Differential Coefficients x = 0

54de664d-68ff-4d8a-80b4-8c38babd3ccd 5 4 5 decimal

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fx size CFd fo 1 el fx : i 2 size range fo i el fo i 1 - el x diff : for x fo fo : i 1 size range fo0 i el 0 fo i el eval : 11 line for fo0 51 line :

Coefficients of the Powers

fx size a1 CFP fo 1 el fx : i 2 size range fo i el fo i 1 - el x diff : for x fo fo : i 1 size 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 :

Continuous Fraction Coefficients

aa Sal Cffc conta 1 : i 1 aa length range aa i el 0 ≡ i 1 ≡ fxa conta el aa i el : conta conta 1 + : 21 line continue if fxa conta el aa 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 :

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 :

Continuous Fraction

ss cd x FracCon size ss length : i size 2 range i size ≡ (cd 2 el 0 ≡ (& (cd 3 el 0 ≠ (& Th ss i 1 - el eval x 2^ss i el eval / (+ : Th Th : 21 line i size ≡ (cd 2 el 0 ≠ (& (cd 3 el 0 ≡ (& Th ss i 1 - eval el eval x 2^ss i el eval / (+ : Th Th : 21 line i size ≠ (cd 2 el 0 ≡ (& (cd 3 el 0 ≠ (& Th ss i 1 - eval el 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 x 2^Th / (+ : Th Th : 21 line i 2 ≡ (cd 2 el 0 ≠ (& (cd 3 el 0 ≡ (& Th x Th / (: Th Th : 21 line i size ≡ (cd 2 el 0 ≠ (& (cd 3 el 0 ≠ (& Th ss i 1 - el eval x ss i el eval / (+ : Th Th : 21 line i size ≠ (cd 2 el 0 ≠ (& (cd 3 el 0 ≠ (& Th ss i 1 - eval el eval x Th / (+ : Th Th : 21 line continue if if if if if if if for Th Th : 31 line :

Caution: These algorithms have only been tested for

the functions of e ^x , Sine (x), Cosine (x), and Tangent (x)

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Size n 14 :

Function x f e x^:

Differential Coefficients s x f n CFd :

Coefficients of the Powers s1 x f n a CFP :

Continuous Fraction Coefficients s2 x f n a1 CFP Sal Cffc :

s 11111111111111141 mat

s1 11 0.5 0.1667 0.0417 0.0083 0.0014 0.0002 2.4802 105 -^* 2.7557 106 -^* 2.7557 107 -^* 2.5052 108 -^* 2.0877 109 -^* 1.6059 1010 -^* 141 mat

s2 112 - 3 - 252 - 7 - 292 - 11 - 213141 mat

Construction of the Continuous Fraction x frac_cont_E s2 s x FracCon :

Continuous Fraction x frac_cont_E 1 x 1 x 2 - x 3 - x 2 x 5 x 2 - x 6.99999999999998 - x 2.00000000000001 x 8.99999999999999 x 2 - x 10.9999999999999 - x 1.99999999999899 x 13.0000000000017 / + / + / + / + / + / + / + / + / + / + / + / + / +

Sanity check

x mE x f confrac x convert evalf maple :

x mE 1 x 1 x 2 - x 3 - x 21 x * 5 / + / + / + / + / +

1 frac_cont_E 2.71828182845903

1 exp 2.71828182845905

1 mE 2.71794871794872

n s s1 s2 Clear 1

n 14 :

x f x sin :

s x f n CFd :

s1 x f n a CFP :

s2 x f n a1 CFP Sal Cffc :

s 0101 - 0101 - 0101 - 01141 mat

s1 010 0.1667 - 0 0.0083 0 0.0002 - 0 2.7557 106 -^* 0 2.5052 108 -^* - 0 1.6059 1010 -^* 141 mat

s2 016 1.4286 - 62.3636 - 0.5647 456.1998 0.1744 - 81 mat

x frac_cont_Sin s2 s x FracCon :

x frac_cont_Sin x 1 x 2^6 x 2^1.42857142857143 - x 2^62.3636363636361 - x 2^0.564687581021522 x 2^456.199849737035 x 2^0.174428178140286 - / + / + / + / + / + / + /

Sanity check

x mSin x f confrac x convert evalf maple :

x mSin x 1 x 2^67 x 2^* 10 / - / + /

1 frac_cont_Sin 0.841470984807329

1 sin 0.841470984807896

1 mSin 0.841269841269841

n s s1 s2 Clear 1

n 14 :

x f x cos :

s x f n CFd :

s1 x f n a CFP :

s2 x f n a1 CFP Sal Cffc :

s 101 - 0101 - 0101 - 010141 mat

s1 10 0.5 - 0 0.0417 0 0.0014 - 0 2.4802 105 -^* 0 2.7557 107 -^* - 0 2.0877 109 -^* 0141 mat

s2 12 - 6 - 3.3333 29.0769 1.9096 - 139.1557 - 71 mat

x frac_cont_Cos s2 s x FracCon :

x frac_cont_Cos 1 x 2^2 - x 2^6 - x 2^3.33333333333333 x 2^29.0769230769232 x 2^1.90960451977401 - x 2^139.155663234404 - / + / + / + / + / + / +

Sanity check

x mCos x f confrac x convert evalf maple :

x mCos 1 x 2^2 - 1666666667 x 2^* 10000000000 / - / +

1 frac_cont_Cos 0.540302305855419

1 cos 0.54030230586814

1 mCos 0.538461538468639

n s s1 s2 Clear 1

n 11 :

x f x tan :

s x f n CFd :

s1 x f n a CFP :

s2 x f n a1 CFP Sal Cffc :

s 01020160272079360111 mat

s1 010 0.3333 0 0.1333 0 0.054 0 0.0219 0111 mat

s2 013 - 57 - 961 mat

x frac_cont_Tan s2 s x FracCon :

x frac_cont_Tan x 1 x 2^3 - x 2^4.99999999999999 x 2^7 - x 2^9.00000000000002 / + / + / + / + /

Sanity check

x mTan x f confrac x convert evalf maple :

x mTan x 1 x 2^3 - 1 x 2^* 5 / + / + /

1 frac_cont_Tan 1.55740740740741

1 tan 1.5574077246549

1 mTan 1.55555555555556