Decimal to Fraction for the HP-41
Overview
-The 3 following programs find the best approximations of a decimal
number x by a fraction p/q.
-"DF" & "DF2" are based on continued fractions:
-We define (pk) & (qk) by q -1 = 0 , q0 = 1 and qk = yk.qk-1 + qk-2
where yk = INT(xk) with x0 = x , xk+1 = 1/FRC(xk) we have then pk = RND(x.qk) [ RND must be executed in FIX 0 ]
-"DF3" uses a more direct - but much slower - approach.
1°) Program#1
-This program displays the successive pk/qk
and stops when the fraction is exactly equal to x
-Synthetic register M may be replaced by any data register.
Data Registers: /
Flag: F29
Subroutines: /
-Line 29 is append /
-If you don't have an X-Functions module, delete lines 38-39
this will change the final outputs, but not the displayed fractions.
| 01 LBL "DF"
02 SIGN 03 0 04 LASTX 05 ENTER^ 06 LBL 01 07 X<> L 08 FRC |
09 1/X
10 INT 11 R^ 12 ST* Y 13 X<> T 14 ST+ Y 15 X<> L 16 RDN |
17 STO M
18 X<>Y 19 ST* Y 20 X<>Y 21 FIX 0 22 CF 29 23 RND 24 R^ |
25 SIGN
26 X<> M 27 CLA 28 ARCL Y 29 "~/" 30 ARCL X 31 FIX 4 32 SF 29 |
33 AVIEW
34 ST/ Y 35 RDN 36 X#Y? 37 GTO 01 38 R^ 39 ANUM 40 END |
( 67 bytes / SIZE 000 )
| STACK | INPUTS | when AVIEWs | OUTPUTS |
| T | / | qk-1 | qn-1 |
| Z | / | x | x |
| Y | / | pk | qn |
| X | x | qk | pn |
Examples:
a) PI XEQ "DF" the hp-41 successively displays "22/7" "333/106" "355/113" "104348/33215"
-When the program stops, X-register = 104348 & Y-register = 33215
b) 1 E^X R/S >>> "3/1" "8/3" "11/4"
"19/7" "87/32" "106/39" "193/71" "1264/465"
"1457/536"
"2721/1001" "23225/8544" "25946/9545" "49171/18089"
"173459/63812"
and eventually, X = 173459 & Y = 63812
-Some numbers produce many continued fractions:
-A famous example is the golden ratio phi = (1+sqrt(5))/2
"DF" displays 22 fractions before it stops!
-The last one is 75025/46368
-If you set flag F21, the calculator will stop at each AVIEW
-The numerators are then in Y-register and the denominators in X-register
-At the end, X = the numerator, Y = the denominator.
2°) Program#2
-The following routine stops when the rounded difference
( x - p/q ) equals 0 in the current display format.
-Synthetic registers M N O P may be replaced by any data registers.
Data Registers: /
Flags: /
Subroutines: /
-Line 04 may be replaced by RCLFLAG
-Line 28 may be replaced by STOFLAG
| 01 LBL "DF2"
02 STO M 03 STO N 04 RCL d 05 STO O 06 CLST 07 SIGN 08 ENTER^ |
09 LBL 01
10 X<> N 11 FRC 12 1/X 13 STO N 14 INT 15 RCL Y 16 * |
17 R^
18 + 19 RCL M 20 RCL Y 21 * 22 FIX 0 23 RND 24 STO P |
25 RCL Y
26 / 27 RCL O 28 STO d 29 CLX 30 RCL M 31 - 32 RND |
33 X#0?
34 GTO 01 35 X<> M 36 X<> L 37 STO Z 38 X<> P 39 CLA 40 END |
( 68 bytes / SIZE 000 )
| STACK | INPUTS | OUTPUTS |
| T | / | qn-1 |
| Z | / | pn/qn - x |
| Y | / | qn |
| X | x | pn |
| L | / | x |
Examples:
FIX 6 PI XEQ "DF2" >>>>
355 RDN 113 RDN 2.66 E-7
whence PI ~ 355/113 and 355/113 - PI =
2.66 E-7
FIX 6 1 E^X
R/S >>>> 2721 RDN 1001 RDN
-1.10 E-7 whence e ~ 2721/1001 and
2721/1001 - e = -1.1 E-7
T = the next to last denominator
The decimal number x is saved in L-register.
3°) Program#3
-The continued fractions give the most interesting results, and moreover,
very fast.
-However, occasionaly, you might need to know the best fractional approximation
p/q of a ( positive ) real x where q is between 2 given integers
q1 & q2
-The program hereunder will give you the answer, but not very fast
if q is "great"
Data Registers: R00 = x
R01 = p R03 = | x - p/q |
R05 = q2
R02 = q R04 = q1 , 1+q1 ,
..... , q2
Flags: /
Subroutines: /
| 01 LBL "DF3"
02 STO 00 03 X<>Y 04 STO 05 05 1 06 STO 03 07 R^ 08 LBL 01 09 STO 04 10 RCL 00 |
11 *
12 INT 13 ENTER^ 14 ENTER^ 15 SIGN 16 LASTX 17 ST+ Y 18 RCL 04 19 ST/ Z 20 / |
21 RCL 00
22 ST- Z 23 - 24 ABS 25 X<=Y? 26 GTO 02 27 X<>Y 28 ISG Z 29 LBL 02 30 RCL 03 |
31 X<=Y?
32 GTO 03 33 X<>Y 34 STO 03 35 R^ 36 STO 01 37 RCL 04 38 STO 02 39 LBL 03 40 RCL 05 |
41 RCL 04
42 1 43 + 44 X<=Y? 45 GTO 01 46 RCL 03 47 RCL 02 48 RCL 01 49 END |
( 64 bytes / SIZE 006 )
| STACK | INPUTS | OUTPUTS |
| Z | q1 | | x - p/q | |
| Y | q2 | q |
| X | x | p |
-We must have x > 0 and 0 < q1 <= q2
Example:
50 ENTER^
100 ENTER^
PI XEQ "DF3" >>>> p = 311
( execution time = 45 seconds )
RDN q = 99
RDN 1.79 E-4 so 311/99
approximates PI with an error of 1.79 E-4 in magnitude.
-This approximation is better than 22/7 but of course less accurate
than 333/106