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
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