11/9/19

# Evaluating Arbitrary Precision Integer Expressions in Julia using Metaprogramming

While watching the Mathologer masterclass on power sums

I came across a challenge to evaluate the following sum
$1^{10}+2^{10}+\cdots+1000^{10}$

This can be easily evaluated via brute force in Julia to yield

function power_cal(n)
a=big(0)
for i=1:n
a+=big(i)^10
end
a
end

julia> power_cal(1000)
91409924241424243424241924242500

Note the I had to use big to makes sure we are using BigInt in the computation. Without that, we would be quickly running in into an overflow issue and we will get a very wrong number.

In the comment section of the video I found a very elegant solution to the above sum, expressed as

(1/11) * 1000^11 + (1/2) * 1000^10 + (5/6) * 1000^9 – 1000^7 + 1000^5-(1/2) * 1000^3 + (5/66) * 1000 = 91409924241424243424241924242500

If I try to plug this into the Julia, I get

julia> (1/11) * 1000^11 + (1/2) * 1000^10 + (5/6) * 1000^9 - 1000^7 + r1000^5-(1/2) * 1000^3 + (5/66) * 1000
-6.740310541071357e18

This negative answer is not surprising at all, because we obviously ran into an overflow. We can, of course, go through that expression and modify all instances of Int64 with BigInt by wrapping it in the big function. But that would be cumbersome to do by hand.

## The power of Metaprogramming

In Julia, metaprogramming allows you to write code that creates code, the idea here to manipulate the abstract syntax tree (AST) of a Julia expression. We start to by “quoting” our original mathematical expressing into a Julia expression. In the at form it is not evaluated yet, however we can always evaluate it via eval.

julia> ex1=:((1/11) * 1000^11 + (1/2) * 1000^10 + (5/6) * 1000^9 - 1000^7 + 1000^5-(1/2) * 1000^3 + (5/66) * 1000)
:((((((1 / 11) * 1000 ^ 11 + (1 / 2) * 1000 ^ 10 + (5 / 6) * 1000 ^ 9) - 1000 ^ 7) + 1000 ^ 5) - (1 / 2) * 1000 ^ 3) + (5 / 66) * 1000)

julia> dump(ex1)
Expr
args: Array{Any}((3,))
1: Symbol +
2: Expr
args: Array{Any}((3,))
1: Symbol -
2: Expr
args: Array{Any}((3,))
1: Symbol +
2: Expr
args: Array{Any}((3,))
1: Symbol -
2: Expr
3: Expr
3: Expr
args: Array{Any}((3,))
1: Symbol ^
2: Int64 1000
3: Int64 5
3: Expr
args: Array{Any}((3,))
1: Symbol *
2: Expr
args: Array{Any}((3,))
1: Symbol /
2: Int64 1
3: Int64 2
3: Expr
args: Array{Any}((3,))
1: Symbol ^
2: Int64 1000
3: Int64 3
3: Expr
args: Array{Any}((3,))
1: Symbol *
2: Expr
args: Array{Any}((3,))
1: Symbol /
2: Int64 5
3: Int64 66
3: Int64 1000

julia> eval(ex1)
-6.740310541071357e18

The output of dump show the follow AST in all its glory (…well almost the depth is a bit truncated). Notice that here all our numbers are interpreted as Int64.
Now we walk through the is AST and change all occurrences of Int64 with BigInt by using the big function.

function makeIntBig!(ex::Expr)
args=ex.args
for i in eachindex(args)
if args[i] isa Int64
args[i]=big(args[i])
end
if args[i] isa Expr
makeIntBig!(args[i])
end
end
end

julia> ex2=copy(ex1)
:((((((1 / 11) * 1000 ^ 11 + (1 / 2) * 1000 ^ 10 + (5 / 6) * 1000 ^ 9) - 1000 ^ 7) + 1000 ^ 5) - (1 / 2) * 1000 ^ 3) + (5 / 66) * 1000)

julia> makeIntBig!(ex2)

julia> eval(ex2)
9.14099242414242434242419242425000000000000000000000000000000000000000000000014e+31

We see an improvement here, but the results are not very satisfactory. The divisions yield BigFloat results, which had a tiny bit of floating point errors. Can we do better?
Julia has support for Rational expressions baked in. We can use that improve the results. We just need to search for call expressions the / symbol and replace it by the // symbol. For safety we just have to makes sure the operands are as subtype of Integer.

function makeIntBig!(ex::Expr)
args=ex.args
if ex.head == :call && args[1]==:/ &&
length(args)==3 &&
all(x->typeof(x) <: Integer,args[2:end])
args[1]=://
args[2]=big(args[2])
args[3]=big(args[3])
else
for i in eachindex(args)
if args[i] isa Int64
args[i]=big(args[i])
end
if args[i] isa Expr
makeIntBig!(args[i])
end
end
end
end

julia> ex2=copy(ex1);

julia> makeIntBig!(ex2)

julia> eval(ex2)
91409924241424243424241924242500//1

Now that is much better! We have not lost any precision and we ended us with a Rational expression.

Finally, we can build a macro so the if we run into such expressions in the future and we want to evaluate them, we could just conveniently call it.

macro eval_bigInt(ex)
makeIntBig!(ex)
ex
end

and we can now simply evaluate our original expression as

julia> @eval_bigInt (1/11) * 1000^11 + (1/2) * 1000^10 + (5/6) * 1000^9 - 1000^7 + 1000^5-(1/2) * 1000^3 + (5/66) * 1000
91409924241424243424241924242500//1

07/28/18

# Exploring left truncatable primes

Recently I came across a fascinating Numberphile video on truncatable primes

I immediately thought it would be cool to whip a quick Julia code to get the full enumeration of all left truncatable primes, count the number of branches and also get the largest left truncatable prime.

using Primes

function get_left_primes(s::String)
p_arr=Array{String,1}()
for i=1:9
number_s="$i$s"
if isprime(parse(BigInt, number_s))
push!(p_arr,number_s)
end
end
p_arr
end

function get_all_left_primes(l)
r_l= Array{String,1}()
n_end_points=0
for i in l
new_l=get_left_primes(i)
isempty(new_l) && (n_end_points+=1)
append!(r_l,new_l)
next_new_l,new_n=get_all_left_primes(new_l)
n_end_points+=new_n # counting the chains
append!(r_l,next_new_l)
end
r_l, n
end

The first function just prepends a number (expressed in String for convenience) and checks for it possible primes that can emerge from a single digit prepending. For example:

julia> get_left_primes("17")
2-element Array{String,1}:
"317"
"617"

The second function, just makes extensive use of the first to get all left truncatable primes and also count the number of branches.

julia> all_left_primes, n_branches=get_all_left_primes([""])
(String["2", "3", "5", "7", "13", "23", "43", "53", "73", "83"  …  "6435616333396997", "6633396997", "76633396997", "963396997", "16396997", "96396997", "616396997", "916396997", "396396997", "4396396997"], 1442)

julia> n_branches
1442

julia> all_left_primes
4260-element Array{String,1}:
"2"
"3"
"5"
"7"
"13"
"23"
⋮
"96396997"
"616396997"
"916396997"
"396396997"
"4396396997"

So we the full list of possible left truncatable primes with a length 4260. Also the total number of branches came to 1442.

We now get the largest left truncatable primes with the following one liner:

julia> largest_left_prime=length.(all_left_primes)|>indmax|> x->all_left_primes[x]
"357686312646216567629137"

After this fun exploration, I found an implementation in Julia for just getting the largest left truncatable prime for any base in Rosseta Code.

04/1/18

# Iterating with Dates and Time in Julia

Julia has good documentation on dealing with Dates and Time, however that is often in the context constructing and Date and Time objects. In this post, I am focus on the ability to iterate over Dates and Times. This is very useful in countless application.

We start of by capturing this moment and moving ahead into the future

julia> this_moment=now()
2018-04-01T23:13:33.437

In one hour that will be

julia> this_moment+Dates.Hour(1)
2018-04-02T00:13:33.437

Notice that Julia was clever enough properly interpret that we will be on the in another day after exactly one hour. Thanks to it multiple dispatch of the DateTime type to be able to do TimeType period arithmatic.

You can then write a nice for loop that does something every four hours for the next two days.

julia> for t=this_moment:Dates.Hour(4):this_moment+Dates.Day(2)
println(t)
#or somethings special with that time
end
2018-04-01T23:13:33.437
2018-04-02T03:13:33.437
2018-04-02T07:13:33.437
2018-04-02T11:13:33.437
2018-04-02T15:13:33.437
2018-04-02T19:13:33.437
2018-04-02T23:13:33.437
2018-04-03T03:13:33.437
2018-04-03T07:13:33.437
2018-04-03T11:13:33.437
2018-04-03T15:13:33.437
2018-04-03T19:13:33.437
2018-04-03T23:13:33.437

Often we are not so interested in the full dates. For example if we are reading a video file and we want to get a frame every 5 seconds while using VideoIO.jl. We can deal here with the simpler Time type.

julia> video_start=Dates.Time(0,5,20)
00:05:20

Here we are interested in starting 5 minutes and 20 seconds into the video.
Now we can make a nice loop from the start to finish

for t=video_start:Dates.Second(5):video_start+Dates.Hour(2)
h=Dates.Hour(t).value
m=Dates.Minute(t).value
s=Dates.Second(t).value
ms=Dates.Millisecond(t).value
# Do something interesting with ffmpeg seek on the video
end
02/9/18

# When Julia is faster than C, digging deeper

In my earlier post I showed an example where Julia is significantly faster than c. I got this insightful response

So I decided to dig deeper. Basically the standard c rand() is not that good. So instead I searched for the fastest Mersenne Twister there is. I downloaded the latest code and compiled it in the fastest way for my architecture.

/* eurler2.c */
#include <stdio.h>      /* printf, NULL */
#include <stdlib.h>     /* srand, rand */
#include "SFMT.h"       /* fast Mersenne Twister */

sfmt_t sfmt;

double r2()
{
return sfmt_genrand_res53(&sfmt);

}

double euler(long int n)
{
long int m=0;
long int i;
for(i=0; i<n; i++){
double the_sum=0;
while(1) {
m++;
the_sum+=r2();
if(the_sum>1.0) break;
}
}
return (double)m/(double)n;
}

int main ()
{
sfmt_init_gen_rand(&sfmt,123456);
printf ("Euler : %2.5f\n", euler(1000000000));

return 0;
}

I had to compile with a whole bunch of flags which I induced from SFMT‘s Makefile to get faster performance.

gcc -O3 -finline-functions -fomit-frame-pointer -DNDEBUG -fno-strict-aliasing --param max-inline-insns-single=1800  -Wall -std=c99 -msse2 -DHAVE_SSE2 -DSFMT_MEXP=1279 -ISFMT-src-1.5.1 -o eulerfast SFMT.c euler2.c

And after all that trouble we got the performance down to 18 seconds. Still slower that Julia‘s 16 seconds.

$time ./eulerfast Euler : 2.71824 real 0m18.075s user 0m18.085s sys 0m0.001s Probably, we could do a bit better with more tweaks, and probably exceed Julia‘s performance with some effort. But at that point, I got tired of pushing this further. The thing I love about Julia is how well it is engineered and hassle free. It is quite phenomenal the performance you get out of it, with so little effort. And for basic technical computing things, like random number generation, you don’t have to dig hard for a better library. The “batteries included” choices in the Julia‘s standard library are pretty good. 02/8/18 # When Julia is faster than C On e-day, I came across this cool tweet from Fermat’s library So I spend a few minutes coding this into Julia function euler(n) m=0 for i=1:n the_sum=0.0 while true m+=1 the_sum+=rand() (the_sum>1.0) && break; end end m/n end Timing this on my machine, I got julia> @time euler(1000000000) 15.959913 seconds (5 allocations: 176 bytes) 2.718219862 Gave a little under 16 seconds. Tried a c implementation #include <stdio.h> /* printf, NULL */ #include <stdlib.h> /* srand, rand */ #include <time.h> /* time */ double r2() { return (double)rand() / (double)((unsigned)RAND_MAX + 1); } double euler(long int n) { long int m=0; long int i; for(i=0; i<n; i++){ double the_sum=0; while(1) { m++; the_sum+=r2(); if(the_sum>1.0) break; } } return (double)m/(double)n; } int main () { printf ("Euler : %2.5f\n", euler(1000000000)); return 0; } and compiling with either gcc gcc -Ofast euler.c or clang clang -Ofast euler.c gave a timing twice as long $ time ./a.out
Euler : 2.71829

real    0m36.213s
user    0m36.238s
sys 0m0.004s

For the curios, I am using this version of Julia

julia> versioninfo()
Julia Version 0.6.3-pre.0
Commit 93168a6 (2017-12-18 07:11 UTC)
Platform Info:
OS: Linux (x86_64-linux-gnu)
CPU: Intel(R) Core(TM) i7-4770HQ CPU @ 2.20GHz
WORD_SIZE: 64
BLAS: libopenblas (USE64BITINT DYNAMIC_ARCH NO_AFFINITY Haswell)
LAPACK: libopenblas64_
LIBM: libopenlibm
LLVM: libLLVM-3.9.1 (ORCJIT, haswell)

Now one should not put too much emphasis on such micro benchmarks. However, I found this a very curious examples when a high level language like Julia could be twice as fast a c. The Julia language authors must be doing some amazing mojo.

01/2/18

# Visualizing the Inscribed Circle and Square Puzzle

Recently, I watched a cool mind your decsions video on an inscribed circle and rectangle puzzle. In the video they showed a diagram that was not scale. I wanted to get a sense of how these differently shaped areas will match.

There was a cool ratio between the outer and inner circle radii that is expressed as

$\frac{R}{r}=\sqrt{\frac{\pi-2}{4-\pi}}$.

I used Compose.jl to rapidly do that.

using Compose
set_default_graphic_size(20cm, 20cm)
ϕ=sqrt((pi -2)/(4-pi))
R=10
r=R/ϕ
ctx=context(units=UnitBox(-10, -10, 20, 20))
composition = compose(ctx,
(ctx, rectangle(-r/√2,-r/√2,r*√2,r*√2),fill("white")),
(ctx,circle(0,0,r),fill("blue")),
(ctx,circle(0,0,R),fill("white")),
(ctx,rectangle(-10,-10,20,20),fill("red")))
composition |> SVG("inscribed.svg")

08/6/17

# Solving the code lock riddle with Julia

I came across a neat math puzzle involving counting the number of unique combinations in a hypothetical lock where digit order does not count. Before you continue, please watch at least the first minute of following video:

The rest of the video describes two related approaches for carrying out the counting. Often when I run into complex counting problems, I like to do a sanity check using brute force computation to make sure I have not missed anything. Julia is fantastic choice for doing such computation. It has C like speed, and with an expressiveness that rivals many other high level languages.

Without further ado, here is the Julia code I used to verify my solution the problem.

1. function unique_combs(n=4)
2.     pat_lookup=Dict{String,Bool}()
3.     for i=0:10^n-1
4.         d=digits(i,10,n) # The digits on an integer in an array with padding
5.         ds=d |> sort |> join # putting the digits in a string after sorting
6.         get(pat_lookup,ds,false) || (pat_lookup[ds]=true)
7.     end
8.     println("The number of unique digits is $(length(pat_lookup))") 9. end In line 2 we create a dictionary that we will be using to check if the number fits a previously seen pattern. The loop starting in line 3, examines all possible ordered combinations. The digits function in line 4 takes any integer and generate an array of its constituent digits. We generate the unique digit string in line 5 using pipes, by first sorting the integer array of digits and then combining them in a string. In line 6 we check if the pattern of digits was seen before and make use of quick short short-circuit evaluation to avoid an if-then statement. 07/14/17 # Julia calling C: A more minimal example Earlier I presented a minimal example of Julia calling C. It mimics how one would go about writing C code, wrapping it a library and then calling it from Julia. Today I came across and even more minimal way of doing that while reading an excellent blog on Julia’s syntactic loop fusion. Associated with the blog was notebook that explores the matter further. Basically, you an write you C in a string and pass it directly to the compiler. It goes something like using Libdl C_code= raw""" double mean(double a, double b) { return (a+b) / 2; } """ const Clib=tempname() open(gcc -fPIC -O3 -xc -shared -o$(Clib * "." * Libdl.dlext) -, "w") do f
print(f, C_code)
end

The tempname function generate a unique temporary file path. On my Linux system Clib will be string like "/tmp/juliaivzRkT". That path is used to generate a library name "/tmp/juliaivzRkT.so" which will then used in the ccall:

meanc(a,b)=ccall((:mean,Clib),Float64,(Float64,Float64),a,b)
julia> meanc(3,4)
3.5

This approach would not be recommended if you are writing anything sophisticated in C. However, it is fun to experiment with for short bits of C code that you might like to call from Julia. Saves you the hassle of creating a Makefile, compiling, etc…

06/29/17

# Solving the Fish Riddle with JuMP

Recently I came across a nice Ted-Ed video presenting a Fish Riddle.

I thought it would be fun to try solving it using Julia’s award winning JuMP package. Before we get started, please watch the above video-you might want to pause at 2:24 if you want to solve it yourself.

To attempt this problem in Julia, you will have to install the JuMP package.

julia> Pkg.add("JuMP")

JuMP provides an algebraic modeling language for dealing with mathematical optimization problems. Basically, that allows you to focus on describing your problem in a simple syntax and it would then take care of transforming that description in a form that can be handled by any number of solvers. Those solvers can deal with several types of optimization problems, and some solvers are more generic than others. It is important to pick the right solver for the problem that you are attempting.

The problem premises are:
1. There are 50 creatures in total. That includes sharks outside the tanks and fish
2. Each SECTOR has anywhere from 1 to 7 sharks, with no two sectors having the same number of sharks.
3. Each tank has an equal number of fish
4. In total, there are 13 or fewer tanks
5. SECTOR ALPHA has 2 sharks and 4 tanks
6. SECTOR BETA has 4 sharsk and 2 tanks
We want to find the number of tanks in sector GAMMA!

Here we identify the problem as mixed integer non-linear program (MINLP). We know that because the problem involves an integer number of fish tanks, sharks, and number of fish inside each tank. It also non-linear (quadratic to be exact) because it involves multiplying two two of the problem variables to get the total number or creatures. Looking at the table of solvers in the JuMP manual. pick the Bonmin solver from AmplNLWriter package. This is an open source solver, so installation should be hassle free.

julia> Pkg.add("AmplNLWriter")

We are now ready to write some code.

using JuMP, AmplNLWriter

# Solve model
m = Model(solver=BonminNLSolver())

# Number of fish in each tank
@variable(m, n>=1, Int)

# Number of sharks in each sector
@variable(m, s[i=1:3], Int)

# Number of tanks in each sector
@variable(m, nt[i=1:3]>=0, Int)

@constraints m begin
# Constraint 2
sharks[i=1:3], 1 <= s[i] <= 7
numfish[i=1:3], 1 <= nt[i]
# Missing uniqueness in restriction
# Constraint 4
sum(nt) <= 13
# Constraint 5
s[1] == 2
nt[1] == 4
# Constraint 6
s[2] == 4
nt[2] == 2
end

# Constraints 1 & 3
@NLconstraint(m, s[1]+s[2]+s[3]+n*(nt[1]+nt[2]+nt[3]) == 50)

# Solve it
status = solve(m)

sharks_in_each_sector=getvalue(s)
fish_in_each_tank=getvalue(n)
tanks_in_each_sector=getvalue(nt)

@printf("We have %d fishes in each tank.\n", fish_in_each_tank)
@printf("We have %d tanks in sector Gamma.\n",tanks_in_each_sector[3])
@printf("We have %d sharks in sector Gamma.\n",sharks_in_each_sector[3])

In that representation we could not capture the restriction that “no two sectors having the same number of sharks”. We end up with the following output:

We have 4 fishes in each tank.
We have 4 tanks in sector Gamma.
We have 4 sharks in sector Gamma.

Since the problem domain is limited, we can possible fix that by adding a constrain that force the number of sharks in sector Gamma to be greater than 4.

@constraint(m,s[3]>=5)

This will result in an answer that that does not violate any of the stated constraints.

We have 3 fishes in each tank.
We have 7 tanks in sector Gamma.
We have 5 sharks in sector Gamma.

However, this seems like a bit of kludge. The proper way go about it is represent the number of sharks in the each sector as binary array, with only one value set to 1.

# Number of sharks in each sector
@variable(m, s[i=1:3,j=1:7], Bin)

We will have to modify our constraint block accordingly

@constraints m begin
# Constraint 2
sharks[i=1:3], sum(s[i,:]) == 1
u_sharks[j=1:7], sum(s[:,j]) <=1 # uniquness
# Constraint 4
sum(nt) <= 13
# Constraint 5
s[1,2] == 1
nt[1] == 4
# Constraint 6
s[2,4] == 1
nt[2] == 2
end

We invent a new variable array st to capture the number of sharks in each sector. This simply obtained by multiplying the binary array by the vector $[1,2,\ldots,7]^\top$

@variable(m,st[i=1:3],Int)
@constraint(m, st.==s*collect(1:7))

We rewrite our last constraint as

# Constraints 1 & 3
@NLconstraint(m, st[1]+st[2]+st[3]+n*(nt[1]+nt[2]+nt[3]) == 50)

After the model has been solved, we extract our output for the number of sharks.

sharks_in_each_sector=getvalue(st)

…and we get the correct output.

This problem might have been an overkill for using a full blown mixed integer non-linear optimizer. It can be solved by a simple table as shown in the video. However, we might not alway find ourselves in such a fortunate position. We could have also use mixed integer quadratic programming solver such as Gurobi which would be more efficient for that sort of problem. Given the small problem size, efficiency hardly matters here.

06/12/17

# Reading DataFrames with non-UTF8 encoding in Julia

Recently I ran into problem where I was trying to read a CSV files from a Scandinavian friend into a DataFrame. I was getting errors it could not properly parse the latin1 encoded names.

I tried running

using DataFrames
dataT=readtable("example.csv", encoding=:latin1)

but the got this error

ArgumentError: Argument 'encoding' only supports ':utf8' currently.

The solution make use of (StringEncodings.jl)[https://github.com/nalimilan/StringEncodings.jl] to wrap the file data stream before presenting it to the readtable function.

f=open("example.csv","r")
s=StringDecoder(f,"LATIN1", "UTF-8")
close(f)
The StringDecoder generates an IO stream that appears to be utf8 for the readtable function.