hyperbolics

Hyperbolic Functions for the HP-41

Overview

-This program computes the hyperbolic sine, cosine, tangent, the Gudermannian function and their inverses.
-I've used E^X-1 & LN1+X instead of E^X & LNX in order to achieve a good accuracy for small arguments too.

Formulae:

   Sh x = Sinh x = [ ( e^x - 1 ) - ( e^-x - 1 ) ]/2                                Ash x = Asinh x = Ln [ 1 + x + x² / ( 1 + ( 1 + x²)^1/2 ) ]
   Ch x = Cosh x = ( e^x + e^-x )/2                                                 Ach x = Acosh x = Ln [ 1 + ( x - 1 ) + ( ( x + 1 )( x - 1 ) )^1/2 ]
   Th x = Tanh x = ( e^2x - 1 )/( e^2x + 1 )                                       Ath x = Atanh x = (1/2).Ln ( 1 + 2x/( 1 - x ) )
   Gd x = 2.Arctan ( e^x ) - Pi/2 = 2.Arctan ( tanh x/2 )                 Agd x = Ln Tan ( x/2 + Pi/4 ) = 2.Atanh ( tan x/2 )

Program Listing

01 LBL "SH"
02 E^X-1
03 LASTX
04 CHS
05 E^X-1
06 -
07 2
08 /
09 RTN
10 LBL "CH"
11 E^X
12 LASTX
13 CHS
14 E^X
15 +
16 2
17 /
18 RTN

19 LBL "TH"
20 ST+ X
21 LBL 01
22 E^X-1
23 RCL X
24 2
25 +
26 /
27 RTN
28 LBL "GD"
29 XEQ 01
30 RAD
31 ATAN
32 DEG
33 ST+ X
34 RTN
35 LBL "ASH"
36 SIGN

37 LASTX
38 ABS
39 ENTER^
40 X^2
41 1
42 +
43 SQRT
44 1
45 +
46 RCL Y
47 X^2
48 X<>Y
49 /
50 +
51 LN1+X
52 *
53 RTN
54 LBL "ACH"

55 1
56 -
57 ENTER^
58 STO Z
59 2
60 +
61 *
62 SQRT
63 +
64 LN1+X
65 RTN
66 LBL "ATH"
67 XEQ 02
68 2
69 /
70 RTN
71 LBL "AGD"
72 2

73 /
74 RAD
75 TAN
76 DEG
77 LBL 02
78 SIGN
79 LASTX
80 ABS
81 ENTER^
82 ST+ Y
83 CHS
84 1
85 +
86 /
87 LN1+X
88 *
89 END

( 145 bytes / SIZE 000 )

STACK	INPUT	OUTPUT
X	x	hyp(x)

Examples:

2 XEQ "SH"   >>>> Sinh 2 = 3.626860408
2 XEQ "CH" >>>> Cosh 2 = 3.762195691
2 XEQ "TH" >>>>   Tanh 2 = 0.964027580
2 XEQ "GD" >>>>   Gd 2   = 1.301760336

14   XEQ "ASH" >>>> Asinh 14 = 3.333477587
14   XEQ "ACH" >>>> Acosh 14 = 3.330926553
0.7 XEQ "ATH" >>>> Atanh 0.7 = 0.867300528
0.7 XEQ "AGD" >>>> Agd 0.7   = 0.765350459

Remarks:

-If the function f is Asinh , Atanh or Agd , and if x < 0 , this program calculates -f(-x)
to avoid a loss of accuracy for large negative arguments.
-For instance, Asinh 12000 = 10.08580911 whence Asinh (-12000) = -10.08580911
-Otherwise, it would give -10.12663110 ( like the Math Pac ) by cancellation of significant digits.

-One may also write several variations, for example:

Hyperbolic cosine for small arguments:

LBL "CHX-1"
E^X-1
LASTX
CHS
E^X-1
+
2
/
RTN

Hyperbolic tangent for large positive arguments:

LBL "1-THX"
ST+ X
E^X
1
+ for example: 12 XEQ "1-THX" >>>> 7.550269089 10^-11 whence Tanh 12 = 0.999,999,999,924,497,309,11
2
X<>Y
/
RTN

Inverse hyperbolic cosine for small arguments:

-Simply add LBL "ACH1+X" after line 56

Inverse hyperbolic tangent for arguments close to 1:

LBL "ATH1-X"
ENTER^
CHS
1
+
ST+ X for example: EEX 12 CHS XEQ "ATH1-X" >>>> Atanh ( 0.999999999999 ) = 14.16208415
X<>Y
/
LN1+X
2
/
RTN

... and so on ...

hp41programs

Hyperbolic Functions for the HP-41