weird error

[JoshCheek]

def fact!(n)
  (1..n).reduce(1.0, :*)
end
alias ! fact!

# e is the sum of the inverses of the integers
0.upto(100).sum { |i| 1.0 / fact!(i) } # => 
Math::E                                # => 

Math::E ** 2                            # => 
0.upto(100).sum { |i| 2**i / fact!(i) } # => 

def sin(x)
  1.step(to: 100, by: 4).sum { |i|
    x**(i) / fact!(i) - x**(i+2) / fact!(i+2)
  }
  1.step(to: 100, by: 2).sum { |i|
    x**i / !(i) / (1-i[1]*2)
  }
end

(1).step(to: 10, by: 2) { |i|
  (-1)**(i%4/2) # => 
  (i+2)%4-2     # => 
  1-2*i[1]      # => 
  1-i[1]*2      # => 
  (-i>>1&1)*2-1 # => 
}

sin      Math::PI / 10 # => 
Math.sin Math::PI / 10 # => 

# 0/0! + x/1! - 0x^2/2! - x^3/3! + 0x^4/4! + x^5/5!
# 1x/1! - x^3/3! + x^5/5! - x^7/7!

# !> /Users/josh/.gem/ruby/2.5.0/gems/seeing_is_believing-3.6.0/lib/seeing_is_believing/event_stream/producer.rb:79:in `record_exception': private method `!' called for nil:NilClass (NoMethodError)
# !> \tfrom /Users/josh/.gem/ruby/2.5.0/gems/seeing_is_believing-3.6.0/lib/seeing_is_believing/the_matrix.rb:81:in `block in <top (required)>'
# !> /var/folders/7g/mbft22555w3_2nqs_h1kbglw0000gn/T/seeing_is_believing_temp_dir20180520-40010-cajhs3/program.rb:18:in `block in sin': private method `!' called for 1:Integer (NoMethodError)
# !> \tfrom /var/folders/7g/mbft22555w3_2nqs_h1kbglw0000gn/T/seeing_is_believing_temp_dir20180520-40010-cajhs3/program.rb:17:in `step'
# !> \tfrom /var/folders/7g/mbft22555w3_2nqs_h1kbglw0000gn/T/seeing_is_believing_temp_dir20180520-40010-cajhs3/program.rb:17:in `each'
# !> \tfrom /var/folders/7g/mbft22555w3_2nqs_h1kbglw0000gn/T/seeing_is_believing_temp_dir20180520-40010-cajhs3/program.rb:17:in `sum'
# !> \tfrom /var/folders/7g/mbft22555w3_2nqs_h1kbglw0000gn/T/seeing_is_believing_temp_dir20180520-40010-cajhs3/program.rb:17:in `sin'
# !> \tfrom /var/folders/7g/mbft22555w3_2nqs_h1kbglw0000gn/T/seeing_is_believing_temp_dir20180520-40010-cajhs3/program.rb:30:in `<main>'

__END__
def divvy_up(jewels)
  unsplit = jewels.chars
  totals  = unsplit.group_by(&:itself).map { |j, js| [j, js.count] }.to_h
  counts  = totals.transform_values { 0 }
  first   = 0
  last    = unsplit.length / 2
  (first...last).each { |i| counts[unsplit[i]] += 1 }
  until counts.all? { |j, c| totals[j] == c*2 }
    counts[jewels[first]] -= 1
    counts[jewels[last]]  += 1
    first += 1
    last  += 1
  end
  [ unsplit[0...first],
    unsplit[first...last],
    unsplit[last..-1],
  ].reject(&:empty?)
end

chars = "RRRRGGGGGG".chars
10.times do
  str = chars.shuffle.join
  [str, divvy_up(str)]
end

def r(n=1, max)
  return n if n == max
  n + 1.0 / r(n+1, max)
end

r 1   # => 1
r 2   # => 1.5
r 3   # => 1.4285714285714286
r 4   # => 1.4333333333333333
r 100 # => 1.4331274267223117

def r(max) # ~> ArgumentError: wrong number of arguments (given 2, expected 1)
  n   = 1
  sum = 0
  while n <= max
    n + 1.0 / r(n+1, max)
    n += 1
  end
end

r 1 # => 
r 2 # => 
r 3 # => 

# ~> ArgumentError
# ~> wrong number of arguments (given 2, expected 1)
# ~>
# ~> /var/folders/7g/mbft22555w3_2nqs_h1kbglw0000gn/T/seeing_is_believing_temp_dir20180520-39264-owea8a/program.rb:57:in `r'
# ~> /var/folders/7g/mbft22555w3_2nqs_h1kbglw0000gn/T/seeing_is_believing_temp_dir20180520-39264-owea8a/program.rb:61:in `r'
# ~> /var/folders/7g/mbft22555w3_2nqs_h1kbglw0000gn/T/seeing_is_believing_temp_dir20180520-39264-owea8a/program.rb:66:in `<main>'

__END__
1 + 1.0/2 # => 1.5
1 + 1.0/(2+1.0/3) # => 1.4285714285714286
1 + 1.0/(2+1.0/(3+1.0/4)) # => 1.4333333333333333
1 + 1.0/(2+1.0/(3+1.0/(4+1.0/5))) # => 1.4331210191082802
1 + 1.0/(2+1.0/(3+1.0/(4+1.0/(5+1.0/6)))) # => 1.433127572016461

Math.sqrt 2 # => 1.4142135623730951

__END__
r = c + n/d
r-c = n/d
d = n/(r-c)

d_old = n/(r_old-c)

r(n)   = n + 1/(n+1)
(r(n)-n)(n+1) = 1
n+1 = 1/(r(n)-n)

n+1/(n+1) = 1/(r(n)-n)
(r(n)-n)(n+1)/(n+1) = 1
(r(n)-n)(n+1) = (n+1)

n + 1/d = r
nd + 1 = rd
1 = (rd - nd)
1 = d(r-n)
1 = ((n+1)+1/d2)(r-n)

r(n+1) = 

d+m = n/(r-c)
(d+m)(r-c) = n
r-c = n/(d+m)
r = c+n/(d+m)