Addition-chain exponentiation: Difference between revisions
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[[Category:Matrices]]
{{draft task|Logic}}{{wikipedia|Addition-chain exponentiation}}
In cases of special objects (such as with [[wp:Matrix (mathematics)|matrices]]) the operation of multiplication can be excessively expensive. In these cases the operation of multiplication should be avoided or reduced to a minimum.
Line 89 ⟶ 90:
=={{header|C}}==
Using complex instead of matrix. Requires [[Addition-chain exponentiation/Achain.c|Achain.c]]. It takes a long while to compute the shortest addition chains, such that if you don't have the chain lengths precomputed and stored somewhere, you are probably better off with a binary chain (normally not shortest but very simple to calculate) whatever you intend to use the chains for.
<
#include "achain.c" /* not common practice */
Line 159 ⟶ 160:
printf("%14d %7d %7d\n", i, seq_len(i), bin_len(i));
return 0;
}</
output
<pre>...
Line 189 ⟶ 190:
Calculating A ^ (m * n) from scratch using this method would take 'for ever' so I've calculated it instead as (A ^ m) ^ n.
<
import (
Line 592 ⟶ 593:
mx2 := mxs[0].pow(n, false)
mx2.print()
}</
{{out}}
Line 662 ⟶ 663:
I think it should nevertheless be retained as it is an interesting approach and there are other solutions to this task which are based on it.
<
Continued fraction addition chains, as described in "Efficient computation
of addition chains" by F. Bergeron, J. Berstel, and S. Brlek, published in
Line 869 ⟶ 870:
}
return m3
}</
Output (manually wrapped at 80 columns.)
<pre>
Line 916 ⟶ 917:
1.00002550055251^27182: 1.99999999997876
count of multiplies: 18
</pre>
=={{header|Haskell}}==
Generators of a nearly-optimal addition chains for a given number.
<syntaxhighlight lang="haskell">dichotomicChain :: Integral a => a -> [a]
dichotomicChain n
| n == 3 = [3, 2, 1]
| n == 2 ^ log2 n = takeWhile (> 0) $ iterate (`div` 2) n
| otherwise = let k = n `div` (2 ^ ((log2 n + 1) `div` 2))
in chain n k
where
chain n1 n2
| n2 <= 1 = dichotomicChain n1
| otherwise = case n1 `divMod` n2 of
(q, 0) -> dichotomicChain q `mul` dichotomicChain n2
(q, r) -> [r] `add` (dichotomicChain q `mul` chain n2 r)
c1 `mul` c2 = map (head c2 *) c1 ++ tail c2
c1 `add` c2 = map (head c2 +) c1 ++ c2
log2 = floor . logBase 2 . fromIntegral
binaryChain :: Integral a => a -> [a]
binaryChain 1 = [1]
binaryChain n | even n = n : binaryChain (n `div` 2)
| odd n = n : binaryChain (n - 1)</syntaxhighlight>
<pre>λ> dichotomicChain 31415
[31415,31360,15680,7840,3920,1960,980,490,245,220,110,55,50,25,20,10,5,4,2,1]
λ> length $ dichotomicChain 31415
20
λ> length $ binaryChain 31415
25
λ> length $ dichotomicChain (31415*27182)
38
λ> length $ binaryChain (31415*27182)
45</pre>
Universal monoid multiplication that uses additive chain
<syntaxhighlight lang="haskell">times :: (Monoid p, Integral a) => a -> p -> p
0 `times` _ = mempty
n `times` x = res
where
(res:_, _, _) = foldl f ([x], 1, 0) $ tail ch
ch = reverse $ dichotomicChain n
f (p:ps, c1, i) c2 = let Just j = elemIndex (c2-c1) ch
in ((p <> ((p:ps) !! (i-j))):p:ps, c2, i+1)</syntaxhighlight>
<pre>λ> 31 `times` "a"
"aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa"
λ> getSum $ 31 `times` 2
62
λ> getProduct $ 31 `times` 2
2147483648</pre>
Implementation of matrix multiplication as monoidal operation.
<syntaxhighlight lang="haskell">data M a = M [[a]] | I deriving Show
instance Num a => Semigroup (M a) where
I <> m = m
m <> I = m
M m1 <> M m2 = M $ map (\r -> map (\c -> r `dot` c) (transpose m2)) m1
where dot a b = sum $ zipWith (*) a b
instance Num a => Monoid (M a) where
mempty = I</syntaxhighlight>
<pre>-- matrix multiplication
λ> M [[2,4],[5,7]] <> M [[4,1],[6,2]]
M [[32,10],[62,19]]
-- matrix exponentiation
λ> 2 `times` M [[2,4],[5,7]]
M [[24,36],[45,69]]
-- calculation of large Fibonacci numbers
λ> 2 `times` M [[1,1],[1,0]]
M [[2,1],[1,1]]
λ> 3 `times` M [[1,1],[1,0]]
M [[3,2],[2,1]]
λ> 4 `times` M [[1,1],[1,0]]
M [[5,3],[3,2]]
λ> 20 `times` M [[1,1],[1,0]]
M [[10946,6765],[6765,4181]]
λ> 200 `times` M [[1,1],[1,0]]
M [[453973694165307953197296969697410619233826,280571172992510140037611932413038677189525],[280571172992510140037611932413038677189525,173402521172797813159685037284371942044301]]</pre>
The task
<pre>λ> :{
Main:> | let a = a = let s = sqrt (1/2) in M [[s,0,s,0,0,0]
Main:> | ,[0,s,0,s,0,0]
Main:> | ,[0,s,0,-s,0,0]
Main:> | ,[-s,0,s,0,0,0]
Main:> | ,[0,0,0,0,0,1]
Main:> | ,[0,0,0,0,1,0]]
Main:> | :}
λ> 31415 `times` a
M [[0.7071067811883202,0.0,0.0,-0.7071067811883202,0.0,0.0],
[0.0,0.7071067811883202,0.7071067811883202,0.0,0.0,0.0],
[0.7071067811883202,0.0,0.0,0.7071067811883202,0.0,0.0],
[0.0,0.7071067811883202,-0.7071067811883202,0.0,0.0,0.0],
[0.0,0.0,0.0,0.0,0.0,1.0],
[0.0,0.0,0.0,0.0,1.0,0.0]]
λ> 27182 `times` a
M [[-0.5000000000010233,-0.5000000000010233,-0.5000000000010233,0.5000000000010233,0.0,0.0],
[0.5000000000010233,-0.5000000000010233,-0.5000000000010233,-0.5000000000010233,0.0,0.0],
[-0.5000000000010233,-0.5000000000010233,0.5000000000010233,-0.5000000000010233,0.0,0.0],
[0.5000000000010233,-0.5000000000010233,0.5000000000010233,0.5000000000010233,0.0,0.0],
[0.0,0.0,0.0,0.0,1.0,0.0],
[0.0,0.0,0.0,0.0,0.0,1.0]]
λ> (31415*27182) `times` a
M [[-0.500000030038797,0.5000000300387971,-0.5000000300387971,0.5000000300387973,0.0,0.0],
[-0.5000000300387971,-0.5000000300387972,-0.5000000300387969,-0.5000000300387969,0.0,0.0],
[-0.5000000300387972,-0.5000000300387971,0.5000000300387969,0.5000000300387969,0.0,0.0],
[0.5000000300387971,-0.500000030038797,-0.5000000300387973,0.5000000300387971,0.0,0.0],
[0.0,0.0,0.0,0.0,1.0,0.0],
[0.0,0.0,0.0,0.0,0.0,1.0]]</pre>
=={{header|Julia}}==
Uses an iterator to generate A003313 (the first 100 values). Uses the Knuth path method for the larger integers. This gives a chain length of 18 for both 27182 and 31415.
<syntaxhighlight lang="julia">import Base.iterate
mutable struct AdditionChains{T}
chains::Vector{Vector{T}}
work_chain::Int
work_element::Int
AdditionChains{T}() where T = new{T}([[one(T)]], 1, 1)
end
function Base.iterate(acs::AdditionChains, state = 1)
i, j = acs.work_chain, acs.work_element
newchain = [acs.chains[i]; acs.chains[i][end] + acs.chains[i][j]]
push!(acs.chains, newchain)
if j == length(acs.chains[i])
acs.work_chain += 1
acs.work_element = 1
else
acs.work_element += 1
end
return newchain, state + 1
end
function findchain!(acs::AdditionChains, n)
@assert n > 0
n == 1 && return [one(eltype(first(acs.chains)))]
idx = findfirst(a -> a[end] == n, acs.chains)
if idx == nothing
for (i, chain) in enumerate(acs)
chain[end] == n && return chain
end
end
return acs.chains[idx]
end
""" memoization for knuth_path """
const knuth_path_p, knuth_path_lvl = Dict(1 => 0), [[1]]
""" knuth path method for addition chains """
function knuth_path(n)::Vector{Int}
iszero(n) && return Int[]
while !haskey(knuth_path_p, n)
q = Int[]
for x in first(knuth_path_lvl), y in knuth_path(x)
if !haskey(knuth_path_p, x + y)
knuth_path_p[x + y] = x
push!(q, x + y)
end
end
knuth_path_lvl[begin] = q
end
return push!(knuth_path(knuth_path_p[n]), n)
end
function pow(x, chain)
p, products = 0, Dict{Int, typeof(x)}(0 => one(x), 1 => x)
for i in chain
products[i] = products[p] * products[i - p]
p = i
end
return products[chain[end]]
end
function test_addition_chains()
additionchain = AdditionChains{Int}()
println("First one hundred addition chain lengths:")
for i in 1:100
print(rpad(length(findchain!(additionchain, i)) -1, 3), i % 10 == 0 ? "\n" : "")
end
println("\nKnuth chains for addition chains of 31415 and 27182:")
expchains = Dict(i => knuth_path(i) for i in [31415, 27182])
for (n, chn) in expchains
println("Exponent: ", rpad(n, 10), "\n Addition Chain: $(chn[begin:end-1]))")
end
println("\n1.00002206445416^31415 = ", pow(1.00002206445416, expchains[31415]))
println("1.00002550055251^27182 = ", pow(1.00002550055251, expchains[27182]))
println("1.00002550055251^(27182 * 31415) = ", pow(BigFloat(pow(1.00002550055251, expchains[27182])), expchains[31415]))
println("(1.000025 + 0.000058i)^27182 = ", pow(Complex(1.000025, 0.000058), expchains[27182]))
println("(1.000022 + 0.000050i)^31415 = ", pow(Complex(1.000022, 0.000050), expchains[31415]))
x = sqrt(1/2)
matrixA = [x 0 x 0 0 0; 0 x 0 x 0 0; 0 x 0 -x 0 0; -x 0 x 0 0 0; 0 0 0 0 0 1; 0 0 0 0 1 0]
println("matrix A ^ 27182 = ")
display(pow(matrixA, expchains[27182]))
println("matrix A ^ 31415 = ")
display(round.(pow(matrixA, expchains[31415]), digits=6))
println("(matrix A ^ 27182) ^ 31415 = ")
display(pow(pow(matrixA, expchains[27182]), expchains[31415]))
end
test_addition_chains()
</syntaxhighlight>{{out}}
<pre>
First one hundred addition chain lengths:
0 1 2 2 3 3 4 3 4 4
5 4 5 5 5 4 5 5 6 5
6 6 6 5 6 6 6 6 7 6
7 5 6 6 7 6 7 7 7 6
7 7 7 7 7 7 8 6 7 7
7 7 8 7 8 7 8 8 8 7
8 8 8 6 7 7 8 7 8 8
9 7 8 8 8 8 8 8 9 7
8 8 8 8 8 8 9 8 9 8
9 8 9 9 9 7 8 8 8 8
Knuth chains for addition chains of 31415 and 27182:
Exponent: 27182
Addition Chain: [1, 2, 3, 5, 7, 14, 21, 35, 70, 140, 143, 283, 566, 849, 1698, 3396, 6792, 6799, 13591])
Exponent: 31415
Addition Chain: [1, 2, 3, 5, 7, 14, 28, 56, 61, 122, 244, 488, 976, 1952, 3904, 7808, 15616, 15677, 31293])
1.00002206445416^31415 = 1.9999999998924638
1.00002550055251^27182 = 1.9999999999792688
1.00002550055251^(27182 * 31415) = 7.199687435551025768365237017391630520475805934721292725697031724530209692195819e+9456
(1.000025 + 0.000058i)^27182 = -0.01128636963542673 + 1.9730308496660347im
(1.000022 + 0.000050i)^31415 = 0.00016144681325535107 + 1.9960329014194498im
matrix A ^ 27182 =
6×6 Matrix{Float64}:
-0.5 -0.5 -0.5 0.5 0.0 0.0
0.5 -0.5 -0.5 -0.5 0.0 0.0
-0.5 -0.5 0.5 -0.5 0.0 0.0
0.5 -0.5 0.5 0.5 0.0 0.0
0.0 0.0 0.0 0.0 1.0 0.0
0.0 0.0 0.0 0.0 0.0 1.0
matrix A ^ 31415 =
6×6 Matrix{Float64}:
0.707107 0.0 -0.0 -0.707107 0.0 0.0
-0.0 0.707107 0.707107 -0.0 0.0 0.0
0.707107 -0.0 -0.0 0.707107 0.0 0.0
0.0 0.707107 -0.707107 -0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 1.0
0.0 0.0 0.0 0.0 1.0 0.0
(matrix A ^ 27182) ^ 31415 =
6×6 Matrix{Float64}:
-0.5 0.5 -0.5 0.5 0.0 0.0
-0.5 -0.5 -0.5 -0.5 0.0 0.0
-0.5 -0.5 0.5 0.5 0.0 0.0
0.5 -0.5 -0.5 0.5 0.0 0.0
0.0 0.0 0.0 0.0 1.0 0.0
0.0 0.0 0.0 0.0 0.0 1.0
</pre>
=={{header|Nim}}==
{{trans|Go}}
<
const
Line 1,253 ⟶ 1,530:
echo "Exponent: $1 x $2 = $3".format(m, n, m * n)
echo "A ^ $1 = (A ^ $2) ^ $3:\n".format(m * n, m, n)
echo mxs[0].pow(n, false)</
{{out}}
Line 1,349 ⟶ 1,626:
in the cases that I tried, somewhat faster than the method on that page.<br>
If you know the A003313 number, you can throw that at it and wait (for several billion years) or get the
power tree length and loop trying to find a path one shorter (and wait several trillion years).
path()
Note that "tries" overflows (crashes) at 1073741824, which I kept in as a deliberate limiter.
<!--<
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #008080;">constant</span> <span style="color: #000000;">p</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">new_dict</span><span style="color: #0000FF;">({{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">0</span><span style="color: #0000FF;">}})</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">lvl</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">}</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">path</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #0000FF;">{}</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">while</span> <span style="color: #7060A8;">getd_index</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">)=</span><span style="color: #004600;">NULL</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">q</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{}</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">lvl</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">x</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">lvl</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">px</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">path</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">px</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">y</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">+</span><span style="color: #000000;">px</span><span style="color: #0000FF;">[</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">getd_index</span><span style="color: #0000FF;">(</span><span style="color: #000000;">y</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">)!=</span><span style="color: #004600;">NULL</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #7060A8;">setd</span><span style="color: #0000FF;">(</span><span style="color: #000000;">y</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">q</span> <span style="color: #0000FF;">&=</span> <span style="color: #000000;">y</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #000000;">lvl</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">q</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">path</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">getd</span><span style="color: #0000FF;">(</span><span style="color: #000000;">n</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">))&</span><span style="color: #000000;">n</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">t1</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">time</span><span style="color: #0000FF;">()+</span><span style="color: #000000;">1</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">tries</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
Line 1,363 ⟶ 1,664:
<span style="color: #008080;">if</span> <span style="color: #000000;">last</span><span style="color: #0000FF;">=</span><span style="color: #000000;">target</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">chosen</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">l</span><span style="color: #0000FF;">=</span><span style="color: #000000;">len</span> <span style="color: #008080;">then</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">platform</span><span style="color: #0000FF;">()!=</span><span style="color: #004600;">JS</span> <span style="color: #008080;">and</span> <span style="color: #7060A8;">time</span><span style="color: #0000FF;">()></span><span style="color: #000000;">t1</span> <span style="color: #008080;">then</span>
<span style="color: #0000FF;">?{</span><span style="color: #008000;">"addition_chain"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">chosen</span><span style="color: #0000FF;">,</span><span style="color: #000000;">tries</span><span style="color: #0000FF;">}</span>
<span style="color: #000000;">t1</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">time</span><span style="color: #0000FF;">()+</span><span style="color: #000000;">1</span>
Line 1,371 ⟶ 1,672:
<span style="color: #004080;">integer</span> <span style="color: #000000;">next</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">last</span><span style="color: #0000FF;">+</span><span style="color: #000000;">chosen</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">next</span><span style="color: #0000FF;"><=</span><span style="color: #000000;">target</span> <span style="color: #008080;">then</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">addition_chain</span><span style="color: #0000FF;">(</span><span style="color: #000000;">target</span><span style="color: #0000FF;">,</span><span style="color: #000000;">len</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">deep_copy</span><span style="color: #0000FF;">(</span><span style="color: #000000;">chosen</span><span style="color: #0000FF;">)&</span><span style="color: #000000;">next</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">res</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">res</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
Line 1,396 ⟶ 1,697:
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"addition_chain(31,8):%s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #7060A8;">sprint</span><span style="color: #0000FF;">(</span><span style="color: #000000;">addition_chain</span><span style="color: #0000FF;">(</span><span style="color: #000000;">31</span><span style="color: #0000FF;">,</span><span style="color: #000000;">8</span><span style="color: #0000FF;">))})</span>
<!--</
{{out}}
Line 1,450 ⟶ 1,751:
--1.00003 ^ 27182 (20) = 1.9999999999727968238298855
--1.00003 ^ 27182 (19) = 1.9999999999727968238298527</pre>
=={{header|Python}}==
<syntaxhighlight lang="python">''' Rosetta Code task Addition-chain_exponentiation '''
from math import sqrt
from numpy import array
from mpmath import mpf
class AdditionChains:
''' two methods of calculating addition chains '''
def __init__(self):
''' memoization for knuth_path '''
self.chains, self.idx, self.pos = [[1]], 0, 0
self.pat, self.lvl = {1: 0}, [[1]]
def add_chain(self):
''' method 1: add chains depth then breadth first until done '''
newchain = self.chains[self.idx].copy()
newchain.append(self.chains[self.idx][-1] +
self.chains[self.idx][self.pos])
self.chains.append(newchain)
if self.pos == len(self.chains[self.idx])-1:
self.idx += 1
self.pos = 0
else:
self.pos += 1
return newchain
def find_chain(self, nexp):
''' method 1 interface: search for chain ending with n, adding more as needed '''
assert nexp > 0
if nexp == 1:
return [1]
chn = next((a for a in self.chains if a[-1] == nexp), None)
if chn is None:
while True:
chn = self.add_chain()
if chn[-1] == nexp:
break
return chn
def knuth_path(self, ngoal):
''' method 2: knuth method, uses memoization to search for a shorter chain '''
if ngoal < 1:
return []
while not ngoal in self.pat:
new_lvl = []
for i in self.lvl[0]:
for j in self.knuth_path(i):
if not i + j in self.pat:
self.pat[i + j] = i
new_lvl.append(i + j)
self.lvl[0] = new_lvl
returnpath = self.knuth_path(self.pat[ngoal])
returnpath.append(ngoal)
return returnpath
def cpow(xbase, chain):
''' raise xbase by an addition exponentiation chain for what becomes x**chain[-1] '''
pows, products = 0, {0: 1, 1: xbase}
for i in chain:
products[i] = products[pows] * products[i - pows]
pows = i
return products[chain[-1]]
if __name__ == '__main__':
# test both addition chain methods
acs = AdditionChains()
print('First one hundred addition chain lengths:')
for k in range(1, 101):
print(f'{len(acs.find_chain(k))-1:3}', end='\n'if k % 10 == 0 else '')
print('\nKnuth chains for addition chains of 31415 and 27182:')
chns = {m: acs.knuth_path(m) for m in [31415, 27182]}
for (num, cha) in chns.items():
print(f'Exponent: {num:10}\n Addition Chain: {cha[:-1]}')
print('\n1.00002206445416^31415 =', cpow(1.00002206445416, chns[31415]))
print('1.00002550055251^27182 =', cpow(1.00002550055251, chns[27182]))
print('1.000025 + 0.000058i)^27182 =',
cpow(complex(1.000025, 0.000058), chns[27182]))
print('1.000022 + 0.000050i)^31415 =',
cpow(complex(1.000022, 0.000050), chns[31415]))
sq05 = mpf(sqrt(0.5))
mat = array([[sq05, 0, sq05, 0, 0, 0], [0, sq05, 0, sq05, 0, 0], [0, sq05, 0, -sq05, 0, 0],
[-sq05, 0, sq05, 0, 0, 0], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 1, 0]])
print('matrix A ^ 27182 =')
print(cpow(mat, chns[27182]))
print('matrix A ^ 31415 =')
print(cpow(mat, chns[31415]))
print('(matrix A ** 27182) ** 31415 =')
print(cpow(cpow(mat, chns[27182]), chns[31415]))
</syntaxhighlight>{{out}}
<pre>
First one hundred addition chain lengths:
0 1 2 2 3 3 4 3 4 4
5 4 5 5 5 4 5 5 6 5
6 6 6 5 6 6 6 6 7 6
7 5 6 6 7 6 7 7 7 6
7 7 7 7 7 7 8 6 7 7
7 7 8 7 8 7 8 8 8 7
8 8 8 6 7 7 8 7 8 8
9 7 8 8 8 8 8 8 9 7
8 8 8 8 8 8 9 8 9 8
9 8 9 9 9 7 8 8 8 8
Knuth chains for addition chains of 31415 and 27182:
Exponent: 31415
Addition Chain: [1, 2, 3, 5, 7, 14, 28, 56, 61, 122, 244, 488, 976, 1952, 3904, 7808, 15616, 15677, 31293]
Exponent: 27182
Addition Chain: [1, 2, 3, 5, 7, 14, 21, 35, 70, 140, 143, 283, 566, 849, 1698, 3396, 6792, 6799, 13591]
1.00002206445416^31415 = 1.9999999998924638
1.00002550055251^27182 = 1.9999999999792688
1.000025 + 0.000058i)^27182 = (-0.01128636963542673+1.9730308496660347j)
1.000022 + 0.000050i)^31415 = (0.00016144681325535107+1.9960329014194498j)
matrix A ^ 27182 =
[[mpf('5.0272320351359433e-4092') 0 mpf('5.0272320351359433e-4092') 0 0 0]
[0 mpf('5.0272320351359433e-4092') 0 mpf('5.0272320351359433e-4092') 0 0]
[0 mpf('5.0272320351359433e-4092') 0 mpf('5.0272320351359433e-4092') 0 0]
[mpf('5.0272320351359433e-4092') 0 mpf('5.0272320351359433e-4092') 0 0 0]
[0 0 0 0 0 1]
[0 0 0 0 1 0]]
matrix A ^ 31415 =
[[mpf('3.7268602513250562e-4729') 0 mpf('3.7268602513250562e-4729') 0 0 0]
[0 mpf('3.7268602513250562e-4729') 0 mpf('3.7268602513250562e-4729') 0 0]
[0 mpf('3.7268602513250562e-4729') 0 mpf('-3.7268602513250562e-4729') 0
0]
[mpf('-3.7268602513250562e-4729') 0 mpf('3.7268602513250562e-4729') 0 0
0]
[0 0 0 0 0 1]
[0 0 0 0 1 0]]
(matrix A ** 27182) ** 31415 =
[[mpf('1.77158544475749e-128528148') 0 mpf('1.77158544475749e-128528148')
0 0 0]
[0 mpf('1.77158544475749e-128528148') 0
mpf('1.77158544475749e-128528148') 0 0]
[0 mpf('1.77158544475749e-128528148') 0
mpf('1.77158544475749e-128528148') 0 0]
[mpf('1.77158544475749e-128528148') 0 mpf('1.77158544475749e-128528148')
0 0 0]
[0 0 0 0 0 1]
[0 0 0 0 1 0]]
</pre>
=={{header|Racket}}==
Line 1,457 ⟶ 1,908:
The addition chains correspond to binary exponentiation.
<
#lang racket
(define (chain n)
Line 1,498 ⟶ 1,949:
(test 1.00002206445416 31415)
(test 1.00002550055251 27182)
</syntaxhighlight>
Output:
<
Chain for 31415
((31415 31414 1) (31414 15707 15707) (15707 15706 1) (15706 7853 7853) (7853 7852 1) (7852 3926 3926) (3926 1963 1963) (1963 1962 1) (1962 981 981) (981 980 1) (980 490 490) (490 245 245) (245 244 1) (244 122 122) (122 61 61) (61 60 1) (60 30 30) (30 15 15) (15 14 1) (14 7 7) (7 6 1) (6 3 3) (3 2 1) (2 1 1))
Line 1,510 ⟶ 1,961:
1.00002550055251 ^ 27182 = 1.9999999999774538
Multiplications: 21
</syntaxhighlight>
=={{header|Raku}}==
{{trans|Python}}
<syntaxhighlight lang="raku" line># 20230327 Raku programming solution
use Math::Matrix;
class AdditionChains { has ( $.idx, $.pos, @.chains, @.lvl, %.pat ) is rw;
method add_chain {
# method 1: add chains depth then breadth first until done
return gather given self {
take my $newchain = .chains[.idx].clone.append:
.chains[.idx][*-1] + .chains[.idx][.pos];
.chains.append: $newchain;
.pos == +.chains[.idx]-1 ?? ( .idx += 1 && .pos = 0 ) !! .pos += 1;
}
}
method find_chain(\nexp where nexp > 0) {
# method 1 interface: search for chain ending with n, adding more as needed
return ([1],) if nexp == 1;
my @chn = self.chains.grep: *.[*-1] == nexp // [];
unless @chn.Bool {
repeat { @chn = self.add_chain } until @chn[*-1][*-1] == nexp
}
return @chn
}
method knuth_path(\ngoal) {
# method 2: knuth method, uses memoization to search for a shorter chain
return [] if ngoal < 1;
until self.pat{ngoal}:exists {
self.lvl[0] = [ gather for self.lvl[0].Array -> \i {
for self.knuth_path(i).Array -> \j {
unless self.pat{i + j}:exists {
self.pat{i + j} = i and take i + j
}
}
} ]
}
return self.knuth_path(self.pat{ngoal}).append: ngoal
}
multi method cpow(\xbase, \chain) {
# raise xbase by an addition exponentiation chain for what becomes x**chain[-1]
my ($pows, %products) = 0, 1 => xbase;
%products{0} = xbase ~~ Math::Matrix
?? Math::Matrix.new([ [ 1 xx xbase.size[1] ] xx xbase.size[0] ]) !! 1;
for chain.Array -> \i {
%products{i} = %products{$pows} * %products{i - $pows};
$pows = i
}
return %products{ chain[*-1] }
}
}
my $chn = AdditionChains.new( idx => 0, pos => 0,
chains => ([1],), lvl => ([1],), pat => {1=>0} );
say 'First one hundred addition chain lengths:';
.Str.say for ( (1..100).map: { +$chn.find_chain($_)[0] - 1 } ).rotor: 10;
my %chns = (31415, 27182).map: { $_ => $chn.knuth_path: $_ };
say "\nKnuth chains for addition chains of 31415 and 27182:";
say "Exponent: $_\n Addition Chain: %chns{$_}[0..*-2]" for %chns.keys;
say '1.00002206445416^31415 = ', $chn.cpow(1.00002206445416, %chns{31415});
say '1.00002550055251^27182 = ', $chn.cpow(1.00002550055251, %chns{27182});
say '(1.000025 + 0.000058i)^27182 = ', $chn.cpow: 1.000025+0.000058i, %chns{27182};
say '(1.000022 + 0.000050i)^31415 = ', $chn.cpow: 1.000022+0.000050i, %chns{31415};
my \sq05 = 0.5.sqrt;
my \mat = Math::Matrix.new( [[sq05, 0, sq05, 0, 0, 0],
[ 0, sq05, 0, sq05, 0, 0],
[ 0, sq05, 0, -sq05, 0, 0],
[-sq05, 0, sq05, 0, 0, 0],
[ 0, 0, 0, 0, 0, 1],
[ 0, 0, 0, 0, 1, 0]] );
say 'matrix A ^ 27182 =';
say my $res27182 = $chn.cpow(mat, %chns{27182});
say 'matrix A ^ 31415 =';
say $chn.cpow(mat, %chns{31415});
say '(matrix A ** 27182) ** 31415 =';
say $chn.cpow($res27182, %chns{31415});
</syntaxhighlight>
{{out}}
<pre>First one hundred addition chain lengths:
0 1 2 2 3 3 4 3 4 4
5 4 5 5 5 4 5 5 6 5
6 6 6 5 6 6 6 6 7 6
7 5 6 6 7 6 7 7 7 6
7 7 7 7 7 7 8 6 7 7
7 7 8 7 8 7 8 8 8 7
8 8 8 6 7 7 8 7 8 8
9 7 8 8 8 8 8 8 9 7
8 8 8 8 8 8 9 8 9 8
9 8 9 9 9 7 8 8 8 8
Knuth chains for addition chains of 31415 and 27182:
Exponent: 27182
Addition Chain: 1 2 3 5 7 14 21 35 70 140 143 283 566 849 1698 3396 6792 6799 13591
Exponent: 31415
Addition Chain: 1 2 3 5 7 14 28 56 61 122 244 488 976 1952 3904 7808 15616 15677 31293
1.00002206445416^31415 = 1.9999999998924638
1.00002550055251^27182 = 1.999999999974053
(1.000025 + 0.000058i)^27182 = -0.01128636963542673+1.9730308496660347i
(1.000022 + 0.000050i)^31415 = 0.00016144681325535107+1.9960329014194498i
matrix A ^ 27182 =
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 1
0 0 0 0 1 0
matrix A ^ 31415 =
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 -0 0 0
-0 0 0 0 0 0
0 0 0 0 0 1
0 0 0 0 1 0
(matrix A ** 27182) ** 31415 =
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 1
0 0 0 0 1 0</pre>
=={{header|Tcl}}==
Line 1,517 ⟶ 2,100:
Using code at [[Matrix multiplication#Tcl]] and [[Matrix Transpose#Tcl]] (not shown here).
{{trans|Go}}
<
# of addition chains" by F. Bergeron, J. Berstel, and S. Brlek, published in
# Journal de théorie des nombres de Bordeaux, 6 no. 1 (1994), p. 21-38,
Line 1,604 ⟶ 2,187:
puts "A**$mn"; set countMult 0
print_matrix [apply $pow_mn $A] %6.3f
puts "$countMult matrix multiplies"</
{{out}}
<pre>A**31415
Line 1,630 ⟶ 2,213:
0.000 0.000 0.000 0.000 0.000 1.000
37 matrix multiplies</pre>
=={{header|Wren}}==
{{trans|Go}}
{{libheader|Wren-dynamic}}
{{libheader|Wren-fmt}}
This is very slow (12 mins 50 secs) compared to Go (25 seconds) but, given the amount of calculation involved, not too bad for the Wren interpreter.
<syntaxhighlight lang="wren">import "./dynamic" for Struct
import "./fmt" for Fmt
var Save = Struct.create("Save", ["p", "i", "v"])
var N = 32
var NMAX = 40000
var u = List.filled(N, 0) // upper bounds
var l = List.filled(N, 0) // lower bounds
var out = List.filled(N, 0)
var sum = List.filled(N, 0)
var tail = List.filled(N, 0)
var cache = List.filled(NMAX + 1, 0)
var known = 2
var stack = 0
var undo = List.filled(N * N, null)
for (i in 0..1) l[i] = u[i] = i + 1
for (i in 0...N*N) undo[i] = Save.new(null, 0, 0)
cache[2] = 1
var replace = Fn.new { |x, i, n|
undo[stack].p = x
undo[stack].i = i
undo[stack].v = x[i]
x[i] = n
stack = stack + 1
}
var restore = Fn.new { |n|
while (stack > n) {
stack = stack - 1
undo[stack].p[undo[stack].i] = undo[stack].v
}
}
/* lower and upper bounds */
var lower = Fn.new { |n, up|
if (n <= 2 || (n <= NMAX && cache[n] != 0)) {
if (up.count > 0) up[0] = cache[n]
return cache[n]
}
var i = -1
var o = 0
while (n != 0) {
if ((n&1) != 0) o = o + 1
n = n >> 1
i = i + 1
}
if (up.count > 0) {
i = i - 1
up[0] = o + i
}
while (true) {
i = i + 1
o = o >> 1
if (o == 0) break
}
o = 2
while (o * o < n) {
if (n%o != 0) {
o = o + 1
continue
}
var q = cache[o] + cache[(n/o).floor]
if (q < up[0]) {
up[0] = q
if (q == i) break
}
o = o + 1
}
if (n > 2) {
if (up[0] > cache[n-2] + 1) up[0] = cache[n-1] + 1
if (up[0] > cache[n-2] + 1) up[0] = cache[n-2] + 1
}
return i
}
var insert = Fn.new { |x, pos|
var save = stack
if (l[pos] > x || u[pos] < x) return false
if (l[pos] != x) {
replace.call(l, pos, x)
var i = pos - 1
while (u[i]*2 < u[i+1]) {
var t = l[i+1] + 1
if (t * 2 > u[i]) {
restore.call(save)
return false
}
replace.call(l, i, t)
i = i - 1
}
i = pos + 1
while (l[i] <= l[i-1]) {
var t = l[i-1] + 1
if (t > u[i]) {
restore.call(save)
return false
}
replace.call(l, i, t)
i = i + 1
}
}
if (u[pos] == x) return true
replace.call(u, pos, x)
var i = pos - 1
while (u[i] >= u[i+1]) {
var t = u[i+1] - 1
if (t < l[i]) {
restore.call(save)
return false
}
replace.call(u, i, t)
i = i - 1
}
i = pos + 1
while (u[i] > u[i-1]*2) {
var t = u[i-1] * 2
if (t < l[i]) {
restore.call(save)
return false
}
replace.call(u, i, t)
i = i + 1
}
return true
}
var try // forward declaration
var seqRecur = Fn.new { |le|
var n = l[le]
if (le < 2) return true
var limit = n - 1
if (out[le] == 1) limit = n - tail[sum[le]]
if (limit > u[le-1]) limit = u[le-1]
// Try to break n into p + q, and see if we can insert p, q into
// list while satisfying bounds.
var p = limit
var q = n - p
while (q <= p) {
if (try.call(p, q, le)) return true
q = q + 1
p = p -1
}
return false
}
try = Fn.new { |p, q, le|
var pl = cache[p]
if (pl >= le) return false
var ql = cache[q]
if (ql >= le) return false
while (pl < le && u[pl] < p) pl = pl + 1
var pu = pl - 1
while (pu < le-1 && u[pu+1] >= p) pu = pu + 1
while (ql < le && u[ql] < q) ql = ql + 1
var qu = ql - 1
while (qu < le-1 && u[qu+1] >= q) qu = qu + 1
if (p != q && pl <= ql) pl = ql + 1
if (pl > pu || ql > qu || ql > pu) return false
if (out[le] == 0) {
pu = le - 1
pl = pu
}
var ps = stack
while (pu >= pl) {
if (!insert.call(p, pu)) {
pu = pu - 1
continue
}
out[pu] = out[pu] + 1
sum[pu] = sum[pu] + le
if (p != q) {
var qs = stack
var j = qu
if (j >= pu) j = pu - 1
while (j >= ql) {
if (!insert.call(q, j)) {
j = j - 1
continue
}
out[j] = out[j] + 1
sum[j] = sum[j] + le
tail[le] = q
if (seqRecur.call(le - 1)) return true
restore.call(qs)
out[j] = out[j] - 1
sum[j] = sum[j] - le
j = j - 1
}
} else {
out[pu] = out[pu] + 1
sum[pu] = sum[pu] + le
tail[le] = p
if (seqRecur.call(le - 1)) return true
out[pu] = out[pu] - 1
sum[pu] = sum[pu] - le
}
out[pu] = out[pu] - 1
sum[pu] = sum[pu] - le
restore.call(ps)
pu = pu - 1
}
return false
}
var seq // forward declaration
var seqLen // recursive function
seqLen = Fn.new { |n|
if (n <= known) return cache[n]
// Need all lower n to compute sequence.
while (known+1 < n) seqLen.call(known + 1)
var ub = 0
var pub = [ub]
var lb = lower.call(n, pub)
ub = pub[0]
while (lb < ub && seq.call(n, lb, []) == 0) {
lb = lb + 1
}
known = n
if (n&1023 == 0) System.print("Cached %(known)")
cache[n] = lb
return lb
}
seq = Fn.new { |n, le, buf|
if (le == 0) le = seqLen.call(n)
stack = 0
l[le] = u[le] = n
for (i in 0..le) out[i] = sum[i] = 0
var i = 2
while (i < le) {
l[i] = l[i-1] + 1
u[i] = u[i-1] * 2
i = i + 1
}
i = le - 1
while (i > 2) {
if (l[i]*2 < l[i+1]) {
l[i] = ((1 + l[i+1]) / 2).floor
}
if (u[i] >= u[i+1]) {
u[i] = u[i+1] - 1
}
i = i - 1
}
if (!seqRecur.call(le)) return 0
if (buf.count > 0) {
for (i in 0..le) buf[i] = u[i]
}
return le
}
var binLen = Fn.new { |n|
var r = -1
var o = -1
while (n != 0) {
if (n&1 != 0) o = o + 1
n = n >> 1
r = r + 1
}
return r + o
}
var mul = Fn.new { |m1, m2|
var rows1 = m1.count
var rows2 = m2.count
var cols1 = m1[0].count
var cols2 = m2[0].count
if (cols1 != rows2) Fiber.abort("Matrices cannot be multiplied.")
var result = List.filled(rows1, null)
for (i in 0...rows1) {
result[i] = List.filled(cols2, 0)
for (j in 0...cols2) {
for (k in 0...rows2) {
result[i][j] = result[i][j] + m1[i][k] * m2[k][j]
}
}
}
return result
}
var pow = Fn.new { |m, n, printout|
var e = List.filled(N, 0)
var v = List.filled(N, null)
for (i in 0...N) v[i] = List.filled(N, 0)
var le = seq.call(n, 0, e)
if (printout) {
System.print("Addition chain:")
for (i in 0..le) {
var c = (i == le) ? "\n" : " "
System.write("%(e[i])%(c)")
}
}
v[0] = m
v[1] = mul.call(m, m)
var i = 2
while (i <= le) {
var j = i - 1
while (j != 0) {
for (k in j..0) {
if (e[k]+e[j] < e[i]) break
if (e[k]+e[j] > e[i]) continue
v[i] = mul.call(v[j], v[k])
j = 1
break
}
j = j - 1
}
i = i + 1
}
return v[le]
}
var print = Fn.new { |m|
for (v in m) Fmt.print("$9.6f", v)
System.print()
}
var m = 27182
var n = 31415
System.print("Precompute chain lengths:")
seqLen.call(n)
var rh = (0.5).sqrt
var mx = [
[rh, 0, rh, 0, 0, 0],
[0, rh, 0, rh, 0, 0],
[0, rh, 0, -rh, 0, 0],
[-rh, 0, rh, 0, 0, 0],
[0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, 1, 0]
]
System.print("\nThe first 100 terms of A003313 are:")
for (i in 1..100) {
Fmt.write("$d ", seqLen.call(i))
if (i%10 == 0) System.print()
}
var exs = [m, n]
var mxs = List.filled(2, null)
var i = 0
for (ex in exs) {
System.print("\nExponent: %(ex)")
mxs[i] = pow.call(mx, ex, true)
Fmt.print("A ^ $d:-\n", ex)
print.call(mxs[i])
System.print("Number of A/C multiplies: %(seqLen.call(ex))")
System.print(" c.f. Binary multiplies: %(binLen.call(ex))")
i = i + 1
}
Fmt.print("\nExponent: $d x $d = $d", m, n, m*n)
Fmt.print("A ^ $d = (A ^ $d) ^ $d:-\n", m*n, m, n)
var mx2 = pow.call(mxs[0], n, false)
print.call(mx2)</syntaxhighlight>
{{out}}
<pre>
Precompute chain lengths:
Cached 1024
Cached 2048
Cached 3072
....
Cached 28672
Cached 29696
Cached 30720
The first 100 terms of A003313 are:
0 1 2 2 3 3 4 3 4 4
5 4 5 5 5 4 5 5 6 5
6 6 6 5 6 6 6 6 7 6
7 5 6 6 7 6 7 7 7 6
7 7 7 7 7 7 8 6 7 7
7 7 8 7 8 7 8 8 8 7
8 8 8 6 7 7 8 7 8 8
9 7 8 8 8 8 8 8 9 7
8 8 8 8 8 8 9 8 9 8
9 8 9 9 9 7 8 8 8 8
Exponent: 27182
Addition chain:
1 2 4 8 10 18 28 46 92 184 212 424 848 1696 3392 6784 13568 27136 27182
A ^ 27182:-
[-0.500000 -0.500000 -0.500000 0.500000 0.000000 0.000000]
[ 0.500000 -0.500000 -0.500000 -0.500000 0.000000 0.000000]
[-0.500000 -0.500000 0.500000 -0.500000 0.000000 0.000000]
[ 0.500000 -0.500000 0.500000 0.500000 0.000000 0.000000]
[ 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000]
[ 0.000000 0.000000 0.000000 0.000000 0.000000 1.000000]
Number of A/C multiplies: 18
c.f. Binary multiplies: 21
Exponent: 31415
Addition chain:
1 2 4 8 16 17 33 49 98 196 392 784 1568 3136 6272 6289 12561 25122 31411 31415
A ^ 31415:-
[ 0.707107 0.000000 0.000000 -0.707107 0.000000 0.000000]
[ 0.000000 0.707107 0.707107 0.000000 0.000000 0.000000]
[ 0.707107 0.000000 0.000000 0.707107 0.000000 0.000000]
[ 0.000000 0.707107 -0.707107 0.000000 0.000000 0.000000]
[ 0.000000 0.000000 0.000000 0.000000 0.000000 1.000000]
[ 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000]
Number of A/C multiplies: 19
c.f. Binary multiplies: 24
Exponent: 27182 x 31415 = 853922530
A ^ 853922530 = (A ^ 27182) ^ 31415:-
[-0.500000 0.500000 -0.500000 0.500000 0.000000 0.000000]
[-0.500000 -0.500000 -0.500000 -0.500000 0.000000 0.000000]
[-0.500000 -0.500000 0.500000 0.500000 0.000000 0.000000]
[ 0.500000 -0.500000 -0.500000 0.500000 0.000000 0.000000]
[ 0.000000 0.000000 0.000000 0.000000 1.000000 0.000000]
[ 0.000000 0.000000 0.000000 0.000000 0.000000 1.000000]
</pre>
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Line 1,636 ⟶ 2,648:
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