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Avoiding multiple loops with multiple variables in R

I have two sets of data stored in tables, one of them is a set [a, b], and the other is [x, Sx, y, Sy, rho]. I have a probability function ffor which it is required (a, b, x, Sx, y, Sy, rho). In the end, I want to find the sum of the probability results for all of the [x, Sx, y, Sy, rho]first [a, b]. Then find the amount for everyone [x, Sx, y, Sy, rho]in the second [a, b], etc.

I would like to have several hundred lines in a file [x, Sx, y, Sy, rho]and several hundred thousand lines in a file [a, b].

I am wondering if there is a way to do this without using two loops? I tried the following and it does not work the way I want, but I know that it will be too slow.

I don't know if this will help, but I added a function to the code. Sorry that the function itself is a mess and is not formatted properly.

# data  file with (a, b)
data            <- matrix( c(1, 0, 1, 1, 0.5, 0), nrow=3, ncol=2) 
colnames(data)  <- c("a", "b") 
Ndat            <- dim(data)
Ndata           <- Ndat[1]

# data2 file with (x, Sx, y, Sy, rho)
data2           <- matrix( c(1, 0.1, 1, 0.1, 0.002, 2, 0.1, 2, 0.1, 0.000001, 
                             2, 0.1, 1, 0.1, 0.002), nrow=3, ncol=5) 
colnames(data2) <- c("x", "Sx", "y", "Sy", "rho") 
Ndat2           <- dim(data)
Ndata2          <- Ndat[1]

# function requires variables (a, b, s, Sx, y, Sy, rho) 
Prob  <- function(a, b, Xi, sX, Yi, sY, rho) {sqrt(1 + a ^ 2) * (
  exp(-((b + a * Xi - Yi) ^ 2 / (
    2 * ((a ^ 2 * sX ^ 2) - 
         (2 * a * rho * sX * sY) + sY ^ 2)))) * sqrt((
           1 - rho ^ 2) / (
             a ^ 2 * sX ^ 2 - 2 * a * rho *sX *sY + sY ^ 2))/(
               sqrt(2 * pi) * sqrt(1 - rho ^ 2)))
    }

# Here is my weak attempt
Table <- NULL
Table <- for (j in 1:Ndata) { 
   sum (for (i in 1:Ndata2) {
   Datatable[i] = Prob(data[j, a], data[j, b], data2[i, x], 
                 data2[i, Sx], data2[i, y], data2[i, Sy], 
                 data2[i, rho])
   })
}

It is very difficult for me to wrap my head around functions applyand when they can / should be used. I know that I probably did not add enough information, so any suggestions that might help me would be great. I am new to programming as well as to R, so please forgive any inappropriate dictionary or formatting.

There is probably a better way to determine the number or lines in datato get Ndataas global, but these are the first ones that I stumbled upon.

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, data2, a, b data. sum . 2 data, a, b, data2

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f <- function(a,b,x,y,z) a+b+x+y+z
f.new <- function(p1,p2) {
  p1=as.list(p1); p2=as.list(p2)
  f(p1$a,p1$b,p2$x,p2$y,p2$z)
}

data1 <- data.frame(a=1:10,b=11:20)
data2 <- data.frame(x=1:5,y=21:25,z=31:35)
indx  <- expand.grid(indx2=seq(nrow(data2)),indx1=seq(nrow(data1)))
result <- with(indx,f.new(data1[indx1,],data2[indx2,]))
sums   <- aggregate(result,by=list(rep(seq(nrow(data1)),each=nrow(data2))),sum)

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indx, , data1 data2, f.new(...), data1 data2, indx. result, (a,b) (x,y,z). , .

; result ~ 10MM, , .

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