Data commit
This commit is contained in:
Ingy döt Net
parent
7387c8f97b
commit
cb5bb5e222
199093 changed files with 3378972 additions and 0 deletions
2
Task/Check-Machin-like-formulas/00-META.yaml
Normal file
2
Task/Check-Machin-like-formulas/00-META.yaml
Normal file
|
|
@ -0,0 +1,2 @@
|
|||
---
|
||||
from: http://rosettacode.org/wiki/Check_Machin-like_formulas
|
||||
41
Task/Check-Machin-like-formulas/00-TASK.txt
Normal file
41
Task/Check-Machin-like-formulas/00-TASK.txt
Normal file
|
|
@ -0,0 +1,41 @@
|
|||
[[wp:Machin-like_formula|Machin-like formulas]] are useful for efficiently computing numerical approximations for <big><big><math>\pi</math></big></big>
|
||||
|
||||
|
||||
;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''':
|
||||
|
||||
: <math>{\pi\over4} = \arctan{1\over2} + \arctan{1\over3}</math>
|
||||
: <math>{\pi\over4} = 2 \arctan{1\over3} + \arctan{1\over7}</math>
|
||||
: <math>{\pi\over4} = 4 \arctan{1\over5} - \arctan{1\over239}</math>
|
||||
: <math>{\pi\over4} = 5 \arctan{1\over7} + 2 \arctan{3\over79}</math>
|
||||
: <math>{\pi\over4} = 5 \arctan{29\over278} + 7 \arctan{3\over79}</math>
|
||||
: <math>{\pi\over4} = \arctan{1\over2} + \arctan{1\over5} + \arctan{1\over8}</math>
|
||||
: <math>{\pi\over4} = 4 \arctan{1\over5} - \arctan{1\over70} + \arctan{1\over99}</math>
|
||||
: <math>{\pi\over4} = 5 \arctan{1\over7} + 4 \arctan{1\over53} + 2 \arctan{1\over4443}</math>
|
||||
: <math>{\pi\over4} = 6 \arctan{1\over8} + 2 \arctan{1\over57} + \arctan{1\over239}</math>
|
||||
: <math>{\pi\over4} = 8 \arctan{1\over10} - \arctan{1\over239} - 4 \arctan{1\over515}</math>
|
||||
: <math>{\pi\over4} = 12 \arctan{1\over18} + 8 \arctan{1\over57} - 5 \arctan{1\over239}</math>
|
||||
: <math>{\pi\over4} = 16 \arctan{1\over21} + 3 \arctan{1\over239} + 4 \arctan{3\over1042}</math>
|
||||
: <math>{\pi\over4} = 22 \arctan{1\over28} + 2 \arctan{1\over443} - 5 \arctan{1\over1393} - 10 \arctan{1\over11018}</math>
|
||||
: <math>{\pi\over4} = 22 \arctan{1\over38} + 17 \arctan{7\over601} + 10 \arctan{7\over8149}</math>
|
||||
: <math>{\pi\over4} = 44 \arctan{1\over57} + 7 \arctan{1\over239} - 12 \arctan{1\over682} + 24 \arctan{1\over12943}</math>
|
||||
: <math>{\pi\over4} = 88 \arctan{1\over172} + 51 \arctan{1\over239} + 32 \arctan{1\over682} + 44 \arctan{1\over5357} + 68 \arctan{1\over12943}</math>
|
||||
|
||||
and confirm that the following formula is ''incorrect'' by showing '''tan''' (''right hand side)'' is ''not'' '''1''':
|
||||
|
||||
: <math>{\pi\over4} = 88 \arctan{1\over172} + 51 \arctan{1\over239} + 32 \arctan{1\over682} + 44 \arctan{1\over5357} + 68 \arctan{1\over12944}</math>
|
||||
|
||||
These identities are useful in calculating the values:
|
||||
: <math>\tan(a + b) = {\tan(a) + \tan(b) \over 1 - \tan(a) \tan(b)}</math>
|
||||
|
||||
: <math>\tan\left(\arctan{a \over b}\right) = {a \over b}</math>
|
||||
|
||||
: <math>\tan(-a) = -\tan(a)</math>
|
||||
|
||||
<br>
|
||||
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 ''<math>{-3 pi \over 4}</math> < right hand side < <math>{5 pi \over 4}</math>'' due to ''<math>\tan()</math>'' periodicity.
|
||||
|
||||
<br><br>
|
||||
|
||||
|
|
@ -0,0 +1,52 @@
|
|||
(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)))
|
||||
|
|
@ -0,0 +1,71 @@
|
|||
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);
|
||||
}
|
||||
}
|
||||
|
|
@ -0,0 +1,58 @@
|
|||
(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 '👍 f '⟾ 1 )
|
||||
(writeln '❌ f '➽ (reduce f) ))))
|
||||
|
||||
|
|
@ -0,0 +1,42 @@
|
|||
(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)
|
||||
|
||||
👍 (tan (+ (arctan 1/2) (arctan 1/3))) ⟾ 1
|
||||
👍 (tan (+ (* 2 (arctan 1/3)) (arctan 1/7))) ⟾ 1
|
||||
👍 (tan (- (* 4 (arctan 1/5)) (arctan 1/239))) ⟾ 1
|
||||
👍 (tan (+ (* 5 (arctan 1/7)) (* 2 (arctan 3/79)))) ⟾ 1
|
||||
👍 (tan (+ (* 5 (arctan 29/278)) (* 7 (arctan 3/79)))) ⟾ 1
|
||||
👍 (tan (+ (arctan 1/2) (arctan 1/5) (arctan 1/8))) ⟾ 1
|
||||
👍 (tan (+ (* 4 (arctan 1/5)) (* -1 (arctan 1/70)) (arctan 1/99))) ⟾ 1
|
||||
👍 (tan (+ (* 5 (arctan 1/7)) (* 4 (arctan 1/53)) (* 2 (arctan 1/4443)))) ⟾ 1
|
||||
👍 (tan (+ (* 6 (arctan 1/8)) (* 2 (arctan 1/57)) (arctan 1/239))) ⟾ 1
|
||||
👍 (tan (+ (* 8 (arctan 1/10)) (* -1 (arctan 1/239)) (* -4 (arctan 1/515)))) ⟾ 1
|
||||
👍 (tan (+ (* 12 (arctan 1/18)) (* 8 (arctan 1/57)) (* -5 (arctan 1/239)))) ⟾ 1
|
||||
👍 (tan (+ (* 16 (arctan 1/21)) (* 3 (arctan 1/239)) (* 4 (arctan 3/1042)))) ⟾ 1
|
||||
👍 (tan (+ (* 22 (arctan 1/28)) (* 2 (arctan 1/443)) (* -5 (arctan 1/1393)) (* -10 (arctan 1/11018)))) ⟾ 1
|
||||
👍 (tan (+ (* 22 (arctan 1/38)) (* 17 (arctan 7/601)) (* 10 (arctan 7/8149)))) ⟾ 1
|
||||
👍 (tan (+ (* 44 (arctan 1/57)) (* 7 (arctan 1/239)) (* -12 (arctan 1/682)) (* 24 (arctan 1/12943)))) ⟾ 1
|
||||
👍 (tan (+ (* 88 (arctan 1/172)) (* 51 (arctan 1/239)) (* 32 (arctan 1/682))
|
||||
(* 44 (arctan 1/5357)) (* 68 (arctan 1/12943)))) ⟾ 1
|
||||
❌ (tan (+ (* 88 (arctan 1/172)) (* 51 (arctan 1/239)) (* 32 (arctan 1/682))
|
||||
(* 44 (arctan 1/5357)) (* 68 (arctan 1/12944)))) ➽ 0.9999991882257442
|
||||
|
|
@ -0,0 +1,39 @@
|
|||
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
|
||||
|
|
@ -0,0 +1,153 @@
|
|||
' 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
|
||||
|
|
@ -0,0 +1,44 @@
|
|||
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]]);
|
||||
|
|
@ -0,0 +1,64 @@
|
|||
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)))
|
||||
}
|
||||
|
|
@ -0,0 +1,31 @@
|
|||
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)
|
||||
|
|
@ -0,0 +1,34 @@
|
|||
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)
|
||||
|
|
@ -0,0 +1 @@
|
|||
machin =: 1r4p1 = [: +/ ({. * _3 o. %/@:}.)"1@:x:
|
||||
|
|
@ -0,0 +1,68 @@
|
|||
R =: <@:(0&".);._2 ];._2 noun define
|
||||
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
|
||||
------------
|
||||
)
|
||||
|
||||
machin&> R
|
||||
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
|
||||
|
|
@ -0,0 +1,9 @@
|
|||
counterExample=. 12944 (<_1;_1)} >{:R
|
||||
counterExample NB. Same as final test case with 12943 incremented to 12944
|
||||
88 1 172
|
||||
51 1 239
|
||||
32 1 682
|
||||
44 1 5357
|
||||
68 1 12944
|
||||
machin counterExample
|
||||
0
|
||||
|
|
@ -0,0 +1,146 @@
|
|||
import java.io.BufferedReader;
|
||||
import java.io.File;
|
||||
import java.io.FileReader;
|
||||
import java.io.IOException;
|
||||
import java.math.BigInteger;
|
||||
import java.util.ArrayList;
|
||||
import java.util.List;
|
||||
import java.util.regex.Matcher;
|
||||
import java.util.regex.Pattern;
|
||||
|
||||
public class CheckMachinFormula {
|
||||
|
||||
private static String FILE_NAME = "MachinFormula.txt";
|
||||
|
||||
public static void main(String[] args) {
|
||||
try {
|
||||
runPrivate();
|
||||
} catch (Exception e) {
|
||||
e.printStackTrace();
|
||||
}
|
||||
}
|
||||
|
||||
private static void runPrivate() throws IOException {
|
||||
try (BufferedReader reader = new BufferedReader(new FileReader(new File(FILE_NAME)));) {
|
||||
String inLine = null;
|
||||
while ( (inLine = reader.readLine()) != null ) {
|
||||
String[] split = inLine.split("=");
|
||||
System.out.println(tanLeft(split[0].trim()) + " = " + split[1].trim().replaceAll("\\s+", " ") + " = " + tanRight(split[1].trim()));
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
private static String tanLeft(String formula) {
|
||||
if ( formula.compareTo("pi/4") == 0 ) {
|
||||
return "1";
|
||||
}
|
||||
throw new RuntimeException("ERROR 104: Unknown left side: " + formula);
|
||||
}
|
||||
|
||||
private static final Pattern ARCTAN_PATTERN = Pattern.compile("(-{0,1}\\d+)\\*arctan\\((\\d+)/(\\d+)\\)");
|
||||
|
||||
private static Fraction tanRight(String formula) {
|
||||
Matcher matcher = ARCTAN_PATTERN.matcher(formula);
|
||||
List<Term> terms = new ArrayList<>();
|
||||
while ( matcher.find() ) {
|
||||
terms.add(new Term(Integer.parseInt(matcher.group(1)), new Fraction(matcher.group(2), matcher.group(3))));
|
||||
}
|
||||
return evaluateArctan(terms);
|
||||
}
|
||||
|
||||
private static Fraction evaluateArctan(List<Term> terms) {
|
||||
if ( terms.size() == 1 ) {
|
||||
Term term = terms.get(0);
|
||||
return evaluateArctan(term.coefficient, term.fraction);
|
||||
}
|
||||
int size = terms.size();
|
||||
List<Term> left = terms.subList(0, (size+1) / 2);
|
||||
List<Term> right = terms.subList((size+1) / 2, size);
|
||||
return arctanFormula(evaluateArctan(left), evaluateArctan(right));
|
||||
}
|
||||
|
||||
private static Fraction evaluateArctan(int coefficient, Fraction fraction) {
|
||||
//System.out.println("C = " + coefficient + ", F = " + fraction);
|
||||
if ( coefficient == 1 ) {
|
||||
return fraction;
|
||||
}
|
||||
else if ( coefficient < 0 ) {
|
||||
return evaluateArctan(-coefficient, fraction).negate();
|
||||
}
|
||||
if ( coefficient % 2 == 0 ) {
|
||||
Fraction f = evaluateArctan(coefficient/2, fraction);
|
||||
return arctanFormula(f, f);
|
||||
}
|
||||
Fraction a = evaluateArctan(coefficient/2, fraction);
|
||||
Fraction b = evaluateArctan(coefficient - (coefficient/2), fraction);
|
||||
return arctanFormula(a, b);
|
||||
}
|
||||
|
||||
private static Fraction arctanFormula(Fraction f1, Fraction f2) {
|
||||
return f1.add(f2).divide(Fraction.ONE.subtract(f1.multiply(f2)));
|
||||
}
|
||||
|
||||
private static class Fraction {
|
||||
|
||||
public static final Fraction ONE = new Fraction("1", "1");
|
||||
|
||||
private BigInteger numerator;
|
||||
private BigInteger denominator;
|
||||
|
||||
public Fraction(String num, String den) {
|
||||
numerator = new BigInteger(num);
|
||||
denominator = new BigInteger(den);
|
||||
}
|
||||
|
||||
public Fraction(BigInteger num, BigInteger den) {
|
||||
numerator = num;
|
||||
denominator = den;
|
||||
}
|
||||
|
||||
public Fraction negate() {
|
||||
return new Fraction(numerator.negate(), denominator);
|
||||
}
|
||||
|
||||
public Fraction add(Fraction f) {
|
||||
BigInteger gcd = denominator.gcd(f.denominator);
|
||||
BigInteger first = numerator.multiply(f.denominator.divide(gcd));
|
||||
BigInteger second = f.numerator.multiply(denominator.divide(gcd));
|
||||
return new Fraction(first.add(second), denominator.multiply(f.denominator).divide(gcd));
|
||||
}
|
||||
|
||||
public Fraction subtract(Fraction f) {
|
||||
return add(f.negate());
|
||||
}
|
||||
|
||||
public Fraction multiply(Fraction f) {
|
||||
BigInteger num = numerator.multiply(f.numerator);
|
||||
BigInteger den = denominator.multiply(f.denominator);
|
||||
BigInteger gcd = num.gcd(den);
|
||||
return new Fraction(num.divide(gcd), den.divide(gcd));
|
||||
}
|
||||
|
||||
public Fraction divide(Fraction f) {
|
||||
return multiply(new Fraction(f.denominator, f.numerator));
|
||||
}
|
||||
|
||||
@Override
|
||||
public String toString() {
|
||||
if ( denominator.compareTo(BigInteger.ONE) == 0 ) {
|
||||
return numerator.toString();
|
||||
}
|
||||
return numerator + " / " + denominator;
|
||||
}
|
||||
}
|
||||
|
||||
private static class Term {
|
||||
|
||||
private int coefficient;
|
||||
private Fraction fraction;
|
||||
|
||||
public Term(int c, Fraction f) {
|
||||
coefficient = c;
|
||||
fraction = f;
|
||||
}
|
||||
}
|
||||
|
||||
}
|
||||
|
|
@ -0,0 +1,37 @@
|
|||
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 = $ans")
|
||||
end
|
||||
end
|
||||
|
||||
runtestmats()
|
||||
|
|
@ -0,0 +1,132 @@
|
|||
// version 1.1.3
|
||||
|
||||
import java.math.BigInteger
|
||||
|
||||
val bigZero = BigInteger.ZERO
|
||||
val bigOne = BigInteger.ONE
|
||||
|
||||
class BigRational : Comparable<BigRational> {
|
||||
|
||||
val num: BigInteger
|
||||
val denom: BigInteger
|
||||
|
||||
constructor(n: BigInteger, d: BigInteger) {
|
||||
require(d != bigZero)
|
||||
var nn = n
|
||||
var dd = d
|
||||
if (nn == bigZero) {
|
||||
dd = bigOne
|
||||
}
|
||||
else if (dd < bigZero) {
|
||||
nn = -nn
|
||||
dd = -dd
|
||||
}
|
||||
val g = nn.gcd(dd)
|
||||
if (g > bigOne) {
|
||||
nn /= g
|
||||
dd /= g
|
||||
}
|
||||
num = nn
|
||||
denom = dd
|
||||
}
|
||||
|
||||
constructor(n: Long, d: Long) : this(BigInteger.valueOf(n), BigInteger.valueOf(d))
|
||||
|
||||
operator fun plus(other: BigRational) =
|
||||
BigRational(num * other.denom + denom * other.num, other.denom * denom)
|
||||
|
||||
operator fun unaryMinus() = BigRational(-num, denom)
|
||||
|
||||
operator fun minus(other: BigRational) = this + (-other)
|
||||
|
||||
operator fun times(other: BigRational) = BigRational(this.num * other.num, this.denom * other.denom)
|
||||
|
||||
fun inverse(): BigRational {
|
||||
require(num != bigZero)
|
||||
return BigRational(denom, num)
|
||||
}
|
||||
|
||||
operator fun div(other: BigRational) = this * other.inverse()
|
||||
|
||||
override fun compareTo(other: BigRational): Int {
|
||||
val diff = this - other
|
||||
return when {
|
||||
diff.num < bigZero -> -1
|
||||
diff.num > bigZero -> +1
|
||||
else -> 0
|
||||
}
|
||||
}
|
||||
|
||||
override fun equals(other: Any?): Boolean {
|
||||
if (other == null || other !is BigRational) return false
|
||||
return this.compareTo(other) == 0
|
||||
}
|
||||
|
||||
override fun toString() = if (denom == bigOne) "$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(")")
|
||||
}
|
||||
}
|
||||
|
|
@ -0,0 +1,21 @@
|
|||
Tan[ArcTan[1/2] + ArcTan[1/3]] == 1
|
||||
Tan[2 ArcTan[1/3] + ArcTan[1/7]] == 1
|
||||
Tan[4 ArcTan[1/5] - ArcTan[1/239]] == 1
|
||||
Tan[5 ArcTan[1/7] + 2 ArcTan[3/79]] == 1
|
||||
Tan[5 ArcTan[29/278] + 7 ArcTan[3/79]] == 1
|
||||
Tan[ArcTan[1/2] + ArcTan[1/5] + ArcTan[1/8]] == 1
|
||||
Tan[4 ArcTan[1/5] - ArcTan[1/70] + ArcTan[1/99]] == 1
|
||||
Tan[5 ArcTan[1/7] + 4 ArcTan[1/53] + 2 ArcTan[1/4443]] == 1
|
||||
Tan[6 ArcTan[1/8] + 2 ArcTan[1/57] + ArcTan[1/239]] == 1
|
||||
Tan[8 ArcTan[1/10] - ArcTan[1/239] - 4 ArcTan[1/515]] == 1
|
||||
Tan[12 ArcTan[1/18] + 8 ArcTan[1/57] - 5 ArcTan[1/239]] == 1
|
||||
Tan[16 ArcTan[1/21] + 3 ArcTan[1/239] + 4 ArcTan[3/1042]] == 1
|
||||
Tan[22 ArcTan[1/28] + 2 ArcTan[1/443] - 5 ArcTan[1/1393] -
|
||||
10 ArcTan[1/11018]] == 1
|
||||
Tan[22 ArcTan[1/38] + 17 ArcTan[7/601] + 10 ArcTan[7/8149]] == 1
|
||||
Tan[44 ArcTan[1/57] + 7 ArcTan[1/239] - 12 ArcTan[1/682] +
|
||||
24 ArcTan[1/12943]] == 1
|
||||
Tan[88 ArcTan[1/172] + 51 ArcTan[1/239] + 32 ArcTan[1/682] +
|
||||
44 ArcTan[1/5357] + 68 ArcTan[1/12943]] == 1
|
||||
Tan[88 ArcTan[1/172] + 51 ArcTan[1/239] + 32 ArcTan[1/682] +
|
||||
44 ArcTan[1/5357] + 68 ArcTan[1/12944]] == 1
|
||||
|
|
@ -0,0 +1,18 @@
|
|||
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);
|
||||
|
|
@ -0,0 +1,265 @@
|
|||
import bignum
|
||||
|
||||
type
|
||||
|
||||
# Description of a term.
|
||||
Term = object
|
||||
factor: int # Multiplier (may be negative).
|
||||
fract: Rat # Argument of arc tangent.
|
||||
|
||||
Expression = seq[Term]
|
||||
|
||||
# Rational 1.
|
||||
let One = newRat(1)
|
||||
|
||||
|
||||
####################################################################################################
|
||||
# Formula parser.
|
||||
|
||||
type
|
||||
|
||||
# Possible tokens for parsing.
|
||||
Token = enum tkPi, tkArctan, tkNumber, tkEqual, tkAdd, tkSub,
|
||||
tkMul, tkDiv, tkLPar, tkRPar, tkError, tkEnd
|
||||
|
||||
# Lexer description.
|
||||
Lexer = object
|
||||
line: string # The line to parse.
|
||||
pos: Natural # Current position of lexer.
|
||||
token: Token # Current token.
|
||||
value: Natural # Associated value (for numbers).
|
||||
|
||||
# Exception raised if an error is found.
|
||||
SyntaxError = object of CatchableError
|
||||
|
||||
#---------------------------------------------------------------------------------------------------
|
||||
|
||||
proc initLexer(line: string): Lexer =
|
||||
## Create and initialize a lexer.
|
||||
result.line = line
|
||||
result.pos = 0
|
||||
|
||||
#---------------------------------------------------------------------------------------------------
|
||||
|
||||
proc parseName(lexer: var Lexer; pos: Natural) =
|
||||
## Parse a name.
|
||||
|
||||
# Build the name.
|
||||
var pos = pos
|
||||
var name = ""
|
||||
while pos < lexer.line.len and (let c = lexer.line[pos]; c) in 'a'..'z':
|
||||
name.add(c)
|
||||
inc pos
|
||||
|
||||
# Update lexer state.
|
||||
lexer.token = if name == "arctan": tkArctan
|
||||
elif name == "pi": tkPi
|
||||
else: tkError
|
||||
lexer.pos = pos
|
||||
|
||||
#---------------------------------------------------------------------------------------------------
|
||||
|
||||
proc parseNumber(lexer: var Lexer; pos: Natural) =
|
||||
## Parse a number.
|
||||
|
||||
# Build the number.
|
||||
var pos = pos
|
||||
var value = 0
|
||||
while pos < lexer.line.len and (let c = lexer.line[pos]; c) in '0'..'9':
|
||||
value = 10 * value + ord(c) - ord('0')
|
||||
inc pos
|
||||
|
||||
# Update lexer state.
|
||||
lexer.token = tkNumber
|
||||
lexer.value = value
|
||||
lexer.pos = pos
|
||||
|
||||
#---------------------------------------------------------------------------------------------------
|
||||
|
||||
proc getNextToken(lexer: var Lexer) =
|
||||
## Find next token.
|
||||
|
||||
var pos = lexer.pos
|
||||
var token: Token
|
||||
while pos < lexer.line.len and lexer.line[pos] == ' ': inc pos
|
||||
if pos == lexer.line.len:
|
||||
# Reached end of string.
|
||||
lexer.pos = pos
|
||||
lexer.token = tkEnd
|
||||
return
|
||||
|
||||
# Find token.
|
||||
case lexer.line[pos]
|
||||
of '=': token = tkEqual
|
||||
of '+': token = tkAdd
|
||||
of '-': token = tkSub
|
||||
of '*': token = tkMul
|
||||
of '/': token = tkDiv
|
||||
of '(': token = tkLPar
|
||||
of ')': token = tkRPar
|
||||
of 'a'..'z':
|
||||
lexer.parseName(pos)
|
||||
return
|
||||
of '0'..'9':
|
||||
lexer.parseNumber(pos)
|
||||
return
|
||||
else: token = tkError
|
||||
|
||||
# Update lexer state.
|
||||
lexer.pos = pos + 1
|
||||
lexer.token = token
|
||||
|
||||
#---------------------------------------------------------------------------------------------------
|
||||
|
||||
template syntaxError(message: string) =
|
||||
## Raise a syntax error exception.
|
||||
raise newException(SyntaxError, message)
|
||||
|
||||
#---------------------------------------------------------------------------------------------------
|
||||
|
||||
proc parseFraction(lexer: var Lexer): Rat =
|
||||
## Parse a fraction: number / number.
|
||||
|
||||
lexer.getNextToken()
|
||||
if lexer.token != tkNumber:
|
||||
syntaxError("number expected.")
|
||||
let num = lexer.value
|
||||
lexer.getNextToken()
|
||||
if lexer.token != tkDiv:
|
||||
syntaxError("“/” expected.")
|
||||
lexer.getNextToken()
|
||||
if lexer.token != tkNumber:
|
||||
syntaxError("number expected")
|
||||
if lexer.value == 0:
|
||||
raise newException(ValueError, "null denominator.")
|
||||
let den = lexer.value
|
||||
result = newRat(num, den)
|
||||
|
||||
#---------------------------------------------------------------------------------------------------
|
||||
|
||||
proc parseTerm(lexer: var Lexer): Term =
|
||||
## Parse a term: factor * arctan(fraction) or arctan(fraction).
|
||||
|
||||
lexer.getNextToken()
|
||||
|
||||
# Parse factor.
|
||||
if lexer.token == tkNumber:
|
||||
result.factor = lexer.value
|
||||
lexer.getNextToken
|
||||
if lexer.token != tkMul:
|
||||
syntaxError("“*” expected.")
|
||||
lexer.getNextToken()
|
||||
else:
|
||||
result.factor = 1
|
||||
|
||||
# Parse arctan.
|
||||
if lexer.token != tkArctan:
|
||||
syntaxError("“arctan” expected.")
|
||||
lexer.getNextToken()
|
||||
if lexer.token != tkLPar:
|
||||
syntaxError("“(” expected.")
|
||||
result.fract = lexer.parseFraction()
|
||||
lexer.getNextToken()
|
||||
if lexer.token != tkRPar:
|
||||
syntaxError("“)” expected.")
|
||||
|
||||
#---------------------------------------------------------------------------------------------------
|
||||
|
||||
proc parse(line: string): Expression =
|
||||
## Parse a formula.
|
||||
|
||||
var lexer = initLexer(line)
|
||||
lexer.getNextToken()
|
||||
|
||||
if lexer.token != tkPi:
|
||||
syntaxError("pi symbol expected.")
|
||||
lexer.getNextToken()
|
||||
|
||||
if lexer.token != tkDiv:
|
||||
syntaxError("'/' expected.")
|
||||
lexer.getNextToken()
|
||||
|
||||
if lexer.token != tkNumber:
|
||||
syntaxError("number expected.")
|
||||
if lexer.value != 4:
|
||||
raise newException(ValueError, "value 4 expected.")
|
||||
lexer.getNextToken()
|
||||
|
||||
if lexer.token != tkEqual:
|
||||
syntaxError("“=” expected.")
|
||||
result.add(lexer.parseTerm())
|
||||
lexer.getNextToken()
|
||||
|
||||
# Parse the next terms.
|
||||
while (let token = lexer.token; token) in {tkAdd, tkSub}:
|
||||
var term = lexer.parseTerm()
|
||||
if token == tkSub:
|
||||
term.factor = -term.factor
|
||||
result.add(term)
|
||||
lexer.getNextToken()
|
||||
|
||||
if lexer.token != tkEnd:
|
||||
syntaxError("invalid characters at end of formula.")
|
||||
|
||||
|
||||
####################################################################################################
|
||||
# Evaluator.
|
||||
|
||||
proc tangent(factor: int; fract: Rat): Rat =
|
||||
## Compute the tangent of "factor * arctan(fract)".
|
||||
|
||||
if factor == 1:
|
||||
return fract
|
||||
if factor < 0:
|
||||
return -tangent(-factor, fract)
|
||||
|
||||
# Split in two parts.
|
||||
let n = factor div 2
|
||||
let a = tangent(n, fract)
|
||||
let b = tangent(factor - n, fract)
|
||||
result = (a + b) / (One - a * b)
|
||||
|
||||
#---------------------------------------------------------------------------------------------------
|
||||
|
||||
proc tangent(expr: Expression): Rat =
|
||||
## Compute the tangent of a sum of terms.
|
||||
|
||||
if expr.len == 1:
|
||||
result = tangent(expr[0].factor, expr[0].fract)
|
||||
else:
|
||||
# Split in two parts.
|
||||
let a = tangent(expr[0..<(expr.len div 2)])
|
||||
let b = tangent(expr[(expr.len div 2)..^1])
|
||||
result = (a + b) / (One - a * b)
|
||||
|
||||
#———————————————————————————————————————————————————————————————————————————————————————————————————
|
||||
|
||||
when isMainModule:
|
||||
|
||||
const Formulas = [
|
||||
"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 formula in Formulas:
|
||||
let expr = formula.parse()
|
||||
let value = tangent(expr)
|
||||
if value == 1:
|
||||
echo "True: ", formula
|
||||
else:
|
||||
echo "False: ", formula
|
||||
echo "Tangent of the right expression is about ", value.toFloat
|
||||
|
|
@ -0,0 +1,51 @@
|
|||
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)])
|
||||
]
|
||||
|
|
@ -0,0 +1,28 @@
|
|||
/*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
|
||||
|
|
@ -0,0 +1,31 @@
|
|||
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]]);
|
||||
|
|
@ -0,0 +1,40 @@
|
|||
use Math::BigRat try=>"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']);
|
||||
|
|
@ -0,0 +1,27 @@
|
|||
(phixonline)-->
|
||||
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
|
||||
<span style="color: #008080;">procedure</span> <span style="color: #000000;">test</span><span style="color: #0000FF;">(</span><span style="color: #004080;">atom</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">3</span><span style="color: #0000FF;">*</span><span style="color: #004600;">PI</span><span style="color: #0000FF;">/</span><span style="color: #000000;">4</span> <span style="color: #0000FF;">>=</span> <span style="color: #000000;">a</span> <span style="color: #008080;">then</span> <span style="color: #0000FF;">?</span><span style="color: #000000;">9</span><span style="color: #0000FF;">/</span><span style="color: #000000;">0</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #000000;">5</span><span style="color: #0000FF;">*</span><span style="color: #004600;">PI</span><span style="color: #0000FF;">/</span><span style="color: #000000;">4</span> <span style="color: #0000FF;"><=</span> <span style="color: #000000;">a</span> <span style="color: #008080;">then</span> <span style="color: #0000FF;">?</span><span style="color: #000000;">9</span><span style="color: #0000FF;">/</span><span style="color: #000000;">0</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
|
||||
<span style="color: #004080;">string</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sprint</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">tan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">))</span>
|
||||
<span style="color: #0000FF;">?</span><span style="color: #000000;">s</span> <span style="color: #000080;font-style:italic;">-- or test for "1.0"/"1", but not 1.0</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
|
||||
<span style="color: #000000;">test</span><span style="color: #0000FF;">(</span> <span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">))</span>
|
||||
<span style="color: #000000;">test</span><span style="color: #0000FF;">(</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">7</span><span style="color: #0000FF;">))</span>
|
||||
<span style="color: #000000;">test</span><span style="color: #0000FF;">(</span> <span style="color: #000000;">4</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">5</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">-</span> <span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">239</span><span style="color: #0000FF;">))</span>
|
||||
<span style="color: #000000;">test</span><span style="color: #0000FF;">(</span> <span style="color: #000000;">5</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">7</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">3</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">79</span><span style="color: #0000FF;">))</span>
|
||||
<span style="color: #000000;">test</span><span style="color: #0000FF;">(</span> <span style="color: #000000;">5</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">29</span><span style="color: #0000FF;">/</span> <span style="color: #000000;">278</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">7</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">3</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">79</span><span style="color: #0000FF;">))</span>
|
||||
<span style="color: #000000;">test</span><span style="color: #0000FF;">(</span> <span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">5</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">8</span><span style="color: #0000FF;">))</span>
|
||||
<span style="color: #000000;">test</span><span style="color: #0000FF;">(</span> <span style="color: #000000;">4</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">5</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">-</span> <span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">70</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">99</span><span style="color: #0000FF;">))</span>
|
||||
<span style="color: #000000;">test</span><span style="color: #0000FF;">(</span> <span style="color: #000000;">5</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">7</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">4</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">53</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">4443</span><span style="color: #0000FF;">))</span>
|
||||
<span style="color: #000000;">test</span><span style="color: #0000FF;">(</span> <span style="color: #000000;">6</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">8</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">57</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">239</span><span style="color: #0000FF;">))</span>
|
||||
<span style="color: #000000;">test</span><span style="color: #0000FF;">(</span> <span style="color: #000000;">8</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">10</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">-</span> <span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">239</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">4</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">515</span><span style="color: #0000FF;">))</span>
|
||||
<span style="color: #000000;">test</span><span style="color: #0000FF;">(</span><span style="color: #000000;">12</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">18</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">8</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">57</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">5</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">239</span><span style="color: #0000FF;">))</span>
|
||||
<span style="color: #000000;">test</span><span style="color: #0000FF;">(</span><span style="color: #000000;">16</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">21</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">239</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">4</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">3</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">1042</span><span style="color: #0000FF;">))</span>
|
||||
<span style="color: #000000;">test</span><span style="color: #0000FF;">(</span><span style="color: #000000;">22</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">28</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">443</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">5</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">1393</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">10</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">11018</span><span style="color: #0000FF;">))</span>
|
||||
<span style="color: #000000;">test</span><span style="color: #0000FF;">(</span><span style="color: #000000;">22</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">38</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">17</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">7</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">601</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span><span style="color: #000000;">10</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">7</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">8149</span><span style="color: #0000FF;">))</span>
|
||||
<span style="color: #000000;">test</span><span style="color: #0000FF;">(</span><span style="color: #000000;">44</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">57</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">7</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">239</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">12</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">682</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">24</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">12943</span><span style="color: #0000FF;">))</span>
|
||||
<span style="color: #000000;">test</span><span style="color: #0000FF;">(</span><span style="color: #000000;">88</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">172</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">51</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">239</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span><span style="color: #000000;">32</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">682</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">44</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">5357</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">68</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">12943</span><span style="color: #0000FF;">))</span>
|
||||
<span style="color: #0000FF;">?</span><span style="color: #008000;">"==="</span>
|
||||
<span style="color: #000000;">test</span><span style="color: #0000FF;">(</span><span style="color: #000000;">88</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">172</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">51</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">239</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">32</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">682</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">44</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">5357</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">68</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">arctan</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">12944</span><span style="color: #0000FF;">))</span>
|
||||
<!--
|
||||
|
|
@ -0,0 +1,99 @@
|
|||
(phixonline)-->
|
||||
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
|
||||
<span style="color: #008080;">include</span> <span style="color: #004080;">mpfr</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span>
|
||||
|
||||
<span style="color: #008080;">function</span> <span style="color: #000000;">tans</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #004080;">sequence</span> <span style="color: #000000;">args</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">))</span>
|
||||
<span style="color: #004080;">mpq</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">aab</span><span style="color: #0000FF;">,</span><span style="color: #000000;">mab</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpq_inits</span><span style="color: #0000FF;">(</span><span style="color: #000000;">4</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #004080;">integer</span> <span style="color: #000000;">h</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">)=</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span>
|
||||
<span style="color: #0000FF;">{</span><span style="color: #004080;">integer</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">mpq</span> <span style="color: #000000;">f</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span>
|
||||
<span style="color: #7060A8;">mpq_set</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">f</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #008080;">return</span> <span style="color: #000000;">a</span>
|
||||
<span style="color: #008080;">elsif</span> <span style="color: #000000;">m</span><span style="color: #0000FF;"><</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span>
|
||||
<span style="color: #000000;">f</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">tans</span><span style="color: #0000FF;">({{-</span><span style="color: #000000;">m</span><span style="color: #0000FF;">,</span><span style="color: #000000;">f</span><span style="color: #0000FF;">}})</span>
|
||||
<span style="color: #7060A8;">mpq_neg</span><span style="color: #0000FF;">(</span><span style="color: #000000;">f</span><span style="color: #0000FF;">,</span><span style="color: #000000;">f</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #008080;">return</span> <span style="color: #000000;">f</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
|
||||
<span style="color: #000000;">h</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">m</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">tans</span><span style="color: #0000FF;">({{</span><span style="color: #000000;">h</span><span style="color: #0000FF;">,</span><span style="color: #000000;">f</span><span style="color: #0000FF;">}})</span>
|
||||
<span style="color: #000000;">b</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">tans</span><span style="color: #0000FF;">({{</span><span style="color: #000000;">m</span><span style="color: #0000FF;">-</span><span style="color: #000000;">h</span><span style="color: #0000FF;">,</span><span style="color: #000000;">f</span><span style="color: #0000FF;">}})</span>
|
||||
<span style="color: #008080;">else</span>
|
||||
<span style="color: #000000;">h</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">)/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">tans</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..</span><span style="color: #000000;">h</span><span style="color: #0000FF;">])</span>
|
||||
<span style="color: #000000;">b</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">tans</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">[</span><span style="color: #000000;">h</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..$])</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
|
||||
<span style="color: #7060A8;">mpq_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">mab</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpq_add</span><span style="color: #0000FF;">(</span><span style="color: #000000;">aab</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpq_set_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpq_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">mab</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpq_div</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">aab</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #0000FF;">{</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">aab</span><span style="color: #0000FF;">,</span><span style="color: #000000;">mab</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpq_free</span><span style="color: #0000FF;">({</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">aab</span><span style="color: #0000FF;">,</span><span style="color: #000000;">mab</span><span style="color: #0000FF;">})</span>
|
||||
<span style="color: #008080;">return</span> <span style="color: #000000;">a</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
|
||||
|
||||
<span style="color: #008080;">function</span> <span style="color: #000000;">parse</span><span style="color: #0000FF;">(</span><span style="color: #004080;">string</span> <span style="color: #000000;">formula</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #000080;font-style:italic;">-- obviously the error handling here is a bit brutal...</span>
|
||||
<span style="color: #000000;">formula</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">substitute</span><span style="color: #0000FF;">(</span><span style="color: #000000;">formula</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" "</span><span style="color: #0000FF;">,</span><span style="color: #008000;">""</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- strip spaces</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #000000;">formula</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..</span><span style="color: #000000;">5</span><span style="color: #0000FF;">]!=</span><span style="color: #008000;">"pi/4="</span> <span style="color: #008080;">then</span> <span style="color: #0000FF;">?</span><span style="color: #000000;">9</span><span style="color: #0000FF;">/</span><span style="color: #000000;">0</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
|
||||
<span style="color: #000000;">formula</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">formula</span><span style="color: #0000FF;">[</span><span style="color: #000000;">6</span><span style="color: #0000FF;">..$]</span>
|
||||
<span style="color: #004080;">sequence</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{},</span> <span style="color: #000000;">r</span>
|
||||
<span style="color: #008080;">while</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">formula</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
|
||||
<span style="color: #004080;">integer</span> <span style="color: #000000;">sgn</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">d</span>
|
||||
<span style="color: #008080;">switch</span> <span style="color: #000000;">formula</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span> <span style="color: #008080;">do</span>
|
||||
<span style="color: #008080;">case</span> <span style="color: #008000;">'-'</span><span style="color: #0000FF;">:</span> <span style="color: #000000;">sgn</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">;</span> <span style="color: #008080;">fallthrough</span>
|
||||
<span style="color: #008080;">case</span> <span style="color: #008000;">'+'</span><span style="color: #0000FF;">:</span> <span style="color: #000000;">formula</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">formula</span><span style="color: #0000FF;">[</span><span style="color: #000000;">2</span><span style="color: #0000FF;">..$]</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">switch</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #000000;">formula</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]=</span><span style="color: #008000;">'a'</span> <span style="color: #008080;">then</span>
|
||||
<span style="color: #000000;">m</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">sgn</span>
|
||||
<span style="color: #008080;">else</span>
|
||||
<span style="color: #000000;">r</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">scanf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">formula</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%d*%s"</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">)!=</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span> <span style="color: #0000FF;">?</span><span style="color: #000000;">9</span><span style="color: #0000FF;">/</span><span style="color: #000000;">0</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
|
||||
<span style="color: #0000FF;">{</span><span style="color: #000000;">m</span><span style="color: #0000FF;">,</span><span style="color: #000000;">formula</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span>
|
||||
<span style="color: #000000;">m</span> <span style="color: #0000FF;">*=</span> <span style="color: #000000;">sgn</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
|
||||
<span style="color: #000000;">r</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">scanf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">formula</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"arctan(%d/%d)%s"</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">)!=</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span> <span style="color: #0000FF;">?</span><span style="color: #000000;">9</span><span style="color: #0000FF;">/</span><span style="color: #000000;">0</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
|
||||
<span style="color: #0000FF;">{</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">d</span><span style="color: #0000FF;">,</span><span style="color: #000000;">formula</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span>
|
||||
<span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">append</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">m</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">mpq_init_set_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">d</span><span style="color: #0000FF;">)})</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
|
||||
<span style="color: #008080;">return</span> <span style="color: #000000;">res</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
|
||||
|
||||
<span style="color: #008080;">procedure</span> <span style="color: #000000;">test</span><span style="color: #0000FF;">(</span><span style="color: #004080;">string</span> <span style="color: #000000;">formula</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #004080;">mpq</span> <span style="color: #000000;">f</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">tans</span><span style="color: #0000FF;">(</span><span style="color: #000000;">parse</span><span style="color: #0000FF;">(</span><span style="color: #000000;">formula</span><span style="color: #0000FF;">))</span>
|
||||
<span style="color: #008080;">if</span> <span style="color: #7060A8;">mpq_cmp_si</span><span style="color: #0000FF;">(</span><span style="color: #000000;">f</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span>
|
||||
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"OK: %s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">formula</span><span style="color: #0000FF;">})</span>
|
||||
<span style="color: #008080;">else</span>
|
||||
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"ERROR: %s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">formula</span><span style="color: #0000FF;">})</span>
|
||||
<span style="color: #004080;">mpz</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">d</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mpz_inits</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpq_get_num</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">f</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">mpq_get_den</span><span style="color: #0000FF;">(</span><span style="color: #000000;">d</span><span style="color: #0000FF;">,</span><span style="color: #000000;">f</span><span style="color: #0000FF;">)</span>
|
||||
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" %s\n\\ %s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #7060A8;">shorten</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">)),</span>
|
||||
<span style="color: #7060A8;">shorten</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">mpz_get_str</span><span style="color: #0000FF;">(</span><span style="color: #000000;">d</span><span style="color: #0000FF;">))})</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
|
||||
|
||||
<span style="color: #008080;">constant</span> <span style="color: #000000;">formulae</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #008000;">"pi/4 = arctan(1/2) + arctan(1/3)"</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #008000;">"pi/4 = 2*arctan(1/3) + arctan(1/7)"</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #008000;">"pi/4 = 4*arctan(1/5) - arctan(1/239)"</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #008000;">"pi/4 = 5*arctan(1/7) + 2*arctan(3/79)"</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #008000;">"pi/4 = 5*arctan(29/278) + 7*arctan(3/79)"</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #008000;">"pi/4 = arctan(1/2) + arctan(1/5) + arctan(1/8)"</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #008000;">"pi/4 = 4*arctan(1/5) - arctan(1/70) + arctan(1/99)"</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #008000;">"pi/4 = 5*arctan(1/7) + 4*arctan(1/53) + 2*arctan(1/4443)"</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #008000;">"pi/4 = 6*arctan(1/8) + 2*arctan(1/57) + arctan(1/239)"</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #008000;">"pi/4 = 8*arctan(1/10) - arctan(1/239) - 4*arctan(1/515)"</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #008000;">"pi/4 = 12*arctan(1/18) + 8*arctan(1/57) - 5*arctan(1/239)"</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #008000;">"pi/4 = 16*arctan(1/21) + 3*arctan(1/239) + 4*arctan(3/1042)"</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #008000;">"pi/4 = 22*arctan(1/28) + 2*arctan(1/443) - 5*arctan(1/1393) - 10*arctan(1/11018)"</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #008000;">"pi/4 = 22*arctan(1/38) + 17*arctan(7/601) + 10*arctan(7/8149)"</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #008000;">"pi/4 = 44*arctan(1/57) + 7*arctan(1/239) - 12*arctan(1/682) + 24*arctan(1/12943)"</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #008000;">"pi/4 = 88*arctan(1/172) + 51*arctan(1/239) + 32*arctan(1/682) + 44*arctan(1/5357) + 68*arctan(1/12943)"</span><span style="color: #0000FF;">,</span>
|
||||
<span style="color: #008000;">"pi/4 = 88*arctan(1/172) + 51*arctan(1/239) + 32*arctan(1/682) + 44*arctan(1/5357) + 68*arctan(1/12944)"</span><span style="color: #0000FF;">}</span>
|
||||
|
||||
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">formulae</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
|
||||
<span style="color: #000000;">test</span><span style="color: #0000FF;">(</span><span style="color: #000000;">formulae</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">])</span>
|
||||
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
|
||||
<!--
|
||||
|
|
@ -0,0 +1,71 @@
|
|||
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))
|
||||
|
|
@ -0,0 +1,6 @@
|
|||
#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))
|
||||
|
|
@ -0,0 +1,34 @@
|
|||
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
|
||||
|
|
@ -0,0 +1,40 @@
|
|||
/*REXX program evaluates some Machin─like formulas and verifies their veracity. */
|
||||
@.=; pi= pi(); numeric digits( length(pi) ) - length(.); numeric fuzz 3
|
||||
say center(' computing with ' digits() " decimal digits ", 110, '═')
|
||||
@.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 "Machin─like" 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. */
|
||||
/*──────────────────────────────────────────────────────────────────────────────────────*/
|
||||
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))
|
||||
/*──────────────────────────────────────────────────────────────────────────────────────*/
|
||||
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
|
||||
/*──────────────────────────────────────────────────────────────────────────────────────*/
|
||||
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
|
||||
|
|
@ -0,0 +1,56 @@
|
|||
#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)
|
||||
|
|
@ -0,0 +1,4 @@
|
|||
Do all correct formulas reduce to 1?
|
||||
#t
|
||||
The incorrect formula reduces to:
|
||||
1009288018000944050967896710431587186456256928584351786643498522649995492271475761189348270710224618853590682465929080006511691833816436374107451368838065354726517908250456341991684635768915704374493675498637876700129004484434187627909285979251682006538817341793224963346197503893270875008524149334251672855130857035205217929335932890740051319216343365800342290782260673215928499123722781078448297609548233999010983373327601187505623621602789012550584784738082074783523787011976757247516095289966708782862528690942242793667539020699840402353522108223/1009288837315638583415701528780402795721935641614456853534313491853293025565940011104051964874275710024625850092154664245109626053906509780125743180758231049920425664246286578958307532545458843067352531217230461290763258378749459637420702619029075083089762088232401888676895047947363883809724322868121990870409574061477638203859217672620508200713073485398199091153535700094640095900731630771349477187594074169815106104524371099618096164871416282464532355211521113449237814080332335526420331468258917484010722587072087349909684004660371264507984339711
|
||||
|
|
@ -0,0 +1,42 @@
|
|||
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]]
|
||||
);
|
||||
|
|
@ -0,0 +1,72 @@
|
|||
$ 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;
|
||||
|
|
@ -0,0 +1,62 @@
|
|||
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)
|
||||
}
|
||||
|
|
@ -0,0 +1,79 @@
|
|||
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"
|
||||
}
|
||||
}
|
||||
|
|
@ -0,0 +1,128 @@
|
|||
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
|
||||
|
|
@ -0,0 +1,72 @@
|
|||
import "/big" for BigRat
|
||||
import "/fmt" for Fmt
|
||||
|
||||
/** represents a term of the form: c * atan(n / d) */
|
||||
class Term {
|
||||
construct new(c, n, d) {
|
||||
_c = c
|
||||
_n = n
|
||||
_d = d
|
||||
}
|
||||
|
||||
c { _c }
|
||||
n { _n }
|
||||
d { _d }
|
||||
|
||||
toString {
|
||||
var a = "atan(%(n)/%(d))"
|
||||
return ((_c == 1) ? " + " :
|
||||
(_c == -1) ? " - " :
|
||||
(_c < 0) ? " - %(-c)*" : " + %(c)*") + a
|
||||
}
|
||||
}
|
||||
|
||||
var tanEval // recursive function
|
||||
tanEval = Fn.new { |c, f|
|
||||
if (c == 1) return f
|
||||
if (c < 0) return -tanEval.call(-c, f)
|
||||
var ca = (c/2).truncate
|
||||
var cb = c - ca
|
||||
var a = tanEval.call(ca, f)
|
||||
var b = tanEval.call(cb, f)
|
||||
return (a + b) / (BigRat.one - (a * b))
|
||||
}
|
||||
|
||||
var tanSum // recursive function
|
||||
tanSum = Fn.new { |terms|
|
||||
if (terms.count == 1) return tanEval.call(terms[0].c, BigRat.new(terms[0].n, terms[0].d))
|
||||
var half = (terms.count/2).floor
|
||||
var a = tanSum.call(terms.take(half).toList)
|
||||
var b = tanSum.call(terms.skip(half).toList)
|
||||
return (a + b) / (BigRat.one - (a * b))
|
||||
}
|
||||
|
||||
var T = Term // type alias
|
||||
|
||||
var termsList = [
|
||||
[T.new(1, 1, 2), T.new(1, 1, 3)],
|
||||
[T.new(2, 1, 3), T.new(1, 1, 7)],
|
||||
[T.new(4, 1, 5), T.new(-1, 1, 239)],
|
||||
[T.new(5, 1, 7), T.new(2, 3, 79)],
|
||||
[T.new(5, 29, 278), T.new(7, 3, 79)],
|
||||
[T.new(1, 1, 2), T.new(1, 1, 5), T.new(1, 1, 8)],
|
||||
[T.new(4, 1, 5), T.new(-1, 1, 70), T.new(1, 1, 99)],
|
||||
[T.new(5, 1, 7), T.new(4, 1, 53), T.new(2, 1, 4443)],
|
||||
[T.new(6, 1, 8), T.new(2, 1, 57), T.new(1, 1, 239)],
|
||||
[T.new(8, 1, 10), T.new(-1, 1, 239), T.new(-4, 1, 515)],
|
||||
[T.new(12, 1, 18), T.new(8, 1, 57), T.new(-5, 1, 239)],
|
||||
[T.new(16, 1, 21), T.new(3, 1, 239), T.new(4, 3, 1042)],
|
||||
[T.new(22, 1, 28), T.new(2, 1, 443), T.new(-5, 1, 1393), T.new(-10, 1, 11018)],
|
||||
[T.new(22, 1, 38), T.new(17, 7, 601), T.new(10, 7, 8149)],
|
||||
[T.new(44, 1, 57), T.new(7, 1, 239), T.new(-12, 1, 682), T.new(24, 1, 12943)],
|
||||
[T.new(88, 1, 172), T.new(51, 1, 239), T.new(32, 1, 682), T.new(44, 1, 5357), T.new(68, 1, 12943)],
|
||||
[T.new(88, 1, 172), T.new(51, 1, 239), T.new(32, 1, 682), T.new(44, 1, 5357), T.new(68, 1, 12944)]
|
||||
]
|
||||
|
||||
for (terms in termsList) {
|
||||
var f = Fmt.swrite("$-5s: 1 == tan(", tanSum.call(terms) == BigRat.one)
|
||||
System.write(f)
|
||||
System.write(terms[0].toString.skip(3).join())
|
||||
for (i in 1...terms.count) System.write(terms[i])
|
||||
System.print(")")
|
||||
}
|
||||
|
|
@ -0,0 +1,96 @@
|
|||
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
|
||||
];
|
||||
]
|
||||
Loading…
Add table
Add a link
Reference in a new issue