Saturday, November 25, 2023

HP 15C, DM42, HP 27S: Floor and Ceiling Functions

HP 15C, DM42, HP 27S:   Floor and Ceiling Functions


(Updated on 12/22/2023 - see note below for the HP 15C versions)


Notes :


HP 15C: includes  HP 15C, Original, Limited, and Collector's Edition, and apps

DM42: includes HP 42S, DM42, Free42, Plus42




Introduction:  Floor and Ceiling Functions


The floor and ceiling functions are two common functions that transfers a number to an integer.   



Floor Function



Floor:   The greatest integer that is less than or equal to x.  A common symbol for floor is |_x_| and a function call is floor(x).


Let int(x) be the integer part function (sometimes labeled intg(x) or lp(x)), and frac(x) be the fractional part function (sometimes labeled fp(x)).


#  x is an integer

If frac(x) = 0, then return floor(x) =  x     


# x is not an integer

If x ≥ 0, then return floor(x) = int(x)

If x < 0, then return floor(x) = int(x) - 1


Per the function.wolfram.com web page: an equivalent using the modulus function for floor(x) is:


floor(x) = x - x mod 1


Examples:


floor(2.8) = 2

floor(6) = 6

floor(-2.8) = -3



Ceiling Function


Ceiling:  The least integer that is greater than or equal to x.  A common symbol for ceiling is |-x-| (except the horizontal lines are the top) and a function call is either ceil(x) or ceiling(x).  


#  x is an integer

If frac(x) = 0, then return ceil(x) =  x     


# x is not an integer

If x ≥ 0, then return ceil(x) = int(x) + 1

If x < 0, then return ceil(x) = int(x) 


Per the function.wolfram.com web page: an equivalent using the modulus function for floor(x) is:


ceil(x) = x + (-x) mod 1


Examples:

ceil(2.8) = 3

ceil(6) = 6

ceil(-2.8) = -2



Calculator Code:  HP 15C, DM 42, HP 27S



HP 15C


This program is NOT accurate for -1 < x < 0 (such as x = -0.5).  Gratitude to Werner for alerting me of this.  (12/22/2023)


Three labels are used:

D:  floor function

E:  ceiling function

1:  used in calculation for both  (roll stack down one extra time when frac(x)≠0)


step #:  key code : key


000:  42, 21, 14:  LBL D

001:  __, __, 36:  ENTER

002:  __, 42, 44:  FRAC

003:  __, 43, 20:  x=0

004:  __, 22, _1:  GTO 1

005:  __, __, 33:  R↓

006:  __, 43, 44:  INT

007:  43, 30, _3:  TEST 3  (x≥0)

008:  __, 43, 32:  RTN

009:  __, __, _1:  1

010:  __, __, 30:  -

011:  __, 43, 32:  RTN


012:  42, 21, 15:  LBL E

013:  __, __, 36:  ENTER

014:  __, 42, 44:  FRAC

015:  __, 43, 20:  x=0

016:  __, 22, _1:  GTO 1

017:  __, __, 33:  R↓

018:  __, 43, 44:  INT

019:  43, 30, _2:  TEST 2 (x<0)

020:  __, 43, 32:  RTN

021:  __, __, _1:  1

022:  __, __, 40:  +

023:  __, 43, 32:  RTN


024:  42, 21, _1:  LBL 1

025:  __, __, 33:  R↓

026:  __, 43, 32:  RTN






DM42/Free 42/HP 42S


00 { 16-Byte Prgm }

01▸LBL "FLOOR"

02 ENTER

03 ENTER

04 1

05 MOD

06 -

07 RTN

08 END


00 { 15-Byte Prgm }

01▸LBL "CEIL"

02 ENTER

03 +/-

04 1

05 MOD

06 +

07 RTN

08 .END.



HP 27S


FLOOR=X-MOD(X:1)


CEIL=X+MOD(-X:1)



Sources


Wolfram Research, Inc.   "Floor Function".  Path:  Integer Functions > Floor[z] > Representations through equivalent functions > With related functions.   Retrieved October 30, 2023.


https://functions.wolfram.com/IntegerFunctions/Floor/27/01/05/



Wolfram Research, Inc.   "Ceiling Function".  Path:  Integer Functions > Ceiling[z] > Representations through equivalent functions > With related functions.   Retrieved October 30, 2023.


https://functions.wolfram.com/IntegerFunctions/Ceiling/27/01/05/




Eddie


All original content copyright, © 2011-2023.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 


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