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reference.bqn
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reference.bqn
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# This file gives reference implementations of BQN primitives assuming
# limited initial functionality. Implementations are designed to be
# simple and not fast.
# Not yet included: complex numbers or comparison tolerance.
# In some cases an operation is defined with limited functionality at
# first and later expanded. For convenience, rather than renaming these
# limited versions, every primitive use refers to the most recent
# definition in source code, as if redefinitions shadowed previous
# primitive definitions.
#⌜
# LAYER 0: Assumed functionality
# IEEE 754, except NaN results cause an error and -0 is converted to 0.
# LIMITED to the stated cases and atomic arguments.
+ # Add
- # Negate Subtract
× # Multiply
÷ # Reciprocal Divide
⋆ # Exponential Power
⌊ # Floor
= # Equals
≤ # Less Than or Equal to
# Other basic functionality that we need to assume
Type # 0 if 𝕩 is an array, 1 if a number, >1 otherwise
! # 𝕩 is 0 or 1; throw an error if it's 0
≢ # LIMITED to monadic case
⥊ # LIMITED to array 𝕩 and (×´𝕨)≡≢𝕩
⊑ # LIMITED to natural number 𝕨 and list 𝕩
_amend # {𝕨˙⌾(𝕗⊸⊑)𝕩}
↕ # LIMITED to number 𝕩
Identity # Left or right identity of function 𝕏
⁼ # Inverse of function 𝔽
Fill # Enclosed fill value for 𝕩
#⌜
# LAYER 1: Foundational operators and functions
# Combinators
◶ ← {𝕨((𝕨𝔽𝕩)⊑𝕘){𝔽}𝕩} # LIMITED to number left operand result
˙ ← {𝕩⋄𝕗}
⊘ ← {𝕨((1˙𝕨)-0)◶𝔽‿𝔾 𝕩}
⊢ ← {𝕩}
⊣ ← {𝕩}⊘{𝕨}
˜ ← {𝕩𝔽𝕨⊣𝕩}
∘ ← {𝔽𝕨𝔾𝕩}
○ ← {(𝔾𝕨)𝔽𝔾𝕩}
⊸ ← {(𝔽𝕨⊣𝕩)𝔾𝕩}
⟜ ← {(𝕨⊣𝕩)𝔽𝔾𝕩}
⍟ ← {𝕨𝔾◶⊢‿𝔽𝕩} # LIMITED to boolean right operand result
IsArray←0=Type
Int←(1=Type)◶⟨0,⌊⊸=⟩
Nat←(1=Type)◶⟨0,0⊸≤×⌊⊸=⟩
≢ ↩ IsArray◶⟨⟩‿≢ # LIMITED to monadic case
# LIMITED to numeric arguments for arithmetic cases
√ ← ⋆⟜(÷2) ⊘ (⋆⟜÷˜)
∧ ← ×
∨ ← (+-×)
¬ ← 1+-
| ← ×⟜× ⊘ {𝕩-𝕨×⌊𝕩÷𝕨}
< ← {⟨⟩⥊⟨𝕩⟩} ⊘ (¬≤˜)
> ← (¬≤)
≥ ← !∘0 ⊘ (≤˜)
≠ ← Length ⊘ (¬=)
= ↩ Rank ⊘ =
× ↩ 0⊸(<->) ⊘ ×
⌊ ↩ ⌊ ⊘ {𝕨{(𝕨>𝕩)⊑𝕨‿𝕩}_perv𝕩}
⌈ ← -∘⌊∘- ⊘ {𝕨{(𝕨<𝕩)⊑𝕨‿𝕩}_perv𝕩}
¨ ← _eachm # LIMITED to monadic case and array 𝕩
´ ← _fold
Rank ← 0⊑≢∘≢
Length ← (0<Rank)◶⟨1⋄0⊑≢⟩
_eachm←{
r←⥊𝕩 ⋄ F←𝔽
E←(≠r)⊸≤◶{r↩r𝕩_amend˜F𝕩⊑r⋄E𝕩+1}‿⊢
E 0 ⋄ (≢𝕩)⥊r
}
{ Identity ↩ 𝕨˙⊸=◶Identity‿𝕩 }´¨ ⟨
×‿1, ¬‿1
⌊‿∞ , ⌈‿¯∞
∨‿0 , ∧‿1
≠‿0 , =‿1
>‿0 , ≥‿1
⟩
_fold←{
! 1==𝕩
l←≠v←𝕩 ⋄ F←𝔽
r←𝕨 (0<l)◶{𝕩⋄Identity f}‿{l↩l-1⋄l⊑𝕩}⊘⊣ 𝕩
{r↩(𝕩⊑v)F r}¨(l-1)⊸-¨↕l
r
}
#⌜
# LAYER 2: Pervasion
# After defining _perv, we apply it to all arithmetic functions,
# making them pervasive. I'm not going to write that out.
ToArray ← IsArray◶<‿⊢
∾ ← {k←≠𝕨⋄k⊸≤◶⟨⊑⟜𝕨⋄-⟜k⊑𝕩˜⟩¨↕k+≠𝕩} # LIMITED to two list arguments
_table←{
m←≠a←⥊𝕨 ⋄ n←≠b←⥊𝕩 ⋄ F←𝔽
r←↕m×n
{𝕩⊸{r↩r((n×𝕨)+𝕩)_amend˜(𝕨⊑a)F(𝕩⊑b)}¨↕n}¨↕m
(𝕨∾○≢𝕩)⥊r
}
_eachd←{
_e←{ # 𝕨 has smaller or equal rank
k←≠p←≢𝕨 ⋄ q←≢𝕩
! ∧´(⊑⟜p=⊑⟜q)¨↕k
l←×´(q⊑˜k⊸+)¨↕q≠⊸-k
a←⥊𝕨 ⋄ b←⥊𝕩
q⥊⥊(≠a) (⊑⟜a𝔽l⊸×⊸+⊑b˙)_table○↕ l
}
(>○=)◶⟨𝔽_e⋄𝔽˜_e˜⟩
}
⌜ ← {(𝔽_eachm)⊘(𝔽_table)○ToArray}
¨ ↩ {(𝔽_eachm)⊘(𝔽_eachd)○ToArray}
_perv←{ # Pervasion
(⊢⊘∨○IsArray)◶⟨𝔽⋄𝔽{𝕨𝔽_perv𝕩}¨⟩
}
#⌜
# LAYER 3: Remove other limits
# Now all implementations are full except ∾; ↕ is monadic only
Deshape←IsArray◶{⟨𝕩⟩}‿⥊
Reshape←{
! 1≥=𝕨
s←Deshape 𝕨
sp←+´p←¬Nat⌜s
! 1≥sp
n←≠d←Deshape 𝕩
l←sp◶(×´)‿{
lp←×´p⊣◶⊢‿1¨𝕩
! 0<lp
I←+´↕∘≠⊸×
t←I e←⟨∘,⌊,⌽,↑⟩=(I p)⊑s
! +´e
a←(2⌊t)◶⟨{!Nat𝕩⋄𝕩},⌊,⌈⟩n÷lp
s↩p⊣◶⊢‿a¨s
{d∾↩(Fill d)⌜↕𝕩-n⋄n↩𝕩}⍟(n⊸<)⍟(3=t)lp×a
} s
s⥊(↕l){!0<n⋄⊑⟜𝕩¨n|𝕨}⍟(l≠n)d
}
Range←{
I←{!Nat𝕩⋄↕𝕩}
M←{!1==𝕩⋄(<⟨⟩)⥊⊸∾⌜´I¨𝕩}
IsArray◶I‿M 𝕩
}
Pick1←{
! 1==𝕨
! 𝕨=○≠s←≢𝕩
! ∧´Int¨𝕨
! ∧´𝕨(≥⟜-∧<)s
𝕨↩𝕨+s×𝕨<0
(⥊𝕩)⊑˜0(⊑⟜𝕨+⊑⟜s×⊢)´-↕⊸¬≠𝕨
}
Pickd←(∨´∘⥊IsArray¨∘⊣)◶Pick1‿{Pickd⟜𝕩¨𝕨}
Pick←IsArray◶⥊‿⊢⊸Pickd
First←(0<≠)◶⟨!∘0,0⊸⊑⟩∘Deshape
match←{¬∘(0⊑𝕨)◶(1⊑𝕨)‿𝕩}´⟨
⟨≠○IsArray , 0⟩
⟨¬IsArray∘⊢, =⟩
⟨≠○= , 0⟩
⟨∨´≠○≢ , 0⟩
{∧´⥊𝕨Match¨𝕩}
⟩
Depth←IsArray◶0‿{1+0⌈´Depth¨⥊𝕩}
⊑ ↩ First ⊘ Pick
⥊ ↩ Deshape ⊘ Reshape
↕ ↩ Range
◶ ↩ {𝕨((𝕨𝔽𝕩)⊑𝕘){𝔽}𝕩} # Same definition, new Pick
≡ ← Depth ⊘ Match
≢ ↩ ≢ ⊘ (¬Match)
#⌜
# LAYER 4: Operators
> ↩ Merge⍟IsArray ⊘ >
≍ ← >∘Pair
⎉ ← _rankOp_
⚇ ← _depthOp_
⍟ ↩ _repeat_
˘ ← ⎉¯1
˝ ← _insert
` ← _scan
DropV← {⊑⟜𝕩¨𝕨+↕𝕨-˜≠𝕩}
Cell ← DropV⟜≢
Pair ← {⟨𝕩⟩} ⊘ {⟨𝕨,𝕩⟩}
Merge←(0<≠∘⥊)◶((∾○≢⥊⊢)⟜Fill)‿{
c←≢⊑𝕩
! ∧´⥊(c≡≢)¨𝕩
𝕩⊑⟜ToArray˜⌜↕c
}
ValidateRanks←{
! 1≥=𝕩
𝕩↩⥊𝕩
! (1⊸≤∧≤⟜3)≠𝕩
! ∧´Int¨𝕩
𝕩
}
_ranks ← {⟨2⟩⊘⟨1,0⟩ ((⊣-1+|)˜⟜≠⊑¨<∘⊢) ValidateRanks∘𝔽}
_depthOp_←{
neg←0>n←𝕨𝔾_ranks𝕩 ⋄ F←𝔽
_d←{
R←(𝕗+neg)_d
𝕨(2⥊(neg∧𝕗≥0)∨(0⌈𝕗)≥Pair○≡)◶(⟨R¨⋄R⟜𝕩¨∘⊣⟩≍⟨(𝕨R⊢)¨∘⊢⋄F⟩)𝕩
}
𝕨 n _d 𝕩
}
_rankOp_←{
k←𝕨(Pair○= (0≤⊢)◶⟨⌊⟜-,0⌈-⟩¨ 𝔾_ranks)𝕩
Enc←{
f←⊑⟜(≢𝕩)¨↕𝕨
c←×´s←𝕨Cell𝕩
f⥊⊑⟜(⥊𝕩)¨∘((s⥊↕c)+c×⊢)¨↕×´f
}
Enc↩(>⟜0×1+≥⟜=)◶⟨<⊢,Enc,<⌜⊢⟩
> ((⊑k)Enc𝕨) 𝔽¨ ((1-˜≠)⊸⊑k)Enc𝕩
}
_insert←{
! 1≤=𝕩
𝕨 𝔽´ <˘𝕩
}
_scan←{
! IsArray 𝕩
! 1≤=𝕩
F←𝔽
cs←1 Cell 𝕩
! (cs≡≢)𝕨
l←≠r←⥊𝕩
𝕨 (0<l)◶⊢‿{
c←×´cs
{r↩≥⟜c◶⟨⊑⟜(⥊𝕩)⊸F⋄⊢⟩⟜(⊑⟜r)¨↕l}𝕨
{r↩r𝕩_amend˜𝕨F○(⊑⟜r)𝕩}⟜(c⊸+)¨↕l-c
(≢𝕩)⥊r
} 𝕩
}
_repeat_←{
n←𝕨𝔾𝕩
f←⊑𝕨⟨𝔽⟩⊘⟨𝕨𝔽⊢⟩𝕩
l←u←0
{!Int𝕩⋄l↩l⌊𝕩⋄u↩u⌈𝕩}⚇0 n
b←𝕨{𝕏⊣}˙⊘{𝕨˙{𝔽𝕏⊣}}0
i←⟨𝕩⟩⋄P←B⊸{𝕎`i∾↕𝕩}
pos←𝕗 P u
neg←𝕗 0⊸<◶⟨i,{𝕏⁼}⊸P⟩ -l
(|⊑<⟜0⊑pos‿neg˙)⚇0 n
}
#⌜
# LAYER 5: Structural functions
⊏ ← 0⊸Select ⊘ Select
↑ ← Prefixes ⊘ Take
↓ ← Suffixes ⊘ Drop
↕ ↩ ↕ ⊘ Windows
» ← Nudge ⊘ ShiftBefore
« ← NudgeBack ⊘ ShiftAfter
⌽ ← Reverse ⊘ Rotate
/ ← Indices ⊘ Replicate
_onAxes_←{
F←𝔽
(𝔾<≡)∘⊣◶{ # One axis
! 1≤=𝕩
𝕨F𝕩
}‿{ # Multiple axes
! 1≥=𝕨
! 𝕨≤○≠≢𝕩
R←{(⊑𝕨)F(1 DropV 𝕨)⊸R˘𝕩}⍟{0<≠𝕨}
𝕨R𝕩
}⟜ToArray
}
SelSub←{
! IsArray 𝕨
! ∧´⥊Int¨ 𝕨
! ∧´⥊ 𝕨 (≥⟜-∧<) ≠𝕩
𝕨↩𝕨+(≠𝕩)×𝕨<0
c←×´s←1 Cell 𝕩
⊑⟜(⥊𝕩)¨(c×𝕨)+⌜s⥊↕c
}
Select←ToArray⊸(SelSub _onAxes_ 1)
JoinTo←{
s←𝕨Pair○≢𝕩
a←1⌈´k←≠¨s
! ∧´1≥a-k
c←(k¬a)+⟜(↕a-1)⊸⊏¨s
! ≡´c
l←+´(a=k)⊣◶1‿(⊑⊢)¨s
(⟨l⟩∾⊑c)⥊𝕨∾○⥊𝕩
}
Take←{
T←{
! Int 𝕨
l←≠𝕩
i←(l+1)|¯1⌈l⌊((𝕨<0)×𝕨+l)+↕|𝕨
i⊏JoinTo⟜(1⊸Cell⥊Fill)⍟(∨´l=i)𝕩
}
𝕨 T _onAxes_ 0 (⟨1⟩⥊˜0⌈𝕨-○≠⊢)⊸∾∘≢⊸⥊𝕩
}
Drop←{
s←(≠𝕨)(⊣↑⊢∾˜1⥊˜0⌈-⟜≠)≢𝕩
((sׯ1⋆𝕨>0)+(-s)⌈s⌊𝕨)↑𝕩
}
Prefixes ← {!1≤=𝕩 ⋄ (↕1+≠𝕩)Take¨<𝕩}
Suffixes ← {!1≤=𝕩 ⋄ (↕1+≠𝕩)Drop¨<𝕩}
ShiftBefore ← {!𝕨1⊸⌈⊸≤○=𝕩 ⋄ ( ≠𝕩) ↑ 𝕨 JoinTo 𝕩}
ShiftAfter ← {!𝕨1⊸⌈⊸≤○=𝕩 ⋄ (-≠𝕩) ↑ 𝕩 JoinTo 𝕨}
Nudge ← (1↑0↑⊢)⊸ShiftBefore
NudgeBack ← (1↑0↑⊢)⊸ShiftAfter
Windows←{
! 1≥=𝕨
! 𝕨≤○≠≢𝕩
! ∧´Nat¨⥊𝕨
s←(≠𝕨)↑≢𝕩
! ∧´𝕨≤1+s
𝕨{(∾⟜(𝕨≠⊸↓≢𝕩)∘≢⥊>)<¨⊸⊏⟜𝕩¨s(¬+⌜○↕⊢)⥊𝕨}⍟(0<≠𝕨)𝕩
}
Reverse ← {!1≤=𝕩 ⋄ (-↕⊸¬≠𝕩)⊏𝕩}
Rotate ← {!Int𝕨 ⋄ l←≠𝕩⋄(l|𝕨+↕l)⊏𝕩} _onAxes_ 0
Indices←{
! 1==𝕩
! ∧´Nat¨𝕩
⟨⟩∾´𝕩⥊¨↕≠𝕩
}
Rep ← Indices⊸⊏
Replicate ← {0<=𝕨}◶(⥊˜⟜≠Rep⊢)‿{!𝕨=○≠𝕩⋄𝕨Rep𝕩} _onAxes_ (1-0=≠)
#⌜
# LAYER 6: Everything else
∾ ↩ Join ⊘ JoinTo
⊔ ← GroupInds ⊘ Group
⍉ ← Transpose ⊘ ReorderAxes
∊ ← MarkFirst ⊘ (IndexOf˜<≠∘⊢)
⍷ ← ∊⊸/ ⊘ Find
⊐ ← ⍷⊸IndexOf ⊘ IndexOf
⍋ ← Cmp _grade ⊘ ( Cmp _bins)
⍒ ← -∘Cmp _grade ⊘ (-∘Cmp _bins)
∧ ↩ ⍋⊸⊏ ⊘ ∧
∨ ↩ ⍒⊸⊏ ⊘ ∨
⊒ ← OccurrenceCount⊘ ProgressiveIndexOf
{ Identity ↩ 𝕨˙⊸=◶Identity‿𝕩 }´¨ ⟨ ∨‿0 , ∧‿1 ⟩
Join←(0<=)◶{!IsArray𝕩⋄>𝕩}‿{
C←(<⟨⟩)⥊⊸∾⌜´⊢ # Cartesian array product
! IsArray 𝕩
s←≢¨𝕩
d←≠⊑s
! ∧´⥊d=≠¨s
! d≥=𝕩
l←(≢𝕩){(𝕩⊑⟜≢a⊑˜(j=𝕩)⊸×)¨↕𝕨}¨j←↕r←=a←𝕩
! (r↑¨s)≡C l
i←C{p←+´¨↑𝕩⋄(↕⊑⌽p)-𝕩/¯1↓p}¨l
>i<¨⊸⊏¨l/𝕩
}⍟(0<≠∘⥊)
Group←{
! IsArray 𝕩
𝕨↩Pair∘ToArray⍟(2>≡)𝕨
! 1==𝕨
{!∧´Int¨𝕩⋄!∧´¯1≤𝕩}∘⥊¨𝕨
n←+´r←=¨𝕨
! n≤=𝕩
ld←(∾≢⌜𝕨)-n↑≢𝕩
! ∧´(0⊸≤∧≤⟜(r/1=r))ld
dr←r⌊(0»+`r)⊏ld∾⟨0⟩
s←dr⊣◶⟨0,¯1⊸⊑⟩¨𝕨
𝕨↩dr(⥊¯1⊸↓⍟⊣)¨𝕨
s⌈↩1+¯1⌈´¨𝕨
𝕩↩((≠¨𝕨)∾n↓≢𝕩)⥊𝕩
(𝕨⊸=/𝕩˙)¨↕s
}
GroupInds←{
! 1==𝕩
𝕩 ⊔ ↕ (1<≡)◶≠‿(∾≢¨) 𝕩
}
# Searching
IndexOf←{
c←1-˜=𝕨
! 0≤c
! c≤=𝕩
𝕨 ∧○(0<≠)⟜⥊◶⟨0⥊˜c-⊸↓≢∘⊢, (+˝∧`)≢⎉c⎉c‿∞⟩ ToArray 𝕩
}
MarkFirst←{
! 1≤=𝕩
u←0↑𝕩
(0<≠)◶⟨⟨⟩,{⊑𝕩∊u}◶{u↩u∾𝕩⋄1}‿0˘⟩𝕩
}
Find←{
r←=𝕨
! r≤=𝕩
𝕨 ≡⎉r ((1+r-⊸↑≢𝕩)⌊≢𝕨)⊸↕⎉r 𝕩
}○ToArray
ReorderAxes←{
𝕩↩<⍟(0=≡)𝕩
! 1≥=𝕨
𝕨↩⥊𝕨
! 𝕨≤○≠≢𝕩
! ∧´Nat¨⥊𝕨
r←(=𝕩)-+´¬∊𝕨
! ∧´𝕨<r
𝕨↩𝕨∾𝕨(¬∘∊˜/⊢)↕r
(𝕨⊸⊏⊑𝕩˙)¨↕⌊´¨𝕨⊔≢𝕩
}
Transpose←(0<=)◶⟨ToArray,(=-1˙)⊸ReorderAxes⟩
# Sorting
Cmp ← ⌈○IsArray◶{ # No arrays
𝕨(>-<)𝕩 # Assume they're numbers
}‿{ # At least one array
e←𝕨-˜○(∨´0=≢)𝕩
𝕨(e=0)◶e‿{
c←𝕨×∘-○(IsArray+=)𝕩
s←≢𝕨 ⋄ t←≢𝕩 ⋄ r←𝕨⌊○=𝕩
l←s{i←+´∧`𝕨=𝕩⋄m←×´i↑𝕨⋄{c↩×-´𝕩⋄m↩m×⌊´𝕩}∘(⊑¨⟜𝕨‿𝕩)⍟(r⊸>)i⋄m}○(r↑⌽)t
a←⥊𝕨⋄b←⥊𝕩
Trav←(=⟜l)◶{Trav∘(1+𝕩)⍟(0⊸=)a Cmp○(𝕩⊸⊑)b}‿c
Trav 0
}𝕩
}
_grade←{
! 1≤=𝕩
i⊐˜+´˘(𝔽⎉∞‿¯1⎉¯1‿∞˜𝕩)(⌈⟜0+=⟜0⊸×)>⌜˜i←↕≠𝕩
}
_bins←{
c←1-˜=𝕨
! 0≤c
! c≤=𝕩
LE←𝔽⎉c≤0˜
! (0<≠)◶⟨1,∧´·LE˝˘2↕⊢⟩𝕨
𝕨 (0<≠𝕨)◶⟨0⎉c∘⊢,+˝LE⎉¯1‿∞⟩ 𝕩
}
OccurrenceCount ← ⊐˜(⊢-⊏)⍋∘⍋
ProgressiveIndexOf ← {𝕨⊐○(((≢∾2˙)⥊≍˘⟜OccurrenceCount∘⥊)𝕨⊸⊐)𝕩}