New Trigonometric Functions Program for the New 12C Platinum



#4

Here is a new trigonometric functions program for the new 12C Platinum. Are there another 
designed specifically for this calculator? This is based on MiniMax Polynomial Approximation
(Thanks to Valentin Albillo who's shed some light on the subject) and drastic range
reductions ([0..pi/12] for arctangent function). The program is focused on accuracy and easy
of use. Arguments are entered in degrees and there is an entry-point for each function.
There are neither constants to be previously stored nor initialization routines. Placing the
constants directly into the program rather than recalling them from registers slows down
program execution but considering the new 12C Platinum is faster, this should not be a
problem. Actually this program has not been tested on the new 12C Platinum. Instead, it has
been tested with a very close version suitable for the 15C. Hopefully, I have made no
mistakes in adapting it to the Platinum. I just hope the new Platinum is at least five times
faster than the golden 12C so no function takes longer than two to three seconds to run.
Cos(x) is calculated as Sin(90-x) for the sake of accuracy. As the program is more than 255
lines long it will not work on the old Platinum unless some modifications are made (such as
replacing the last two or three constants in the program for registers recalls).

The accuracy is comparable with that of the HP-35. Cos(x) is calculated as Sin(90-x) for the
sake of accuracy. The program gives at least nine significant figures, many times matching
the 15C results. Also, the stack register X is always saved. So, the following expression

asin(acos(atan(tan(cos(sen(9))))))

may be evaluated as:

9 R/S GTO 090 R/S GTO 100 R/S GTO 153 R/S GTO 137 R/S GTO 119 R/S

The result in the 15C with the equivalent program is 8.999661629

That's all for the while,

Gerson.


TRIGONOMETRIC FUNCTIONS ON THE 12C PLATINUM

001 2 055 . 109 x<>y 163 1 217 1
002 STO 1 056 8 110 RCL 4 164 g LASTx 218 4
003 Rv 057 1 111 GTO 093 165 g x<=y 219 2
004 x<>y 058 7 112 x<>y 166 GTO 169 220 8
005 STO 4 059 7 113 RCL 4 167 1/x 221 3
006 x<>y 060 6 114 x<>y 168 9 222 7
007 STO 2 061 4 115 / 169 0 223 9
008 g x^2 062 1 116 RCL 3 170 STO 1 224 6
009 ENTER 063 7 117 x<>y 171 x<>y 225 -
010 ENTER 064 3 118 GTO 000 172 2 226 *
011 ENTER 065 EEX 119 x<>y 173 ENTER 227 .
012 4 066 3 120 STO 4 174 3 228 1
013 . 067 CHS 121 x<>y 175 SQRT 229 9
014 4 068 + 122 ENTER 176 STO 3 230 9
015 5 069 RCL 2 123 g x^2 177 - 231 9
016 0 070 * 124 1 178 x<>y 232 9
017 2 071 ENTER 125 - 179 g x<=y 233 9
018 CHS 072 g x^2 126 g x=0 180 GTO 192 234 8
019 EEX 073 4 127 GTO 132 181 ENTER 235 3
020 2 074 * 128 CHS 182 ENTER 236 2
021 0 075 CHS 129 SQRT 183 RCL 3 237 +
022 CHS 076 3 130 / 184 * 238 *
023 * 077 + 131 GTO 156 185 1 239 3
024 5 078 * 132 x<>y 186 - 240 1/x
025 . 079 RCL 1 133 9 187 x<>y 241 -
026 5 080 2 134 0 188 RCL 3 242 *
027 5 081 g x<=y 135 * 189 + 243 1
028 3 082 GTO 087 136 GTO 268 190 / 244 +
029 8 083 Rv 137 x<>y 191 3 245 RCL 3
030 3 084 g x=0 138 STO 4 192 0 246 *
031 9 085 GTO 112 139 x<>y 193 STO 2 247 5
032 EEX 086 GTO 107 140 g x=0 194 x<>y 248 7
033 1 087 Rv 141 GTO 150 195 STO 3 249 .
034 4 088 x<>y 142 SQRT 196 g x^2 250 2
035 CHS 089 GTO 268 143 g x^2 197 ENTER 251 9
036 + 090 2 144 g x^2 198 ENTER 252 5
037 * 091 STO 1 145 1/x 199 ENTER 253 7
038 3 092 Rv 146 1 200 . 254 7
039 . 093 g x^2 147 - 201 0 255 9
040 2 094 SQRT 148 SQRT 202 7 256 5
041 8 095 CHS 149 GTO 156 203 8 257 1
042 1 096 9 150 9 204 4 258 *
043 8 097 0 151 0 205 CHS 259 RCL 2
044 3 098 + 152 GTO 268 206 * 260 +
045 7 099 GTO 004 153 x<>y 207 . 261 RCL 1
046 5 100 STO 4 154 STO 4 208 1 262 g x=0
047 8 101 x<>y 155 x<>y 209 1 263 g x<>y
048 1 102 STO 3 156 ENTER 210 0 264 g x<>y
049 EEX 103 1 157 g x^2 211 3 265 -
050 8 104 STO 1 158 SQRT 212 5 266 RCL 0
051 CHS 105 RCL 4 159 g x=0 213 1 267 *
052 - 106 GTO 007 160 GTO 268 214 + 268 RCL 4
053 * 107 CLx 161 / 215 * 269 x<>y
054 5 108 STO 1 162 STO 0 216 . 270 GTO 000


SIN: R/S (-90 <= x <= 90)
COS: GTO 090 R/S (-180 <= x <= 180)
TAN: GTO 100 R/S (-90 <= x <= 90)
ASIN: GTO 119 R/S (-1 <= x <= 1)
ACOS: GTO 137 R/S ( 0 <= x <= 1) (just a program limitation!)
ATAN: GTO 153 R/S (-9.99...E49 <= x <= 9.99...E49)



#5

Hi Gerson, yes it looks like it will indeed run on the new 12c pt. The new one is not 5 times faster than the golden one ;-) But its accuracy may surprise you as it seems to have 12 sig. digits under the hood. 3 [1/x] shows 0.333333333 but then if we remove 6 of the 3s with .333333 [-] and multiply by E6 we see 0.333333000 - another 6 3s.
Cheers,
Tony


#6

Hi Tony,

Thanks for the good news! When I was adjusting the MiniMax coefficients for the arctangent
function (the lowest power coefficients don't require so many significant figures), I considered
1/3 as 0.333333333333 (that's the constant in line 239) in my test spreadsheet. I correctly
guessed the 12C Platinum might have some extra guarding digits. By the way, that constant
should be 0.333333333089303 or 0.3333333331 to ten places but I wouldn't write it this way
just because of a '1' in the leftmost position. So I used 1/3 to 12 places and adjusted the
other constants with help of a spreadsheet and a graphics.

Using the constants explicitly in the program have significantly slowed down the execution time
as more steps have to be run. The constants could have been stores in registers, but then an
initialization routine would have been needed to avoid having to enter them by hand. Anyway, I
haven't calculated whether there would have been free registers left since the program uses
five registers already. In short, the gain in speed obtained by using only three coefficients
in the sine aproximation and only four in the arctangent approximating is lost when the constants
are built into the program. But the easy of use may compensate for this. Notice that the constants
beginning in lines 54 and 247 are pi/540 and 180/pi, respectively.

Reading again my post, I realized the example I provided was out of context. What I meant is that,
calculations like the following are easily done, since the latest computation is saved on the stack:

((sin(60) + tan(30)) * 6/5) ^ 2 :

60 R/S 30 GTO 100 R/S + 6 * 5 / g x^2 => 3.000000001

Thanks again for your remarks.

Cheers,

Gerson.


Correction:

Checking again my test spreadsheet, I discovered I had approximated 1/3 to only 10 significant figures, although I had previously thought of using 12 digits. As a consequence, the '2' in line 236 should be a '3'. Anyway, '2' implies in a maximum absolute error of 5.15E-12 for arguments between 0 and 1, while '3' brings the maximum error down to 4.10E-12.

-------------------------------------------------------------
In the tables below, the HP-15C column shows results obtained
with the built-in HP-15 functions, all of them correct to 10
significant figures, whereas the 12C Platinum shows results
obtained with the equivalent program run on the HP-15C. According
to Tony observations, the results on the real Platinum should
vary slightly, hopefully for better. The HP-35 shows the results
obtained on a bugless HP-35 (version 3). Like the program, its
only angular mode was Degrees.


Sin(x):

x (deg) HP-15C 12C Platinum HP-35
---------------------------------------------------------------
0.00000 0.0000000000 0.0000000000 0.0000000000
0.00001 1.745329252E-07 1.745329252E-07 1.745000000E-07
0.00011 1.919862177E-06 1.919862177E-06 1.919800000E-06
0.02200 3.839724260E-04 3.839724261E-04 3.839723910E-04
3.330000 5.808674960E-02 5.808674960E-02 5.808674960E-02
14.44000 0.2493660251 0.2493660251 0.2493660250
25.55000 0.4312985870 0.4312985869 0.4312985871
36.66000 0.5970652564 0.5970652561 0.5970652561
47.77000 0.7404527827 0.7404527825 0.7404527828
58.88000 0.8560867283 0.8560867282 0.8560867285
69.99000 0.9396329127 0.9396329129 0.9396329127
81.11000 0.9879868528 0.9879868528 0.9879868527
88.88000 0.9998089500 0.9998089502 0.9998089499
89.99000 0.9999999848 0.9999999850 0.9999999848
89.99990 1.0000000000 0.9999999998 1.0000000000
90.00000 1.0000000000 1.0000000000 1.0000000000

Tan(x):

x (deg) HP-15C 12C Platinum HP-35
---------------------------------------------------------------
0.00000 0.0000000000 0.0000000000 0.0000000000
0.00001 1.745329252E-07 1.745329252E-07 1.745000000E-07
0.00011 1.919862177E-06 1.919862177E-06 1.919800000E-06
0.02200 3.839724543E-04 3.839724543E-04 3.839724542E-04
3.330000 5.818499267E-02 5.818499266E-02 5.818499260E-02
14.44000 0.2575006491 0.2575006491 0.2575006490
25.55000 0.4780471798 0.4780471797 0.4780471798
36.66000 0.7442915883 0.7442915881 0.7442915880
47.77000 1.101686578 1.101686577 1.101686578
58.88000 1.656411391 1.656411390 1.656411391
69.99000 2.745986117 2.745986118 2.745986119
81.11000 6.393166451 6.393166452 6.393166426
88.88000 51.15042993 51.15042993 51.15042860
89.99000 5729.577893 5729.577895 5729.569869
89.99990 572957.7951 572957.7950 573019.3057
90.00000 9.999999999E+99 Error 0 9.999999999E+99

ArcTan(x):

x HP-15C 12C Platinum HP-35
---------------------------------------------------------------
0.00000 0.0000000000 0.0000000000 0.0000000000
0.00011 6.302535721E-03 6.302535723E-03 6.302535688E-03
0.15500 8.810732986 8.810732984 8.810732984
0.26795 15.00004318 15.00004318 15.00004317
0.41421 22.49982578 22.49982579 22.49982579
0.57735 29.99998843 29.99998844 29.99998843
0.77700 37.84720677 37.84720678 37.84720676
0.88800 41.60507646 41.60507648 41.60507646
1.00000 45.00000000 45.00000000 45.00000000
1.22200 50.70548702 50.70548700 50.70548702
1.48880 56.11145723 56.11145722 56.11145722
2.11100 64.65265735 64.65265734 64.65265735
4.88800 78.43782359 78.43782359 78.43782360
7.55500 82.46000683 82.46000683 82.46000679
99.9990 89.42705557 89.42705557 89.42705555
3333333 89.99998281 89.99998281 89.99998281


Edited: 12 Nov 2005, 3:12 p.m.


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