# [QUIZ] Digits of e (#226)

D

#### Daniel Moore

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## Digits of e (#226)

Wayumbe Rubyists,

The mathematical constant e is the unique real number such that the
value of the derivative (slope of the tangent line) of the function
f(x) =3D e^x at the point x =3D 0 is exactly 1. The function e^x so
defined is called the exponential function, and its inverse is the
natural logarithm, or logarithm to base e.[1]

e is one of the most important numbers in mathematics, alongside the
additive and multiplicative identities 0 and 1, the constant =F0, and
the imaginary unit i. These are the five constants appearing in one
formulation of [Euler's identity][2].

This week=A2s quiz is to write a Ruby program that can compute the first
100,000 digits of e.

Have fun!

[1]: http://en.wikipedia.org/wiki/E_(mathematical_constant)
[2]: http://en.wikipedia.org/wiki/Euler's_identity

--=20
-Daniel
http://rubyquiz.strd6.com

J

#### Jack Rouse

Daniel said:
This week¢s quiz is to write a Ruby program that can compute the first
100,000 digits of e.

Have fun!

[1]: http://en.wikipedia.org/wiki/E_(mathematical_constant)
[2]: http://en.wikipedia.org/wiki/Euler's_identity

I used the fast converging continued fraction series from the first
Wikipedia article:

# compute e*10**digits as a rounded integer
def calc_e(digits)
limit = 10**((digits+3)/2)
p = [2, 3]
q = [2, 1]
n = 1
begin
if p.length.even?
a = 4*(4*n-1)
else
a = 4*n+1
n += 1
end
p << a*p[-1] + p[-2]
q << a*q[-1] + q[-2]
end while p.last <= limit
(p.last*10**(digits+3)/q.last+500)/1000
end

if __FILE__ == $0 p calc_e(if ARGV.length.zero? then 100_000 else Integer(ARGV[0]) end) end J #### Jean-Julien Fleck Hello, 2010/1/8 Daniel Moore said: This week=92s quiz is to write a Ruby program that can compute the first 100,000 digits of e. Have fun! Well I had ) I got inspired from Quiz Digits of Pi (#202) and tried to estimate how long the standard library call could take (I didn't want to waste too much CPU on this): require 'bigdecimal' require 'bigdecimal/math' include Math include BigMath d =3D [Time.now] 1000.step(20000,1000) do |i| E(i) d << Time.now puts i.to_s + " needing #{d[-1] - d[-2]} s" end which gives 1000 needing 0.147869 s 2000 needing 0.956326 s 3000 needing 2.993516 s 4000 needing 6.738271 s 5000 needing 12.78967 s 6000 needing 21.584826 s 7000 needing 34.154156 s 8000 needing 49.223259 s 9000 needing 69.57902 s 10000 needing 94.507271 s 11000 needing 123.327777 s 12000 needing 158.839553 s 13000 needing 198.976605 s 14000 needing 246.62736 s 15000 needing 300.471167 s 16000 needing 364.013845 s ^C almost scaling as the cube of the digit number you ask for. So that it should take around 10**5 s to compute E(100_000). I then tried my own implementation using the definition exp(x) =3D \Sum{n=3D0}{\infty} x^n/n! applied in x=3D1 and checking from E(number) that it was correct. It's converging pretty fast (in term of iteration number, only 34_000 to get 100_000 digits) but it's quite exactly 10 times slower than the standard method (and I didn't want to wait 10**6 s to get all the 100_000 digits ). The needed number of iterations is guessed using Stirling formula and I discovered that each step does not take the same time (it gets slower as time goes) whereas it does not depend on the initial number of digits asked for (that is the 1000 first iterations take approximately the same time when you ask for 10_000 digits or for 100_000 digits) which I couldn't really understand. I first thought that the slowing-down was due to the fact that for more digits you need bigger integers to play with but the latter observations seems to invalidate this argument.. If anybody have an explanation, I will be happy to hear it. require 'rational' require 'enumerator' require 'bigdecimal' require 'bigdecimal/math' include Math include BigMath d1 =3D Time.now precision =3D (ARGV[0] || 1000).to_i iterations =3D precision /(log10(precision)) + precision/(log10(precision))*log10(log10(precision)) puts 'Approximate number of iterations needed: ' + iterations.to_i.to_s accuracy =3D 10**precision.to_i fact =3D 1 final =3D accuracy other =3D false d =3D [d1] 1.upto(iterations) do |i| if i%100 =3D=3D 0 d << Time.now puts i.to_s + " needing #{d[-1] - d[-2]} s" end fact *=3D i final_old =3D final final +=3D Rational(accuracy,fact) end d2 =3D Time.now puts "Time elapsed in computation: " + (d2 - d1).to_s + ' s' puts "Computation terminated: computing E(#{precision}) now." puts "Delta * 10**(#{precision}): " + (final.round - E(precision)*accuracy).to_s d3 =3D Time.now puts "Time elapsed calling E(#{precision}): " + (d3 - d2).to_s + ' s' Last but not least, the same method could be called as one line in irb using inject and getting the result in the form of a Rational: fact =3D 1 ; (1..34_000).inject(1) {|sum,i| fact *=3D i ; sum + Rational(1,= fact)} but you will rather want to try with a smaller number of steps (say 500 to get 1000 digits, remember: 34_000 steps will take around 10**6 s...) Cheers, --=20 JJ Fleck PCSI1 Lyc=E9e Kl=E9ber T #### Thorsten Hater Hello, based on the spigot algorithm of Rabonitz and Wagon [1]: #!/usr/bin/ruby def spigot n e = [] arr = [2] + Array.new(n+1,1) (n-1).times do |i| (n+1).downto 0 do |j| arr[j] *= 10 end q = 0 (n+1).downto 0 do |j| arr[j] += q q = arr[j]/(j+1) arr[j] %= j+1 end e << q end e[0]/= 10.0 puts e.join('') end if$0 == __FILE__
spigot ARGV[0].to_i
end

[1] http://www.mathpropress.com/stan/bibliography/spigot.pdf : citing
the easier case of e

Am 08.01.2010 06:26, schrieb Daniel Moore:
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-

The three rules of Ruby Quiz:

1. Please do not post any solutions or spoiler discussion for this
quiz until 48 hours have elapsed from the time this message was
sent.

2. Support Ruby Quiz by submitting ideas and responses
as often as you can.

3. Enjoy!

Suggestion: A [QUIZ] in the subject of emails about the problem
the original quiz message, if you can.

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## Digits of e (#226)

Wayumbe Rubyists,

The mathematical constant e is the unique real number such that the
value of the derivative (slope of the tangent line) of the function
f(x) = e^x at the point x = 0 is exactly 1. The function e^x so
defined is called the exponential function, and its inverse is the
natural logarithm, or logarithm to base e.[1]

e is one of the most important numbers in mathematics, alongside the
additive and multiplicative identities 0 and 1, the constant ð, and
the imaginary unit i. These are the five constants appearing in one
formulation of [Euler's identity][2].

This week¢s quiz is to write a Ruby program that can compute the first
100,000 digits of e.

Have fun!

[1]: http://en.wikipedia.org/wiki/E_(mathematical_constant)
[2]: http://en.wikipedia.org/wiki/Euler's_identity

--
Thorsten Hater
Institut fuer Theoretische Physik I
Ruhr-Universitaet Bochum
Tel.: 0234/32-23-441

T

#### Thorsten Hater

Sorry, Rabonitz meant Rabinowitz, ugly typo.

Am 11.01.2010 14:57, schrieb Thorsten Hater:
Hello,

based on the spigot algorithm of Rabonitz and Wagon [1]:

#!/usr/bin/ruby

def spigot n
e = []
arr = [2] + Array.new(n+1,1)
(n-1).times do |i|
(n+1).downto 0 do |j|
arr[j] *= 10
end
q = 0
(n+1).downto 0 do |j|
arr[j] += q
q = arr[j]/(j+1)
arr[j] %= j+1
end
e << q
end
e[0]/= 10.0
puts e.join('')
end

if $0 == __FILE__ spigot ARGV[0].to_i end [1] http://www.mathpropress.com/stan/bibliography/spigot.pdf : citing the easier case of e Am 08.01.2010 06:26, schrieb Daniel Moore: -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- The three rules of Ruby Quiz: 1. Please do not post any solutions or spoiler discussion for this quiz until 48 hours have elapsed from the time this message was sent. 2. Support Ruby Quiz by submitting ideas and responses as often as you can. 3. Enjoy! Suggestion: A [QUIZ] in the subject of emails about the problem helps everyone on Ruby Talk follow the discussion. Please reply to the original quiz message, if you can. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - RSS Feed: http://rubyquiz.strd6.com/quizzes.rss Suggestions?: http://rubyquiz.strd6.com/suggestions -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- ## Digits of e (#226) Wayumbe Rubyists, The mathematical constant e is the unique real number such that the value of the derivative (slope of the tangent line) of the function f(x) = e^x at the point x = 0 is exactly 1. The function e^x so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e.[1] e is one of the most important numbers in mathematics, alongside the additive and multiplicative identities 0 and 1, the constant ð, and the imaginary unit i. These are the five constants appearing in one formulation of [Euler's identity][2]. This week¢s quiz is to write a Ruby program that can compute the first 100,000 digits of e. Have fun! [1]: http://en.wikipedia.org/wiki/E_(mathematical_constant) [2]: http://en.wikipedia.org/wiki/Euler's_identity -- Thorsten Hater Institut fuer Theoretische Physik I Ruhr-Universitaet Bochum E-Mail: (e-mail address removed) Tel.: 0234/32-23-441 W #### William Rutiser After recovering some vaguely remembered math, and trying not to look to closely at any code on the net, here is my solution. The reference data for testing, from Project Gutenberg, has been truncated for this posting. The program will report "Incorrect" for more than a couple of hundred digits. Bill Rutiser wrutiser AT gmail DOT com # Produce digits of e # wrutiser AT gmail DOT com # # The Loop Invariant requires the following conceptual numbers to sum to _e_. # (1) The number represented by the decimal digits previously generated. # # (2) An intermediate fraction having the value of the sum of several # terms taken from the infinite series expansion that have yet to be # included in (1) # # (3) The tail of the infinite series expansion. # # The generated digits are contained in the variable _z_ but are not used # in the generation of the subsequent bits. # # The current term of the infinite series is represented by the variable _i_. # The term itself, given by the expression "1/factorial( i )", is never actually # computed. # # The sum of the intermediate terms is represented by the variables _nn_, # _dd_, and _mm_. While the sum itself is never actually computed, it is # equivalent to the expression "nn / (mm * dd)". # # _mm_ is a power of 10, increasing as each digit is extracted. # _dd_ is maintained equal to factorial( i ). # # _nn_ is the numerator of the conceptional rational number. # # Each loop interation: # removes the first remaining term from the series # # adds the removed term to the intermediate fraction # # computes the leading two digits of the fraction # # if these digits are the same as those computed for the # previous term, one digit is extracted. # def digits_int( n_terms ) nn = 0 dd = 1 mm = 1 q1 = -1 z = "2.\n" digits = 0 2.upto(n_terms) do |i| dd *= i nn *= i nn += mm q2, r2 = (nn * 100).divmod( dd ) if( q1 == q2 ) q, r = (nn * 10).divmod( dd ) z << q.to_s digits += 1 z << " " if 0 == digits % 10 z << "\n" if 0 == digits % 50 mm *= 10 nn = r q1 = -1 else q1 = q2 end end puts( "\ndigits: #{digits} n_terms: #{n_terms}" \ " terms/digit: #{Float(n_terms) /digits}" ) puts( "dd.size: #{dd.size} nn.size: #{nn.size}" ) puts z if compare?($gutenberg_digits, z )
puts "Correct!!"
else
puts "Not correct."
end
end

def compare?( ssx, ssy )
sx = ssx.delete( " \n" )
sy = ssy.delete( " \n" )
sx.start_with?( sy )
end

# Digits from http://www.gutenberg.org/files/127/127.txt
<http://www.gutenberg.org/files/127/127.txt>
# Computed by Robert Nemiroff and Jerry Bonnell.

$gutenberg_digits = <<HERE 2. 7182818284590452353602874713526624977572470936999595749669676277240766303535 47594571382178525166427427466391932003059921817413596629043572900334295260595630 73813232862794349076323382988075319525101901157383418793070215408914993488416750 #### thousands of digits removed from email posting #### HERE$gutenberg_digits = $gutenberg_digits.delete( " \n" ) puts "gutenberg_digits.length: #{$gutenberg_digits.length}"

J

#### Jay Anderson

## Digits of e (#226)
This weekï¿½s quiz is to write a Ruby program that can compute the first
100,000 digits of e.

$cat e.rb digits = 100000 fudge = 10 unity = 10**(digits + fudge) e = unity n = unity i = 0 while (n>0) i += 1 n /= i e += n end e /= 10**fudge p e$ time ruby e.rb > /dev/null

real 0m4.023s
user 0m3.940s
sys 0m0.087s

This uses the taylor series definition of e which actually converges
quite fast.

-----Jay

D

#### David Springer

I decided to work on an old quiz.
----------------------
This week's quiz is to write a Ruby program that can compute the first
100,000 digits of e.

Here is what I came up with:
---------------------------------------------------------
PLACES = 100_000

euler = 0
big_one = 10**(PLACES+5)
nxt_one = big_one
n = 1

while (nxt_one != 0)
n += 1
nxt_one /= n
euler += nxt_one
end

puts "2."+euler.to_s[0..PLACES-1]

B

D

#### David Springer

Math::E**Math:I-Math:I ###
=> 19.9990999791895