This package includes an example Recurrent Neural Network. The package is loaded using:
library(rnn)We can view the code of the main rnn() function by calling it without the parathesis.
rnn## function(Y, X1, X2, binary_dim, alpha, input_dim, hidden_dim, output_dim, silent = FALSE) {
##
## # check what largest possible number is
## largest_number = 2^binary_dim
##
## # initialize neural network weights
## synapse_0 = matrix(stats::runif(n = input_dim*hidden_dim, min=-1, max=1), nrow=input_dim)
## synapse_1 = matrix(stats::runif(n = hidden_dim*output_dim, min=-1, max=1), nrow=hidden_dim)
## synapse_h = matrix(stats::runif(n = hidden_dim*hidden_dim, min=-1, max=1), nrow=hidden_dim)
##
## synapse_0_update = matrix(0, nrow = input_dim, ncol = hidden_dim)
## synapse_1_update = matrix(0, nrow = hidden_dim, ncol = output_dim)
## synapse_h_update = matrix(0, nrow = hidden_dim, ncol = hidden_dim)
##
## # training logic
## for (j in 1:length(Y)) {
##
## if(!silent && j %% 1000 == 0) {
## print(paste('Iteration:', j))
## }
##
## # generate a simple addition problem (a + b = c)
## a_int = X1[j] # int version
## a = int2binary(a_int, binary_dim) # binary encoding
##
## b_int = X2[j] # int version
## b = int2binary(b_int, binary_dim)
##
## # true answer
## c_int = Y[j]
## c = int2binary(c_int, binary_dim)
##
## # where we'll store our best guesss (binary encoded)
## d = matrix(0, nrow = 1, ncol = binary_dim)
##
## overallError = 0
##
## layer_2_deltas = matrix(0)
## layer_1_values = matrix(0, nrow=1, ncol = hidden_dim)
## # layer_1_values = rbind(layer_1_values, matrix(0, nrow=1, ncol=hidden_dim))
##
## # moving along the positions in the binary encoding
## for (position in 0:(binary_dim-1)) {
##
## # generate input and output
## X = cbind(a[binary_dim - position],b[binary_dim - position])
## y = c[binary_dim - position]
##
## # hidden layer (input ~+ prev_hidden)
## layer_1 = sigmoid((X%*%synapse_0) + (layer_1_values[dim(layer_1_values)[1],] %*% synapse_h))
##
## # output layer (new binary representation)
## layer_2 = sigmoid(layer_1 %*% synapse_1)
##
## # did we miss?... if so, by how much?
## layer_2_error = y - layer_2
## layer_2_deltas = rbind(layer_2_deltas, layer_2_error * sigmoid_output_to_derivative(layer_2))
## overallError = overallError + abs(layer_2_error)
##
## # decode estimate so we can print it out
## d[binary_dim - position] = round(layer_2)
##
## # store hidden layer so we can print it out
## layer_1_values = rbind(layer_1_values, layer_1)
##
## if(!silent && j %% 1000 == 0) {
## print(paste('x1:', a[binary_dim - position]))
## print(paste('x2:', b[binary_dim - position], '+'))
## print('-------')
## print(paste('y: ', c[binary_dim - position]))
## print(paste('y^:', d[binary_dim - position]))
## print('=======')
## }
## }
##
## future_layer_1_delta = matrix(0, nrow = 1, ncol = hidden_dim)
##
## for (position in 0:(binary_dim-1)) {
##
## X = cbind(a[position+1], b[position+1])
## layer_1 = layer_1_values[dim(layer_1_values)[1]-position,]
## prev_layer_1 = layer_1_values[dim(layer_1_values)[1]-(position+1),]
##
## # error at output layer
## layer_2_delta = layer_2_deltas[dim(layer_2_deltas)[1]-position,]
## # error at hidden layer
## layer_1_delta = (future_layer_1_delta %*% t(synapse_h) + layer_2_delta %*% t(synapse_1)) *
## sigmoid_output_to_derivative(layer_1)
##
## # let's update all our weights so we can try again
## synapse_1_update = synapse_1_update + matrix(layer_1) %*% layer_2_delta
## synapse_h_update = synapse_h_update + matrix(prev_layer_1) %*% layer_1_delta
## synapse_0_update = synapse_0_update + t(X) %*% layer_1_delta
##
## future_layer_1_delta = layer_1_delta
## }
##
## synapse_0 = synapse_0 + ( synapse_0_update * alpha )
## synapse_1 = synapse_1 + ( synapse_1_update * alpha )
## synapse_h = synapse_h + ( synapse_h_update * alpha )
##
## synapse_0_update = synapse_0_update * 0
## synapse_1_update = synapse_1_update * 0
## synapse_h_update = synapse_h_update * 0
##
## # print out progress
## if(!silent && j %% 1000 == 0) {
## print(paste('Error:', overallError))
## print(paste('X1:', paste(a, collapse = ' '), ' ', '(', a_int, ')'))
## print(paste('X2:', paste(b, collapse = ' '), '+', '(', b_int, ')'))
## print('-----------------------------')
## print(paste('Y: ', paste(c, collapse = ' '), ' ', '(', c_int, ')'))
## out = 0
## for (x in 1:length(d)) {
## out[x] = rev(d)[x]*2^(x-1) }
## print(paste('Y^:', paste(d, collapse = ' '), ' ', '(', sum(out), ')'))
## print('=============================')
## }
## }
## }
## <environment: namespace:rnn>As can be seen from the above, the model relies on three other functions that are included with the package.
The first one is int2binary, which stands for integer to binary, and converts an integer number to its binary representation:
int2binary(146, length=8)## [1] 1 0 0 1 0 0 1 0The code for this function is:
int2binary## function(x, length) {
## utils::tail(rev(as.integer(intToBits(x))), length) }
## <environment: namespace:rnn>The second function is sigmoid(), which converts an integer to its sigmoid value.
(a <- sigmoid(3))## [1] 0.9525741The code for the sigmoid() function is:
sigmoid## function(x) {
## output = 1 / (1+exp(-x))
## return(output) }
## <environment: namespace:rnn>The final function converts the sigmoid value of a number to its derivative.
sigmoid_output_to_derivative(a) # a was created above using sigmoid()## [1] 0.04517666Finally, we can inspect this code using:
sigmoid_output_to_derivative## function(output) {
## return( output*(1-output) ) }
## <environment: namespace:rnn>By setting a seed for the random number generator, we ensure replicability.
set.seed(123)An example is included in the help file.
help('rnn')First we generate the data:
# create sample inputs
X1 = sample(0:127, 5000, replace=TRUE)
X2 = sample(0:127, 5000, replace=TRUE)
# create sample output
Y <- X1 + X2This example is:
rnn(Y,
X1,
X2,
binary_dim = 8,
alpha = 0.1,
input_dim = 2,
hidden_dim = 10,
output_dim = 1)## [1] "Iteration: 1000"
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "Error: 3.79861131785405"
## [1] "X1: 0 0 0 0 1 1 0 1 ( 13 )"
## [1] "X2: 0 1 1 1 0 0 1 1 + ( 115 )"
## [1] "-----------------------------"
## [1] "Y: 1 0 0 0 0 0 0 0 ( 128 )"
## [1] "Y^: 0 0 0 0 0 0 0 0 ( 0 )"
## [1] "============================="
## [1] "Iteration: 2000"
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "Error: 4.35601700739721"
## [1] "X1: 0 0 1 1 1 0 0 1 ( 57 )"
## [1] "X2: 0 1 0 1 0 1 1 1 + ( 87 )"
## [1] "-----------------------------"
## [1] "Y: 1 0 0 1 0 0 0 0 ( 144 )"
## [1] "Y^: 1 1 1 1 1 1 1 1 ( 255 )"
## [1] "============================="
## [1] "Iteration: 3000"
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "Error: 4.23322205952535"
## [1] "X1: 0 1 0 1 0 1 1 1 ( 87 )"
## [1] "X2: 0 0 1 0 1 1 1 1 + ( 47 )"
## [1] "-----------------------------"
## [1] "Y: 1 0 0 0 0 1 1 0 ( 134 )"
## [1] "Y^: 1 1 1 1 1 1 1 0 ( 254 )"
## [1] "============================="
## [1] "Iteration: 4000"
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "Error: 3.94852213025003"
## [1] "X1: 0 0 1 1 0 1 1 0 ( 54 )"
## [1] "X2: 0 1 1 0 0 1 0 0 + ( 100 )"
## [1] "-----------------------------"
## [1] "Y: 1 0 0 1 1 0 1 0 ( 154 )"
## [1] "Y^: 0 1 0 0 1 0 0 0 ( 72 )"
## [1] "============================="
## [1] "Iteration: 5000"
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "Error: 2.88197887989984"
## [1] "X1: 0 0 0 0 0 0 0 1 ( 1 )"
## [1] "X2: 0 1 1 1 0 0 1 1 + ( 115 )"
## [1] "-----------------------------"
## [1] "Y: 0 1 1 1 0 1 0 0 ( 116 )"
## [1] "Y^: 0 1 1 1 0 1 1 0 ( 118 )"
## [1] "============================="It is interesting to vary the number of hidden units and iterations.
# create sample inputs
X1 = sample(0:127, 20000, replace=TRUE)
X2 = sample(0:127, 20000, replace=TRUE)
# create sample output
Y <- X1 + X2
rnn(Y,
X1,
X2,
binary_dim = 8,
alpha = 0.1,
input_dim = 2,
hidden_dim = 3,
output_dim = 1)## [1] "Iteration: 1000"
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "Error: 4.00581060883197"
## [1] "X1: 0 0 0 1 0 0 1 1 ( 19 )"
## [1] "X2: 0 0 1 1 1 1 1 0 + ( 62 )"
## [1] "-----------------------------"
## [1] "Y: 0 1 0 1 0 0 0 1 ( 81 )"
## [1] "Y^: 1 1 1 1 1 1 1 1 ( 255 )"
## [1] "============================="
## [1] "Iteration: 2000"
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "Error: 4.05519613296592"
## [1] "X1: 0 0 1 1 1 0 1 0 ( 58 )"
## [1] "X2: 0 0 1 1 1 1 1 1 + ( 63 )"
## [1] "-----------------------------"
## [1] "Y: 0 1 1 1 1 0 0 1 ( 121 )"
## [1] "Y^: 0 0 0 0 0 0 0 0 ( 0 )"
## [1] "============================="
## [1] "Iteration: 3000"
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "Error: 4.00309087882978"
## [1] "X1: 0 1 1 0 1 0 1 1 ( 107 )"
## [1] "X2: 0 1 0 0 0 1 0 0 + ( 68 )"
## [1] "-----------------------------"
## [1] "Y: 1 0 1 0 1 1 1 1 ( 175 )"
## [1] "Y^: 0 0 0 0 1 0 0 0 ( 8 )"
## [1] "============================="
## [1] "Iteration: 4000"
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "Error: 3.9979157964333"
## [1] "X1: 0 0 0 1 0 1 1 1 ( 23 )"
## [1] "X2: 0 1 0 0 1 0 1 0 + ( 74 )"
## [1] "-----------------------------"
## [1] "Y: 0 1 1 0 0 0 0 1 ( 97 )"
## [1] "Y^: 1 0 0 1 0 1 0 0 ( 148 )"
## [1] "============================="
## [1] "Iteration: 5000"
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "Error: 4.02559985626535"
## [1] "X1: 0 0 0 1 0 0 0 0 ( 16 )"
## [1] "X2: 0 0 1 0 0 0 1 1 + ( 35 )"
## [1] "-----------------------------"
## [1] "Y: 0 0 1 1 0 0 1 1 ( 51 )"
## [1] "Y^: 0 0 0 0 0 0 0 0 ( 0 )"
## [1] "============================="
## [1] "Iteration: 6000"
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "Error: 3.98188869279542"
## [1] "X1: 0 0 0 1 1 0 1 1 ( 27 )"
## [1] "X2: 0 0 0 0 1 1 1 0 + ( 14 )"
## [1] "-----------------------------"
## [1] "Y: 0 0 1 0 1 0 0 1 ( 41 )"
## [1] "Y^: 0 0 0 0 1 0 0 0 ( 8 )"
## [1] "============================="
## [1] "Iteration: 7000"
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "Error: 3.97125001336543"
## [1] "X1: 0 0 0 1 0 0 1 0 ( 18 )"
## [1] "X2: 0 0 1 1 0 1 0 0 + ( 52 )"
## [1] "-----------------------------"
## [1] "Y: 0 1 0 0 0 1 1 0 ( 70 )"
## [1] "Y^: 0 0 0 0 1 0 0 0 ( 8 )"
## [1] "============================="
## [1] "Iteration: 8000"
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "Error: 3.95976138728772"
## [1] "X1: 0 1 0 0 1 1 1 1 ( 79 )"
## [1] "X2: 0 1 0 0 1 0 1 1 + ( 75 )"
## [1] "-----------------------------"
## [1] "Y: 1 0 0 1 1 0 1 0 ( 154 )"
## [1] "Y^: 1 0 0 1 0 1 1 0 ( 150 )"
## [1] "============================="
## [1] "Iteration: 9000"
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "Error: 3.9000311712921"
## [1] "X1: 0 1 0 1 1 0 0 1 ( 89 )"
## [1] "X2: 0 0 0 0 0 1 1 0 + ( 6 )"
## [1] "-----------------------------"
## [1] "Y: 0 1 0 1 1 1 1 1 ( 95 )"
## [1] "Y^: 0 0 0 0 1 1 0 1 ( 13 )"
## [1] "============================="
## [1] "Iteration: 10000"
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "Error: 3.77216128486592"
## [1] "X1: 0 0 0 1 1 1 1 1 ( 31 )"
## [1] "X2: 0 0 0 1 0 1 1 0 + ( 22 )"
## [1] "-----------------------------"
## [1] "Y: 0 0 1 1 0 1 0 1 ( 53 )"
## [1] "Y^: 0 0 0 0 1 1 1 1 ( 15 )"
## [1] "============================="
## [1] "Iteration: 11000"
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "Error: 3.53622411158228"
## [1] "X1: 0 0 0 1 1 0 0 0 ( 24 )"
## [1] "X2: 0 1 1 1 0 1 0 0 + ( 116 )"
## [1] "-----------------------------"
## [1] "Y: 1 0 0 0 1 1 0 0 ( 140 )"
## [1] "Y^: 1 1 1 0 1 0 0 0 ( 232 )"
## [1] "============================="
## [1] "Iteration: 12000"
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "Error: 3.63749232770687"
## [1] "X1: 0 0 0 0 1 1 0 1 ( 13 )"
## [1] "X2: 0 0 1 1 1 0 0 1 + ( 57 )"
## [1] "-----------------------------"
## [1] "Y: 0 1 0 0 0 1 1 0 ( 70 )"
## [1] "Y^: 0 0 1 1 0 1 1 0 ( 54 )"
## [1] "============================="
## [1] "Iteration: 13000"
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "Error: 3.53987923342722"
## [1] "X1: 0 1 0 1 0 1 0 0 ( 84 )"
## [1] "X2: 0 1 0 0 0 0 0 1 + ( 65 )"
## [1] "-----------------------------"
## [1] "Y: 1 0 0 1 0 1 0 1 ( 149 )"
## [1] "Y^: 1 0 0 1 0 1 1 1 ( 151 )"
## [1] "============================="
## [1] "Iteration: 14000"
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "Error: 3.06002694096972"
## [1] "X1: 0 1 0 1 0 1 1 0 ( 86 )"
## [1] "X2: 0 1 0 1 0 1 1 0 + ( 86 )"
## [1] "-----------------------------"
## [1] "Y: 1 0 1 0 1 1 0 0 ( 172 )"
## [1] "Y^: 1 0 1 0 1 0 0 0 ( 168 )"
## [1] "============================="
## [1] "Iteration: 15000"
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "Error: 3.59314580597317"
## [1] "X1: 0 1 0 0 1 0 1 1 ( 75 )"
## [1] "X2: 0 1 0 1 1 1 1 1 + ( 95 )"
## [1] "-----------------------------"
## [1] "Y: 1 0 1 0 1 0 1 0 ( 170 )"
## [1] "Y^: 1 0 1 0 0 0 0 0 ( 160 )"
## [1] "============================="
## [1] "Iteration: 16000"
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "Error: 3.52772271488862"
## [1] "X1: 0 0 0 1 0 0 1 1 ( 19 )"
## [1] "X2: 0 1 1 1 1 1 0 0 + ( 124 )"
## [1] "-----------------------------"
## [1] "Y: 1 0 0 0 1 1 1 1 ( 143 )"
## [1] "Y^: 1 1 0 0 1 1 1 1 ( 207 )"
## [1] "============================="
## [1] "Iteration: 17000"
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "Error: 1.01627141527645"
## [1] "X1: 0 1 0 1 0 0 0 0 ( 80 )"
## [1] "X2: 0 0 1 0 0 1 0 0 + ( 36 )"
## [1] "-----------------------------"
## [1] "Y: 0 1 1 1 0 1 0 0 ( 116 )"
## [1] "Y^: 0 1 1 1 0 1 0 0 ( 116 )"
## [1] "============================="
## [1] "Iteration: 18000"
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "Error: 3.12585151531538"
## [1] "X1: 0 1 0 1 0 1 1 1 ( 87 )"
## [1] "X2: 0 1 0 1 1 0 0 0 + ( 88 )"
## [1] "-----------------------------"
## [1] "Y: 1 0 1 0 1 1 1 1 ( 175 )"
## [1] "Y^: 1 0 1 0 1 1 1 0 ( 174 )"
## [1] "============================="
## [1] "Iteration: 19000"
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "Error: 3.76283074446683"
## [1] "X1: 0 1 1 1 0 1 1 1 ( 119 )"
## [1] "X2: 0 0 0 1 0 0 0 0 + ( 16 )"
## [1] "-----------------------------"
## [1] "Y: 1 0 0 0 0 1 1 1 ( 135 )"
## [1] "Y^: 1 1 0 0 1 1 1 1 ( 207 )"
## [1] "============================="
## [1] "Iteration: 20000"
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 1 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 0"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 1"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 1"
## [1] "y^: 1"
## [1] "======="
## [1] "x1: 0"
## [1] "x2: 0 +"
## [1] "-------"
## [1] "y: 0"
## [1] "y^: 0"
## [1] "======="
## [1] "Error: 2.73563618581195"
## [1] "X1: 0 1 0 1 1 1 0 0 ( 92 )"
## [1] "X2: 0 0 0 1 1 1 1 0 + ( 30 )"
## [1] "-----------------------------"
## [1] "Y: 0 1 1 1 1 0 1 0 ( 122 )"
## [1] "Y^: 0 1 1 0 0 0 1 0 ( 98 )"
## [1] "============================="