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Gerson W. Barbosa · 03-31-2013, 12:59 AM


HP-12C


01-   3	                21-  1/x

02-   /		        22-   +

03- ENTER		23-   2

04- ENTER               24-   /	 

05-   4                 25-   -

06-  y^x                26- ENTER

07-  12/                27- ENTER

08- ENTER    		28-   *

09- ENTER               29-   4

10-   *			30-   *

11-   1			31-   3

12-   4			32-   -

13-   0			33-   *

14-   /                 34- ENTER

15-   +  		35- ENTER  

16-   2			36-   *

17-   + 		37-  CHS

18-  x<>y               38-   1

19-  e^x                39-   +

20- ENTER               40-  SQRT

                        41- GTO 00

Usage:
Enter angle in radians ( 0 <= x <= pi )

 R/S  ->  sin(x)

x<>y  ->  cos(x)

  /   ->  tan(x)

Examples:

   x  |    sin(x)   |    cos(x)   |    tan(x)   |

------+-------------+-------------+-------------+

 3.14 | 0.001592545 |-0.999998732 |-0.001592547 |

 3.02 | 0.121293236 |-0.992616719 |-0.122195439 |

 2.50 | 0.598472141 |-0.801143618 |-0.747022290 |

 2.00 | 0.909297427 |-0.416146836 |-2.185039869 |

 1.65 | 0.996865028 |-0.079120890 |-12.59926467 |

 1.58 | 0.999999647 |-0.009203545 |-108.6491891 |

 1.57 | 0.999999683 | 0.000796324 | 1255.769712 |

 1.50 | 0.997494987 | 0.070737197 | 14.10142097 |

 1.00 | 0.841470986 | 0.540302303 | 1.557407735 |

 0.50 | 0.479425548 | 0.877582557 | 0.546302504 |

 0.05 | 0.049979231 | 0.998750257 | 0.050041770 |

 0.00 | 0.000000000 | 1.000000000 | 0.000000000 |

   x  |sin(x*pi/180)|cos(x*pi/180)|tan(x*pi/180)|

------+-------------+-------------+-------------+

180.0 | 0.000000000 |-1.000000000 | 0.000000000 |

179.0 | 0.017452387 |-0.999847696 |-0.017455045 |

150.0 | 0.499999998 | 0.866025405 |-0.577350265 |

135.0 | 0.707106777 |-0.707106785 |-0.999999989 |

120.0 | 0.866025404 |-0.500000000 |-1.732050807 |

 90.1 | 0.999998477 |-0.001745334 |-572.9552528 |

 90.0 | 1.000000000 | 0.000000002 | 577350260.0*|

 89.9 | 0.999998477 | 0.001745326 | 572.9581364 |

 60.0 | 0.866025403 | 0.500000002 | 1.732050799 |

 45.0 | 0.707106784 | 0.707106779 | 1.000000006 |

 30.0 | 0.499999999 | 0.866025404 | 0.577350268 |

  1.0 | 0.017452857 | 0.999847687 | 0.017455515 |

(*) tan(90°) = oo 

    pi = 3.141592654   

Formulas:

  cos(x) ~ x^4/12 + (x^4/12)^2/140 - cosh(x) + 2

  sin(x) = sqrt(1 - (cos(x))^2)

cos(3*x) = cos(x)*(4*(cos(x))^2 - 3)

More details at Mardi Gras Basic Trigs (HP-12C).

HP 12c Platinum


001-   3	        020-  1/x

002-   /		021-   +

003- ENTER		022-   2

004- ENTER              023-   /	 

005-  x^2               024-   -

006-  x^2               025- ENTER

007-  12/               026-  x^2

008- ENTER    		027-   4

009-  x^2               028-   *

010-   1		029-   3

011-   4		030-   -

012-   0		031-   *

013-   /                032- ENTER

014-   +  		033-  x^2  

015-   2		034-  CHS

016-   + 		035-   1

017-  x<>y              036-   +

018-  e^x               037-  SQRT

019- ENTER              038- GTO 000

                        
Examples:

   x  |    sin(x)   |    cos(x)   |    tan(x)   |

------+-------------+-------------+-------------+

 3.14 | 0.001592628 |-0.999998732 |-0.001592630 |

 3.02 | 0.121293239 |-0.992616719 |-0.122195442 |

 2.50 | 0.598472143 |-0.801143617 |-0.747022294 |

 2.00 | 0.909297427 |-0.416146837 |-2.185039863 |

 1.65 | 0.996865028 |-0.079120889 |-12.59926479 |

 1.58 | 0.999999646 |-0.009203543 |-108.6492026 |

 1.57 | 0.999999683 | 0.000796327 | 1255.765641 |

 1.50 | 0.997494987 | 0.070737202 | 14.10141994 |

 1.00 | 0.841470985 | 0.540302306 | 1.557407725 |

 0.50 | 0.479425539 | 0.877582562 | 0.546302490 |

 0.05 | 0.049979169 | 0.998750260 | 0.050041708 |

 0.00 | 0.000000000 | 1.000000000 | 0.000000000 |

   x  |sin(x*pi/180)|cos(x*pi/180)|tan(x*pi/180)|

------+-------------+-------------+-------------+

180.0 | 0.000000000 |-1.000000000 | 0.000000000 |

179.0 | 0.017452383 |-0.999847696 |-0.017455041 |

150.0 | 0.499999997 | 0.866025406 |-0.577350264 |

135.0 | 0.707106780 |-0.707106785 |-1.000000003 |

120.0 | 0.866025404 |-0.500000000 |-1.732050806 |

 90.1 | 0.999998477 |-0.001745329 |-572.9571280 |

 90.0 | 1.000000000 |-2.684678-10 |-3724840447* |

 89.9 | 0.999998477 | 0.001745328 | 572.9572886 |

 60.0 | 0.866025404 | 0.500000000 | 1.732050808 |

(*) tan(90°) = oo

_{Edited by mistake. No changes.}

Edited: 1 Apr 2013, 2:48 p.m. after one or more responses were posted

Namir · 03-31-2013, 08:53 AM

Gerson,

Thank you soooo much for the listing and its algorithm. This has come at a time when I am also studying the algorithms than can be used to calculate the sine function. So I have added your method to the list. The accuracy results of your method are very impressive. So far your algorithm is at the top of the hit parade, so to speak! Using the trig identities for cos(3x) and sin(2x) I can use your method to cover a wide range of angles WHILE STILL maintaining very low errors!

Namir

Edited: 31 Mar 2013, 8:54 a.m.

Namir · 03-31-2013, 10:18 AM

It seems the presence of the cosh(x) function in your formula for calculating cos(x) is unusual and yet probably renders the formula stable enough to yield good results. My approach has been to find and use polynomials (either regular ones or rational ones like with the Pade approximants) and avoid as much as possible relying on functions like exp(x), ln(x), and so on.

The Pade approximants yield rational polynomials (of a while variety of orders to pick and choose from) that generally offer stable results, especially when the approximated function has a Taylor series with consecutive terms that alternate in signs causing the series to converge very slowly. Such slow convergence also requires a large number of terms to achieve small errors over a significant range of x values.

Edited: 31 Mar 2013, 10:19 a.m.

Gerson W. Barbosa · 03-31-2013, 11:50 AM

Hello Namir,

I am glad you have liked this approximation. I find it particularly suitable for the HP-12C, since it has e^x and divide by 12 is available as a single instruction. Even with only the term x^4/12 it is good enough for practical purposes in the range [-pi/4..pi/4]:

plot cos(x) - (x^4/12 - cosh(x) + 2), x=-pi/4..pi/4 (WolframAlpha)

Adding a second term significantly improves the approximation:

plot cos(x) - (x^4/12 + (x^4/12)^2/140 - cosh(x) + 2), x=-pi/4..pi/4 (WolframAlpha)

One additional term would perhaps allow for the evaluation of the function in the range [-pi/2..pi/2] to 10 places without range reduction.

Best regards,

Gerson.

P.S.: plot cos(x) - (x^4/12 + (x^4/12)^2/140 + (x^4/12)^3/138600 - cosh(x) + 2), x=-pi/2..pi/2

Edited: 31 Mar 2013, 1:40 p.m.

Namir · 03-31-2013, 01:42 PM

Adding the cubic term does improve accuracy a bit, and I welcome that!!!

Thanks for adding the plots!!!!

I have also tested other approximations for sin(x) that I found on the web. You methods puts them all to shame!!

Namir

Edited: 31 Mar 2013, 1:44 p.m.

Gerson W. Barbosa · 03-31-2013, 01:59 PM

Thanks, but please keep in mind that this only works when we have e^x readily available, otherwise the computation of e^x alone would not compensate at all.

Gerson.

Namir · 04-01-2013, 07:21 AM

I am thinking about expanding cosh(x) into a Taylor series polynomial (or even better yet a Pade approximant rattional polynomials) so that the whole evaluation of cos(x) (and then sin(x)) relies on basic operations only.

Namir

Namir · 04-01-2013, 08:05 AM

Using the following Pade approximation for cosh(x) works well:

cosh(x) = (39251520 + 18471600 x^2 + 1075032 x^4 + 14615 x^6)/(39251520 - 

 1154160 x^2 + 16632 x^4 - 127 x^6)

The calculated cos and sin have a very good accuracy from 0 to 89 degrees. At 90 degrees this (and many other) approximations suffer!

Namir

Edited: 1 Apr 2013, 1:53 p.m.

Thomas Klemm · 03-31-2013, 04:39 PM

Quote:
especially when the approximated function has a Taylor series with consecutive terms that alternate in signs causing the series to converge very slowly.

Maybe I do not understand you correctly, but why should an alternating series converge slowly? On the contrary: in alternating series, the Euler transformation can be used for convergence acceleration.

Kind regards

Thomas

Edited: 1 Apr 2013, 9:15 a.m. after one or more responses were posted

Namir · 04-01-2013, 08:10 AM

Maybe you are correct. I will check it.

Namir

Thomas Klemm · 03-31-2013, 03:54 PM

May I suggest the following change:

16-  x<>y

17-  e^x 

18- ENTER

29-  1/x

20-   +

21-   2

22-   /  

23-   2  ; or LSTx

24-   -

25-   -

The rationale behind this change is that 2 - cosh(x) < 1 for x # 0, thus we gain one digit. For small values of x this makes a difference:

   x  |    sin(x)   |    cos(x)   |    tan(x)   |

------+-------------+-------------+-------------+

 0.05 | 0.049979164 | 0.998750261 | 0.050041703 |

Kind regards

Thomas

Edited: 31 Mar 2013, 4:00 p.m.

Gerson W. Barbosa · 03-31-2013, 04:46 PM

Thanks for the improvement!

I've just tried the following on the fast HP-12C+:

01 STO 0

02 0

03 STO 1

04 STO 2

05 RCL 0

06 RCL 1

07 y^x

08 LASTx

09 n!

10 /

11 STO+ 2

12 4

13 STO+ 1

14 2

15 0

16 RCL 1

17 x<=y?

18 GTO 05

19 RCL 2

20 2

21 *

22 RCL 0

23 e^x

24 ENTER

25 1/x

26 + 

27 2

28 /

29 -

30 ENTER

31 ENTER

32 *

33 CHS

34 1

35 +

36 SQRT

37 GTO 00
For your example, I get

   x  |    sin(x)   |    cos(x)   |    tan(x)   |

------+-------------+-------------+-------------+

 0.05 | 0.049979164 | 0.998750261 | 0.050041736 |

With 20 in lines 14 and 15 (6 terms of the Taylor expansion) the valid range for cos(x) appears to go from -pi to pi without range reduction (this have to be checked). The hp 12 Platinum might give more accurate results, but I don't know if it will be fast enough. There are many rounding errors involved, but perhaps your method should apply here. I have to leave now, so I'll see this later.

Cheers,

Gerson.

P.S.: There was a typo in line 19 (RCL 1 instead of RCL 2);

It takes about four seconds on the hp 12 Platinum:

   x  |    sin(x)   |    cos(x)   |    tan(x)   |

------+-------------+-------------+-------------+

 0.05 |0.04997916917| 0.9987502604|0.05004170837|

Formula:

Edited: 31 Mar 2013, 7:36 p.m.

Namir · 03-31-2013, 09:29 PM

Ah! I was going to ask you for the equation that generates the polynomial part.

Namir

Gerson W. Barbosa · 04-01-2013, 12:21 AM

This is better explained in this thread, message #5:

http://www.hpmuseum.org/cgi-sys/cgiwrap/hpmuseum/archv020.cgi?read=179837

For testing purpose, the following should give at least 10 correct digits on the HP-50g, in the range [-pi..pi].

%%HP: T(3)A(R)F(.);

\<< \-> x

  \<< '2*\GS(k=0,5,x^(4*k)/(4*k)!)-COSH(x)' EVAL

  \>>

\>>

Gerson.

Gerson W. Barbosa · 04-01-2013, 09:54 PM

Thomas,

I'll accept your suggestion. Listings and examples below. The results comparisons have been made against the HP-15C.

Cheers,

Gerson.

HP-12C


01-   3	                21-   2

02-   /		        22-   /

03- ENTER		23-  LSTx

04- ENTER               24-   -	 

05-   4                 25-   -

06-  y^x                26- ENTER

07-  12/                27- ENTER

08- ENTER    		28-   *

09- ENTER               29-   4

10-   *			30-   *

11-   1			31-   3

12-   4			32-   -

13-   0			33-   *

14-   /                 34- ENTER

15-   +  		35- ENTER  

16-  x<>y		36-   *

17-  e^x 		37-  CHS

18-  ENTER              38-   1

19-  1/x                39-   +

20-   +                 40-  SQRT

                        41- GTO 00

Usage:
Enter angle in radians ( 0 <= x <= pi )

 R/S  ->  sin(x)


x<>y  ->  cos(x)


  /   ->  tan(x)

Examples:

   x  |      sin(x)      |      cos(x)      |      tan(x)       |

------+------------------+------------------+-------------------+

 3.14 | 0.001592545(653) |-0.999998732      |-0.001592547(655)  | 

 3.02 | 0.121293238(55)  |-0.992616719(7)   |-0.122195441(59)   |

 2.50 | 0.598472142(4)   |-0.801143617(6)   |-0.747022293(7)    |

 2.00 | 0.909297427      |-0.416146836(7)   |-2.185039866(3)    |

 1.65 | 0.996865029      |-0.079120888(9)   |-12.59926494(79)   |

 1.58 | 0.999999647      |-0.009203547(3)   |-108.6491585(2036) |

 1.57 | 0.999999683      | 0.000796325(7)   | 1255.768346(5591) |

 1.50 | 0.997494987      | 0.070737197(202) | 14.10142097(1995) |

 1.00 | 0.841470986(5)   | 0.540302304(6)   | 1.557407731(25)   |

 0.50 | 0.479425542(39)  | 0.877582560(2)   | 0.546302495(0)    |

 0.05 | 0.049979164(9)   | 0.998750261(0)   | 0.050041703(8)    |

 0.00 | 0.000000000      | 1.000000000      | 0.000000000       |

   x  |  sin(x*pi/180)   |  cos(x*pi/180)   |  tan(x*pi/180)    |

------+------------------+------------------+-------------------+

180.0 | 0.000002000(0000)|-1.000000000      | 0.000002000(0000) |

179.0 | 0.017452392(406) |-0.999847695      |-0.017455051(65)   |

150.0 | 0.499999997(50..)|-0.866025406(4)   |-0.577350265(9)    |

135.0 | 0.707106778(81)  |-0.707106784(1)   |-0.999999989(100..)|

120.0 | 0.866025404      |-0.500000000      |-1.732050810(08)   |

 90.1 | 0.999998477      |-0.001745333(28)  |-572.9558211(72134)|

 90.0 | 1.000000000      | 0.000000002(0)   | 1154700538 ( +oo )|

 89.9 | 0.999998477      | 0.001745327(8)   | 572.9575675(2134) |

 60.0 | 0.866025405(4)   | 0.499999999(50..)| 1.732050814(08)   |

 45.0 | 0.707106781      | 0.707106782(1)   | 0.999999999(10..) |

 30.0 | 0.500000007(0)   | 0.866025400(4)   | 0.577350280(69)   |

  1.0 | 0.017452793(406) | 0.999847688(95)  | 0.017455452(065)  |

   pi = 3.141592654

HP 12c Platinum


001-   3	        020-   2

002-   /		021-   /

003- ENTER		022-  LSTx

004- ENTER              023-   -	 

005-  x^2               024-   -

006-  x^2               025- ENTER

007-  12/               026-  x^2

008- ENTER    		027-   4

009-  x^2               028-   *

010-   1		029-   3

011-   4		030-   -

012-   0		031-   *

013-   /                032- ENTER

014-   +  		033-  x^2  

015-  x<>y		034-  CHS

016-  e^x		035-   1

017-  ENTER             036-   +

018-   1/X              037-  SQRT

019-    +               038- GTO 000
Examples:

   x  |      sin(x)      |      cos(x)      |      tan(x)       |

------+------------------+------------------+-------------------+

 3.14 | 0.001592628(53)  |-0.999998732      |-0.001592630(55)   | 

 3.02 | 0.121293239(55)  |-0.992616719(7)   |-0.122195442(59)   |

 2.50 | 0.598472143(4)   |-0.801143617(6)   |-0.747022294(7)    |

 2.00 | 0.909297427      |-0.416146837      |-2.185039862(3)    |

 1.65 | 0.996865028(9)   |-0.079120889      |-12.59926479       |

 1.58 | 0.999999646(7)   |-0.009203543      |-108.6492029(36)   |

 1.57 | 0.999999683      | 0.000796327      | 1255.765628(591)  |

 1.50 | 0.997494987      | 0.070737202      | 14.10141994(5)    |

 1.00 | 0.841470985      | 0.540302306      | 1.557407725       |

 0.50 | 0.479425539      | 0.877582562      | 0.546302490       |

 0.05 | 0.049979169      | 0.998750260      | 0.050041708       |

 0.00 | 0.000000000      | 1.000000000      | 0.000000000       |

   x  |  sin(x*pi/180)   |  cos(x*pi/180)   |  tan(x*pi/180)    |

------+------------------+------------------+-------------------+

180.0 | 0.000000000      |-1.000000000      | 0.000000000       |

179.0 | 0.017452382(406) |-0.999847696(6)   |-0.017455041(65)   |

150.0 | 0.499999997(50..)|-0.866025406(4)   |-0.577350264(9)    |

135.0 | 0.707106780(1)   |-0.707106782(1)   |-0.999999997(10..) |

120.0 | 0.866025404      |-0.500000000      |-1.732050805(8)    |

 90.1 | 0.999998477      |-0.001745329(8)   |-572.9571394(2134) |

 90.0 | 1.000000000      |-0.00000000026(+0)|-3849001795 ( +oo )|

 89.9 | 0.999998477      | 0.001745328      | 572.9572858(134)  |

 60.0 | 0.866025404      | 0.500000000      | 1.732050808       |

 45.0 | 0.707106781      | 0.707106781      | 1.000000000       |

 30.0 | 0.500000000      | 0.866025404      | 0.577350269       |

  1.0 | 0.017452408(6)   | 0.999847695      | 0.017455067(5)    |

   pi = 3.141592654

BShoring · 03-31-2013, 05:54 PM

Thank you for this nice program. It also works on the HP-38E and HP-38C which happens to be my favorite calculator, and which was the predeccessor to the HP-12C.

Gerson W. Barbosa · 03-31-2013, 07:56 PM

Which do you have, the HP-38E or HP-38C? There is another trigs program on the software library, using another approach (minimax polynomial approximation). It's very inconvenient on the HP-38E however (99 steps and 7 constants in the registers).

http://www.hpmuseum.org/software/38ctrig.htm

Regards,

Gerson.

BShoring · 04-01-2013, 01:31 AM

Gerson, I had the C, for 20 great years of heavy use. Died about 10 years ago as has my HP41CV. Fortunately I have a 38C emulator (and many others) that works perfectly.

Thanks,
Bob

Namir · 04-01-2013, 01:55 PM

Gerson,

How did the cosh(x) term end up in the approximation for cos(x)????

A very curious Namir

Gerson W. Barbosa · 04-01-2013, 03:14 PM

Just sum the respective Taylor series together and then isolate cos x:

Regards,

Gerson.

_{Edited to fix a typo and to correct a mistake in the fourth paragraph, per Thomas Klemm's observation below}

Edited: 4 Apr 2013, 1:29 p.m. after one or more responses were posted

Namir · 04-01-2013, 04:13 PM

Very elegant and very simple!!!

:-)

Namir

Les Koller · 04-01-2013, 09:50 PM

That's the way I like my math!

Gerson W. Barbosa · 04-01-2013, 10:16 PM

I fixed a missing multiplication by 2 in the fourth line soon after posting, but forgot to do the same on the third line. Sorry for the mistake!

_{It's been corrected now!}

Edited: 4 Apr 2013, 1:42 p.m.

Namir · 04-01-2013, 04:20 PM

One can use a similar approach for sin(x):

sin(x) = x (2 + y/60 + y^2/181440 + y^3/3113510400 + 

         y^4/177843714048000 + y^5/25545471085854720000 + 

         y^6/7755605021665492992000000 + 

         y^7/4420880996869850977271808000000) - sinh(x)
Where y = x^4

Namir

Edited: 1 Apr 2013, 4:21 p.m.

Gerson W. Barbosa · 04-02-2013, 07:23 PM

Which of course works nicely :-)

Thomas Klemm · 04-04-2013, 11:41 AM

Quote:

The factor 2 is missing:

But of course we still get the idea.

Kind regards

Thomas

Gerson W. Barbosa · 04-04-2013, 01:41 PM

The last line was right, nonetheless. The previous line has now been corrected, thanks!

Cheers,

Gerson.

Gerson W. Barbosa · 04-03-2013, 09:17 PM

The following gives about 10 to 11 significant digits in the range [0..+/-pi], in case anyone wants to try. They should work in other HP-17B models as well.

SIN=2*SIGMA(K:1:21:4:X^K/FACT(K))-(EXP(X)-EXP(-X))/2
COS=2*SIGMA(K:0:20:4:X^K/FACT(K))-(EXP(X)+EXP(-X))/2

TAN=(4*SIGMA(K:1:21:4:X^K/FACT(K))-EXP(X)+EXP(-X))/(4*SIGMA(K:0:20:4:X^K/FACT(K))-EXP(X)-EXP(-X)) 

X in radians.

Eddie W. Shore · 04-04-2013, 07:53 AM

Gerson,

Simple, elegant, and accurate (to at least 6 digits). Beautiful algorithm!

Eddie

Gerson W. Barbosa · 04-04-2013, 02:19 PM

Thanks, Eddie!

Quote:
accurate (to at least 6 digits)

The next term in the Taylor series, 2*(x)^16/16!, would contribute with only 2e-13, at most (in the extremes of the [-pi..pi] range, when x = pi/3). The lost of accuracy is greater for the sine function (and tangent), because it is computed as sqrt(1 - cos²x). The lack of at least two guard digits is also responsible for many accumulated errors.

Anyway, 6 digits are quite an improvement after almost seven centuries :-)

"In 1342, Levi (ben Gerson) wrote On Sines, Chords and Arcs, which examined trigonometry, in particular proving the sine law for plane triangles and giving five-figure sine tables."

(From Wikipedia)

Cheers,

Gerson.

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