Vocabulary/ltco

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<: y

Decrement

Rank 0 -- operates on individual atoms of y, producing a result of the same shape -- WHY IS THIS IMPORTANT?

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Equivalent to y-1

   <: _5 1 0 5
_6 0 _1 4

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Common uses

1. Simplify a phrase which decrements a given value

   y=. <:y             NB. decrement y

   z=: '[a,b]'
   (<: z i. ',') { z   NB. pick the letter BEFORE the comma
a

2. Simplify an unwieldy phrase

   z=: 10
   (100-z)-1
89
   <: 100-z
89

3. Compile a simpler tacit definition from a given explicit (verb) definition

the verb below computes the common mathematical expression: (y-1)/y

   13 : '(y-1)%y'
] %~ 1 -~ ]
   13 : '(<:y)%y'
<: % ]

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Related Primitives

Increment >: y

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x <: y

Less Or Equal

Rank 0 0 -- operates on individual atoms of x and y, producing a result of the same shape -- WHY IS THIS IMPORTANT?

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Compares x and y atom-by-atom. See Equal (=) for the details.

Wherever x is less than, or equal to, y, returns 1 in that position, else 0.

   z=: 7 8 9
   z <: 8
1 1 0

x and y must both be numeric, else you get a domain error

   'a' <: 9
|domain error
|   'a'    <:9
|[-0]

This differs from what happens with [[Vocabulary/eq|Equal]] (`=`)

{{{   'a' = 9
0

Less Or Equal (<:) uses tolerant comparison in the same way as Equal (=).

To require exact comparison, use (<:!.0) in place of (<:) to temporarily set the comparison tolerance to zero

   2.00000000000001 <: 2
1
   2.00000000000001 (<:!.0) 2
0

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Common uses

1. Make a conditional phrase in a verb definition (explicit definition)

if. x<:y do.  'x y'=. y;x  end.

2. Make a mask of the same shape as array, to process array in some chosen way

   array=: 3 1 4 1 5 9 2 6
   mask=: array <: 5

   array ,: mask

3 1 4 1 5 9 2 6
1 1 1 1 1 0 1 0

   mask # array       NB. keep only the atoms that are <: 5
3 1 4 1 5 2

NOTE: The Boolean atoms of mask are not only truth values but also valid numbers, 0 or 1. Therefore you can use mask directly to process array like this:

   array + 3 * mask   NB. add 3 to the atoms that are <: 5
6 4 7 4 8 9 5 6

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Related Primitives

Equal (x = y), Less Than (x < y), Larger Than (x > y), Larger Or Equal (x >: y), Not-Equal (x ~: y)

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More Information

1. If an argument to x <: y is complex, its imaginary part must be tolerantly equal to 0, using the default tolerance of 2^_44 even if the comparison x <: y itself uses a different tolerance.

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Performance Note

Comparisons are fastest when the precisions are identical. If an atom is compared against an array, it helps to choose the right precision for the atom.

   0. <: %: i. 10  NB. float list: faster than 0 <:
   00 <: i. 10  NB.  integer list: faster than 0 <:

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Use These Combinations

Combinations using x <: y that have exceptionally good performance include those shown in Searching and Matching Items: Fast List Operations (FLOs), as well as the following:

What it does	Type; Precisions; Ranks	Syntax	Variants; Restrictions	Benefits; Bug Warnings
Boolean reductions along diagonals	Boolean	+.//. y	*. = ~: < <: > >: in place of +.	avoids building argument cells

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