HP 15C: Fibonacci Numbers
(Updated, bug with HP 15C LE?)
(edited 3/23/2017)
Approximation
Formula
This program calculates the nth Fibonacci number.
Store n in memory register 0 before running the program.
Formula: ( (1 + √5)^n – (1 - √5)^n ) / (2^n * √5)
Caution:
Due to internal calculator calculation, you may not get an integer answer.
You might have to round the answer manually. From Joe Horn, with thanks
and appreciation for letting me post his comment:
"Hi, Eddie! I just keyed up your HP-15C program for
Fibonacci numbers, and noticed that it gets wrong answers due to round-off
errors much of the time. For example, with an input of 5, it should output 5,
but it outputs 4.9999999998. Even rounding to the nearest integer doesn't fix
the problem for inputs of 40 through 49. The only way to get the right answers
for all inputs is with a loop (the usual method)." - Joe Horn
Therefore, I would recommend using this program for n <
40. You may need to round results to the
nearest integer.
|
Step
|
Key
|
Code
|
|
001
|
LBL D
|
42, 21, 14
|
|
002
|
1
|
1
|
|
003
|
ENTER
|
36
|
|
004
|
5
|
5
|
|
005
|
√
|
11
|
|
006
|
+
|
40
|
|
007
|
RCL 0
|
45, 0
|
|
008
|
Y^X
|
14
|
|
009
|
1
|
1
|
|
010
|
ENTER
|
36
|
|
011
|
5
|
5
|
|
012
|
√
|
11
|
|
013
|
-
|
30
|
|
014
|
RCL 0
|
45, 0
|
|
015
|
Y^X
|
14
|
|
016
|
-
|
30
|
|
017
|
2
|
2
|
|
018
|
RCL 0
|
45, 0
|
|
019
|
Y^X
|
14
|
|
020
|
÷
|
10
|
|
021
|
5
|
5
|
|
022
|
√
|
11
|
|
023
|
÷
|
10
|
|
024
|
RTN
|
43, 32
|
Example: R0 = n = 16, Output: 987
Loop
Method – Joe Horn
Joe Horn provided a more accurate program which uses loops. It is slower, however should speed should not
be a problem if a HP 15C Limited Edition or emulator is used. Full credit and thanks goes to Joe Horn for
providing this program.
|
Step
|
Key
|
Code
|
|
001
|
LBL A
|
42, 21, 11
|
|
002
|
STO 0
|
44, 0
|
|
003
|
1
|
1
|
|
004
|
ENTER
|
36
|
|
005
|
0
|
0
|
|
006
|
LBL 1
|
42, 21, 1
|
|
007
|
+
|
40
|
|
008
|
LST X
|
43, 36
|
|
009
|
X<>Y
|
34
|
|
010
|
DSE 0
|
42, 5, 0
|
|
011
|
GTO 1
|
22, 1
|
|
012
|
RTN
|
43, 32
|
A nice part is that you don’t have to pre-store n in memory
0. This method is accurate for n ≤ 49.
Comparing
Results
Here are some results of some selected n:
|
n
|
Approximation
|
Loop Method
|
|
6
|
8
|
8
|
|
15
|
610
|
610
|
|
16
|
987
|
987
|
|
22
|
17710.9999
|
17711
|
|
29
|
514228.9979
|
514229
|
|
36
|
14930351.92
|
14930352
|
|
40
|
102334154.4
|
102334155
|
|
44
|
701408728.7
|
701408733
|
|
49
|
7778741992
|
7778742049
|
Bug?
I manually calculated the
approximation formula on my HP 15C LE (Limited Edition) and HP Prime for n = 44
and n = 49.
n = 44
HP15C LE: 701408728.7
HP Prime: 701408733.002
n = 49
HP 15C LE: 7778741992
HP Prime: 7778742049.02
I think we may have found
a bug on the HP 15C LE.
Thanks to Joe Horn for the comments, much appreciated.
This blog is property of Edward Shore, 2017.
HP 15C: Fibonacci Numbers
(Updated, bug with HP 15C LE?)