math functions Taylor Series of functions can be used to implement math routines not available in the programming language you are using. Some will require range reduction of X iterate(sqrt(x))= Y = X/2 loop 20 times Y = ((X/Y)+Y)/2 end loop could exit if |(Y-previous Y)| < 1.0E-16 return Y very accurate sqrt(x) taylor(sqrt(1+x),x,0,9)= 2 3 4 5 6 7 8 9 X X X 5 X 7 X 21 X 33 X 429 X 715 X 1 + - - -- + -- - ---- + ---- - ----- + ----- - ------ + ------ + . . . 2 8 16 128 256 1024 2048 32768 65536 taylor(exp(x),x,0,10)= 2 3 4 5 6 7 8 9 10 X X X X X X X X X 1 + X + -- + -- + -- + --- + --- + ---- + ----- + ------ + ------- . . . 2 6 24 120 720 5040 40320 362880 3628800 2! 3! 4! 5! 6! 7! 8! 9! 10! taylor(log(1+x),x,0,7)= 2 3 4 5 6 7 X X X X X X X - -- + -- - -- + -- - -- + -- + . . . 2 3 4 5 6 7 taylor(log(x),x,1,7)= 2 3 4 5 6 7 (X - 1) (X - 1) (X - 1) (X - 1) (X - 1) (X - 1) X - 1 - -------- + -------- - -------- + -------- - -------- + -------- 2 3 4 5 6 7 taylor(sin(x),x,0,13)= needs x range reduction -pi/2 <= x <= pi/2 3 5 7 9 11 13 X X X X X X X - -- + --- - ---- + ------ - -------- + ---------- . . . 6 120 5040 362880 39916800 6227020800 3! 5! 7! 9! 11! 13! taylor(cos(x),x,0,12)= needs x range reduction -pi/2 <= x <= pi/2 2 4 6 8 10 12 X X X X X X 1 - -- + -- - --- + ----- - ------- + --------- . . . 2 24 720 40320 3628800 479001600 2! 4! 6! 8! 10! 12! taylor(tan(x),x,0,9)= or sin(x)/cos(x) 3 5 7 9 X 2 X 17 X 62 X X + -- + ---- + ----- + ----- + . . . 3 15 315 2835 taylor(tan(x+%pi/2),x,0,5)= 3 5 1 X X 2 X - - + - + -- + ---- + . . . X 3 45 945 taylor(cot(x),x,0,9)= or 1/tan(x) or cos(x)/sin(x) 3 5 7 9 1 X X 2 X X 2 X - - - - -- - ---- - ---- - ----- + . . . X 3 45 945 4725 93555 taylor(cot(%PI/2+x),x,0,9)= 3 5 7 9 X 2 X 17 X 62 X - X - -- - ---- - ----- - ----- + . . . 3 15 315 2835 taylor(cot(x+%pi),x,0,5)= 3 5 1 X X 2 X - - - - -- - ---- + . . . X 3 45 945 taylor(asin(x),x,0,9)= 3 5 7 9 X 3 X 5 X 35 X X + -- + ---- + ---- + ----- + . . . 6 40 112 1152 3 5 7 9 X 3 X 3*5 X 3*5*7 X X + ---- + ----- + ------- + --------- + . . . 2*3 2*4*5 2*4*6*7 2*4*6*8*9 taylor(asin(1-x),x,0,9)= 3/2 5/2 %PI SQRT(2) X (3 SQRT(2)) X --- - SQRT(2) SQRT(X) - ------------ - ---------------- 2 12 160 taylor(acos(x),x,0,9)= or pi/2 - asin(x) 3 5 7 9 %PI X 3 X 5 X 35 X --- - X - -- - ---- - ---- - ----- + . . . 2 6 40 112 1152 taylor(atan(x),x,0,9)= 3 5 7 9 X X X X |X| < 1 X - -- + -- - -- + -- + . . . 3 5 7 9 taylor(atan(x),x,0,7)= pi 1 1 1 1 X <= -1 - -- - - + --- - --- + --- . . . 2 X 3 5 7 3X 5X 7X taylor(atan(x),x,0,7)= pi 1 1 1 1 X >= 1 -- - - + --- - --- + --- . . . 2 X 3 5 7 3X 5X 7X taylor(atan(x+1),x,0,9)= 2 3 5 6 7 9 %PI X X X X X X X --- + - - -- + -- - -- + -- - --- + --- + . . . 4 2 4 12 40 48 112 288 taylor(atan(x-1),x,0,6)= 2 3 5 6 %PI X X X X X - --- + - + -- + -- - -- - -- + . . . 4 2 4 12 taylor(acot(x),x,0,9)= or 1/atan(x) Is X positive or negative? positive; 3 5 7 9 %PI X X X X --- - X + -- - -- + -- - -- + . . . 2 3 5 7 9 taylor(acot(x),x,0,9)= Is X positive or negative? negative; 3 5 7 9 %PI X X X X - --- - X + -- - -- + -- - -- + . . . 2 3 5 7 9 taylor(acot(1+x),x,0,9)= 2 3 5 6 7 9 %PI X X X X X X X --- - - + -- - -- + -- - -- + --- - --- + . . . 4 2 4 12 40 48 112 288 taylor(acot(x-1),x,0,9)= 2 3 5 6 7 9 %PI X X X X X X X - --- - - - -- - -- + -- + -- + --- - --- + . . . 4 2 4 12 40 48 112 288 taylor(acot(1/x),x,0,5)= 3 5 X X X - -- + -- + . . . 3 5 taylor(sinh(x),x,0,9)= or (exp(x)-exp(-x))/2 3 5 7 9 X X X X X + -- + --- + ---- + ------ + . . . 6 120 5040 362880 3! 5! 7! 9! taylor(cosh(x),x,0,9)= or (exp(x)+exp(-x))/2 2 4 6 8 X X X X 1 + -- + -- + --- + ----- + . . . 2 24 720 40320 2! 4! 6! 8! taylor(tanh(x),x,0,9)= or sinh(x)/cosh(x) 3 5 7 9 X 2 X 17 X 62 X X - -- + ---- - ----- + ----- + . . . 3 15 315 2835 taylor(coth(x),x,0,9)= or 1/tanh(x) 3 5 7 9 1 X X 2 X X 2 X - + - - -- + ---- - ---- + ----- + . . . X 3 45 945 4725 93555 taylor(coth(1/x),x,0,5); TAYLOR encountered an essential singularity in above taylor(asinh(x),x,0,9)= 3 5 7 9 X 3 X 5 X 35 X X - -- + ---- - ---- + ----- + . . . 6 40 112 1152 taylor(asinh(x-1),x,0,3)= 2 3 SQRT(2) X SQRT(2) X SQRT(2) X LOG(SQRT(2) - 1) + --------- + ---------- + ---------- + . . . 2 8 48 taylor(asinh(1/x),x,0,4)= 2 4 2 X 3 X LOG(-) + -- - ---- + . . . X 4 32 taylor(acosh(x),x,0,9)= 3 5 7 9 %I %PI %I X (3 %I) X (5 %I) X (35 %I) X - ------ + %I X + ----- + --------- + --------- + ---------- + . . . 2 6 40 112 1152 taylor(acosh(1+x),x,0,3)= 3/2 5/2 SQRT(2) X (3 SQRT(2)) X SQRT(2) SQRT(X) - ------------ + ---------------- + . . . 12 160 taylor(atanh(x),x,0,9)= 3 5 7 9 X X X X X + -- + -- + -- + -- + . . . 3 5 7 9 taylor(atanh(1+x),x,0,5)= 2 LOG(- -) 2 3 4 5 X X X X X X -------- + - - -- + -- - --- + --- + . . . 2 4 16 48 128 320 taylor(atanh(x-1),x,0,5)= X LOG(-) 2 3 4 5 2 X X X X X ------ + - + -- + -- + --- + --- + . . . 2 4 16 48 128 320 taylor(acoth(1+x),x,0,9)= 2 LOG(-) 2 3 4 5 6 7 8 9 X X X X X X X X X X ------ + - - -- + -- - --- + --- - --- + ---- - ---- + ---- + . . . 2 4 16 48 128 320 768 1792 4096 9216 taylor(acoth(x-1),x,0,9)= X LOG(- -) 2 3 4 5 6 7 8 9 2 X X X X X X X X X -------- + - + -- + -- + --- + --- + --- + ---- + ---- + ---- + . . . 2 4 16 48 128 320 768 1792 4096 9216 taylor(acoth(1/x),x,0,5)= 3 5 X X X + -- + -- + . . . 3 5