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 HP15C program for
Fibonacci numbers, and noticed that it gets wrong answers due to roundoff
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 prestore 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?)
The only reason the "Approximation Method" is approximate is due to the calculator's representation of √5, isn't it (other bugs notwithstanding)? It is the iterative version of the recursive function and should produce exact results if done by hand (or when evaluating √5 exactly) . When I wrote my own UserRPL version for the 48GX back in '94/'95, I had to round the result, hence the "0 RND" in the UserRPL code below.
ReplyDeleteI have the same program now on my 50G. I ran a version without rounding, however, and, using the input set from above, got the following results:
6 > 8
15 > 610.000000002
16 > 987.000000003
22 > 17711
29 > 514228.999999
36 > 14930352
40 > 102334155
44 > 701408733
49 > 7778742049.04
UserRPL is:
\<< 5 \v/ \> n k
\<< 1 k + 2 / n ^
1 k  2 / n ^
 k / 0 RND
\>>
\>>
Cheers!