<p fontName="Arial">Continuous Fraction algorithm</p>

3917ff8f-23b5-4bfa-9243-028c8702d72f 18 4 5 true true 0 decimal

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fx n x FracCon1 fx size CFd fo 1 el fx : i 2 n range fo i el fo i 1 - el x diff : for x fo fo : i 1 n range fo0 i el 0 fo i el eval : 11 line for fo0 51 line : s fx n CFd : fx n a1 CFP fo 1 el fx : i 2 n range fo i el fo i 1 - el x diff : for x fo fo : i 1 n range fo0 i el 0 fo i el eval : fo1 i el fo0 i el 1 i 1 - (! / * : 21 line for a1 fo1 : 51 line : s1 fx n a1 CFP : ac b a f b 0 ≠ i 1 b range cont ac : j 1 b range d i j el a cont 1 + el : cont cont 1 + : 21 line for ac ac 1 + : 31 line for d 111 mat : if d det 21 line : s1 Sal Cffc conta 1 : i 1 s1 length range s1 i el 0 ≡ i 1 ≡ fxa conta el s1 i el : conta conta 1 + : 21 line continue if fxa conta el s1 i el : conta conta 1 + : 21 line if 11 line for C1 1 el fxa 1 el : C1 2 el 1 fxa 2 el / : 12 mat i 3 fxa length range it i 1 - : it 2 mod 0 ≡ C1 i el 1 it 2 / fxa f (2^2 it 2 / 1 - fxa f 2 it 2 / fxa f * / - : C1 i el 2 it 1 - 2 / fxa f (2^1 it 1 - 2 / fxa f 11 i 1 - 2 / + fxa f * / : if 21 line for Sal C1 : 51 line : cd s : ss s1 Sal Cffc : size ss length : i size 2 range i size ≡ (cd 2 el 0 ≡ (& (cd 3 el 0 ≠ (& Th ss i 1 - eval el eval 4 round eval x 2^ss i el eval 4 round eval / (+ : Th Th : 21 line i size ≡ (cd 2 el 0 ≠ (& (cd 3 el 0 ≡ (& Th ss i 1 - eval el eval 4 round eval x 2^ss i el eval 4 round eval / (+ : Th Th : 21 line i size ≠ (cd 2 el 0 ≡ (& (cd 3 el 0 ≠ (& Th ss i 1 - eval el eval 4 round eval x 2^Th / (+ : Th Th : 21 line i size ≠ (i 2 ≠ (& (cd 2 el 0 ≠ (& (cd 3 el 0 ≡ (& Th ss i 1 - eval el eval 4 round eval x 2^Th / (+ : Th Th : 21 line i 2 ≡ (cd 2 el 0 ≠ (& (cd 3 el 0 ≡ (& (cd 1 el 0 ≡ (& Th x Th / (: Th Th : 21 line i 2 ≡ (cd 2 el 0 ≠ (& (cd 3 el 0 ≡ (& (cd 1 el 0 ≠ (& Th ss i 1 - eval el eval 4 round eval x Th / (+ : Th Th : 21 line i size ≡ (cd 2 el 0 ≠ (& (cd 3 el 0 ≠ (& Th ss i 1 - eval el eval 4 round eval x ss i el 4 round eval / (+ : Th Th : 21 line i size ≠ (cd 2 el 0 ≠ (& (cd 3 el 0 ≠ (& Th ss i 1 - eval el eval 4 round eval x Th / (+ : Th Th : 21 line continue if if if if if if if if for Th Th : 31 line 91 line :

Size n 12 :

Optimiz OFF [None]

set 'n' the length of expansion

x sin n x FracCon1 x 1 x 2^6 x 2^1.4286 - x 2^62.3636 - x 2^0.5647 x 2^456.1998 / + / + / + / + / + /

x g x sin confrac x n convert maple : x 1 x 2^6 x 2^107 / - x 2^68611 / - x 2^21783857 / 1331 x 2^* 607202 / + / + / + / + / + /

x cos n x FracCon1 1 x 2^2 - x 2^6 - x 2^3.3333 x 2^29.0769 x 2^1.9096 - / + / + / + / + / +

x g x cos confrac x n convert maple : 1 x 2^2 - x 2^6 - x 2^103 / x 2^37813 / 177 x 2^* 338 / - / + / + / + / +

x tan 5 x FracCon1 x 1 x 2^3 - / + /

x g x tan confrac x 5 convert maple : x 1 x 2^3 / - /

x exp n x FracCon1 1 x 1 x 2 - x 3 - x 2 x 5 x 2 - x 7 - x 2 x 9 x 2 - x 11 - / + / + / + / + / + / + / + / + / + / + / +

x g x exp confrac x n convert maple : 1 x 1 x 2 - x 3 - x 2 x 5 x 2 - x 7 - x 2 x 9 x 2 - x 11 / - / + / + / + / + / + / + / + / + / + / +

x atan n x FracCon1 x 1 x 2^3 x 2^1.25 x 2^3.1111 x 2^1.2656 x 2^3.1289 / + / + / + / + / + /

x Clear

Thiele continued fraction x g x atan confrac x n convert maple : (x 1 x 2^3 x 2^54 / x 2^289 / x 2^8164 / 225 x 2^* 704 / + / + / + / + / + / ≡

x asin 8 x FracCon1 x 1 x 2^6 - x 2^0.5882 x 2^7.3698 - / + / + / + /

x g x asin confrac x 6 convert maple : (x 1 x 2^6 - x 2^1017 / 549 x 2^* 4046 / - / + / + / ≡

x acos 10 x FracCon1 1.5708 x 1 - x 2^6 x 2^0.5882 - x 2^7.3698 x 2^0.5135 - / + / + / + / + / +

x g x acos confrac x n convert maple : (π 2 / x 1 - x 2^6 x 2^1017 / - x 2^4046549 / x 2^6028021173833 / - 12889271025 x 2^* 104890816822 / + / + / + / + / + / + ≡

x LnC 1 x + ln n x FracCon1 :

x LnC 0 x 1 x 2 x 3 x 1 x 5 x 0.6667 x 7 x 0.5 x 9 x 0.4 x 11 / + / + / + / + / + / + / + / + / + / + / +

x Ln 1 x + ln confrac x n convert maple : x 1 x 2 x 3 x 1 x 5 x 23 / x 7 x 12 / x 9 x 25 / x 11 / + / + / + / + / + / + / + / + / + / + /

x f x sqrt :

x SQRTC 1 x + f n x FracCon1 :

x SQRTC 1 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 / + / + / + / + / + / + / + / + / + / + / +

Thiele continued fraction x SQRT 1 x + f confrac x n convert maple : 1 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 / + / + / + / + / + / + / + / + / + / + / +

x λ 10 x ≤ (x 4 ≤ (& cases :

x 1 - Ln x λ * x ln 2 1 sys x 1 - SQRT x λ * x sqrt 2 1 sys x 1 - LnC x λ * x ln 2 1 sys x 1 - SQRTC x λ * x sqrt 2 1 sys