� >Coef
@� Calculate polynomial approximations to arbitrary functions
B� in an arbitrary range, using Chebyshev Forced Oscillation of
(4� Error method to derive a near minimax solution
2�
<� �
FE� "Please enter function to be approximated {eg SIN(x*PI/2)} ";f$
PV� "Approximation range [a,b] {eg [0,1]}"'"Please enter a ";a$;"Please enter b ";b$
Z<� "Please enter degree of approximation (1 to 100) ";n%'
da = � a$
nb = � b$
xu = 2/(b-a)
�v = (a+b)/(a-b)
�:� f(101), c(101), t(100, 100), p(100), s(100), t2(100)
�� i%=0 � n%+1
�) x = 0.5*((b-a)*�(i%*�/(n%+1))+(a+b))
� f(i%) = � f$
��
�� j%=0 � n%+1
�= � (j%�1) sum = (f(0)-f(n%+1))/2 � sum = (f(0)+f(n%+1))/2
� � i%=1 � n%
�$  sum += f(i%)*�(i%*j%*�/(n%+1))
� �
� c(j%) = sum/(n%+1)
� � j%<>0 c(j%) = 2*c(j%)

�

E = �(c(n%+1)/2)

H� '"Writing z = ";u;"*x + ";v;", polynomial approximation then is:"'

"� i%=0 � n%

,# � "T[";i%;"](z) *",c(i%);�27);

6 � i%<n% � "+" � �

@�

J� '"where T[0](z)   = 1;"

T�  "      T[1](z)   = z;"

^9�  "and   T[n+1](z) = 2zT[n](z)-T[n-1](z), for n>=1"'

h$� Now we need to gen the T polys

rC� note we use array t(i,j) to hold the coef of z^j in poly T[i]

|t(0,0)=1

�t(1,1)=1

�i%=1

�ȕ i%<n%

� t(i%+1,0) = -t(i%-1,0)

� � j% = 1 � i%+1

�*  t(i%+1,j%) = 2*t(i%,j%-1)-t(i%-1,j%)

� �

�
 i%+=1

��

�B� Then we expand them to give an approximating poly array p(j)

�8� where each element is the coef of z^j in that poly

�� i%=0 � n%

� sum = 0
 � j%=i% � n%
  sum += t(j%, i%)*c(j%)
 �
& p(i%) = sum
0�
:/� And now we make the substitution z = ux+v
DC� to get our final poly in argument x (poly stored in array t2)
Nt2(0)=p(0)+p(1)*v
Xt2(1)=p(1)*u
b
s(0)=v
l
s(1)=u
vi%=2
�
ȕ i%<=n%
� � j%=i% � 1 � -1
�  s(j%)=v*s(j%)+u*s(j%-1)
� �
� s(0)=v*s(0)
� � j%=0 � i%
�  t2(j%)+=p(i%)*s(j%)
� �
�
 i%+=1
��
�$� '"Which may also be written:"'
�� i%=0 � n%
�! � "x^";i%;"  *",t2(i%);�27);
 � i%<n% � "+" � �
�
maxerror = 0
 � x=a � b � (b-a)/1000
* y = t2(n%)
4 � i%=n%-1 � 0 � -1
>  y = y*x+t2(i%)
H �
R e = �(y - � f$)
\ � e>maxerror maxerror=e
f�
pE� ''"For this approximation an upper bound on error is ";maxerror
zY�   "Also we now know minimax error p (the error made by the best possible order ";n%
�$�   "polynomial) lies between:"'
�� E;"  and  ";maxerror;"."'
�<� "This indicates how close our approximation is to it."
�V� ''"Finally, if only need say 16 bit accuracy, note that our poly's coefficients"
�>�   "multiplied by 2^20 & written in hex are as follows:"'
�� i%=n% � 0 � -1
� y = t2(i%)
� � y<0 s$="-":y=-y � s$="+"
� y2 = y*1048576+0.5
�1 � "x^";i%;"  *",s$;�"00000000"+�~y2,8);�27);
� � i%>0 � "+" � �
��
�

00000000  0d 00 0a 0b f4 20 3e 43  6f 65 66 0d 00 14 40 f4  |..... >Coef...@.|
00000010  20 43 61 6c 63 75 6c 61  74 65 20 70 6f 6c 79 6e  | Calculate polyn|
00000020  6f 6d 69 61 6c 20 61 70  70 72 6f 78 69 6d 61 74  |omial approximat|
00000030  69 6f 6e 73 20 74 6f 20  61 72 62 69 74 72 61 72  |ions to arbitrar|
00000040  79 20 66 75 6e 63 74 69  6f 6e 73 0d 00 1e 42 f4  |y functions...B.|
00000050  20 69 6e 20 61 6e 20 61  72 62 69 74 72 61 72 79  | in an arbitrary|
00000060  20 72 61 6e 67 65 2c 20  75 73 69 6e 67 20 43 68  | range, using Ch|
00000070  65 62 79 73 68 65 76 20  46 6f 72 63 65 64 20 4f  |ebyshev Forced O|
00000080  73 63 69 6c 6c 61 74 69  6f 6e 20 6f 66 0d 00 28  |scillation of..(|
00000090  34 f4 20 45 72 72 6f 72  20 6d 65 74 68 6f 64 20  |4. Error method |
000000a0  74 6f 20 64 65 72 69 76  65 20 61 20 6e 65 61 72  |to derive a near|
000000b0  20 6d 69 6e 69 6d 61 78  20 73 6f 6c 75 74 69 6f  | minimax solutio|
000000c0  6e 0d 00 32 05 f4 0d 00  3c 07 eb 20 eb 0d 00 46  |n..2....<.. ...F|
000000d0  45 e8 20 22 50 6c 65 61  73 65 20 65 6e 74 65 72  |E. "Please enter|
000000e0  20 66 75 6e 63 74 69 6f  6e 20 74 6f 20 62 65 20  | function to be |
000000f0  61 70 70 72 6f 78 69 6d  61 74 65 64 20 7b 65 67  |approximated {eg|
00000100  20 53 49 4e 28 78 2a 50  49 2f 32 29 7d 20 22 3b  | SIN(x*PI/2)} ";|
00000110  66 24 0d 00 50 56 e8 20  22 41 70 70 72 6f 78 69  |f$..PV. "Approxi|
00000120  6d 61 74 69 6f 6e 20 72  61 6e 67 65 20 5b 61 2c  |mation range [a,|
00000130  62 5d 20 7b 65 67 20 5b  30 2c 31 5d 7d 22 27 22  |b] {eg [0,1]}"'"|
00000140  50 6c 65 61 73 65 20 65  6e 74 65 72 20 61 20 22  |Please enter a "|
00000150  3b 61 24 3b 22 50 6c 65  61 73 65 20 65 6e 74 65  |;a$;"Please ente|
00000160  72 20 62 20 22 3b 62 24  0d 00 5a 3c e8 20 22 50  |r b ";b$..Z<. "P|
00000170  6c 65 61 73 65 20 65 6e  74 65 72 20 64 65 67 72  |lease enter degr|
00000180  65 65 20 6f 66 20 61 70  70 72 6f 78 69 6d 61 74  |ee of approximat|
00000190  69 6f 6e 20 28 31 20 74  6f 20 31 30 30 29 20 22  |ion (1 to 100) "|
000001a0  3b 6e 25 27 0d 00 64 0c  61 20 3d 20 a0 20 61 24  |;n%'..d.a = . a$|
000001b0  0d 00 6e 0c 62 20 3d 20  a0 20 62 24 0d 00 78 0f  |..n.b = . b$..x.|
000001c0  75 20 3d 20 32 2f 28 62  2d 61 29 0d 00 82 13 76  |u = 2/(b-a)....v|
000001d0  20 3d 20 28 61 2b 62 29  2f 28 61 2d 62 29 0d 00  | = (a+b)/(a-b)..|
000001e0  8c 3a de 20 66 28 31 30  31 29 2c 20 63 28 31 30  |.:. f(101), c(10|
000001f0  31 29 2c 20 74 28 31 30  30 2c 20 31 30 30 29 2c  |1), t(100, 100),|
00000200  20 70 28 31 30 30 29 2c  20 73 28 31 30 30 29 2c  | p(100), s(100),|
00000210  20 74 32 28 31 30 30 29  0d 00 96 11 e3 20 69 25  | t2(100)..... i%|
00000220  3d 30 20 b8 20 6e 25 2b  31 0d 00 a0 29 20 78 20  |=0 . n%+1...) x |
00000230  3d 20 30 2e 35 2a 28 28  62 2d 61 29 2a 9b 28 69  |= 0.5*((b-a)*.(i|
00000240  25 2a af 2f 28 6e 25 2b  31 29 29 2b 28 61 2b 62  |%*./(n%+1))+(a+b|
00000250  29 29 0d 00 aa 11 20 66  28 69 25 29 20 3d 20 a0  |)).... f(i%) = .|
00000260  20 66 24 0d 00 b4 05 ed  0d 00 be 11 e3 20 6a 25  | f$.......... j%|
00000270  3d 30 20 b8 20 6e 25 2b  31 0d 00 c8 3d 20 e7 20  |=0 . n%+1...= . |
00000280  28 6a 25 80 31 29 20 73  75 6d 20 3d 20 28 66 28  |(j%.1) sum = (f(|
00000290  30 29 2d 66 28 6e 25 2b  31 29 29 2f 32 20 8b 20  |0)-f(n%+1))/2 . |
000002a0  73 75 6d 20 3d 20 28 66  28 30 29 2b 66 28 6e 25  |sum = (f(0)+f(n%|
000002b0  2b 31 29 29 2f 32 0d 00  d2 10 20 e3 20 69 25 3d  |+1))/2.... . i%=|
000002c0  31 20 b8 20 6e 25 0d 00  dc 24 20 20 73 75 6d 20  |1 . n%...$  sum |
000002d0  2b 3d 20 66 28 69 25 29  2a 9b 28 69 25 2a 6a 25  |+= f(i%)*.(i%*j%|
000002e0  2a af 2f 28 6e 25 2b 31  29 29 0d 00 e6 06 20 ed  |*./(n%+1)).... .|
000002f0  0d 00 f0 17 20 63 28 6a  25 29 20 3d 20 73 75 6d  |.... c(j%) = sum|
00000300  2f 28 6e 25 2b 31 29 0d  00 fa 1c 20 e7 20 6a 25  |/(n%+1).... . j%|
00000310  3c 3e 30 20 63 28 6a 25  29 20 3d 20 32 2a 63 28  |<>0 c(j%) = 2*c(|
00000320  6a 25 29 0d 01 04 05 ed  0d 01 0e 14 45 20 3d 20  |j%).........E = |
00000330  94 28 63 28 6e 25 2b 31  29 2f 32 29 0d 01 18 48  |.(c(n%+1)/2)...H|
00000340  f1 20 27 22 57 72 69 74  69 6e 67 20 7a 20 3d 20  |. '"Writing z = |
00000350  22 3b 75 3b 22 2a 78 20  2b 20 22 3b 76 3b 22 2c  |";u;"*x + ";v;",|
00000360  20 70 6f 6c 79 6e 6f 6d  69 61 6c 20 61 70 70 72  | polynomial appr|
00000370  6f 78 69 6d 61 74 69 6f  6e 20 74 68 65 6e 20 69  |oximation then i|
00000380  73 3a 22 27 0d 01 22 0f  e3 20 69 25 3d 30 20 b8  |s:"'..".. i%=0 .|
00000390  20 6e 25 0d 01 2c 23 20  f1 20 22 54 5b 22 3b 69  | n%..,# . "T[";i|
000003a0  25 3b 22 5d 28 7a 29 20  2a 22 2c 63 28 69 25 29  |%;"](z) *",c(i%)|
000003b0  3b 8a 32 37 29 3b 0d 01  36 16 20 e7 20 69 25 3c  |;.27);..6. . i%<|
000003c0  6e 25 20 f1 20 22 2b 22  20 8b 20 f1 0d 01 40 05  |n% . "+" . ...@.|
000003d0  ed 0d 01 4a 1d f1 20 27  22 77 68 65 72 65 20 54  |...J.. '"where T|
000003e0  5b 30 5d 28 7a 29 20 20  20 3d 20 31 3b 22 0d 01  |[0](z)   = 1;"..|
000003f0  54 1d f1 20 20 22 20 20  20 20 20 20 54 5b 31 5d  |T..  "      T[1]|
00000400  28 7a 29 20 20 20 3d 20  7a 3b 22 0d 01 5e 39 f1  |(z)   = z;"..^9.|
00000410  20 20 22 61 6e 64 20 20  20 54 5b 6e 2b 31 5d 28  |  "and   T[n+1](|
00000420  7a 29 20 3d 20 32 7a 54  5b 6e 5d 28 7a 29 2d 54  |z) = 2zT[n](z)-T|
00000430  5b 6e 2d 31 5d 28 7a 29  2c 20 66 6f 72 20 6e 3e  |[n-1](z), for n>|
00000440  3d 31 22 27 0d 01 68 24  f4 20 4e 6f 77 20 77 65  |=1"'..h$. Now we|
00000450  20 6e 65 65 64 20 74 6f  20 67 65 6e 20 74 68 65  | need to gen the|
00000460  20 54 20 70 6f 6c 79 73  0d 01 72 43 f4 20 6e 6f  | T polys..rC. no|
00000470  74 65 20 77 65 20 75 73  65 20 61 72 72 61 79 20  |te we use array |
00000480  74 28 69 2c 6a 29 20 74  6f 20 68 6f 6c 64 20 74  |t(i,j) to hold t|
00000490  68 65 20 63 6f 65 66 20  6f 66 20 7a 5e 6a 20 69  |he coef of z^j i|
000004a0  6e 20 70 6f 6c 79 20 54  5b 69 5d 0d 01 7c 0c 74  |n poly T[i]..|.t|
000004b0  28 30 2c 30 29 3d 31 0d  01 86 0c 74 28 31 2c 31  |(0,0)=1....t(1,1|
000004c0  29 3d 31 0d 01 90 08 69  25 3d 31 0d 01 9a 0c c8  |)=1....i%=1.....|
000004d0  95 20 69 25 3c 6e 25 0d  01 a4 1b 20 74 28 69 25  |. i%<n%.... t(i%|
000004e0  2b 31 2c 30 29 20 3d 20  2d 74 28 69 25 2d 31 2c  |+1,0) = -t(i%-1,|
000004f0  30 29 0d 01 ae 14 20 e3  20 6a 25 20 3d 20 31 20  |0).... . j% = 1 |
00000500  b8 20 69 25 2b 31 0d 01  b8 2a 20 20 74 28 69 25  |. i%+1...*  t(i%|
00000510  2b 31 2c 6a 25 29 20 3d  20 32 2a 74 28 69 25 2c  |+1,j%) = 2*t(i%,|
00000520  6a 25 2d 31 29 2d 74 28  69 25 2d 31 2c 6a 25 29  |j%-1)-t(i%-1,j%)|
00000530  0d 01 c2 06 20 ed 0d 01  cc 0a 20 69 25 2b 3d 31  |.... ..... i%+=1|
00000540  0d 01 d6 05 ce 0d 01 e0  42 f4 20 54 68 65 6e 20  |........B. Then |
00000550  77 65 20 65 78 70 61 6e  64 20 74 68 65 6d 20 74  |we expand them t|
00000560  6f 20 67 69 76 65 20 61  6e 20 61 70 70 72 6f 78  |o give an approx|
00000570  69 6d 61 74 69 6e 67 20  70 6f 6c 79 20 61 72 72  |imating poly arr|
00000580  61 79 20 70 28 6a 29 0d  01 ea 38 f4 20 77 68 65  |ay p(j)...8. whe|
00000590  72 65 20 65 61 63 68 20  65 6c 65 6d 65 6e 74 20  |re each element |
000005a0  69 73 20 74 68 65 20 63  6f 65 66 20 6f 66 20 7a  |is the coef of z|
000005b0  5e 6a 20 69 6e 20 74 68  61 74 20 70 6f 6c 79 0d  |^j in that poly.|
000005c0  01 f4 0f e3 20 69 25 3d  30 20 b8 20 6e 25 0d 01  |.... i%=0 . n%..|
000005d0  fe 0c 20 73 75 6d 20 3d  20 30 0d 02 08 11 20 e3  |.. sum = 0.... .|
000005e0  20 6a 25 3d 69 25 20 b8  20 6e 25 0d 02 12 1c 20  | j%=i% . n%.... |
000005f0  20 73 75 6d 20 2b 3d 20  74 28 6a 25 2c 20 69 25  | sum += t(j%, i%|
00000600  29 2a 63 28 6a 25 29 0d  02 1c 06 20 ed 0d 02 26  |)*c(j%).... ...&|
00000610  10 20 70 28 69 25 29 20  3d 20 73 75 6d 0d 02 30  |. p(i%) = sum..0|
00000620  05 ed 0d 02 3a 2f f4 20  41 6e 64 20 6e 6f 77 20  |....:/. And now |
00000630  77 65 20 6d 61 6b 65 20  74 68 65 20 73 75 62 73  |we make the subs|
00000640  74 69 74 75 74 69 6f 6e  20 7a 20 3d 20 75 78 2b  |titution z = ux+|
00000650  76 0d 02 44 43 f4 20 74  6f 20 67 65 74 20 6f 75  |v..DC. to get ou|
00000660  72 20 66 69 6e 61 6c 20  70 6f 6c 79 20 69 6e 20  |r final poly in |
00000670  61 72 67 75 6d 65 6e 74  20 78 20 28 70 6f 6c 79  |argument x (poly|
00000680  20 73 74 6f 72 65 64 20  69 6e 20 61 72 72 61 79  | stored in array|
00000690  20 74 32 29 0d 02 4e 15  74 32 28 30 29 3d 70 28  | t2)..N.t2(0)=p(|
000006a0  30 29 2b 70 28 31 29 2a  76 0d 02 58 10 74 32 28  |0)+p(1)*v..X.t2(|
000006b0  31 29 3d 70 28 31 29 2a  75 0d 02 62 0a 73 28 30  |1)=p(1)*u..b.s(0|
000006c0  29 3d 76 0d 02 6c 0a 73  28 31 29 3d 75 0d 02 76  |)=v..l.s(1)=u..v|
000006d0  08 69 25 3d 32 0d 02 80  0d c8 95 20 69 25 3c 3d  |.i%=2...... i%<=|
000006e0  6e 25 0d 02 8a 15 20 e3  20 6a 25 3d 69 25 20 b8  |n%.... . j%=i% .|
000006f0  20 31 20 88 20 2d 31 0d  02 94 1d 20 20 73 28 6a  | 1 . -1....  s(j|
00000700  25 29 3d 76 2a 73 28 6a  25 29 2b 75 2a 73 28 6a  |%)=v*s(j%)+u*s(j|
00000710  25 2d 31 29 0d 02 9e 06  20 ed 0d 02 a8 10 20 73  |%-1).... ..... s|
00000720  28 30 29 3d 76 2a 73 28  30 29 0d 02 b2 10 20 e3  |(0)=v*s(0).... .|
00000730  20 6a 25 3d 30 20 b8 20  69 25 0d 02 bc 19 20 20  | j%=0 . i%....  |
00000740  74 32 28 6a 25 29 2b 3d  70 28 69 25 29 2a 73 28  |t2(j%)+=p(i%)*s(|
00000750  6a 25 29 0d 02 c6 06 20  ed 0d 02 d0 0a 20 69 25  |j%).... ..... i%|
00000760  2b 3d 31 0d 02 da 05 ce  0d 02 e4 24 f1 20 27 22  |+=1........$. '"|
00000770  57 68 69 63 68 20 6d 61  79 20 61 6c 73 6f 20 62  |Which may also b|
00000780  65 20 77 72 69 74 74 65  6e 3a 22 27 0d 02 ee 0f  |e written:"'....|
00000790  e3 20 69 25 3d 30 20 b8  20 6e 25 0d 02 f8 21 20  |. i%=0 . n%...! |
000007a0  f1 20 22 78 5e 22 3b 69  25 3b 22 20 20 2a 22 2c  |. "x^";i%;"  *",|
000007b0  74 32 28 69 25 29 3b 8a  32 37 29 3b 0d 03 02 16  |t2(i%);.27);....|
000007c0  20 e7 20 69 25 3c 6e 25  20 f1 20 22 2b 22 20 8b  | . i%<n% . "+" .|
000007d0  20 f1 0d 03 0c 05 ed 0d  03 16 10 6d 61 78 65 72  | ..........maxer|
000007e0  72 6f 72 20 3d 20 30 0d  03 20 1a e3 20 78 3d 61  |ror = 0.. .. x=a|
000007f0  20 b8 20 62 20 88 20 28  62 2d 61 29 2f 31 30 30  | . b . (b-a)/100|
00000800  30 0d 03 2a 0f 20 79 20  3d 20 74 32 28 6e 25 29  |0..*. y = t2(n%)|
00000810  0d 03 34 17 20 e3 20 69  25 3d 6e 25 2d 31 20 b8  |..4. . i%=n%-1 .|
00000820  20 30 20 88 20 2d 31 0d  03 3e 14 20 20 79 20 3d  | 0 . -1..>.  y =|
00000830  20 79 2a 78 2b 74 32 28  69 25 29 0d 03 48 06 20  | y*x+t2(i%)..H. |
00000840  ed 0d 03 52 14 20 65 20  3d 20 94 28 79 20 2d 20  |...R. e = .(y - |
00000850  a0 20 66 24 29 0d 03 5c  1c 20 e7 20 65 3e 6d 61  |. f$)..\. . e>ma|
00000860  78 65 72 72 6f 72 20 6d  61 78 65 72 72 6f 72 3d  |xerror maxerror=|
00000870  65 0d 03 66 05 ed 0d 03  70 45 f1 20 27 27 22 46  |e..f....pE. ''"F|
00000880  6f 72 20 74 68 69 73 20  61 70 70 72 6f 78 69 6d  |or this approxim|
00000890  61 74 69 6f 6e 20 61 6e  20 75 70 70 65 72 20 62  |ation an upper b|
000008a0  6f 75 6e 64 20 6f 6e 20  65 72 72 6f 72 20 69 73  |ound on error is|
000008b0  20 22 3b 6d 61 78 65 72  72 6f 72 0d 03 7a 59 f1  | ";maxerror..zY.|
000008c0  20 20 20 22 41 6c 73 6f  20 77 65 20 6e 6f 77 20  |   "Also we now |
000008d0  6b 6e 6f 77 20 6d 69 6e  69 6d 61 78 20 65 72 72  |know minimax err|
000008e0  6f 72 20 70 20 28 74 68  65 20 65 72 72 6f 72 20  |or p (the error |
000008f0  6d 61 64 65 20 62 79 20  74 68 65 20 62 65 73 74  |made by the best|
00000900  20 70 6f 73 73 69 62 6c  65 20 6f 72 64 65 72 20  | possible order |
00000910  22 3b 6e 25 0d 03 84 24  f1 20 20 20 22 70 6f 6c  |";n%...$.   "pol|
00000920  79 6e 6f 6d 69 61 6c 29  20 6c 69 65 73 20 62 65  |ynomial) lies be|
00000930  74 77 65 65 6e 3a 22 27  0d 03 8e 1f f1 20 45 3b  |tween:"'..... E;|
00000940  22 20 20 61 6e 64 20 20  22 3b 6d 61 78 65 72 72  |"  and  ";maxerr|
00000950  6f 72 3b 22 2e 22 27 0d  03 98 3c f1 20 22 54 68  |or;"."'...<. "Th|
00000960  69 73 20 69 6e 64 69 63  61 74 65 73 20 68 6f 77  |is indicates how|
00000970  20 63 6c 6f 73 65 20 6f  75 72 20 61 70 70 72 6f  | close our appro|
00000980  78 69 6d 61 74 69 6f 6e  20 69 73 20 74 6f 20 69  |ximation is to i|
00000990  74 2e 22 0d 03 a2 56 f1  20 27 27 22 46 69 6e 61  |t."...V. ''"Fina|
000009a0  6c 6c 79 2c 20 69 66 20  6f 6e 6c 79 20 6e 65 65  |lly, if only nee|
000009b0  64 20 73 61 79 20 31 36  20 62 69 74 20 61 63 63  |d say 16 bit acc|
000009c0  75 72 61 63 79 2c 20 6e  6f 74 65 20 74 68 61 74  |uracy, note that|
000009d0  20 6f 75 72 20 70 6f 6c  79 27 73 20 63 6f 65 66  | our poly's coef|
000009e0  66 69 63 69 65 6e 74 73  22 0d 03 ac 3e f1 20 20  |ficients"...>.  |
000009f0  20 22 6d 75 6c 74 69 70  6c 69 65 64 20 62 79 20  | "multiplied by |
00000a00  32 5e 32 30 20 26 20 77  72 69 74 74 65 6e 20 69  |2^20 & written i|
00000a10  6e 20 68 65 78 20 61 72  65 20 61 73 20 66 6f 6c  |n hex are as fol|
00000a20  6c 6f 77 73 3a 22 27 0d  03 b6 14 e3 20 69 25 3d  |lows:"'..... i%=|
00000a30  6e 25 20 b8 20 30 20 88  20 2d 31 0d 03 c0 0f 20  |n% . 0 . -1.... |
00000a40  79 20 3d 20 74 32 28 69  25 29 0d 03 ca 1f 20 e7  |y = t2(i%).... .|
00000a50  20 79 3c 30 20 73 24 3d  22 2d 22 3a 79 3d 2d 79  | y<0 s$="-":y=-y|
00000a60  20 8b 20 73 24 3d 22 2b  22 0d 03 d4 17 20 79 32  | . s$="+".... y2|
00000a70  20 3d 20 79 2a 31 30 34  38 35 37 36 2b 30 2e 35  | = y*1048576+0.5|
00000a80  0d 03 de 31 20 f1 20 22  78 5e 22 3b 69 25 3b 22  |...1 . "x^";i%;"|
00000a90  20 20 2a 22 2c 73 24 3b  c2 22 30 30 30 30 30 30  |  *",s$;."000000|
00000aa0  30 30 22 2b c3 7e 79 32  2c 38 29 3b 8a 32 37 29  |00"+.~y2,8);.27)|
00000ab0  3b 0d 03 e8 15 20 e7 20  69 25 3e 30 20 f1 20 22  |;.... . i%>0 . "|
00000ac0  2b 22 20 8b 20 f1 0d 03  f2 05 ed 0d ff           |+" . ........|
00000acd