans(x[h+1..$])
end if
return frac_div(frac_add(a,b) , frac_sub(frac_new(1),frac_mul(a,b)))
end function
function parse(string formula)
-- obviously the error handling here is a bit brutal...
sequence res = {}, r
integer m,n,d
formula = substitute(formula," ","") -- strip spaces
if formula[1..5]!="pi/4=" then ?9/0 end if
formula = formula[6..$]
res = {}
while length(formula) do
integer sgn = +1
switch formula[1] do
case '-': sgn = -1; fallthrough
case '+': formula = formula[2..$]
end switch
if formula[1]='a' then
m = sgn
else
r = scanf(formula,"%d*%s")
if length(r)!=1 then ?9/0 end if
{m,formula} = r[1]
m *= sgn
end if
r = scanf(formula,"arctan(%d/%d)%s")
if length(r)!=1 then ?9/0 end if
{n,d,formula} = r[1]
res = append(res,{m,frac_new(n,d)})
end while
return res
end function
procedure test(string formula)
frac f = tans(parse(formula))
if frac_eq(f,frac_one) then
printf(1,"OK: %s\n",{formula})
else
printf(1,"ERROR: %s\n",{formula})
printf(1," %s\n\\ %s\n",frac_sprint(f,asPair:=true))
end if
end procedure
constant formulae = {"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)"}
for i=1 to length(formulae) do
test(formulae[i])
end for
{{out}} Last line manually edited (both numerator and denominator were 550-digit numbers).
OK: pi/4 = arctan(1/2) + arctan(1/3)
OK: pi/4 = 2*arctan(1/3) + arctan(1/7)
OK: pi/4 = 4*arctan(1/5) - arctan(1/239)
OK: pi/4 = 5*arctan(1/7) + 2*arctan(3/79)
OK: pi/4 = 5*arctan(29/278) + 7*arctan(3/79)
OK: pi/4 = arctan(1/2) + arctan(1/5) + arctan(1/8)
OK: pi/4 = 4*arctan(1/5) - arctan(1/70) + arctan(1/99)
OK: pi/4 = 5*arctan(1/7) + 4*arctan(1/53) + 2*arctan(1/4443)
OK: pi/4 = 6*arctan(1/8) + 2*arctan(1/57) + arctan(1/239)
OK: pi/4 = 8*arctan(1/10) - arctan(1/239) - 4*arctan(1/515)
OK: pi/4 = 12*arctan(1/18) + 8*arctan(1/57) - 5*arctan(1/239)
OK: pi/4 = 16*arctan(1/21) + 3*arctan(1/239) + 4*arctan(3/1042)
OK: pi/4 = 22*arctan(1/28) + 2*arctan(1/443) - 5*arctan(1/1393) - 10*arctan(1/11018)
OK: pi/4 = 22*arctan(1/38) + 17*arctan(7/601) + 10*arctan(7/8149)
OK: pi/4 = 44*arctan(1/57) + 7*arctan(1/239) - 12*arctan(1/682) + 24*arctan(1/12943)
OK: pi/4 = 88*arctan(1/172) + 51*arctan(1/239) + 32*arctan(1/682) + 44*arctan(1/5357) + 68*arctan(1/12943)
ERROR: pi/4 = 88*arctan(1/172) + 51*arctan(1/239) + 32*arctan(1/682) + 44*arctan(1/5357) + 68*arctan(1/12944)
10092880180009440509678967104315871864562569285843 ... 82862528690942242793667539020699840402353522108223
\ 10092888373156385834157015287804027957219356416144 ... 84010722587072087349909684004660371264507984339711
Python
This example parses the [http://rosettacode.org/mw/index.php?title=Check_Machin-like_formulas&oldid=146749 original] equations to form an intermediate representation then does the checks.
Function tans and tanEval are translations of the Haskel functions of the same names.
import re
from fractions import Fraction
from pprint import pprint as pp
equationtext = '''\
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)
'''
def parse_eqn(equationtext=equationtext):
eqn_re = re.compile(r"""(?mx)
(?P<lhs> ^ \s* pi/4 \s* = \s*)? # LHS of equation
(?: # RHS
\s* (?P<sign> [+-])? \s*
(?: (?P<mult> \d+) \s* \*)?
\s* arctan\( (?P<numer> \d+) / (?P<denom> \d+)
)""")
found = eqn_re.findall(equationtext)
machins, part = [], []
for lhs, sign, mult, numer, denom in eqn_re.findall(equationtext):
if lhs and part:
machins.append(part)
part = []
part.append( ( (-1 if sign == '-' else 1) * ( int(mult) if mult else 1),
Fraction(int(numer), (int(denom) if denom else 1)) ) )
machins.append(part)
return machins
def tans(xs):
xslen = len(xs)
if xslen == 1:
return tanEval(*xs[0])
aa, bb = xs[:xslen//2], xs[xslen//2:]
a, b = tans(aa), tans(bb)
return (a + b) / (1 - a * b)
def tanEval(coef, f):
if coef == 1:
return f
if coef < 0:
return -tanEval(-coef, f)
ca = coef // 2
cb = coef - ca
a, b = tanEval(ca, f), tanEval(cb, f)
return (a + b) / (1 - a * b)
if __name__ == '__main__':
machins = parse_eqn()
#pp(machins, width=160)
for machin, eqn in zip(machins, equationtext.split('\n')):
ans = tans(machin)
print('%5s: %s' % ( ('OK' if ans == 1 else 'ERROR'), eqn))
{{out}}
OK: pi/4 = arctan(1/2) + arctan(1/3)
OK: pi/4 = 2*arctan(1/3) + arctan(1/7)
OK: pi/4 = 4*arctan(1/5) - arctan(1/239)
OK: pi/4 = 5*arctan(1/7) + 2*arctan(3/79)
OK: pi/4 = 5*arctan(29/278) + 7*arctan(3/79)
OK: pi/4 = arctan(1/2) + arctan(1/5) + arctan(1/8)
OK: pi/4 = 4*arctan(1/5) - arctan(1/70) + arctan(1/99)
OK: pi/4 = 5*arctan(1/7) + 4*arctan(1/53) + 2*arctan(1/4443)
OK: pi/4 = 6*arctan(1/8) + 2*arctan(1/57) + arctan(1/239)
OK: pi/4 = 8*arctan(1/10) - arctan(1/239) - 4*arctan(1/515)
OK: pi/4 = 12*arctan(1/18) + 8*arctan(1/57) - 5*arctan(1/239)
OK: pi/4 = 16*arctan(1/21) + 3*arctan(1/239) + 4*arctan(3/1042)
OK: pi/4 = 22*arctan(1/28) + 2*arctan(1/443) - 5*arctan(1/1393) - 10*arctan(1/11018)
OK: pi/4 = 22*arctan(1/38) + 17*arctan(7/601) + 10*arctan(7/8149)
OK: pi/4 = 44*arctan(1/57) + 7*arctan(1/239) - 12*arctan(1/682) + 24*arctan(1/12943)
OK: pi/4 = 88*arctan(1/172) + 51*arctan(1/239) + 32*arctan(1/682) + 44*arctan(1/5357) + 68*arctan(1/12943)
ERROR: pi/4 = 88*arctan(1/172) + 51*arctan(1/239) + 32*arctan(1/682) + 44*arctan(1/5357) + 68*arctan(1/12944)
'''Note:''' the [http://kodos.sourceforge.net/ Kodos] tool was used in developing the regular expression.
R
#lang R
library(Rmpfr)
prec <- 1000 # precision in bits
`%:%` <- function(e1, e2) '/'(mpfr(e1, prec), mpfr(e2, prec)) # operator %:% for high precision division
# function for checking identity of tan of expression and 1, making use of high precision division operator %:%
tanident_1 <- function(x) identical(round(tan(eval(parse(text = gsub("/", "%:%", deparse(substitute(x)))))), (prec/10)), mpfr(1, prec))
{{out}}
tanident_1( 1*atan(1/2) + 1*atan(1/3) )
## [1] TRUE
tanident_1( 2*atan(1/3) + 1*atan(1/7))
## [1] TRUE
tanident_1( 4*atan(1/5) + -1*atan(1/239))
## [1] TRUE
tanident_1( 5*atan(1/7) + 2*atan(3/79))
## [1] TRUE
tanident_1( 5*atan(29/278) + 7*atan(3/79))
## [1] TRUE
tanident_1( 1*atan(1/2) + 1*atan(1/5) + 1*atan(1/8) )
## [1] TRUE
tanident_1( 4*atan(1/5) + -1*atan(1/70) + 1*atan(1/99) )
## [1] TRUE
tanident_1( 5*atan(1/7) + 4*atan(1/53) + 2*atan(1/4443))
## [1] TRUE
tanident_1( 6*atan(1/8) + 2*atan(1/57) + 1*atan(1/239))
## [1] TRUE
tanident_1( 8*atan(1/10) + -1*atan(1/239) + -4*atan(1/515))
## [1] TRUE
tanident_1(12*atan(1/18) + 8*atan(1/57) + -5*atan(1/239))
## [1] TRUE
tanident_1(16*atan(1/21) + 3*atan(1/239) + 4*atan(3/1042))
## [1] TRUE
tanident_1(22*atan(1/28) + 2*atan(1/443) + -5*atan(1/1393) + -10*atan(1/11018))
## [1] TRUE
tanident_1(22*atan(1/38) + 17*atan(7/601) + 10*atan(7/8149))
## [1] TRUE
tanident_1(44*atan(1/57) + 7*atan(1/239) + -12*atan(1/682) + 24*atan(1/12943))
## [1] TRUE
tanident_1(88*atan(1/172) + 51*atan(1/239) + 32*atan(1/682) + 44*atan(1/5357) + 68*atan(1/12943))
## [1] TRUE
tanident_1(88*atan(1/172) + 51*atan(1/239) + 32*atan(1/682) + 44*atan(1/5357) + 68*atan(1/12944))
## [1] FALSE
Racket
#lang racket
(define (reduce e)
(match e
[(? number? a) a]
[(list '+ (? number? a) (? number? b)) (+ a b)]
[(list '- (? number? a) (? number? b)) (- a b)]
[(list '- (? number? a)) (- a)]
[(list '* (? number? a) (? number? b)) (* a b)]
[(list '/ (? number? a) (? number? b)) (/ a b)]
[(list '+ a b) (reduce `(+ ,(reduce a) ,(reduce b)))]
[(list '- a b) (reduce `(- ,(reduce a) ,(reduce b)))]
[(list '- a) (reduce `(- ,(reduce a)))]
[(list '* a b) (reduce `(* ,(reduce a) ,(reduce b)))]
[(list '/ a b) (reduce `(/ ,(reduce a) ,(reduce b)))]
[(list 'tan (list 'arctan a)) (reduce a)]
[(list 'tan (list '- a)) (reduce `(- ,(reduce `(tan ,a))))]
[(list 'tan (list '+ a b)) (reduce `(/ (+ (tan ,a) (tan ,b))
(- 1 (* (tan ,a) (tan ,b)))))]
[(list 'tan (list '+ a b c ...)) (reduce `(tan (+ ,a (+ ,b ,@c))))]
[(list 'tan (list '- a b)) (reduce `(/ (+ (tan ,a) (tan (- ,b)))
(- 1 (* (tan ,a) (tan (- ,b))))))]
[(list 'tan (list '* 1 a)) (reduce `(tan ,a))]
[(list 'tan (list '* (? number? n) a))
(cond [(< n 0) (reduce `(- (tan (* ,(- n) ,a))))]
[(= n 0) 0]
[(even? n) (reduce `(tan (+ (* ,(/ n 2) ,a) (* ,(/ n 2) ,a))))]
[else (reduce `(tan (+ ,a (* ,(- n 1) ,a))))])]))
(define correct-formulas
'((tan (+ (arctan 1/2) (arctan 1/3)))
(tan (+ (* 2 (arctan 1/3)) (arctan 1/7)))
(tan (- (* 4 (arctan 1/5)) (arctan 1/239)))
(tan (+ (* 5 (arctan 1/7)) (* 2 (arctan 3/79))))
(tan (+ (* 5 (arctan 29/278)) (* 7 (arctan 3/79))))
(tan (+ (arctan 1/2) (arctan 1/5) (arctan 1/8)))
(tan (+ (* 4 (arctan 1/5)) (* -1 (arctan 1/70)) (arctan 1/99)))
(tan (+ (* 5 (arctan 1/7)) (* 4 (arctan 1/53)) (* 2 (arctan 1/4443))))
(tan (+ (* 6 (arctan 1/8)) (* 2 (arctan 1/57)) (arctan 1/239)))
(tan (+ (* 8 (arctan 1/10)) (* -1 (arctan 1/239)) (* -4 (arctan 1/515))))
(tan (+ (* 12 (arctan 1/18)) (* 8 (arctan 1/57)) (* -5 (arctan 1/239))))
(tan (+ (* 16 (arctan 1/21)) (* 3 (arctan 1/239)) (* 4 (arctan 3/1042))))
(tan (+ (* 22 (arctan 1/28)) (* 2 (arctan 1/443)) (* -5 (arctan 1/1393)) (* -10 (arctan 1/11018))))
(tan (+ (* 22 (arctan 1/38)) (* 17 (arctan 7/601)) (* 10 (arctan 7/8149))))
(tan (+ (* 44 (arctan 1/57)) (* 7 (arctan 1/239)) (* -12 (arctan 1/682)) (* 24 (arctan 1/12943))))
(tan (+ (* 88 (arctan 1/172)) (* 51 (arctan 1/239)) (* 32 (arctan 1/682))
(* 44 (arctan 1/5357)) (* 68 (arctan 1/12943))))))
(define wrong-formula
'(tan (+ (* 88 (arctan 1/172)) (* 51 (arctan 1/239)) (* 32 (arctan 1/682))
(* 44 (arctan 1/5357)) (* 68 (arctan 1/12944)))))
(displayln "Do all correct formulas reduce to 1?")
(for/and ([f correct-formulas]) (= 1 (reduce f)))
(displayln "The incorrect formula reduces to:")
(reduce wrong-formula)
Output:
Do all correct formulas reduce to 1?
#t
The incorrect formula reduces to:
1009288018000944050967896710431587186456256928584351786643498522649995492271475761189348270710224618853590682465929080006511691833816436374107451368838065354726517908250456341991684635768915704374493675498637876700129004484434187627909285979251682006538817341793224963346197503893270875008524149334251672855130857035205217929335932890740051319216343365800342290782260673215928499123722781078448297609548233999010983373327601187505623621602789012550584784738082074783523787011976757247516095289966708782862528690942242793667539020699840402353522108223/1009288837315638583415701528780402795721935641614456853534313491853293025565940011104051964874275710024625850092154664245109626053906509780125743180758231049920425664246286578958307532545458843067352531217230461290763258378749459637420702619029075083089762088232401888676895047947363883809724322868121990870409574061477638203859217672620508200713073485398199091153535700094640095900731630771349477187594074169815106104524371099618096164871416282464532355211521113449237814080332335526420331468258917484010722587072087349909684004660371264507984339711
REXX
Note: REXX doesn't have many math functions, so a few of them are included here.
Noticed: the test arguments specified for this Rosetta Code task need only '''nine''' decimal digits for verification, '''eight''' decimal digits is ''not'' enough to catch the "bad" equation. With this in mind, the REXX's ''decimal digit precision'' was increased to the number of decimal digits specified for the variable '''pi''' (which, for these cases, is a bit of overkill, but the difference in execution times were barely noticeable).
An extra formula was added to stress test the near exactness of a value.
/*REXX program evaluates some Machinβlike formulas and verifies their veracity. */
@.=; pi= pi(); numeric digits( length(pi) ) - length(.); numeric fuzz 3
@.1 = 'pi/4 = atan(1/2) + atan(1/3)'
@.2 = 'pi/4 = 2*atan(1/3) + atan(1/7)'
@.3 = 'pi/4 = 4*atan(1/5) - atan(1/239)'
@.4 = 'pi/4 = 5*atan(1/7) + 2*atan(3/79)'
@.5 = 'pi/4 = 5*atan(29/278) + 7*atan(3/79)'
@.6 = 'pi/4 = atan(1/2) + atan(1/5) + atan(1/8)'
@.7 = 'pi/4 = 4*atan(1/5) - atan(1/70) + atan(1/99)'
@.8 = 'pi/4 = 5*atan(1/7) + 4*atan(1/53) + 2*atan(1/4443)'
@.9 = 'pi/4 = 6*atan(1/8) + 2*atan(1/57) + atan(1/239)'
@.10= 'pi/4 = 8*atan(1/10) - atan(1/239) - 4*atan(1/515)'
@.11= 'pi/4 = 12*atan(1/18) + 8*atan(1/57) - 5*atan(1/239)'
@.12= 'pi/4 = 16*atan(1/21) + 3*atan(1/239) + 4*atan(3/1042)'
@.13= 'pi/4 = 22*atan(1/28) + 2*atan(1/443) - 5*atan(1/1393) - 10*atan(1/11018)'
@.14= 'pi/4 = 22*atan(1/38) + 17*atan(7/601) + 10*atan(7/8149)'
@.15= 'pi/4 = 44*atan(1/57) + 7*atan(1/239) - 12*atan(1/682) + 24*atan(1/12943)'
@.16= 'pi/4 = 88*atan(1/172) + 51*atan(1/239) + 32*atan(1/682) + 44*atan(1/5357) + 68*atan(1/12943)'
@.17= 'pi/4 = 88*atan(1/172) + 51*atan(1/239) + 32*atan(1/682) + 44*atan(1/5357) + 68*atan(1/12944)'
@.18= 'pi/4 = 88*atan(1/172) + 51*atan(1/239) + 32*atan(1/682) + 44*atan(1/5357) + 67.9999999994*atan(1/12943)'
do j=1 while @.j\=='' /*evaluate each "Machinβlike" formulas.*/
interpret 'answer=' "(" @.j ')' /*where REXX does the heavy lifting. */
say right( word( 'bad OK', answer+1), 3)": " @.j
end /*j*/
exit /*stick a fork in it, we're all done. */
/*ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ*/
pi: return 3.141592653589793238462643383279502884197169399375105820974944592307816406286
Acos: procedure; parse arg x; return pi() * .5 - Asin(x)
Atan: procedure; arg x; if abs(x)=1 then return pi()/4*sign(x); return Asin(x/sqrt(1+x*x))
/*ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ*/
Asin: procedure; parse arg x 1 z 1 o 1 p; a=abs(x); aa=a*a
if a>=sqrt(2)*.5 then return sign(x) * Acos( sqrt(1 - aa) )
do j=2 by 2 until p=z; p=z; o=o*aa*(j-1)/j; z=z+o/(j+1); end /*j*/; return z
/*ββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ*/
sqrt: procedure; parse arg x; if x=0 then return 0; d=digits(); m.=9; h=d+6; numeric form
numeric digits; parse value format(x,2,1,,0) 'E0' with g 'E' _ .; g=g *.5'e'_ % 2
do j=0 while h>9; m.j=h; h=h%2+1; end /*j*/
do k=j+5 to 0 by -1; numeric digits m.k; g=(g+x/g)*.5; end /*k*/; return g
{{out|output|text= when using the internal default input:}}
OK: pi/4 = atan(1/2) + atan(1/3)
OK: pi/4 = 2*atan(1/3) + atan(1/7)
OK: pi/4 = 4*atan(1/5) - atan(1/239)
OK: pi/4 = 5*atan(1/7) + 2*atan(3/79)
OK: pi/4 = 5*atan(29/278) + 7*atan(3/79)
OK: pi/4 = atan(1/2) + atan(1/5) + atan(1/8)
OK: pi/4 = 4*atan(1/5) - atan(1/70) + atan(1/99)
OK: pi/4 = 5*atan(1/7) + 4*atan(1/53) + 2*atan(1/4443)
OK: pi/4 = 6*atan(1/8) + 2*atan(1/57) + atan(1/239)
OK: pi/4 = 8*atan(1/10) - atan(1/239) - 4*atan(1/515)
OK: pi/4 = 12*atan(1/18) + 8*atan(1/57) - 5*atan(1/239)
OK: pi/4 = 16*atan(1/21) + 3*atan(1/239) + 4*atan(3/1042)
OK: pi/4 = 22*atan(1/28) + 2*atan(1/443) - 5*atan(1/1393) - 10*atan(1/11018)
OK: pi/4 = 22*atan(1/38) + 17*atan(7/601) + 10*atan(7/8149)
OK: pi/4 = 44*atan(1/57) + 7*atan(1/239) - 12*atan(1/682) + 24*atan(1/12943)
OK: pi/4 = 88*atan(1/172) + 51*atan(1/239) + 32*atan(1/682) + 44*atan(1/5357) + 68*atan(1/12943)
bad: pi/4 = 88*atan(1/172) + 51*atan(1/239) + 32*atan(1/682) + 44*atan(1/5357) + 68*atan(1/12944)
bad: pi/4 = 88*atan(1/172) + 51*atan(1/239) + 32*atan(1/682) + 44*atan(1/5357) + 67.9999999994*atan(1/12943)
Seed7
$ include "seed7_05.s7i";
include "bigint.s7i";
include "bigrat.s7i";
const type: mTerms is array array bigInteger;
const array mTerms: testCases is [] (
[] ([] ( 1_, 1_, 2_), [] ( 1_, 1_, 3_)),
[] ([] ( 2_, 1_, 3_), [] ( 1_, 1_, 7_)),
[] ([] ( 4_, 1_, 5_), [] (-1_, 1_, 239_)),
[] ([] ( 5_, 1_, 7_), [] ( 2_, 3_, 79_)),
[] ([] ( 1_, 1_, 2_), [] ( 1_, 1_, 5_), [] ( 1_, 1_, 8_)),
[] ([] ( 4_, 1_, 5_), [] (-1_, 1_, 70_), [] ( 1_, 1_, 99_)),
[] ([] ( 5_, 1_, 7_), [] ( 4_, 1_, 53_), [] ( 2_, 1_, 4443_)),
[] ([] ( 6_, 1_, 8_), [] ( 2_, 1_, 57_), [] ( 1_, 1_, 239_)),
[] ([] ( 8_, 1_, 10_), [] (-1_, 1_, 239_), [] ( -4_, 1_, 515_)),
[] ([] (12_, 1_, 18_), [] ( 8_, 1_, 57_), [] ( -5_, 1_, 239_)),
[] ([] (16_, 1_, 21_), [] ( 3_, 1_, 239_), [] ( 4_, 3_, 1042_)),
[] ([] (22_, 1_, 28_), [] ( 2_, 1_, 443_), [] ( -5_, 1_, 1393_), [] (-10_, 1_, 11018_)),
[] ([] (22_, 1_, 38_), [] (17_, 7_, 601_), [] ( 10_, 7_, 8149_)),
[] ([] (44_, 1_, 57_), [] ( 7_, 1_, 239_), [] (-12_, 1_, 682_), [] ( 24_, 1_, 12943_)),
[] ([] (88_, 1_, 172_), [] (51_, 1_, 239_), [] ( 32_, 1_, 682_), [] ( 44_, 1_, 5357_), [] (68_, 1_, 12943_)),
[] ([] (88_, 1_, 172_), [] (51_, 1_, 239_), [] ( 32_, 1_, 682_), [] ( 44_, 1_, 5357_), [] (68_, 1_, 12944_))
);
const func bigRational: tanEval (in bigInteger: coef, in bigRational: f) is func
result
var bigRational: tanEval is bigRational.value;
local
var bigRational: a is bigRational.value;
var bigRational: b is bigRational.value;
begin
if coef = 1_ then
tanEval := f;
elsif coef < 0_ then
tanEval := -tanEval(-coef, f);
else
a := tanEval(coef div 2_, f);
b := tanEval(coef - coef div 2_, f);
tanEval := (a + b) / (1_/1_ - a * b);
end if;
end func;
const func bigRational: tans (in mTerms: terms) is func
result
var bigRational: tans is bigRational.value;
local
var bigRational: a is bigRational.value;
var bigRational: b is bigRational.value;
begin
if length(terms) = 1 then
tans := tanEval(terms[1][1], terms[1][2] / terms[1][3]);
else
a := tans(terms[.. length(terms) div 2]);
b := tans(terms[succ(length(terms) div 2) ..]);
tans := (a + b) / (1_/1_ - a * b);
end if;
end func;
const proc: main is func
local
var integer: index is 0;
var array bigInteger: term is 0 times 0_;
begin
for key index range testCases do
write(tans(testCases[index]) = 1_/1_ <& ": pi/4 = ");
for term range testCases[index] do
write([0] ("+", "-")[ord(term[1] < 0_)] <& abs(term[1]) <& "*arctan(" <& term[2] <& "/" <& term[3] <& ")");
end for;
writeln;
end for;
end func;
{{out}}
TRUE: pi/4 = +1*arctan(1/2)+1*arctan(1/3)
TRUE: pi/4 = +2*arctan(1/3)+1*arctan(1/7)
TRUE: pi/4 = +4*arctan(1/5)-1*arctan(1/239)
TRUE: pi/4 = +5*arctan(1/7)+2*arctan(3/79)
TRUE: pi/4 = +1*arctan(1/2)+1*arctan(1/5)+1*arctan(1/8)
TRUE: pi/4 = +4*arctan(1/5)-1*arctan(1/70)+1*arctan(1/99)
TRUE: pi/4 = +5*arctan(1/7)+4*arctan(1/53)+2*arctan(1/4443)
TRUE: pi/4 = +6*arctan(1/8)+2*arctan(1/57)+1*arctan(1/239)
TRUE: pi/4 = +8*arctan(1/10)-1*arctan(1/239)-4*arctan(1/515)
TRUE: pi/4 = +12*arctan(1/18)+8*arctan(1/57)-5*arctan(1/239)
TRUE: pi/4 = +16*arctan(1/21)+3*arctan(1/239)+4*arctan(3/1042)
TRUE: pi/4 = +22*arctan(1/28)+2*arctan(1/443)-5*arctan(1/1393)-10*arctan(1/11018)
TRUE: pi/4 = +22*arctan(1/38)+17*arctan(7/601)+10*arctan(7/8149)
TRUE: pi/4 = +44*arctan(1/57)+7*arctan(1/239)-12*arctan(1/682)+24*arctan(1/12943)
TRUE: pi/4 = +88*arctan(1/172)+51*arctan(1/239)+32*arctan(1/682)+44*arctan(1/5357)+68*arctan(1/12943)
FALSE: pi/4 = +88*arctan(1/172)+51*arctan(1/239)+32*arctan(1/682)+44*arctan(1/5357)+68*arctan(1/12944)
Sidef
{{trans|Python}}
var equationtext = <<'EOT'
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)
EOT
func parse_eqn(equation) {
static eqn_re = %r{
(^ \s* pi/4 \s* = \s* )? # LHS of equation
(?: # RHS
\s* ( [-+] )? \s*
(?: ( \d+ ) \s* \*)?
\s* arctan\((.*?)\)
)}x
gather {
for lhs,sign,mult,rat in (equation.findall(eqn_re)) {
take([
[+1, -1][sign == '-'] * (mult ? Num(mult) : 1),
Num(rat)
])
}
}
}
func tanEval(coef, f) {
return f if (coef == 1)
return -tanEval(-coef, f) if (coef < 0)
var ca = coef>>1
var cb = (coef - ca)
var (a, b) = (tanEval(ca, f), tanEval(cb, f))
(a + b) / (1 - a*b)
}
func tans(xs) {
var xslen = xs.len
return tanEval(xs[0]...) if (xslen == 1)
var (aa, bb) = xs.part(xslen>>1)
var (a, b) = (tans(aa), tans(bb))
(a + b) / (1 - a*b)
}
var machins = equationtext.lines.map(parse_eqn)
for machin,eqn in (machins ~Z equationtext.lines) {
var ans = tans(machin)
printf("%5s: %s\n", (ans == 1 ? 'OK' : 'ERROR'), eqn)
}
{{out}}
OK: pi/4 = arctan(1/2) + arctan(1/3)
OK: pi/4 = 2*arctan(1/3) + arctan(1/7)
OK: pi/4 = 4*arctan(1/5) - arctan(1/239)
OK: pi/4 = 5*arctan(1/7) + 2*arctan(3/79)
OK: pi/4 = 5*arctan(29/278) + 7*arctan(3/79)
OK: pi/4 = arctan(1/2) + arctan(1/5) + arctan(1/8)
OK: pi/4 = 4*arctan(1/5) - arctan(1/70) + arctan(1/99)
OK: pi/4 = 5*arctan(1/7) + 4*arctan(1/53) + 2*arctan(1/4443)
OK: pi/4 = 6*arctan(1/8) + 2*arctan(1/57) + arctan(1/239)
OK: pi/4 = 8*arctan(1/10) - arctan(1/239) - 4*arctan(1/515)
OK: pi/4 = 12*arctan(1/18) + 8*arctan(1/57) - 5*arctan(1/239)
OK: pi/4 = 16*arctan(1/21) + 3*arctan(1/239) + 4*arctan(3/1042)
OK: pi/4 = 22*arctan(1/28) + 2*arctan(1/443) - 5*arctan(1/1393) - 10*arctan(1/11018)
OK: pi/4 = 22*arctan(1/38) + 17*arctan(7/601) + 10*arctan(7/8149)
OK: pi/4 = 44*arctan(1/57) + 7*arctan(1/239) - 12*arctan(1/682) + 24*arctan(1/12943)
OK: pi/4 = 88*arctan(1/172) + 51*arctan(1/239) + 32*arctan(1/682) + 44*arctan(1/5357) + 68*arctan(1/12943)
ERROR: pi/4 = 88*arctan(1/172) + 51*arctan(1/239) + 32*arctan(1/682) + 44*arctan(1/5357) + 68*arctan(1/12944)
Tcl
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"
}
}
{{out}}
Yes! 'pi/4 = arctan(1/2) + arctan(1/3)' is true
Yes! 'pi/4 = 2*arctan(1/3) + arctan(1/7)' is true
Yes! 'pi/4 = 4*arctan(1/5) - arctan(1/239)' is true
Yes! 'pi/4 = 5*arctan(1/7) + 2*arctan(3/79)' is true
Yes! 'pi/4 = 5*arctan(29/278) + 7*arctan(3/79)' is true
Yes! 'pi/4 = arctan(1/2) + arctan(1/5) + arctan(1/8)' is true
Yes! 'pi/4 = 4*arctan(1/5) - arctan(1/70) + arctan(1/99)' is true
Yes! 'pi/4 = 5*arctan(1/7) + 4*arctan(1/53) + 2*arctan(1/4443)' is true
Yes! 'pi/4 = 6*arctan(1/8) + 2*arctan(1/57) + arctan(1/239)' is true
Yes! 'pi/4 = 8*arctan(1/10) - arctan(1/239) - 4*arctan(1/515)' is true
Yes! 'pi/4 = 12*arctan(1/18) + 8*arctan(1/57) - 5*arctan(1/239)' is true
Yes! 'pi/4 = 16*arctan(1/21) + 3*arctan(1/239) + 4*arctan(3/1042)' is true
Yes! 'pi/4 = 22*arctan(1/28) + 2*arctan(1/443) - 5*arctan(1/1393) - 10*arctan(1/11018)' is true
Yes! 'pi/4 = 22*arctan(1/38) + 17*arctan(7/601) + 10*arctan(7/8149)' is true
Yes! 'pi/4 = 44*arctan(1/57) + 7*arctan(1/239) - 12*arctan(1/682) + 24*arctan(1/12943)' is true
Yes! 'pi/4 = 88*arctan(1/172) + 51*arctan(1/239) + 32*arctan(1/682) + 44*arctan(1/5357) + 68*arctan(1/12943)' is true
No! 'pi/4 = 88*arctan(1/172) + 51*arctan(1/239) + 32*arctan(1/682) + 44*arctan(1/5357) + 68*arctan(1/12944)' not true
Visual Basic .NET
'''BigRat''' class based on the Arithmetic/Rational#C here at Rosetta Code.
The parser here allows for some flexibility in the input text. Case is ignored, and a variable number of spaces are allowed. Atan(), arctan(), atn() are all recognized as valid. If one of those three are not found, a warning will appear. The coefficient need not have a multiplication sign between it and the "arctan()". The left side of the equation must be pi / 4, otherwise a warning will appear.
Imports System.Numerics
Public Class BigRat ' Big Rational Class constructed with BigIntegers
Implements IComparable
Public nu, de As BigInteger
Public Shared Zero = New BigRat(BigInteger.Zero, BigInteger.One),
One = New BigRat(BigInteger.One, BigInteger.One)
Sub New(bRat As BigRat)
nu = bRat.nu : de = bRat.de
End Sub
Sub New(n As BigInteger, d As BigInteger)
If d = BigInteger.Zero Then _
Throw (New Exception(String.Format("tried to set a BigRat with ({0}/{1})", n, d)))
Dim bi As BigInteger = BigInteger.GreatestCommonDivisor(n, d)
If bi > BigInteger.One Then n /= bi : d /= bi
If d < BigInteger.Zero Then n = -n : d = -d
nu = n : de = d
End Sub
Shared Operator -(x As BigRat) As BigRat
Return New BigRat(-x.nu, x.de)
End Operator
Shared Operator +(x As BigRat, y As BigRat)
Return New BigRat(x.nu * y.de + x.de * y.nu, x.de * y.de)
End Operator
Shared Operator -(x As BigRat, y As BigRat) As BigRat
Return x + (-y)
End Operator
Shared Operator *(x As BigRat, y As BigRat) As BigRat
Return New BigRat(x.nu * y.nu, x.de * y.de)
End Operator
Shared Operator /(x As BigRat, y As BigRat) As BigRat
Return New BigRat(x.nu * y.de, x.de * y.nu)
End Operator
Public Function CompareTo(obj As Object) As Integer Implements IComparable.CompareTo
Dim dif As BigRat = New BigRat(nu, de) - obj
If dif.nu < BigInteger.Zero Then Return -1
If dif.nu > BigInteger.Zero Then Return 1
Return 0
End Function
Shared Operator =(x As BigRat, y As BigRat) As Boolean
Return x.CompareTo(y) = 0
End Operator
Shared Operator <>(x As BigRat, y As BigRat) As Boolean
Return x.CompareTo(y) <> 0
End Operator
Overrides Function ToString() As String
If de = BigInteger.One Then Return nu.ToString
Return String.Format("({0}/{1})", nu, de)
End Function
Shared Function Combine(a As BigRat, b As BigRat) As BigRat
Return (a + b) / (BigRat.One - (a * b))
End Function
End Class
Public Structure Term ' coefficent, BigRational construction for each term
Dim c As Integer, br As BigRat
Sub New(cc As Integer, bigr As BigRat)
c = cc : br = bigr
End Sub
End Structure
Module Module1
Function Eval(c As Integer, x As BigRat) As BigRat
If c = 1 Then Return x Else If c < 0 Then Return Eval(-c, -x)
Dim hc As Integer = c \ 2
Return BigRat.Combine(Eval(hc, x), Eval(c - hc, x))
End Function
Function Sum(terms As List(Of Term)) As BigRat
If terms.Count = 1 Then Return Eval(terms(0).c, terms(0).br)
Dim htc As Integer = terms.Count / 2
Return BigRat.Combine(Sum(terms.Take(htc).ToList), Sum(terms.Skip(htc).ToList))
End Function
Function ParseLine(ByVal s As String) As List(Of Term)
ParseLine = New List(Of Term) : Dim t As String = s.ToLower, p As Integer, x As New Term(1, BigRat.Zero)
While t.Contains(" ") : t = t.Replace(" ", "") : End While
p = t.IndexOf("pi/4=") : If p < 0 Then _
Console.WriteLine("warning: tan(left side of equation) <> 1") : ParseLine.Add(x) : Exit Function
t = t.Substring(p + 5)
For Each item As String In t.Split(")")
If item.Length > 5 Then
If (Not item.Contains("tan") OrElse item.IndexOf("a") < 0 OrElse
item.IndexOf("a") > item.IndexOf("tan")) AndAlso Not item.Contains("atn") Then
Console.WriteLine("warning: a term is mising a valid arctangent identifier on the right side of the equation: [{0})]", item)
ParseLine = New List(Of Term) : ParseLine.Add(New Term(1, BigRat.Zero)) : Exit Function
End If
x.c = 1 : x.br = New BigRat(BigRat.One)
p = item.IndexOf("/") : If p > 0 Then
x.br.de = UInt64.Parse(item.Substring(p + 1))
item = item.Substring(0, p)
p = item.IndexOf("(") : If p > 0 Then
x.br.nu = UInt64.Parse(item.Substring(p + 1))
p = item.IndexOf("a") : If p > 0 Then
Integer.TryParse(item.Substring(0, p).Replace("*", ""), x.c)
If x.c = 0 Then x.c = 1
If item.Contains("-") AndAlso x.c > 0 Then x.c = -x.c
End If
ParseLine.Add(x)
End If
End If
End If
Next
End Function
Sub Main(ByVal args As String())
Dim nl As String = vbLf
For Each item In ("pi/4 = ATan(1 / 2) + ATan(1/3)" & nl &
"pi/4 = 2Atan(1/3) + ATan(1/7)" & nl &
"pi/4 = 4ArcTan(1/5) - ATan(1 / 239)" & nl &
"pi/4 = 5arctan(1/7) + 2 * atan(3/79)" & nl &
"Pi/4 = 5ATan(29/278) + 7*ATan(3/79)" & nl &
"pi/4 = atn(1/2) + ATan(1/5) + ATan(1/8)" & nl &
"PI/4 = 4ATan(1/5) - Atan(1/70) + ATan(1/99)" & nl &
"pi /4 = 5*ATan(1/7) + 4 ATan(1/53) + 2ATan(1/4443)" & nl &
"pi / 4 = 6ATan(1/8) + 2arctangent(1/57) + ATan(1/239)" & nl &
"pi/ 4 = 8ATan(1/10) - ATan(1/239) - 4ATan(1/515)" & nl &
"pi/4 = 12ATan(1/18) + 8ATan(1/57) - 5ATan(1/239)" & nl &
"pi/4 = 16 * ATan(1/21) + 3ATan(1/239) + 4ATan(3/1042)" & nl &
"pi/4 = 22ATan(1/28) + 2ATan(1/443) - 5ATan(1/1393) - 10 ATan( 1 / 11018 )" & nl &
"pi/4 = 22ATan(1/38) + 17ATan(7/601) + 10ATan(7 / 8149)" & nl &
"pi/4 = 44ATan(1/57) + 7ATan(1/239) - 12ATan(1/682) + 24ATan(1/12943)" & nl &
"pi/4 = 88ATan(1/172) + 51ATan(1/239) + 32ATan(1/682) + 44ATan(1/5357) + 68ATan(1/12943)" & nl &
"pi/4 = 88ATan(1/172) + 51ATan(1/239) + 32ATan(1/682) + 44ATan(1/5357) + 68ATan(1/12944)").Split(nl)
Console.WriteLine("{0}: {1}", If(Sum(ParseLine(item)) = BigRat.One, "Pass", "Fail"), item)
Next
End Sub
End Module
{{out}}
Pass: pi/4 = ATan(1 / 2) + ATan(1/3)
Pass: pi/4 = 2Atan(1/3) + ATan(1/7)
Pass: pi/4 = 4ArcTan(1/5) - ATan(1 / 239)
Pass: pi/4 = 5arctan(1/7) + 2 * atan(3/79)
Pass: pi/4 = 5ATan(29/278) + 7*ATan(3/79)
Pass: pi/4 = atn(1/2) + ATan(1/5) + ATan(1/8)
Pass: pi/4 = 4ATan(1/5) - Atan(1/70) + ATan(1/99)
Pass: pi /4 = 5*ATan(1/7) + 4 ATan(1/53) + 2ATan(1/4443)
Pass: pi / 4 = 6ATan(1/8) + 2arctangent(1/57) + ATan(1/239)
Pass: pi/ 4 = 8ATan(1/10) - ATan(1/239) - 4ATan(1/515)
Pass: pi/4 = 12ATan(1/18) + 8ATan(1/57) - 5ATan(1/239)
Pass: pi/4 = 16 * ATan(1/21) + 3ATan(1/239) + 4ATan(3/1042)
Pass: pi/4 = 22ATan(1/28) + 2ATan(1/443) - 5ATan(1/1393) - 10 ATan( 1 / 11018 )
Pass: pi/4 = 22ATan(1/38) + 17ATan(7/601) + 10ATan(7 / 8149)
Pass: pi/4 = 44ATan(1/57) + 7ATan(1/239) - 12ATan(1/682) + 24ATan(1/12943)
Pass: pi/4 = 88ATan(1/172) + 51ATan(1/239) + 32ATan(1/682) + 44ATan(1/5357) + 68ATan(1/12943)
Fail: pi/4 = 88ATan(1/172) + 51ATan(1/239) + 32ATan(1/682) + 44ATan(1/5357) + 68ATan(1/12944)
XPL0
code ChOut=8, Text=12; \intrinsic routines
int Number(18); \numbers from equations
def LF=$0A; \ASCII line feed (end-of-line character)
func Parse(S); \Convert numbers in string S to binary in Number array
char S;
int I, Neg;
proc GetNum; \Get number from string S
int N;
[while S(0)<^0 ! S(0)>^9 do S:= S+1;
N:= S(0)-^0; S:= S+1;
while S(0)>=^0 & S(0)<=^9 do
[N:= N*10 + S(0) - ^0; S:= S+1];
Number(I):= N; I:= I+1;
];
[while S(0)#^= do S:= S+1; \skip to "="
I:= 0;
loop [Neg:= false; \assume positive term
loop [S:= S+1; \next char
case S(0) of
LF: [Number(I):= 0; return S+1]; \mark end of array
^-: Neg:= true; \term is negative
^a: [Number(I):= 1; I:= I+1; quit] \no coefficient so use 1
other if S(0)>=^0 & S(0)<=^9 then \if digit
[S:= S-1; GetNum; quit]; \backup and get number
];
GetNum; \numerator
if Neg then Number(I-1):= -Number(I-1); \tan(-a) = -tan(a)
GetNum; \denominator
];
];
func GCD(U, V); \Return the greatest common divisor of U and V
int U, V;
int T;
[while V do \Euclid's method
[T:= U; U:= V; V:= rem(T/V)];
return abs(U);
];
proc Verify; \Verify that tangent of equation = 1 (i.e: E = F)
int E, F, I, J;
proc Machin(A, B, C, D);
int A, B, C, D;
int Div;
\tan(a+b) = (tan(a) + tan(b)) / (1 - tan(a)*tan(b))
\tan(arctan(A/B) + arctan(C/D))
\ = (tan(arctan(A/B)) + tan(arctan(C/D))) / (1 - tan(arctan(A/B))*tan(arctan(C/D)))
\ = (A/B + C/D) / (1 - A/B*C/D)
\ = (A*D/B*D + B*C/B*D) / (B*D/B*D - A*C/B*D)
\ = (A*D + B*C) / (B*D - A*C)
[E:= A*D + B*C; F:= B*D - A*C;
Div:= GCD(E, F); \keep integers from getting too big
E:= E/Div; F:= F/Div;
];
[E:= 0; F:= 1; I:= 0;
while Number(I) do
[for J:= 1 to Number(I) do
Machin(E, F, Number(I+1), Number(I+2));
I:= I+3;
];
Text(0, if E=F then "Yes " else "No ");
];
char S, SS; int I;
[S:= "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)
"; \Python version of equations (thanks!)
for I:= 1 to 17 do
[SS:= S; \save start of string line
S:= Parse(S); \returns start of next line
Verify; \correct Machin equation? Yes or No
repeat ChOut(0, SS(0)); SS:= SS+1 until SS(0)=LF; ChOut(0, LF); \show equation
];
]
{{out}}
Yes pi/4 = arctan(1/2) + arctan(1/3)
Yes pi/4 = 2*arctan(1/3) + arctan(1/7)
Yes pi/4 = 4*arctan(1/5) - arctan(1/239)
Yes pi/4 = 5*arctan(1/7) + 2*arctan(3/79)
Yes pi/4 = 5*arctan(29/278) + 7*arctan(3/79)
Yes pi/4 = arctan(1/2) + arctan(1/5) + arctan(1/8)
Yes pi/4 = 4*arctan(1/5) - arctan(1/70) + arctan(1/99)
Yes pi/4 = 5*arctan(1/7) + 4*arctan(1/53) + 2*arctan(1/4443)
Yes pi/4 = 6*arctan(1/8) + 2*arctan(1/57) + arctan(1/239)
Yes pi/4 = 8*arctan(1/10) - arctan(1/239) - 4*arctan(1/515)
Yes pi/4 = 12*arctan(1/18) + 8*arctan(1/57) - 5*arctan(1/239)
Yes pi/4 = 16*arctan(1/21) + 3*arctan(1/239) + 4*arctan(3/1042)
Yes pi/4 = 22*arctan(1/28) + 2*arctan(1/443) - 5*arctan(1/1393) - 10*arctan(1/11018)
Yes pi/4 = 22*arctan(1/38) + 17*arctan(7/601) + 10*arctan(7/8149)
Yes pi/4 = 44*arctan(1/57) + 7*arctan(1/239) - 12*arctan(1/682) + 24*arctan(1/12943)
Yes pi/4 = 88*arctan(1/172) + 51*arctan(1/239) + 32*arctan(1/682) + 44*arctan(1/5357) + 68*arctan(1/12943)
No pi/4 = 88*arctan(1/172) + 51*arctan(1/239) + 32*arctan(1/682) + 44*arctan(1/5357) + 68*arctan(1/12944)