Easter Sunday Trigs ( rpn38-CX)

[bshoring]

Gerson,
For what it's worth, I was able to eliminate one program step on the latest and greatest trig program.

I replaced the 2nd & 3rd steps:
02 - 31 ENTER
03 - 61 ×

with a single "RCL X 8" as that squares the value in X register. Having previously removed the line that swapped X<>Y at step 30, the program is now 97 steps.

Regards,
Bob

[Gerson W. Barbosa]

(04-02-2016 11:12 PM)bshoring Wrote:  Gerson,
For what it's worth, I was able to eliminate one program step on the latest and greatest trig program.

I replaced the 2nd & 3rd steps:
02 - 31 ENTER
03 - 61 ×

with a single "RCL X 8" as that squares the value in X register. Having previously removed the line that swapped X<>Y at step 30, the program is now 97 steps.

Regards,
Bob

Great! Also, the same can be done twice around 60 steps further in the program. Since I prefer sine in X, I can spare yet another step:

Code:



Trigonometric Functions

R.0: 5.817764173314432e-03
R.1: 0.199999779
R.2: 0.142841665
R.3: 1.107161127e-01
R.4: 0.086263068
R.5: 0.05051923
R.6: 3.2818376136867e-08
R.7: 5.553916059e-14
R.8: 4.4756602e-20
R.9: 2.0935e-26


Angles in DEGREES

-90 =< x <= 90

R/S               --> sin(x)
R/S x<>y      --> cos(x)
R/S x<>y /    --> tan(x)

GTO 30 R/S --> asin(x)       0 =< x <= 1
GTO 41 R/S --> acos(x)      0 =< x <= 1
GTO 49 R/S --> atan(x)   1e-50 < x < 1e42

by Gerson W. Barbosa - Apr/2016


01 - 21 8         STO 8
02 - 22 61 8      RCL × 8
03 - 31           ENTER
04 - 31           ENTER
05 - 31           ENTER
06 - 86 61 9      RCL × .9
07 - 86 41 8      RCL − .8
08 - 61           ×
09 - 86 51 7      RCL + .7
10 - 61           ×
11 - 86 41 6      RCL − .6
12 - 61           ×
13 - 86 51 0      RCL + .0
14 - 22 61 8      RCL × 8
15 - 21 8         STO 8
16 - 21 51 8      STO + 8
17 - 3            3
18 - 22 8         RCL 8
19 - 22 61 8      RCL × 8
20 - 41           −
21 - 61           ×
22 - 21 8         STO 8
23 - 22 61 8      RCL × 8
24 - 32           CHS
25 - 1            1
26 - 51           +
27 - 24 21        √x
28 - 22 8         RCL 8
29 - 25 7 00      GTO 00
30 - 25 6         x=0
31 - 25 7 00      GTO 00
32 - 31           ENTER
33 - 61           ×
34 - 24 71        1/x
35 - 1            1
36 - 41           −
37 - 25 6         x=0
38 - 22 73 9      RCL .9
39 - 24 71        1/x
40 - 25 7 48      GTO 48
41 - 25 6         x=0
42 - 25 7 38      GTO 38
43 - 31           ENTER
44 - 61           ×
45 - 24 71        1/x
46 - 1            1
47 - 41           −
48 - 24 21        √x
49 - 25 6         x=0
50 - 25 7 00      GTO 00
51 - 1            1
52 - 33           x≷y
53 - 25 5         x≤y
54 - 25 7 57      GTO 57
55 - 24 71        1/x
56 - 9            9
57 - 0            0
58 - 21 9         STO 9
59 - 25 33        R↓
60 - 24 71        1/x
61 - 21 8         STO 8
62 - 22 61 8      RCL × 8
63 - 51           +
64 - 24 21        √x
65 - 22 41 8      RCL − 8
66 - 21 8         STO 8
67 - 22 61 8      RCL × 8
68 - 31           ENTER
69 - 31           ENTER
70 - 31           ENTER
71 - 86 61 5      RCL × .5
72 - 86 41 4      RCL − .4
73 - 61           ×
74 - 86 51 3      RCL + .3
75 - 61           ×
76 - 86 41 2      RCL − .2
77 - 61           ×
78 - 86 51 1      RCL + .1
79 - 61           ×
80 - 3            3
81 - 24 71        1/x
82 - 41           −
83 - 61           ×
84 - 1            1
85 - 51           +
86 - 22 61 8      RCL × 8
87 - 8            8
88 - 25 12        12÷
89 - 61           ×
90 - 86 71 0      RCL ÷ .0
91 - 22 9         RCL 9
92 - 25 6         x=0
93 - 33           x≷y
94 - 33           x≷y
95 - 41           −
96 - 25 7 00      GTO 00

By storing the constant 1/3 in register 7 and replacing the steps 80 through 82 with a single RCL- 7 instruction, per Willy's idea above, two other steps can be saved, thus leaving us six steps more to play with.

Regards,

Gerson.

[Gerson W. Barbosa]

.
Trigonometric Functions (RPN-38 CX)

Code:


01 - 3            3
02 - 71           ÷
03 - 21 8         STO 8
04 - 22 61 8      RCL × 8
05 - 31           ENTER
06 - 31           ENTER
07 - 31           ENTER
08 - 86 61 9      RCL × .9
09 - 86 41 8      RCL − .8
10 - 61           ×
11 - 86 51 7      RCL + .7
12 - 61           ×
13 - 86 41 6      RCL − .6
14 - 61           ×
15 - 86 51 0      RCL + .0
16 - 22 61 8      RCL × 8
17 - 21 8         STO 8
18 - 21 51 8      STO + 8
19 - 3            3
20 - 22 8         RCL 8
21 - 22 61 8      RCL × 8
22 - 41           −
23 - 61           ×
24 - 21 8         STO 8
25 - 1            1
26 - 22 8         RCL 8
27 - 22 61 8      RCL × 8
28 - 41           −
29 - 24 21        √x
30 - 25 7 00      GTO 00
31 - 86 71 0      RCL ÷ .0
32 - 25 7 00      GTO 00
33 - 86 61 0      RCL × .0
34 - 25 7 00      GTO 00
35 - 25 6         x=0
36 - 25 7 00      GTO 00
37 - 31           ENTER
38 - 61           ×
39 - 24 71        1/x
40 - 1            1
41 - 41           −
42 - 25 6         x=0
43 - 22 73 9      RCL .9
44 - 24 71        1/x
45 - 25 7 53      GTO 53
46 - 25 6         x=0
47 - 25 7 43      GTO 43
48 - 31           ENTER
49 - 61           ×
50 - 24 71        1/x
51 - 1            1
52 - 41           −
53 - 24 21        √x
54 - 25 6         x=0
55 - 25 7 00      GTO 00
56 - 1            1
57 - 33           x≷y
58 - 25 5         x≤y
59 - 25 7 62      GTO 62
60 - 24 71        1/x
61 - 9            9
62 - 0            0
63 - 21 9         STO 9
64 - 25 33        R↓
65 - 24 71        1/x
66 - 21 8         STO 8
67 - 22 61 8      RCL × 8
68 - 51           +
69 - 24 21        √x
70 - 22 41 8      RCL − 8
71 - 21 8         STO 8
72 - 21 51 8      STO + 8
73 - 31           ENTER
74 - 61           ×
75 - 31           ENTER
76 - 31           ENTER
77 - 31           ENTER
78 - 86 61 5      RCL × .5
79 - 86 41 4      RCL − .4
80 - 61           ×
81 - 86 51 3      RCL + .3
82 - 61           ×
83 - 86 41 2      RCL − .2
84 - 61           ×
85 - 86 51 1      RCL + .1
86 - 61           ×
87 - 3            3
88 - 24 71        1/x
89 - 41           −
90 - 61           ×
91 - 1            1
92 - 51           +
93 - 22 61 8      RCL × 8
94 - 86 71 0      RCL ÷ .0
95 - 22 9         RCL 9
96 - 25 6         x=0
97 - 33           x≷y
98 - 33           x≷y
99 - 41           −

------------------------------------

Trigonometric Functions

by Gerson W. Barbosa - Apr/2016

R.0: 1.74532925199433e-2
R.1: 0.199999779
R.2: 0.142841665
R.3: 1.107161127e-01
R.4: 0.086263068
R.5: 0.05051923
R.6: 8.860961556954e-07
R.7: 1.3496016023e-11
R.8: 9.78826891e-17
R.9: 4.12066e-22


Angles in DEGREES

-90 =< x <= 90

R/S      --> cos(x)
R/S x<>y --> sin(x)
R/S /    --> tan(x)

GTO 35 R/S --> asin(x)      0 =< x <= 1
GTO 46 R/S --> acos(x)      0 =< x <= 1
GTO 54 R/S --> atan(x)   1e-50 < x < 1e42, x=0

GTO 31 R/S --> Rad->Deg
GTO 33 R/S --> Deg->Rad

------------------------------------------

Constants:

R.0: 1.74532925199433e-2
R.1: 0.199999779
R.2: 0.142841665
R.3: 1.107161127e-01
R.4: 0.086263068
R.5: 0.05051923
R.6: 8.860961556954e-07
R.7: 1.3496016023e-11
R.8: 9.78826891e-17
R.9: 4.12066e-22


------------------------------------------

Trigonometric Functions:

Angles in DEGREES

-90 =< x <= 90

R/S --> cos(x)
R/S x<>y --> sin(x)
R/S / --> tan(x)

GTO 35 R/S --> asin(x) 0 =< x <= 1
GTO 46 R/S --> acos(x) 0 =< x <= 1
GTO 54 R/S --> atan(x) 1e-50 < x < 1e42, x=0

GTO 31 R/S --> Rad->Deg
GTO 33 R/S --> Deg->Rad

------------------------------------------
Examples:

0.0001 R/S --> 1.000000000 ; cos(0.0001)
x<>y --> 1.745329252E-06 ; sin(0.0001)
/ --> 1.745329252E-06 ; tan(0.0001)

0.9999 GTO 35 R/S --> 89.18960856 ; asin(0.9999)
0.9999 GTO 46 R/S --> 0.8102914371 ; acos(0.9999)
0.9999 GTO 54 R/S --> 44.99713507 ; atan(0.9999)

180 RCL × .0 --> 3.141592654 ; π
180 GTO 33 R/S --> 3.141592654 ; 180° = π rad
3.141592654 ENTER 8 / GTO 31 R/S --> 22.50000000 ; π/8 rad = 22.5°

Other examples:

sin(0.01) = 1.745329243E-04
cos(0.01) = 0.9999999848
tan(0.01) = 1.745329270E-04

sin(30) = 0.5000000000
cos(30) = 0.8660254038
tan(30) = 0.5773502692

sin(60) = 0.8660254038
cos(60) = 0.5000000000
tan(60) = 1.732050808

sin(75) = 0.9659258263
cos(75) = 0.2588190451
tan(75) = 3.732050808

sin(89) = 0.9998476952
cos(89) = 1.745240644E-2
tan(89) = 57.28996163

sin(89.99) = 0.9999999848
cos(89.99) = 1.74532924(0)E-4
tan(89.99) = 5729.577(902)

sin(89.9999) = 1.000000000
cos(89.9999) = 1.7452(81608)E-6
tan(89.9999) = 5729(73.3462)

asin(0) = 0.000000000
acos(0) = 90.00000000
atan(0) = 0.000000000

asin(1) = 90.00000000
acos(1) = 0.000000000
atan(1) = 45.00000000

asin(1e10) = 90.00000000

atan(0.4142135624) = 22.50000000

acos(0.8660254038) = 30.00000000

atan(50) = 88.85423716

------------------------------------------

Forensic result:

9 R/S x<>y R/S R/S / GTO 54 R/S GTO 46 R/S GTO 35 R/S --> 9.000000282

------------------------------------------

The extension of the attached file should be changed to .rpn38

Edited to fix a typo.

[bshoring]

Nice program. Same accuracy as before, but with the added conversion functions DEG->RAD & RAD->DEG. Very well done!

Since lines 05 through 15 look like a polynomial, I have been toying with the idea of trying to achieve the same thing using the Net Present Value (NPV) function, to at least compute a Sine, as NPV is based on a polynomial. Of course R.0 and R.6-R.9 would have to be moved into R0-R4. Maybe more trouble than it's worth, but just an idea, out of curiosity.

Thanks for the great programs!

[Gerson W. Barbosa]

(04-04-2016 04:20 AM)bshoring Wrote:  Since lines 05 through 15 look like a polynomial, I have been toying with the idea of trying to achieve the same thing using the Net Present Value (NPV) function, to at least compute a Sine, as NPV is based on a polynomial. Of course R.0 and R.6-R.9 would have to be moved into R0-R4. Maybe more trouble than it's worth, but just an idea, out of curiosity.

The arctangent polynomial starts at line 75.

Accuracy is granted is limited ranges, as you can see in the W|A plots:

Plot sin(x)-(x-0.166666666665558x^3+0.00833333320429368x^5-0.000198410347967009x^7+0.0000027420184217757x^9),x=0..pi/6

Plot atan(x)-(x-0.333333333333333x^3+0.199999779x^5-0.142841665x^7+0.1107161127x^9-0.086263068x^11+0.05051923x^13),x=0..sqrt(2)-1

Yes, the built-in polynomial solver might be handy, but I've never tried it a program. A program based on this would be interesting.

Manually, on the HP-12C:

sin(x) ~ x - 1/6*x^3 + 8.333333204e-3*x^5 - 1.98410348e-4*x^7 + 2.742018422e-6*x^9

f CLEAR FIN
3.141592654 ENTER 6 / 1 Delta% i
0 g CF0
1 g CFj 0 g CFj 6 1/x CHS g CFj 0 g CFj 8.333333204 EEX CHS 3 g CFj 0 g CFj
1.98410348 CHS EEX CHS 4 g CFj 0 g CFj 2.742018422 EEX CHS 6 g CFj
f NVP --> 0.5000000001 ; sin(pi/6)

Regards,

Gerson.

----------------------------------------

P.S.: Or, using less registers:

y = x^2

sin(x) ~ x*(1 - 1/6*y + 8.333333204e-3*y^2 - 1.98410348e-4*y^3 + 2.742018422e-6*y^4)

f CLEAR FIN
3.141592654 ENTER 6 / STO 5 ENTER * 1 Delta% i
1 g CF0
6 1/x CHS g CFj 8.333333204 EEX CHS 3 g CFj
1.98410348 CHS EEX CHS 4 g CFj 2.742018422 EEX CHS 6 g CFj
f NVP RCL 5 * --> 0.5000000001 ; sin(pi/6)

[bshoring]

Thanks for the insight. This satisfies my curiosity that one can compute Sine, Cosine and Tangent using the Net Present Value function.

Here's my short little program based on your knowledge and hard work!

Trig using NPV

Angles in Degrees

R/S --> cos(x)
R/S x<>y --> sin(x)
R/S / --> tan(x)

Constants:

n: 9
R0: 0
R1: 1
R2: 0
R3: -0.166666666666667
R4: 0
R5: 8.333333204E-3
R6: 0
R7: -1.98410348E-4
R8: 0
R9: 2.742018422E-6
R.9: 1.74532925199433e-2

01 - 86 61 9 RCL × .9
02 - 1 1
03 - 24 23 ∆%
04 - 12 i
05 - 24 13 NPV
06 - 31 ENTER
07 - 31 ENTER
08 - 61 ×
09 - 32 CHS
10 - 1 1
11 - 51 +
12 - 24 21 √x
13 - 25 7 00 GTO 00

On RPN-38 CX, it only takes 12 program steps. On my original HP-38C, with slight modification, it also works quite nicely, with similar results. So far the NPV route isn't producing quite the level of accuracy that your fine programs do, but I do find it exciting to think that a built in financial function can be used this way!

If run on an actual HP-12C or HP-38 E/C, the first line could be replaced with 2 lines:
RCL FV
X
(And the constant 1.745329252E-6 needs to loaded into the FV register).

[Gerson W. Barbosa]

(04-05-2016 04:42 AM)bshoring Wrote:  Thanks for the insight. This satisfies my curiosity that one can compute Sine, Cosine and Tangent using the Net Present Value function.

Here's my short little program based on your knowledge and hard work!

Trig using NPV

Angles in Degrees

R/S --> cos(x)
R/S x<>y --> sin(x)
R/S / --> tan(x)

Constants:

n: 9
R0: 0
R1: 1
R2: 0
R3: -0.166666666666667
R4: 0
R5: 8.333333204E-3
R6: 0
R7: -1.98410348E-4
R8: 0
R9: 2.742018422E-6
R.9: 1.74532925199433e-2

01 - 86 61 9 RCL × .9
02 - 1 1
03 - 24 23 ∆%
04 - 12 i
05 - 24 13 NPV
06 - 31 ENTER
07 - 31 ENTER
08 - 61 ×
09 - 32 CHS
10 - 1 1
11 - 51 +
12 - 24 21 √x
13 - 25 7 00 GTO 00

On RPN-38 CX, it only takes 12 program steps. On my original HP-38C, with slight modification, it also works quite nicely, with similar results. So far the NPV route isn't producing quite the level of accuracy that your fine programs do, but I do find it exciting to think that a built in financial function can be used this way!

If run on an actual HP-12C or HP-38 E/C, the first line could be replaced with 2 lines:
RCL FV
X
(And the constant 1.745329252E-6 needs to loaded into the FV register).

On the HP-12C/38C four coefficients will suffice. Notice range reduction is needed here since the polynomial approximation is accurate enough only from -30 to 30 degrees (the argument is divide by three then a trigonometric relation is used to compute the sine of the original angle). But the program becomes somewhat long: 30 steps and 5 constants.

The computed values of cosine and tangent become poorer and poorer as the angle approaches 90 degrees. In this case, use the complementary angle. For instance, tan 89.9999° = cot(90° - 89.9999°). On the HP-12C:

90 ENTER 89.9999 - R/S x<>y / --> 572.957.7951 ; tan(89.9999°)

75 R/S -> 0.25881904(43) ; cos(75°)
X<>y --> 0.965925826(5) ; sin(75°)
x<> / --> 3.7320508(20) ; tan(75°)

Regards,

Gerson.

Code:


01 - 22 4         RCL 4
02 - 61           ×
03 - 21 5         STO 5
04 - 31           ENTER
05 - 61           ×
06 - 1            1
07 - 24 23        ∆%
08 - 12           i
09 - 3            3
10 - 11           n
11 - 24 13        NPV
12 - 22 5         RCL 5
13 - 61           ×
14 - 31           ENTER
15 - 31           ENTER
16 - 61           ×
17 - 4            4
18 - 61           ×
19 - 32           CHS
20 - 3            3
21 - 51           +
22 - 61           ×
23 - 31           ENTER
24 - 31           ENTER
25 - 61           ×
26 - 32           CHS
27 - 1            1
28 - 51           +
29 - 24 21        √x
30 - 25 7 00      GTO 00

R0: 1
R1: -1.666666667   ;  -1/6
R2: 8.33322556e-03
R3: -0.000197278
R4: 5.817764173e-3 ; pi/540

[bshoring]

Thanks, Gerson for the programs and explanations. It is interesting that we can use the Net Present Value on a financial calculator to compute sine, up to a point, but as you say, the closer one gets to 90 degrees, the less accurate that method is.

I'd say that the best program is the one you posted on 3 April, 2016. It appears to give the best overall accuracy for the 6 main trig functions. This exercise has been quite an enlightenment for me.

Thanks!

[Gerson W. Barbosa]

(04-08-2016 10:42 PM)bshoring Wrote:  It is interesting that we can use the Net Present Value on a financial calculator to compute sine, up to a point, but as you say, the closer one gets to 90 degrees, the less accurate that method is.

These issues are due mostly to cancellation errors. Notice the polynomial cannot be evaluate for zero when using the NPV method, unless this is properly handled. It appears NVP is not as accurate on the HP-12C Platinum as it is on the HP-12C for certain values, judging by a few results which are better on the latter when it should be otherwise.

(04-08-2016 10:42 PM)bshoring Wrote:  I'd say that the best program is the one you posted on 3 April, 2016. It appears to give the best overall accuracy for the 6 main trig functions. This exercise has been quite an enlightenment for me.

That's the one I like best too. However doing calculations with three or four extra digits doesn't prevent cancellation errors from popping up in those critical regions.

(04-08-2016 10:42 PM)bshoring Wrote:  Thanks!

My pleasure! You might also want to take a look at this this HP-17BII equation, in case you haven't seen it yet.

The following program is meant for the REAL HP-38C, but it has been tested only on the HP-12C and the HP-12 Prestige.

Regards,

Gerson.

.
Trigonometric Functions - REAL HP-38C

Code:


01 - 3            3
02 - 21 11        STO n
03 - 71           ÷
04 - 21 14        STO PMT
05 - 25 6         x=0
06 - 25 7 13      GTO 13
07 - 31           ENTER
08 - 61           ×
09 - 1            1
10 - 24 23        ∆%
11 - 21 12        STO i
12 - 24 13        NPV
13 - 22 14        RCL PMT
14 - 61           ×
15 - 31           ENTER
16 - 31           ENTER
17 - 61           ×
18 - 4            4
19 - 61           ×
20 - 32           CHS
21 - 3            3
22 - 51           +
23 - 61           ×
24 - 31           ENTER
25 - 31           ENTER
26 - 61           ×
27 - 32           CHS
28 - 1            1
29 - 51           +
30 - 24 21        √x
31 - 25 7 00      GTO 00
32 - 31           ENTER
33 - 61           ×
34 - 32           CHS
35 - 1            1
36 - 51           +
37 - 25 7 40      GTO 40
38 - 31           ENTER
39 - 61           ×
40 - 24 71        1/x
41 - 1            1
42 - 41           −
43 - 24 21        √x
44 - 25 6         x=0
45 - 25 7 00      GTO 00
46 - 1            1
47 - 33           x≷y
48 - 25 5         x≤y
49 - 25 7 52      GTO 52
50 - 24 71        1/x
51 - 9            9
52 - 0            0
53 - 21 13        STO PV
54 - 25 33        R↓
55 - 24 71        1/x
56 - 21 14        STO PMT
57 - 31           ENTER
58 - 61           ×
59 - 51           +
60 - 24 21        √x
61 - 22 14        RCL PMT
62 - 41           −
63 - 21 14        STO PMT
64 - 31           ENTER
65 - 61           ×
66 - 31           ENTER
67 - 31           ENTER
68 - 31           ENTER
69 - 22 6         RCL 6
70 - 61           ×
71 - 73           .
72 - 1            1
73 - 0            0
74 - 6            6
75 - 51           +
76 - 61           ×
77 - 22 5         RCL 5
78 - 41           −
79 - 61           ×
80 - 22 4         RCL 4
81 - 51           +
82 - 61           ×
83 - 3            3
84 - 24 71        1/x
85 - 41           −
86 - 61           ×
87 - 1            1
88 - 51           +
89 - 22 14        RCL PMT
90 - 61           ×
91 - 2            2
92 - 61           ×
93 - 22 0         RCL 0
94 - 71           ÷
95 - 22 13        RCL PV
96 - 25 6         x=0
97 - 33           x≷y
98 - 33           x≷y
99 - 41           −

Constants:

R0:  1.745329252e-02
R1: -8.86096144e-07
R2:  1.3495798e-11
R3: -9.7284-17
R4:  0.199989464
R5:  0.14247163
R6: -0.0606366

Angles in DEGREES

-90 =< x =< 90

R/S     --> cos(x)
R/S x≷y --> sin(x)
R/S ÷   --> tan(x)

GTO 32 R/S --> asin(x)      0 =< x <  1
GTO 38 R/S --> acos(x)      0  < x <= 1
GTO 44 R/S --> atan(x)   1e-50 < x < 1e50, x=0

RCL 0 ÷  --> Rad->Deg
RCL 0 ×  --> Deg->Rad

------------------------------------------

Constants:

R0: 1.745329252e-02
R1: -8.86096144e-07
R2: 1.3495798e-11
R3: -9.7284-17
R4: 0.199989464
R5: 0.14247163
R6: -0.0606366


------------------------------------------

Trigonometric Functions:

Angles in DEGREES

-90 =< x =< 90

R/S --> cos(x)
R/S x≷y --> sin(x)
R/S ÷ --> tan(x)

GTO 32 R/S --> asin(x) 0 =< x < 1
GTO 38 R/S --> acos(x) 0 < x <= 1
GTO 44 R/S --> atan(x) 1e-50 < x < 1e50, x=0

RCL 0 ÷ --> Rad->Deg
RCL 0 × --> Deg->Rad

------------------------------------------
Examples:

0.0001 R/S --> 1.000000000 ; cos(0.0001) ; [result on the HP-12C Prestige: 1.000000000] {HP-15C built-in function result: 1.000000000}
x≷y --> 1.745329252E-06 ; sin(0.0001) ; [1.745329252E-06] {1.745329252E-06}
x≷y ÷ --> 1.745329252E-06 ; tan(0.0001) ; [1.745329252E-06] {1.745329252E-06}

0.9999 GTO 32 R/S --> 89.18970909 ; asin(0.9999) ; [89.18970856] {89.18970856}
0.9999 GTO 38 R/S --> 0.8102914371 ; acos(0.9999) ; [0.8102914375] {0.8102914371}
0.9999 GTO 44 R/S --> 44.99713507 ; atan(0.9999) ; [44.99713506] {44.99713507}

180 RCL 0 × --> 3.141592654 ; 180° = π rad
3.141592654 ENTER 8 / RCL 0 ÷ --> 22.50000001 ; π/8 rad = 22.5°

Other examples:

sin(0) = 0.000000000 [0.000000000] {0.000000000}
cos(0) = 1.000000000 [1.000000000] {1.000000000}
tan(0) = 0.000000000 [0.000000000] {0.000000000}

sin(1) = 1.745240644E-02 [1.745240644E-02] {1.745240644E-02}
cos(1) = 0.9998476952 [0.9998476952] {0.9998476952}
tan(1) = 1.745506493E-02 [1.745506493E-02] {1.745506493E-02}

sin(15) = 0.2588190451 [0.2588190451] {0.2588190451}
cos(15) = 0.9659258263 [0.9659258263] {0.9659258263}
tan(15) = 0.2679491924 [0.2679491925] {0.2679491924}

sin(0.01) = 1.745329243E-04 [1.745329243E-04] {1.745329243E-04}
cos(0.01) = 0.9999999847 [0.9999999848] {0.9999999848}
tan(0.01) = 1.745329270E-04 [1.745329270E-04] {1.745329270E-04}

sin(30) = 0.5000000002 [0.5000000000] {0.5000000000}
cos(30) = 0.8660254037 [0.8660254038] {0.8660254038}
tan(30) = 0.5773502695 [0.5773502693] {0.5773502692}

sin(60) = 0.8660254034 [0.8660254038] {0.8660254038}
cos(60) = 0.5000000001 [0.5000000000] {0.5000000000}
tan(60) = 1.732050804 [1.732050808] {1.732050808}

sin(75) = 0.9659258263 [0.9659258265] {0.9659258263}
cos(75) = 0.2588190451 [0.2588190445] {0.2588190451}
tan(75) = 3.732050808 [3.732050808] {3.732050808}

sin(89) = 0.9998476951 [0.9998476952] {0.9998476952}
cos(89) = 1.745240958E-2 [1.745240611E-2] {1.745240644E-2}
tan(89) = 57.28995131 [57.28996271] {57.28996163}

sin(89.99) = 0.9999999850 [0.9999999848] {0.9999999848}
cos(89.99) = 1.732050808E-4 [1.745508522E-4] {1.745329243E-4}
tan(89.99) = 5773.502604 [5728.989416] {5729.577893}

sin(89.9999) = 1.000000000 [1.000000000] {1.000000000}
cos(89.9999) = 0.000000000 [2.449489743E-6] {1.745329252E-6}
tan(89.9999) = Error 0 [408248.2905] {572957.7951}

asin(0) = 0.000000000 [0.000000000] {0.000000000}
acos(0) = Error 0 [Error 0] {90.00000000}
atan(0) = 0.000000000 [0.000000000] {0.000000000}

asin(1) = Error 0 [Error 0] {90.00000000}
acos(1) = 0.000000000 [0.000000000] {0.000000000}
atan(1) = 45.99999997 [45.99999999] {45.00000000}

asin(1e10) = 90.00000000 [90.00000000] {89.99999999}

atan(0.4142135624) = 22.49999990 [22.49999994] {22.50000000}

acos(0.8660254038) = 29.99999997 [30.00000001] {30.00000000}

atan(50) = 88.85423719 [88.85423716] {88.85423716}

------------------------------------------

Forensic result:

9 R/S x≷y R/S R/S ÷ GTO 44 R/S GTO 38 R/S GTO 32 R/S --> 9.707005398 [9.008774143] {9.000417403}

------------------------------------------

P.S.: W|A plots for accuracy checking of the polynomials used in this program:

Plot sin(x)-(x-0.1666666645x^3+0.008333198583x^5-0.0001971967922x^7),x=0..pi/6

Plot atan(x)-(x-0.3333333333x^3+0.199989464x^5-0.14247163x^7+0.106x^9-0.0606366x^11),x=0..sqrt(2)-1

P.P.S.: FV in lines 53 and 95 has been replaced with PV. The former is the only register left for the user.

[bshoring]

Thanks. Works well on my actual HP-38C. Results as indicated. When I get time I'll set it up also on RPN-38 CX, to see how they compare.

[Gerson W. Barbosa]

(04-12-2016 04:16 AM)bshoring Wrote:  Thanks. Works well on my actual HP-38C. Results as indicated.

Thanks for testing it on the real thing. Thanks also for the idea of using NPV to evaluate at least one of the polynomials. This has saved memory for a higher order arctangent polynomial.

At least one of the HP-12C results in my previous posting is wrong (copy & paste error) . This is the correct one:

0.9999 GTO 38 R/S --> 0.8102920566 ; acos(0.9999) ; [0.8102914375] {0.8102914371}

(04-12-2016 04:16 AM)bshoring Wrote:  When I get time I'll set it up also on RPN-38 CX, to see how they compare.

Sin(x):

x (deg)......HP-38C.........RPN-38CX......HP-15C (1982)...HP-35 (1972)
------------------------------------------------------------------------
0.000000 0.000000000E+00 0.000000000E+00 0.000000000E+00 0.000000000E+00
0.000010 1.745329252E-07 1.745329252E-07 1.745329252E-07 1.745000000E-07
0.000110 1.919862177E-06 1.919862177E-06 1.919862177E-06 1.919800000E-06
0.022000 3.839724258E-04 3.839724260E-04 3.839724260E-04 3.839723931E-04
3.330000 5.808674960E-02 5.808674960E-02 5.808674960E-02 5.808674961E-02
14.44000 2.493660251E-01 2.493660251E-01 2.493660251E-01 2.493660250E-01
25.55000 4.312985869E-01 4.312985870E-01 4.312985870E-01 4.312985871E-01
36.66000 5.970652564E-01 5.970652564E-01 5.970652564E-01 5.970652561E-01
47.77000 7.404527823E-01 7.404527826E-01 7.404527827E-01 7.404527828E-01
58.88000 8.560867284E-01 8.560867282E-01 8.560867283E-01 8.560867285E-01
69.99000 9.396329131E-01 9.396329128E-01 9.396329127E-01 9.396329127E-01
81.11000 9.879868530E-01 9.879868529E-01 9.879868528E-01 9.879868527E-01
88.88000 9.998089502E-01 9.998089500E-01 9.998089500E-01 9.998089499E-01
89.99000 9.999999850E-01 9.999999848E-01 9.999999848E-01 9.999999848E-01
89.99900 1.000000000E+00 9.999999998E-01 9.999999998E-01 9.999999998E-01
90.00000 1.000000000E+00 1.000000000E+00 1.000000000E+00 1.000000000E+00


Tan(x):

x (deg)......HP-38C.........RPN-38CX......HP-15C (1982)...HP-35 (1972)
------------------------------------------------------------------------
0.000000 0.000000000E+00 0.000000000E+00 0.000000000E+00 0.000000000E+00
0.000010 1.745329252E-07 1.745329252E-07 1.745329252E-07 1.745000000E-07
0.000110 1.919862177E-06 1.919862177E-06 1.919862177E-06 1.919800000E-06
0.022000 3.839724258E-04 3.839724543E-04 3.839724543E-04 3.839724542E-04
3.330000 5.818499267E-02 5.818499267E-02 5.818499267E-02 5.818499266E-02
14.44000 2.575006491E-01 2.575006491E-01 2.575006491E-01 2.575006490E-01
25.55000 4.780471796E-01 4.780471798E-01 4.780471798E-01 4.780471798E-01
36.66000 7.442915883E-01 7.442915882E-01 7.442915883E-01 7.442915880E-01
47.77000 1.101686576E+00 1.101686577E+00 1.101686578E+00 1.101686578E+00
58.88000 1.656411391E+00 1.656411390E+00 1.656411391E+00 1.656411391E+00
69.99000 2.745986127E+00 2.745986118E+00 2.745986117E+00 2.745986119E+00
81.11000 6.393166511E+00 6.393166482E+00 6.393166451E+00 6.393166426E+00
88.88000 5.115045162E+01 5.115043063E+01 5.115042993E+01 5.115042860E+01
89.99000 5.773502604E+04 5.729581914E+04 5.729577951E+05 5.729655162E+04
89.99990 ....Error 0.... 5.729525534E+05 5.729577951E+05 5.730193057E+05
90.00000 ....Error 0.... ....Error 0.... 9.999999999E+99 9.999999999E+99


ArcTan(x):

...x........HP-38C.........RPN-38CX......HP-15C (1982)...HP-35 (1972)
-------------------------------------------------------------------------
0.00000 0.000000000E+00 0.000000000E+00 0.000000000E+00 0.000000000E+00
0.00011 6.302535741E-03 6.302535789E-03 6.302535721E-03 6.302535688E-03
0.15500 8.810733019E+00 8.810732984E+00 8.810732986E+00 8.810732984E+00
0.26795 1.500004317E+01 1.500004315E+01 1.500004317E+01 1.500004317E+01
0.41421 2.249982576E+01 2.249982574E+01 2.249982578E+01 2.249982579E+01
0.57735 2.999998843E+01 2.999998845E+01 2.999998843E+01 2.999998843E+01
0.77700 3.784720679E+01 3.784720676E+01 3.784720677E+01 3.784720676E+01
0.88800 4.160507644E+01 4.160507646E+01 4.160507646E+01 4.160507646E+01
1.00000 4.499999997E+01 4.500000000E+01 4.500000000E+01 4.500000000E+01
1.22200 5.070548705E+01 5.070548704E+01 5.070548702E+01 5.070548702E+01
1.48880 5.611145719E+01 5.611145720E+01 5.611145723E+01 5.611145722E+01
2.11100 6.465265739E+01 6.465265738E+01 6.465265735E+01 6.465265735E+01
4.88800 7.843782363E+01 7.843782359E+01 7.843782359E+01 7.843782360E+01
7.55500 8.246000676E+01 8.246000683E+01 8.246000683E+01 8.246000679E+01
99.9990 8.942705958E+01 8.942705557E+01 8.942705557E+01 8.942705555E+01
7777.77 8.999266614E+01 8.999263339E+01 8.999263339E+01 8.999263337E+01
1.0E+05 9.000000000E+01 8.999942704E+01 8.999942704E+01 8.999942704E+01


The arctangent polynomial constants have been changed slightly:

R4: 0.199991241
R5: 0.14251795
R6: -0.0616468

Also, steps 71 through 74 have been changed:

71- 9
72- .
73- 4
74- 1/x

Plot #1

Plot #2

Regards,

Gerson.

Edited to fix a couple a typos in the tables.

[Gerson W. Barbosa]

In the previous post I forgot to include a couple HP-12C Platinum results to demonstrate the difference two extra digits can make. Here they are:

0.9999 GTO 038 R/S --> 0.8102914375 ; arccos(0.9999 deg)

0.155 GTO 044 R/S --> 8.810732983 ; arctan(0.155)

Compare these with the results obtained on the HP-38C/12C.

Since we're talking 38, just another calculation involving a few instances of 38 on both calculators:

HP-38C/12C:

0.788888888 ENTER 1138.777777 1/x - GTO 38 --> 38.00000006

HP-12C Platinum/Prestige:

0.788888888 ENTER 1138.777777 1/x - GTO 038 --> 37.999999999

Can you guess what the actual 10-digit result might be? :-)

[bshoring]

(04-13-2016 12:54 AM)Gerson W. Barbosa Wrote:  

(04-12-2016 04:16 AM)bshoring Wrote:  Thanks. Works well on my actual HP-38C. Results as indicated.

Thanks for testing it on the real thing. Thanks also for the idea of using NPV to evaluate at least one of the polynomials. This has saved memory for a higher order arctangent polynomial.

At least one of the HP-12C results in my previous posting is wrong (copy & paste error) . This is the correct one:

0.9999 GTO 38 R/S --> 0.8102920566 ; acos(0.9999) ; [0.8102914375] {0.8102914371}

(04-12-2016 04:16 AM)bshoring Wrote:  When I get time I'll set it up also on RPN-38 CX, to see how they compare.

Sin(x):

x (deg)......HP-38C.........RPN-38CX......HP-15C (1982)...HP-35 (1972)
------------------------------------------------------------------------
0.000000 0.000000000E+00 0.000000000E+00 0.000000000E+00 0.000000000E+00
0.000010 1.745329252E-07 1.745329252E-07 1.745329252E-07 1.745000000E-07
0.000110 1.919862177E-06 1.919862177E-06 1.919862177E-06 1.919800000E-06
0.022000 3.839724258E-04 3.839724260E-04 3.839724260E-04 3.839723931E-04
3.330000 5.808674960E-02 5.808674960E-02 5.808674960E-02 5.808674961E-02
14.44000 2.493660251E-01 2.493660251E-01 2.493660251E-01 2.493660250E-01
25.55000 4.312985869E-01 4.312985870E-01 4.312985870E-01 4.312985871E-01
36.66000 5.970652564E-01 5.970652564E-01 5.970652564E-01 5.970652561E-01
47.77000 7.404527823E-01 7.404527826E-01 7.404527827E-01 7.404527828E-01
58.88000 8.560867284E-01 8.560867282E-01 8.560867283E-01 8.560867285E-01
69.99000 9.396329131E-01 9.396329128E-01 9.396329127E-01 9.396329127E-01
81.11000 9.879868530E-01 9.879868529E-01 9.879868528E-01 9.879868527E-01
88.88000 9.998089502E-01 9.998089500E-01 9.998089500E-01 9.998089499E-01
89.99000 9.999999850E-01 9.999999848E-01 9.999999848E-01 9.999999848E-01
89.99900 1.000000000E+00 9.999999998E-01 9.999999998E-01 9.999999998E-01
90.00000 1.000000000E+00 1.000000000E+00 1.000000000E+00 1.000000000E+00


Tan(x):

x (deg)......HP-38C.........RPN-38CX......HP-15C (1982)...HP-35 (1972)
------------------------------------------------------------------------
0.000000 0.000000000E+00 0.000000000E+00 0.000000000E+00 0.000000000E+00
0.000010 1.745329252E-07 1.745329252E-07 1.745329252E-07 1.745000000E-07
0.000110 1.919862177E-06 1.919862177E-06 1.919862177E-06 1.919800000E-06
0.022000 3.839724258E-04 3.839724543E-04 3.839724543E-04 3.839724542E-04
3.330000 5.818499267E-02 5.818499267E-02 5.818499267E-02 5.818499266E-02
14.44000 2.575006491E-01 2.575006491E-01 2.575006491E-01 2.575006490E-01
25.55000 4.780471796E-01 4.780471798E-01 4.780471798E-01 4.780471798E-01
36.66000 7.442915883E-01 7.442915882E-01 7.442915883E-01 7.442915880E-01
47.77000 1.101686576E+00 1.101686577E+00 1.101686578E+00 1.101686578E+00
58.88000 1.656411391E+00 1.656411390E+00 1.656411391E+00 1.656411391E+00
69.99000 2.745986127E+00 2.745986118E+00 2.745986117E+00 2.745986119E+00
81.11000 6.393166511E+00 6.393166482E+00 6.393166451E+00 6.393166426E+00
88.88000 5.115045162E+01 5.115043063E+01 5.115042993E+01 5.115042860E+01
89.99900 5.773502604E+04 5.729581914E+04 5.729577951E+05 5.729655162E+04
89.99990 ....Error 0.... 5.729525534E+05 5.729577951E+05 5.730193057E+05
90.00000 ....Error 0.... ....Error 0.... 9.999999999E+99 9.999999999E+99


ArcTan(x):

...x........HP-38C.........RPN-38CX......HP-15C (1982)...HP-35 (1972)
-------------------------------------------------------------------------
0.00000 0.000000000E+00 0.000000000E+00 0.000000000E+00 0.000000000E+00
0.00011 6.302535741E-03 6.302535789E-03 6.302535721E-03 6.302535688E-03
0.15500 8.810733019E+00 8.810732984E+00 8.810732986E+00 8.810732984E+00
0.26795 1.500004317E+01 1.500004315E+01 1.500004317E+01 1.500004317E+01
0.41421 2.249982576E+01 2.249982574E+01 2.249982578E+01 2.249982579E+01
0.57735 2.999998843E+01 2.999998845E+01 2.999998843E+01 2.999998843E+01
0.77700 3.784720679E+01 3.784720676E+01 3.784720677E+01 3.784720676E+01
0.88800 4.160507644E+01 4.160507646E+01 4.160507646E+01 4.160507646E+01
1.00000 4.499999997E+01 4.500000000E+01 4.500000000E+01 4.500000000E+01
1.22200 5.070548705E+01 5.070548704E+01 5.070548702E+01 5.070548702E+01
1.48880 5.611145719E+01 5.611145720E+01 5.611145723E+01 5.611145722E+01
2.11100 6.465265739E+01 6.465265738E+01 6.465265735E+01 6.465265735E+01
4.88800 7.843782363E+01 7.843782359E+01 7.843782359E+01 7.843782360E+01
7.55500 8.246000676E+01 8.246000683E+01 8.246000683E+01 8.246000679E+01
99.9990 8.942705558E+01 8.942705557E+01 8.942705557E+01 8.942705555E+01
7777.77 8.999266614E+01 8.999263339E+01 8.999263339E+01 8.999263337E+01
1.0E+05 9.000000000E+01 8.999942704E+01 8.999942704E+01 8.999942704E+01


The arctangent polynomial constants have been changed slightly:

R4: 0.199991241
R5: 0.14251795
R6: -0.0616468

Also, steps 71 through 74 have been changed:

71- 9
72- .
73- 4
74- 1/x

Plot #1

Plot #2

Regards,

Gerson.

After updating my HP-38C with the new steps and constants in R4-6, I get
0.9999 GTO 38 R/S --> 0.8102920566 (.810291465 with my HP-35v.4)

TAN:
.00000 I get .01690354059 on HP-38C.
89.999 I get Error 0 (divide by 0) on HP-38C.

ArcTan:
.00000 I get Error 0 on HP-38C
99.999 I get 8.942705958E+01 on HP-38C

I didn't try every one, but of all the rest I tried, I got the same results as you.
Pretty good accuracy, I'd say. Going through this exercise gives me a great respect for the talented minds that created the first HP-35 !

Thanks,
Bob

[Gerson W. Barbosa]

(04-13-2016 10:55 PM)bshoring Wrote:  After updating my HP-38C with the new steps and constants in R4-6, I get
0.9999 GTO 38 R/S --> 0.8102920566 (.810291465 with my HP-35v.4)

I get exactly the same on my HP-12C and my HP-35v.3, respectively.

(04-13-2016 10:55 PM)bshoring Wrote:  TAN:
.00000 I get .01690354059 on HP-38C.
89.999 I get Error 0 (divide by 0) on HP-38C.

The latter was supposed to be 89.99. Fixed. Thanks!

For .00000, I get tan(.00000) = 0, as it should be.

(04-13-2016 10:55 PM)bshoring Wrote:  ArcTan:
.00000 I get Error 0 on HP-38C
99.999 I get 8.942705958E+01 on HP-38C

I get arctan(.00000) = 0.

For arctan(99.999) my HP-12C returns 8.942705958E+01 just like your HP-38C. Another copy & paste error. Fixed. Thanks!

(04-13-2016 10:55 PM)bshoring Wrote:  Going through this exercise gives me a great respect for the talented minds that created the first HP-35 !

My feelings as well. They did milk a stone!

Regards,

Gerson.

[bshoring]

On RPN-38 CX, forensic result is 9.000000330. Nice job !

Maybe you should post some of these programs in the General Software Library of this forum.

[bshoring]

Gerson, if two of your Trig programs could be included in the Sample Programs in the next RPN-38 CX update which ones would you include?

Personally, Id suggest one regular one (not using NPV) & one using NPV.

Willy is interested as he plans to do an update soon.

[Gerson W. Barbosa]

(04-15-2016 08:02 PM)bshoring Wrote:  Gerson, if two of your Trig programs could be included in the Sample Programs in the next RPN-38 CX update which ones would you include?

Personally, Id suggest one regular one (not using NPV) & one using NPV.

Willy is interested as he plans to do an update soon.

Bob,

Your suggestion is ok. At first, I didn't see the point of including the program using NPV, as it is meant for the real HP-38C, but it makes sense as some RPN-38 CX user have the real thing and occasionally might run such program on their vintage calculators. I suspect you are my only customer, though :-)

So the programs I would choose are the one in post #23, written specifically for RPN-38 CX, and the one in post #29 with the modified constants R4 through R6 and modified steps 71 through 74 as listed in post #31. RPN-38 CX is also a great development tool. Thanks, Willy, for its editing capabilities!

If RPN-32 CE is also going to be updated, I would suggest the addition of the conditional tests x<=y and x=0. I haven't explored it yet, but I so far I've seen x=y, x<y and x>y only.

Again, thank you for your suggestions and ideas.

Gerson.

[bshoring]

I think you will be pleased with the next updates of both apps.

[bshoring]

(04-15-2016 11:28 PM)Gerson W. Barbosa Wrote:  

(04-15-2016 08:02 PM)bshoring Wrote:  Gerson, if two of your Trig programs could be included in the Sample Programs in the next RPN-38 CX update which ones would you include?

Personally, Id suggest one regular one (not using NPV) & one using NPV.

Willy is interested as he plans to do an update soon.

Bob,

Your suggestion is ok. At first, I didn't see the point of including the program using NPV, as it is meant for the real HP-38C, but it makes sense as some RPN-38 CX user have the real thing and occasionally might run such program on their vintage calculators. I suspect you are my only customer, though :-)

So the programs I would choose are the one in post #23, written specifically for RPN-38 CX, and the one in post #29 with the modified constants R4 through R6 and modified steps 71 through 74 as listed in post #31. RPN-38 CX is also a great development tool. Thanks, Willy, for its editing capabilities!

If RPN-32 CE is also going to be updated, I would suggest the addition of the conditional tests x<=y and x=0. I haven't explored it yet, but I so far I've seen x=y, x<y and x>y only.

Again, thank you for your suggestions and ideas.

Gerson.

Gerson, your Trig program of 4 April is included as a sample program in the recent update to RPN-38 CX. It's listed as "Trig Functions (High-Precision)." Constants are automatically loaded if you select "Import w/Regs (all)." You may have to press either Add or Reload Sample Programs the first time after updating the app. Also, your issues with both the 38 & 32 apps should have been addressed with this latest update. RCL arithmetic for Last X was added. Also constants are now saved when exporting to Dropbox and (I think) email.

Your program seems to be running perfectly. For ASIN, ACOS, & ATAN, GOTO 35, 46, & 54.

[Gerson W. Barbosa]

(06-05-2016 08:14 PM)bshoring Wrote:  Gerson, your Trig program of 4 April is included as a sample program in the recent update to RPN-38 CX. It's listed as "Trig Functions (High-Precision)." Constants are automatically loaded if you select "Import w/Regs (all)." You may have to press either Add or Reload Sample Programs the first time after updating the app. Also, your issues with both the 38 & 32 apps should have been addressed with this latest update. RCL arithmetic for Last X was added. Also constants are now saved when exporting to Dropbox and (I think) email.

Your program seems to be running perfectly. For ASIN, ACOS, & ATAN, GOTO 35, 46, & 54.

Thanks! I am not sure High-Precision is a good description though. Sure it is more accurate than the original HP-38C program, but we know there are regions where some precision is lost, even with the extra digits provided by the RPN-38CX Simulator. This might disappoint some users.
FWIW, the program for the real HP-38C is now available at the General Software Library:

(HP-12C, 12C Platinum, 38C) Trigonometric Functions

Best regards,

Gerson.