## Wednesday, March 22, 2017

### HP 15C: Fibonacci Numbers

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?)

1. 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.

I 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!

2. Hi I Copied the Loop Method into the HP 41CX and it gives the exact numbers like HP Prime. Thanks Jon, it was fun to program My HP Calc.

### HP 42S/DM 42/Free42: Function Table

HP 42S/DM 42/Free42: Function Table Introduction The program FTAB uses the function defined in FX, with variable “X” to gene...