Nothing
R/galambosExpr.R
In copula: Multivariate Dependence with Copulas
## This file is not to be edited. It is generated by ../inst/docs/mathnb/exprPrep.R
`galambosCopula.cdf.expr` <-
expression(0, exp(log(u1 * u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha))))
`galambosCopula.cdf.algr` <-
expression({
.value <- 0
.grad <- array(0, c(length(.value), 1L), list(NULL, c("s")))
.grad[, "s"] <- 0
attr(.value, "gradient") <- .grad
.value
}, {
.expr2 <- log(u1 * u2)
.expr4 <- log(u2)/.expr2
.expr5 <- -alpha
.value <- exp(.expr2 * (1 - (.expr4^.expr5 + (1 - .expr4)^.expr5)^(-1/alpha)))
.grad <- array(0, c(length(.value), 1L), list(NULL, c("s")))
.grad[, "s"] <- 0
attr(.value, "gradient") <- .grad
.value
})
`galambosCopula.cdfDerWrtArg.expr` <-
expression(0, exp(log(u1 * u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha))) * (u2/(u1 *
u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(-1/alpha)) - log(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)))))))
`galambosCopula.cdfDerWrtArg.algr` <-
expression({
.value <- 0
.grad <- array(0, c(length(.value), 1L), list(NULL, c("s")))
.grad[, "s"] <- 0
attr(.value, "gradient") <- .grad
.value
}, {
.expr1 <- u1 * u2
.expr2 <- log(.expr1)
.expr3 <- log(u2)
.expr4 <- .expr3/.expr2
.expr5 <- -alpha
.expr7 <- 1 - .expr4
.expr9 <- .expr4^.expr5 + .expr7^.expr5
.expr11 <- -1/alpha
.expr13 <- 1 - .expr9^.expr11
.expr16 <- u2/.expr1
.expr20 <- .expr5 - 1
.expr25 <- .expr5 * (.expr3 * .expr16/.expr2^2)
.value <- exp(.expr2 * .expr13) * (.expr16 * .expr13 - .expr2 *
(.expr9^(.expr11 - 1) * (.expr11 * (.expr7^.expr20 *
.expr25 - .expr4^.expr20 * .expr25))))
.grad <- array(0, c(length(.value), 1L), list(NULL, c("s")))
.grad[, "s"] <- 0
attr(.value, "gradient") <- .grad
.value
})
`galambosCopula.cdfDerWrtPar.expr` <-
expression(0, -(exp(log(u1 * u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha))) * (log(u1 *
u2) * (((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(-1/alpha) * (log(((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))) * (1/alpha^2)) -
((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * ((-1/alpha) * ((1 -
(log(u2)/log(u1 * u2)))^(-alpha) * log((1 - (log(u2)/log(u1 *
u2)))) + (log(u2)/log(u1 * u2))^(-alpha) * log((log(u2)/log(u1 *
u2)))))))))
`galambosCopula.cdfDerWrtPar.algr` <-
expression({
.value <- 0
.grad <- array(0, c(length(.value), 1L), list(NULL, c("s")))
.grad[, "s"] <- 0
attr(.value, "gradient") <- .grad
.value
}, {
.expr2 <- log(u1 * u2)
.expr4 <- log(u2)/.expr2
.expr5 <- -alpha
.expr6 <- .expr4^.expr5
.expr7 <- 1 - .expr4
.expr8 <- .expr7^.expr5
.expr9 <- .expr6 + .expr8
.expr11 <- -1/alpha
.expr12 <- .expr9^.expr11
.value <- -(exp(.expr2 * (1 - .expr12)) * (.expr2 * (.expr12 *
(log(.expr9) * (1/alpha^2)) - .expr9^(.expr11 - 1) *
(.expr11 * (.expr8 * log(.expr7) + .expr6 * log(.expr4))))))
.grad <- array(0, c(length(.value), 1L), list(NULL, c("s")))
.grad[, "s"] <- 0
attr(.value, "gradient") <- .grad
.value
})
`galambosCopula.pdf.expr` <-
expression(0, exp(log(u1 * u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha))) * (u1/(u1 *
u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(-1/alpha)) - log(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) - 1) *
((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) *
((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)))))) * (u2/(u1 * u2) * (1 - ((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha)) -
log(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * ((-1/alpha) * ((1 -
(log(u2)/log(u1 * u2)))^((-alpha) - 1) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) - (log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2)))))) + exp(log(u1 * u2) * (1 - ((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha))) *
((1/(u1 * u2) - u2 * u1/(u1 * u2)^2) * (1 - ((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha)) -
u2/(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) + (1 -
(log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) - 1) *
((-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2))))) -
(u1/(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) + (1 -
(log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) - 1) *
((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2))))) + log(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(((-1/alpha) -
1) - 1) * (((-1/alpha) - 1) * ((log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)) -
(1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) * ((-alpha) *
(1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)))) * ((-1/alpha) * ((1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)))) + ((log(u2)/log(u1 * u2))^(-alpha) + (1 -
(log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) - 1) *
((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * ((1/u2 * (u2/(u1 * u2)) + log(u2) *
(1/(u1 * u2) - u2 * u1/(u1 * u2)^2))/log(u1 *
u2)^2 - log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 *
u2) * log(u1 * u2)))/(log(u1 * u2)^2)^2)) - (1 -
(log(u2)/log(u1 * u2)))^(((-alpha) - 1) - 1) *
(((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
((log(u2)/log(u1 * u2))^(((-alpha) - 1) - 1) *
(((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) +
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
((1/u2 * (u2/(u1 * u2)) + log(u2) * (1/(u1 *
u2) - u2 * u1/(u1 * u2)^2))/log(u1 * u2)^2 -
log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 *
u2) * log(u1 * u2)))/(log(u1 * u2)^2)^2)))))))))
`galambosCopula.pdf.algr` <-
expression({
.value <- 0
.grad <- array(0, c(length(.value), 1L), list(NULL, c("s")))
.grad[, "s"] <- 0
attr(.value, "gradient") <- .grad
.value
}, {
.expr1 <- u1 * u2
.expr2 <- log(.expr1)
.expr3 <- log(u2)
.expr4 <- .expr3/.expr2
.expr5 <- -alpha
.expr7 <- 1 - .expr4
.expr9 <- .expr4^.expr5 + .expr7^.expr5
.expr11 <- -1/alpha
.expr13 <- 1 - .expr9^.expr11
.expr15 <- exp(.expr2 * .expr13)
.expr16 <- u1/.expr1
.expr18 <- .expr11 - 1
.expr19 <- .expr9^.expr18
.expr20 <- .expr5 - 1
.expr21 <- .expr4^.expr20
.expr22 <- 1/u2
.expr25 <- .expr2^2
.expr27 <- .expr22/.expr2 - .expr3 * .expr16/.expr25
.expr28 <- .expr5 * .expr27
.expr30 <- .expr7^.expr20
.expr32 <- .expr21 * .expr28 - .expr30 * .expr28
.expr34 <- .expr19 * (.expr11 * .expr32)
.expr38 <- u2/.expr1
.expr40 <- .expr3 * .expr38
.expr42 <- .expr5 * (.expr40/.expr25)
.expr46 <- .expr11 * (.expr30 * .expr42 - .expr21 * .expr42)
.expr47 <- .expr19 * .expr46
.expr55 <- 1/.expr1 - u2 * u1/.expr1^2
.expr75 <- .expr5 * ((.expr22 * .expr38 + .expr3 * .expr55)/.expr25 -
.expr40 * (2 * (.expr16 * .expr2))/.expr25^2)
.expr77 <- .expr20 - 1
.expr79 <- .expr20 * .expr27
.value <- .expr15 * (.expr16 * .expr13 - .expr2 * .expr34) *
(.expr38 * .expr13 - .expr2 * .expr47) + .expr15 * (.expr55 *
.expr13 - .expr38 * .expr34 - (.expr16 * .expr47 + .expr2 *
(.expr9^(.expr18 - 1) * (.expr18 * .expr32) * .expr46 +
.expr19 * (.expr11 * (.expr30 * .expr75 - .expr7^.expr77 *
.expr79 * .expr42 - (.expr4^.expr77 * .expr79 *
.expr42 + .expr21 * .expr75))))))
.grad <- array(0, c(length(.value), 1L), list(NULL, c("s")))
.grad[, "s"] <- 0
attr(.value, "gradient") <- .grad
.value
})
`galambosCopula.pdfDerWrtArg.expr` <-
expression(0, (exp(log(u1 * u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha))) * (u2/(u1 *
u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(-1/alpha)) - log(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)))))) * (u1/(u1 * u2) *
(1 - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(-1/alpha)) - log(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) - 1) *
((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) *
((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)))))) + exp(log(u1 * u2) * (1 - ((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha))) *
((1/(u1 * u2) - u1 * u2/(u1 * u2)^2) * (1 - ((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha)) -
u1/(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) + (1 -
(log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) - 1) *
((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2))))) - (u2/(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2))))) + log(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(((-1/alpha) -
1) - 1) * (((-1/alpha) - 1) * ((1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)))) *
((-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) - 1) *
((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)))) -
((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * ((-1/alpha) *
((log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
(1/u2 * (u2/(u1 * u2))/log(u1 * u2)^2 + (log(u2) *
(1/(u1 * u2) - u1 * u2/(u1 * u2)^2)/log(u1 *
u2)^2 - log(u2) * (u1/(u1 * u2)) * (2 * (u2/(u1 *
u2) * log(u1 * u2)))/(log(u1 * u2)^2)^2))) +
(log(u2)/log(u1 * u2))^(((-alpha) - 1) - 1) *
(((-alpha) - 1) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) * ((-alpha) * (1/u2/log(u1 * u2) -
log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)) +
((1 - (log(u2)/log(u1 * u2)))^(((-alpha) - 1) -
1) * (((-alpha) - 1) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2)) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)) -
(1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) *
((-alpha) * (1/u2 * (u2/(u1 * u2))/log(u1 *
u2)^2 + (log(u2) * (1/(u1 * u2) - u1 *
u2/(u1 * u2)^2)/log(u1 * u2)^2 - log(u2) *
(u1/(u1 * u2)) * (2 * (u2/(u1 * u2) * log(u1 *
u2)))/(log(u1 * u2)^2)^2)))))))))) * (u2/(u1 *
u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(-1/alpha)) - log(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)))))) - exp(log(u1 * u2) *
(1 - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(-1/alpha))) * (u1/(u1 * u2) * (1 - ((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha)) -
log(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * ((-1/alpha) * ((log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 * u2) -
log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 * u2) -
log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)))))) * (u2/(u1 *
u2) * (((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2))))) + u2 * u2/(u1 *
u2)^2 * (1 - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(-1/alpha)) + (u2/(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2))))) + log(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(((-1/alpha) -
1) - 1) * (((-1/alpha) - 1) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)))) * ((-1/alpha) * ((1 -
(log(u2)/log(u1 * u2)))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)))) +
((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * ((-1/alpha) * ((1 -
(log(u2)/log(u1 * u2)))^(((-alpha) - 1) - 1) * (((-alpha) -
1) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (log(u2) * (u2 * u2/(u1 *
u2)^2)/log(u1 * u2)^2 + log(u2) * (u2/(u1 * u2)) * (2 *
(u2/(u1 * u2) * log(u1 * u2)))/(log(u1 * u2)^2)^2)) +
((log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
(log(u2) * (u2 * u2/(u1 * u2)^2)/log(u1 * u2)^2 +
log(u2) * (u2/(u1 * u2)) * (2 * (u2/(u1 * u2) *
log(u1 * u2)))/(log(u1 * u2)^2)^2)) + (log(u2)/log(u1 *
u2))^(((-alpha) - 1) - 1) * (((-alpha) - 1) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)))))))) + (exp(log(u1 *
u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(-1/alpha))) * (u2/(u1 * u2) * (1 - ((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha)) -
log(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * ((-1/alpha) * ((1 -
(log(u2)/log(u1 * u2)))^((-alpha) - 1) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) - (log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2)))))) * ((1/(u1 * u2) - u2 * u1/(u1 *
u2)^2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(-1/alpha)) - u2/(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) - 1) *
((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) *
((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2))))) - (u1/(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) - 1) *
((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) *
((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2))))) + log(u1 * u2) *
(((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(((-1/alpha) - 1) - 1) * (((-1/alpha) -
1) * ((log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
(1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2)))) * ((-1/alpha) * ((1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)))) +
((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * ((-1/alpha) *
((1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) * ((-alpha) *
((1/u2 * (u2/(u1 * u2)) + log(u2) * (1/(u1 *
u2) - u2 * u1/(u1 * u2)^2))/log(u1 * u2)^2 -
log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 * u2) *
log(u1 * u2)))/(log(u1 * u2)^2)^2)) - (1 -
(log(u2)/log(u1 * u2)))^(((-alpha) - 1) - 1) *
(((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
((log(u2)/log(u1 * u2))^(((-alpha) - 1) - 1) *
(((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) +
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
((1/u2 * (u2/(u1 * u2)) + log(u2) * (1/(u1 *
u2) - u2 * u1/(u1 * u2)^2))/log(u1 * u2)^2 -
log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 *
u2) * log(u1 * u2)))/(log(u1 * u2)^2)^2)))))))) -
exp(log(u1 * u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha))) *
((1/(u1 * u2) - u2 * u1/(u1 * u2)^2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) - 1) *
((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2))))) +
(u2/(u1 * u2)^2 + (u2/(u1 * u2)^2 - u2 * u1 * (2 *
(u2 * (u1 * u2)))/((u1 * u2)^2)^2)) * (1 - ((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha)) +
(u2/(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(((-1/alpha) -
1) - 1) * (((-1/alpha) - 1) * ((1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) - (log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)))) * ((-1/alpha) *
((log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
(1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)))) - ((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (1/u2 * (u2/(u1 * u2))/log(u1 *
u2)^2 + (log(u2) * (1/(u1 * u2) - u1 * u2/(u1 *
u2)^2)/log(u1 * u2)^2 - log(u2) * (u1/(u1 * u2)) *
(2 * (u2/(u1 * u2) * log(u1 * u2)))/(log(u1 *
u2)^2)^2))) + (log(u2)/log(u1 * u2))^(((-alpha) -
1) - 1) * (((-alpha) - 1) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2)) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)) +
((1 - (log(u2)/log(u1 * u2)))^(((-alpha) - 1) -
1) * (((-alpha) - 1) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2)) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)) -
(1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) *
((-alpha) * (1/u2 * (u2/(u1 * u2))/log(u1 *
u2)^2 + (log(u2) * (1/(u1 * u2) - u1 *
u2/(u1 * u2)^2)/log(u1 * u2)^2 - log(u2) *
(u1/(u1 * u2)) * (2 * (u2/(u1 * u2) * log(u1 *
u2)))/(log(u1 * u2)^2)^2))))))) - u2 *
u2/(u1 * u2)^2 * (((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)))))) +
((1/(u1 * u2) - u1 * u2/(u1 * u2)^2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2))))) + u1/(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(((-1/alpha) -
1) - 1) * (((-1/alpha) - 1) * ((1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) - (log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)))) * ((-1/alpha) *
((1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) *
((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)))) + ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^(((-alpha) -
1) - 1) * (((-alpha) - 1) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2)) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2 * u2/(u1 * u2)^2)/log(u1 * u2)^2 + log(u2) *
(u2/(u1 * u2)) * (2 * (u2/(u1 * u2) * log(u1 *
u2)))/(log(u1 * u2)^2)^2)) + ((log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2 * u2/(u1 * u2)^2)/log(u1 * u2)^2 + log(u2) *
(u2/(u1 * u2)) * (2 * (u2/(u1 * u2) * log(u1 *
u2)))/(log(u1 * u2)^2)^2)) + (log(u2)/log(u1 *
u2))^(((-alpha) - 1) - 1) * (((-alpha) - 1) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) *
((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)))))) + (u2/(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(((-1/alpha) -
1) - 1) * (((-1/alpha) - 1) * ((log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)) -
(1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) *
((-alpha) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)))) * ((-1/alpha) *
((1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) *
((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)))) + ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * ((1/u2 * (u2/(u1 * u2)) + log(u2) *
(1/(u1 * u2) - u2 * u1/(u1 * u2)^2))/log(u1 *
u2)^2 - log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 *
u2) * log(u1 * u2)))/(log(u1 * u2)^2)^2)) - (1 -
(log(u2)/log(u1 * u2)))^(((-alpha) - 1) - 1) *
(((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
((log(u2)/log(u1 * u2))^(((-alpha) - 1) - 1) *
(((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) +
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
((1/u2 * (u2/(u1 * u2)) + log(u2) * (1/(u1 *
u2) - u2 * u1/(u1 * u2)^2))/log(u1 * u2)^2 -
log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 *
u2) * log(u1 * u2)))/(log(u1 * u2)^2)^2)))))) +
log(u1 * u2) * ((((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((((-1/alpha) -
1) - 1) - 1) * ((((-1/alpha) - 1) - 1) * ((1 -
(log(u2)/log(u1 * u2)))^((-alpha) - 1) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)))) *
(((-1/alpha) - 1) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)))) -
((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(((-1/alpha) - 1) - 1) *
(((-1/alpha) - 1) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (1/u2 * (u2/(u1 * u2))/log(u1 *
u2)^2 + (log(u2) * (1/(u1 * u2) - u1 *
u2/(u1 * u2)^2)/log(u1 * u2)^2 - log(u2) *
(u1/(u1 * u2)) * (2 * (u2/(u1 * u2) * log(u1 *
u2)))/(log(u1 * u2)^2)^2))) + (log(u2)/log(u1 *
u2))^(((-alpha) - 1) - 1) * (((-alpha) -
1) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) * ((-alpha) * (1/u2/log(u1 * u2) -
log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)) +
((1 - (log(u2)/log(u1 * u2)))^(((-alpha) -
1) - 1) * (((-alpha) - 1) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) *
(1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (1/u2 *
(u2/(u1 * u2))/log(u1 * u2)^2 + (log(u2) *
(1/(u1 * u2) - u1 * u2/(u1 * u2)^2)/log(u1 *
u2)^2 - log(u2) * (u1/(u1 * u2)) * (2 *
(u2/(u1 * u2) * log(u1 * u2)))/(log(u1 *
u2)^2)^2))))))) * ((-1/alpha) * ((1 -
(log(u2)/log(u1 * u2)))^((-alpha) - 1) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)))) +
((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(((-1/alpha) - 1) - 1) *
(((-1/alpha) - 1) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) - (1 -
(log(u2)/log(u1 * u2)))^((-alpha) - 1) *
((-alpha) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)))) * ((-1/alpha) *
((1 - (log(u2)/log(u1 * u2)))^(((-alpha) -
1) - 1) * (((-alpha) - 1) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
(1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2 * u2/(u1 *
u2)^2)/log(u1 * u2)^2 + log(u2) * (u2/(u1 *
u2)) * (2 * (u2/(u1 * u2) * log(u1 *
u2)))/(log(u1 * u2)^2)^2)) + ((log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2 * u2/(u1 * u2)^2)/log(u1 * u2)^2 +
log(u2) * (u2/(u1 * u2)) * (2 * (u2/(u1 *
u2) * log(u1 * u2)))/(log(u1 * u2)^2)^2)) +
(log(u2)/log(u1 * u2))^(((-alpha) - 1) -
1) * (((-alpha) - 1) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2)) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2))))) +
(((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(((-1/alpha) - 1) - 1) *
(((-1/alpha) - 1) * ((1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) - (log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)))) * ((-1/alpha) *
((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * ((1/u2 * (u2/(u1 * u2)) +
log(u2) * (1/(u1 * u2) - u2 * u1/(u1 *
u2)^2))/log(u1 * u2)^2 - log(u2) * (u2/(u1 *
u2)) * (2 * (u1/(u1 * u2) * log(u1 * u2)))/(log(u1 *
u2)^2)^2)) - (1 - (log(u2)/log(u1 * u2)))^(((-alpha) -
1) - 1) * (((-alpha) - 1) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)) * ((-alpha) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2)) - ((log(u2)/log(u1 *
u2))^(((-alpha) - 1) - 1) * (((-alpha) -
1) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2)) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) + (log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * ((1/u2 *
(u2/(u1 * u2)) + log(u2) * (1/(u1 * u2) -
u2 * u1/(u1 * u2)^2))/log(u1 * u2)^2 -
log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 *
u2) * log(u1 * u2)))/(log(u1 * u2)^2)^2))))) +
((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * ((-1/alpha) *
((1 - (log(u2)/log(u1 * u2)))^(((-alpha) -
1) - 1) * (((-alpha) - 1) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) *
((1/u2 * (u2/(u1 * u2)) + log(u2) * (1/(u1 *
u2) - u2 * u1/(u1 * u2)^2))/log(u1 *
u2)^2 - log(u2) * (u2/(u1 * u2)) *
(2 * (u1/(u1 * u2) * log(u1 * u2)))/(log(u1 *
u2)^2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * ((log(u2) *
(u2/(u1 * u2)^2 + (u2/(u1 * u2)^2 - u2 *
u1 * (2 * (u2 * (u1 * u2)))/((u1 *
u2)^2)^2)) + 1/u2 * (u2 * u2/(u1 *
u2)^2))/log(u1 * u2)^2 + (1/u2 * (u2/(u1 *
u2)) + log(u2) * (1/(u1 * u2) - u2 *
u1/(u1 * u2)^2)) * (2 * (u2/(u1 * u2) *
log(u1 * u2)))/(log(u1 * u2)^2)^2 + ((log(u2) *
(u2/(u1 * u2)) * (2 * ((1/(u1 * u2) -
u1 * u2/(u1 * u2)^2) * log(u1 * u2) +
u1/(u1 * u2) * (u2/(u1 * u2)))) - log(u2) *
(u2 * u2/(u1 * u2)^2) * (2 * (u1/(u1 *
u2) * log(u1 * u2))))/(log(u1 * u2)^2)^2 -
log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 *
u2) * log(u1 * u2))) * (2 * (2 * (u2/(u1 *
u2) * log(u1 * u2)) * (log(u1 * u2)^2)))/((log(u1 *
u2)^2)^2)^2))) - (((1 - (log(u2)/log(u1 *
u2)))^((((-alpha) - 1) - 1) - 1) * ((((-alpha) -
1) - 1) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) * (((-alpha) - 1) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^(((-alpha) -
1) - 1) * (((-alpha) - 1) * (1/u2 * (u2/(u1 *
u2))/log(u1 * u2)^2 + (log(u2) * (1/(u1 *
u2) - u1 * u2/(u1 * u2)^2)/log(u1 * u2)^2 -
log(u2) * (u1/(u1 * u2)) * (2 * (u2/(u1 *
u2) * log(u1 * u2)))/(log(u1 * u2)^2)^2)))) *
((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^(((-alpha) -
1) - 1) * (((-alpha) - 1) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)) * ((-alpha) * (log(u2) * (u2 *
u2/(u1 * u2)^2)/log(u1 * u2)^2 + log(u2) *
(u2/(u1 * u2)) * (2 * (u2/(u1 * u2) *
log(u1 * u2)))/(log(u1 * u2)^2)^2))) +
((log(u2)/log(u1 * u2))^((-alpha) - 1) *
((-alpha) * ((log(u2) * (u2/(u1 * u2)^2 +
(u2/(u1 * u2)^2 - u2 * u1 * (2 *
(u2 * (u1 * u2)))/((u1 * u2)^2)^2)) +
1/u2 * (u2 * u2/(u1 * u2)^2))/log(u1 *
u2)^2 + (1/u2 * (u2/(u1 * u2)) +
log(u2) * (1/(u1 * u2) - u2 * u1/(u1 *
u2)^2)) * (2 * (u2/(u1 * u2) *
log(u1 * u2)))/(log(u1 * u2)^2)^2 +
((log(u2) * (u2/(u1 * u2)) * (2 *
((1/(u1 * u2) - u1 * u2/(u1 * u2)^2) *
log(u1 * u2) + u1/(u1 * u2) *
(u2/(u1 * u2)))) - log(u2) *
(u2 * u2/(u1 * u2)^2) * (2 * (u1/(u1 *
u2) * log(u1 * u2))))/(log(u1 *
u2)^2)^2 - log(u2) * (u2/(u1 *
u2)) * (2 * (u1/(u1 * u2) * log(u1 *
u2))) * (2 * (2 * (u2/(u1 * u2) *
log(u1 * u2)) * (log(u1 * u2)^2)))/((log(u1 *
u2)^2)^2)^2))) + (log(u2)/log(u1 *
u2))^(((-alpha) - 1) - 1) * (((-alpha) -
1) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) * ((-alpha) * ((1/u2 * (u2/(u1 *
u2)) + log(u2) * (1/(u1 * u2) - u2 *
u1/(u1 * u2)^2))/log(u1 * u2)^2 - log(u2) *
(u2/(u1 * u2)) * (2 * (u1/(u1 * u2) *
log(u1 * u2)))/(log(u1 * u2)^2)^2)) +
((log(u2)/log(u1 * u2))^(((-alpha) -
1) - 1) * (((-alpha) - 1) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)) * ((-alpha) * (log(u2) *
(u2 * u2/(u1 * u2)^2)/log(u1 * u2)^2 +
log(u2) * (u2/(u1 * u2)) * (2 * (u2/(u1 *
u2) * log(u1 * u2)))/(log(u1 *
u2)^2)^2)) + ((log(u2)/log(u1 *
u2))^(((-alpha) - 1) - 1) * (((-alpha) -
1) * (1/u2 * (u2/(u1 * u2))/log(u1 *
u2)^2 + (log(u2) * (1/(u1 * u2) -
u1 * u2/(u1 * u2)^2)/log(u1 * u2)^2 -
log(u2) * (u1/(u1 * u2)) * (2 * (u2/(u1 *
u2) * log(u1 * u2)))/(log(u1 *
u2)^2)^2))) + (log(u2)/log(u1 *
u2))^((((-alpha) - 1) - 1) - 1) *
((((-alpha) - 1) - 1) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) *
(((-alpha) - 1) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2))) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)))))))))))))
`galambosCopula.pdfDerWrtArg.algr` <-
expression({
.value <- 0
.grad <- array(0, c(length(.value), 1L), list(NULL, c("s")))
.grad[, "s"] <- 0
attr(.value, "gradient") <- .grad
.value
}, {
.expr1 <- u1 * u2
.expr2 <- log(.expr1)
.expr3 <- log(u2)
.expr4 <- .expr3/.expr2
.expr5 <- -alpha
.expr7 <- 1 - .expr4
.expr9 <- .expr4^.expr5 + .expr7^.expr5
.expr11 <- -1/alpha
.expr13 <- 1 - .expr9^.expr11
.expr15 <- exp(.expr2 * .expr13)
.expr16 <- u2/.expr1
.expr18 <- .expr11 - 1
.expr19 <- .expr9^.expr18
.expr20 <- .expr5 - 1
.expr21 <- .expr7^.expr20
.expr22 <- .expr3 * .expr16
.expr23 <- .expr2^2
.expr24 <- .expr22/.expr23
.expr25 <- .expr5 * .expr24
.expr27 <- .expr4^.expr20
.expr29 <- .expr21 * .expr25 - .expr27 * .expr25
.expr30 <- .expr11 * .expr29
.expr31 <- .expr19 * .expr30
.expr33 <- .expr16 * .expr13 - .expr2 * .expr31
.expr34 <- .expr15 * .expr33
.expr35 <- u1/.expr1
.expr37 <- 1/u2
.expr39 <- .expr3 * .expr35
.expr41 <- .expr37/.expr2 - .expr39/.expr23
.expr42 <- .expr5 * .expr41
.expr45 <- .expr27 * .expr42 - .expr21 * .expr42
.expr46 <- .expr11 * .expr45
.expr47 <- .expr19 * .expr46
.expr49 <- .expr35 * .expr13 - .expr2 * .expr47
.expr51 <- 1/.expr1
.expr52 <- .expr1^2
.expr54 <- .expr51 - .expr1/.expr52
.expr56 <- .expr35 * .expr31
.expr58 <- .expr16 * .expr47
.expr59 <- .expr18 - 1
.expr60 <- .expr9^.expr59
.expr62 <- .expr60 * (.expr18 * .expr29)
.expr64 <- .expr37 * .expr16
.expr69 <- 2 * (.expr16 * .expr2)
.expr71 <- .expr23^2
.expr74 <- .expr64/.expr23 + (.expr3 * .expr54/.expr23 -
.expr39 * .expr69/.expr71)
.expr75 <- .expr5 * .expr74
.expr77 <- .expr20 - 1
.expr78 <- .expr4^.expr77
.expr79 <- .expr20 * .expr24
.expr80 <- .expr78 * .expr79
.expr83 <- .expr7^.expr77
.expr84 <- .expr83 * .expr79
.expr88 <- .expr27 * .expr75 + .expr80 * .expr42 + (.expr84 *
.expr42 - .expr21 * .expr75)
.expr91 <- .expr62 * .expr46 - .expr19 * (.expr11 * .expr88)
.expr99 <- .expr16 * .expr31
.expr101 <- u2 * u2/.expr52
.expr106 <- .expr3 * .expr101
.expr111 <- .expr5 * (.expr106/.expr23 + .expr22 * .expr69/.expr71)
.expr118 <- .expr11 * (.expr84 * .expr25 - .expr21 * .expr111 +
(.expr27 * .expr111 + .expr80 * .expr25))
.expr120 <- .expr62 * .expr30 + .expr19 * .expr118
.expr126 <- u2 * u1
.expr128 <- .expr51 - .expr126/.expr52
.expr131 <- .expr18 * .expr45
.expr132 <- .expr60 * .expr131
.expr135 <- .expr64 + .expr3 * .expr128
.expr138 <- 2 * (.expr35 * .expr2)
.expr139 <- .expr22 * .expr138
.expr142 <- .expr5 * (.expr135/.expr23 - .expr139/.expr71)
.expr144 <- .expr20 * .expr41
.expr145 <- .expr83 * .expr144
.expr148 <- .expr78 * .expr144
.expr153 <- .expr11 * (.expr21 * .expr142 - .expr145 * .expr25 -
(.expr148 * .expr25 + .expr27 * .expr142))
.expr155 <- .expr132 * .expr30 + .expr19 * .expr153
.expr161 <- u2/.expr52
.expr168 <- .expr161 + (.expr161 - .expr126 * (2 * (u2 *
.expr1))/.expr52^2)
.expr214 <- .expr5 * ((.expr3 * .expr168 + .expr37 * .expr101)/.expr23 +
.expr135 * .expr69/.expr71 + ((.expr22 * (2 * (.expr54 *
.expr2 + .expr35 * .expr16)) - .expr106 * .expr138)/.expr71 -
.expr139 * (2 * (.expr69 * .expr23))/.expr71^2))
.expr217 <- .expr77 - 1
.expr219 <- .expr77 * .expr24
.expr222 <- .expr20 * .expr74
.value <- (.expr34 * .expr49 + .expr15 * (.expr54 * .expr13 -
.expr56 - (.expr58 + .expr2 * .expr91))) * .expr33 -
.expr15 * .expr49 * (.expr99 + .expr101 * .expr13 + (.expr99 +
.expr2 * .expr120)) + (.expr34 * (.expr128 * .expr13 -
.expr58 - (.expr56 + .expr2 * .expr155)) - .expr15 *
(.expr128 * .expr31 + .expr168 * .expr13 + (.expr16 *
.expr91 - .expr101 * .expr47) + (.expr54 * .expr31 +
.expr35 * .expr120 + (.expr16 * .expr155 + .expr2 *
((.expr9^(.expr59 - 1) * (.expr59 * .expr29) * .expr131 -
.expr60 * (.expr18 * .expr88)) * .expr30 + .expr132 *
.expr118 + (.expr62 * .expr153 + .expr19 * (.expr11 *
(.expr84 * .expr142 - .expr21 * .expr214 - ((.expr7^.expr217 *
.expr219 * .expr144 - .expr83 * .expr222) *
.expr25 - .expr145 * .expr111) + (.expr27 *
.expr214 + .expr80 * .expr142 + (.expr148 *
.expr111 + (.expr78 * .expr222 + .expr4^.expr217 *
.expr219 * .expr144) * .expr25))))))))))
.grad <- array(0, c(length(.value), 1L), list(NULL, c("s")))
.grad[, "s"] <- 0
attr(.value, "gradient") <- .grad
.value
})
`galambosCopula.pdfDerWrtPar.expr` <-
expression(0, -(exp(log(u1 * u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha))) * ((1/(u1 *
u2) - u2 * u1/(u1 * u2)^2) * (((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha) * (log(((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))) *
(1/alpha^2)) - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 *
u2)))^(-alpha) * log((1 - (log(u2)/log(u1 * u2)))) + (log(u2)/log(u1 *
u2))^(-alpha) * log((log(u2)/log(u1 * u2)))))) + u2/(u1 *
u2) * ((((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * (log(((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))) *
(1/alpha^2)) - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(((-1/alpha) - 1) - 1) * (((-1/alpha) - 1) *
((1 - (log(u2)/log(u1 * u2)))^(-alpha) * log((1 - (log(u2)/log(u1 *
u2)))) + (log(u2)/log(u1 * u2))^(-alpha) * log((log(u2)/log(u1 *
u2)))))) * ((-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2)))) + ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) - 1) *
(1/alpha^2 * ((log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
(1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2))) - (-1/alpha) * ((log(u2)/log(u1 *
u2))^((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2) + (log(u2)/log(u1 * u2))^((-alpha) -
1) * log((log(u2)/log(u1 * u2))) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)) - ((1 -
(log(u2)/log(u1 * u2)))^((-alpha) - 1) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2) + (1 -
(log(u2)/log(u1 * u2)))^((-alpha) - 1) * log((1 - (log(u2)/log(u1 *
u2)))) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2)))))) + (u1/(u1 * u2) * ((((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * (log(((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))) * (1/alpha^2)) - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(((-1/alpha) - 1) -
1) * (((-1/alpha) - 1) * ((1 - (log(u2)/log(u1 * u2)))^(-alpha) *
log((1 - (log(u2)/log(u1 * u2)))) + (log(u2)/log(u1 * u2))^(-alpha) *
log((log(u2)/log(u1 * u2)))))) * ((-1/alpha) * ((1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)))) + ((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * (1/alpha^2 * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2))) - (-1/alpha) * ((1 -
(log(u2)/log(u1 * u2)))^((-alpha) - 1) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2) + (1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * log((1 - (log(u2)/log(u1 * u2)))) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) - ((log(u2)/log(u1 * u2))^((-alpha) -
1) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2) + (log(u2)/log(u1 *
u2))^((-alpha) - 1) * log((log(u2)/log(u1 * u2))) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)))))) + log(u1 *
u2) * (((((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(((-1/alpha) - 1) - 1) * (log(((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))) *
(1/alpha^2)) - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((((-1/alpha) - 1) - 1) - 1) * ((((-1/alpha) -
1) - 1) * ((1 - (log(u2)/log(u1 * u2)))^(-alpha) * log((1 -
(log(u2)/log(u1 * u2)))) + (log(u2)/log(u1 * u2))^(-alpha) *
log((log(u2)/log(u1 * u2)))))) * (((-1/alpha) - 1) * ((log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)))) + ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(((-1/alpha) - 1) -
1) * (1/alpha^2 * ((log(u2)/log(u1 * u2))^((-alpha) - 1) *
((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) *
((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2))) - ((-1/alpha) - 1) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2) + (log(u2)/log(u1 * u2))^((-alpha) - 1) * log((log(u2)/log(u1 *
u2))) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2)) - ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2) + (1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) * log((1 -
(log(u2)/log(u1 * u2)))) * ((-alpha) * (1/u2/log(u1 * u2) -
log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)))))) * ((-1/alpha) *
((1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) - (log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2)))) + ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(((-1/alpha) - 1) -
1) * (((-1/alpha) - 1) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2)))) * (1/alpha^2 * ((1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2))) - (-1/alpha) *
((1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2) + (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * log((1 - (log(u2)/log(u1 * u2)))) *
((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
((log(u2)/log(u1 * u2))^((-alpha) - 1) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2) + (log(u2)/log(u1 * u2))^((-alpha) -
1) * log((log(u2)/log(u1 * u2))) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2))))) + ((((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * (log(((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))) * (1/alpha^2)) - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(((-1/alpha) - 1) -
1) * (((-1/alpha) - 1) * ((1 - (log(u2)/log(u1 * u2)))^(-alpha) *
log((1 - (log(u2)/log(u1 * u2)))) + (log(u2)/log(u1 * u2))^(-alpha) *
log((log(u2)/log(u1 * u2)))))) * ((-1/alpha) * ((1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * ((1/u2 * (u2/(u1 * u2)) +
log(u2) * (1/(u1 * u2) - u2 * u1/(u1 * u2)^2))/log(u1 * u2)^2 -
log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 * u2) * log(u1 *
u2)))/(log(u1 * u2)^2)^2)) - (1 - (log(u2)/log(u1 * u2)))^(((-alpha) -
1) - 1) * (((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) - ((log(u2)/log(u1 * u2))^(((-alpha) -
1) - 1) * (((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) + (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * ((1/u2 * (u2/(u1 * u2)) + log(u2) * (1/(u1 *
u2) - u2 * u1/(u1 * u2)^2))/log(u1 * u2)^2 - log(u2) * (u2/(u1 *
u2)) * (2 * (u1/(u1 * u2) * log(u1 * u2)))/(log(u1 * u2)^2)^2))))) +
((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * (1/alpha^2 * ((1 -
(log(u2)/log(u1 * u2)))^((-alpha) - 1) * ((-alpha) *
((1/u2 * (u2/(u1 * u2)) + log(u2) * (1/(u1 * u2) - u2 *
u1/(u1 * u2)^2))/log(u1 * u2)^2 - log(u2) * (u2/(u1 *
u2)) * (2 * (u1/(u1 * u2) * log(u1 * u2)))/(log(u1 *
u2)^2)^2)) - (1 - (log(u2)/log(u1 * u2)))^(((-alpha) -
1) - 1) * (((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) - ((log(u2)/log(u1 *
u2))^(((-alpha) - 1) - 1) * (((-alpha) - 1) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) + (log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * ((1/u2 * (u2/(u1 *
u2)) + log(u2) * (1/(u1 * u2) - u2 * u1/(u1 * u2)^2))/log(u1 *
u2)^2 - log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 * u2) *
log(u1 * u2)))/(log(u1 * u2)^2)^2)))) - (-1/alpha) *
((1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) * ((1/u2 *
(u2/(u1 * u2)) + log(u2) * (1/(u1 * u2) - u2 * u1/(u1 *
u2)^2))/log(u1 * u2)^2 - log(u2) * (u2/(u1 * u2)) *
(2 * (u1/(u1 * u2) * log(u1 * u2)))/(log(u1 * u2)^2)^2) +
(1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) * log((1 -
(log(u2)/log(u1 * u2)))) * ((-alpha) * ((1/u2 *
(u2/(u1 * u2)) + log(u2) * (1/(u1 * u2) - u2 *
u1/(u1 * u2)^2))/log(u1 * u2)^2 - log(u2) * (u2/(u1 *
u2)) * (2 * (u1/(u1 * u2) * log(u1 * u2)))/(log(u1 *
u2)^2)^2)) - ((1 - (log(u2)/log(u1 * u2)))^(((-alpha) -
1) - 1) * (((-alpha) - 1) * (1/u2/log(u1 * u2) -
log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2) + ((1 - (log(u2)/log(u1 *
u2)))^(((-alpha) - 1) - 1) * (1/u2/log(u1 * u2) -
log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2) + (1 - (log(u2)/log(u1 *
u2)))^(((-alpha) - 1) - 1) * log((1 - (log(u2)/log(u1 *
u2)))) * (((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2))) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2))) - ((log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((1/u2 * (u2/(u1 * u2)) + log(u2) *
(1/(u1 * u2) - u2 * u1/(u1 * u2)^2))/log(u1 * u2)^2 -
log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 * u2) * log(u1 *
u2)))/(log(u1 * u2)^2)^2) + (log(u2)/log(u1 *
u2))^((-alpha) - 1) * log((log(u2)/log(u1 * u2))) *
((-alpha) * ((1/u2 * (u2/(u1 * u2)) + log(u2) * (1/(u1 *
u2) - u2 * u1/(u1 * u2)^2))/log(u1 * u2)^2 -
log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 * u2) *
log(u1 * u2)))/(log(u1 * u2)^2)^2)) + ((log(u2)/log(u1 *
u2))^(((-alpha) - 1) - 1) * (((-alpha) - 1) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2) + ((log(u2)/log(u1 *
u2))^(((-alpha) - 1) - 1) * (1/u2/log(u1 * u2) -
log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2) + (log(u2)/log(u1 *
u2))^(((-alpha) - 1) - 1) * log((log(u2)/log(u1 *
u2))) * (((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2))) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)))))))))) + exp(log(u1 *
u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(-1/alpha))) * (log(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha) *
(log(((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))) * (1/alpha^2)) - ((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^(-alpha) *
log((1 - (log(u2)/log(u1 * u2)))) + (log(u2)/log(u1 * u2))^(-alpha) *
log((log(u2)/log(u1 * u2))))))) * ((1/(u1 * u2) - u2 * u1/(u1 *
u2)^2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(-1/alpha)) - u2/(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) - 1) *
((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) *
((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2))))) - (u1/(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) - 1) *
((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) *
((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2))))) + log(u1 * u2) *
(((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(((-1/alpha) - 1) - 1) * (((-1/alpha) -
1) * ((log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
(1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2)))) * ((-1/alpha) * ((1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)))) +
((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * ((-1/alpha) *
((1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) * ((-alpha) *
((1/u2 * (u2/(u1 * u2)) + log(u2) * (1/(u1 *
u2) - u2 * u1/(u1 * u2)^2))/log(u1 * u2)^2 -
log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 * u2) *
log(u1 * u2)))/(log(u1 * u2)^2)^2)) - (1 -
(log(u2)/log(u1 * u2)))^(((-alpha) - 1) - 1) *
(((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
((log(u2)/log(u1 * u2))^(((-alpha) - 1) - 1) *
(((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) +
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
((1/u2 * (u2/(u1 * u2)) + log(u2) * (1/(u1 *
u2) - u2 * u1/(u1 * u2)^2))/log(u1 * u2)^2 -
log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 *
u2) * log(u1 * u2)))/(log(u1 * u2)^2)^2)))))))) +
(exp(log(u1 * u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha))) *
(u1/(u1 * u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha)) -
log(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)))))) *
(u2/(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) + (1 -
(log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha) * (log(((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))) *
(1/alpha^2)) - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^(-alpha) *
log((1 - (log(u2)/log(u1 * u2)))) + (log(u2)/log(u1 *
u2))^(-alpha) * log((log(u2)/log(u1 * u2)))))) +
log(u1 * u2) * ((((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * (log(((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))) * (1/alpha^2)) -
((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(((-1/alpha) - 1) - 1) * (((-1/alpha) -
1) * ((1 - (log(u2)/log(u1 * u2)))^(-alpha) *
log((1 - (log(u2)/log(u1 * u2)))) + (log(u2)/log(u1 *
u2))^(-alpha) * log((log(u2)/log(u1 * u2)))))) *
((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)))) + ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * (1/alpha^2 * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2))) - (-1/alpha) * ((1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2) + (1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * log((1 - (log(u2)/log(u1 * u2)))) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
((log(u2)/log(u1 * u2))^((-alpha) - 1) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2) + (log(u2)/log(u1 *
u2))^((-alpha) - 1) * log((log(u2)/log(u1 *
u2))) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2))))))) + (exp(log(u1 * u2) * (1 - ((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha))) *
(u1/(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) + (1 -
(log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha) * (log(((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))) *
(1/alpha^2)) - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^(-alpha) *
log((1 - (log(u2)/log(u1 * u2)))) + (log(u2)/log(u1 *
u2))^(-alpha) * log((log(u2)/log(u1 * u2)))))) +
log(u1 * u2) * ((((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * (log(((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))) * (1/alpha^2)) -
((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(((-1/alpha) - 1) - 1) * (((-1/alpha) -
1) * ((1 - (log(u2)/log(u1 * u2)))^(-alpha) *
log((1 - (log(u2)/log(u1 * u2)))) + (log(u2)/log(u1 *
u2))^(-alpha) * log((log(u2)/log(u1 * u2)))))) *
((-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)))) +
((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * (1/alpha^2 *
((log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
(1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2))) -
(-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2) + (log(u2)/log(u1 *
u2))^((-alpha) - 1) * log((log(u2)/log(u1 *
u2))) * ((-alpha) * (1/u2/log(u1 * u2) -
log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)) -
((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2) + (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * log((1 - (log(u2)/log(u1 *
u2)))) * ((-alpha) * (1/u2/log(u1 * u2) -
log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2))))))) +
exp(log(u1 * u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha))) *
(log(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha) *
(log(((log(u2)/log(u1 * u2))^(-alpha) + (1 -
(log(u2)/log(u1 * u2)))^(-alpha))) * (1/alpha^2)) -
((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * ((-1/alpha) *
((1 - (log(u2)/log(u1 * u2)))^(-alpha) * log((1 -
(log(u2)/log(u1 * u2)))) + (log(u2)/log(u1 *
u2))^(-alpha) * log((log(u2)/log(u1 * u2))))))) *
(u1/(u1 * u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha)) -
log(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2))))))) *
(u2/(u1 * u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha)) -
log(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)))))))))
`galambosCopula.pdfDerWrtPar.algr` <-
expression({
.value <- 0
.grad <- array(0, c(length(.value), 1L), list(NULL, c("s")))
.grad[, "s"] <- 0
attr(.value, "gradient") <- .grad
.value
}, {
.expr1 <- u1 * u2
.expr2 <- log(.expr1)
.expr3 <- log(u2)
.expr4 <- .expr3/.expr2
.expr5 <- -alpha
.expr6 <- .expr4^.expr5
.expr7 <- 1 - .expr4
.expr8 <- .expr7^.expr5
.expr9 <- .expr6 + .expr8
.expr11 <- -1/alpha
.expr12 <- .expr9^.expr11
.expr13 <- 1 - .expr12
.expr15 <- exp(.expr2 * .expr13)
.expr20 <- 1/.expr1 - u2 * u1/.expr1^2
.expr23 <- 1/alpha^2
.expr24 <- log(.expr9) * .expr23
.expr26 <- .expr11 - 1
.expr27 <- .expr9^.expr26
.expr28 <- log(.expr7)
.expr30 <- log(.expr4)
.expr32 <- .expr8 * .expr28 + .expr6 * .expr30
.expr35 <- .expr12 * .expr24 - .expr27 * (.expr11 * .expr32)
.expr37 <- u2/.expr1
.expr39 <- .expr26 - 1
.expr40 <- .expr9^.expr39
.expr43 <- .expr27 * .expr24 - .expr40 * (.expr26 * .expr32)
.expr44 <- .expr5 - 1
.expr45 <- .expr4^.expr44
.expr46 <- 1/u2
.expr48 <- u1/.expr1
.expr50 <- .expr2^2
.expr52 <- .expr46/.expr2 - .expr3 * .expr48/.expr50
.expr53 <- .expr5 * .expr52
.expr55 <- .expr7^.expr44
.expr57 <- .expr45 * .expr53 - .expr55 * .expr53
.expr58 <- .expr11 * .expr57
.expr60 <- .expr23 * .expr57
.expr62 <- .expr45 * .expr30
.expr66 <- .expr55 * .expr28
.expr69 <- .expr45 * .expr52 + .expr62 * .expr53 - (.expr55 *
.expr52 + .expr66 * .expr53)
.expr73 <- .expr43 * .expr58 + .expr27 * (.expr60 - .expr11 *
.expr69)
.expr76 <- .expr3 * .expr37
.expr77 <- .expr76/.expr50
.expr78 <- .expr5 * .expr77
.expr81 <- .expr55 * .expr78 - .expr45 * .expr78
.expr82 <- .expr11 * .expr81
.expr93 <- .expr23 * .expr81 - .expr11 * (.expr55 * .expr77 +
.expr66 * .expr78 - (.expr45 * .expr77 + .expr62 * .expr78))
.expr95 <- .expr43 * .expr82 + .expr27 * .expr93
.expr103 <- .expr26 * .expr57
.expr110 <- .expr40 * .expr103
.expr122 <- (.expr46 * .expr37 + .expr3 * .expr20)/.expr50 -
.expr76 * (2 * (.expr48 * .expr2))/.expr50^2
.expr123 <- .expr5 * .expr122
.expr125 <- .expr44 - 1
.expr126 <- .expr7^.expr125
.expr127 <- .expr44 * .expr52
.expr128 <- .expr126 * .expr127
.expr131 <- .expr4^.expr125
.expr132 <- .expr131 * .expr127
.expr136 <- .expr55 * .expr123 - .expr128 * .expr78 - (.expr132 *
.expr78 + .expr45 * .expr123)
.expr137 <- .expr11 * .expr136
.expr173 <- .expr15 * (.expr2 * .expr35)
.expr175 <- .expr27 * .expr58
.expr178 <- .expr27 * .expr82
.expr190 <- .expr48 * .expr13 - .expr2 * .expr175
.value <- -(.expr15 * (.expr20 * .expr35 + .expr37 * .expr73 +
(.expr48 * .expr95 + .expr2 * (((.expr40 * .expr24 -
.expr9^(.expr39 - 1) * (.expr39 * .expr32)) * .expr103 +
.expr40 * (.expr60 - .expr26 * .expr69)) * .expr82 +
.expr110 * .expr93 + (.expr43 * .expr137 + .expr27 *
(.expr23 * .expr136 - .expr11 * (.expr55 * .expr122 +
.expr66 * .expr123 - (.expr128 * .expr77 + (.expr126 *
.expr52 + .expr126 * .expr28 * .expr127) * .expr78) -
(.expr45 * .expr122 + .expr62 * .expr123 + (.expr132 *
.expr77 + (.expr131 * .expr52 + .expr131 *
.expr30 * .expr127) * .expr78)))))))) + .expr173 *
(.expr20 * .expr13 - .expr37 * .expr175 - (.expr48 *
.expr178 + .expr2 * (.expr110 * .expr82 + .expr27 *
.expr137))) + (.expr15 * .expr190 * (.expr37 * .expr35 +
.expr2 * .expr95) + (.expr15 * (.expr48 * .expr35 + .expr2 *
.expr73) + .expr173 * .expr190) * (.expr37 * .expr13 -
.expr2 * .expr178)))
.grad <- array(0, c(length(.value), 1L), list(NULL, c("s")))
.grad[, "s"] <- 0
attr(.value, "gradient") <- .grad
.value
})
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## This file is not to be edited. It is generated by ../inst/docs/mathnb/exprPrep.R
`galambosCopula.cdf.expr` <-
expression(0, exp(log(u1 * u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha))))
`galambosCopula.cdf.algr` <-
expression({
.value <- 0
.grad <- array(0, c(length(.value), 1L), list(NULL, c("s")))
.grad[, "s"] <- 0
attr(.value, "gradient") <- .grad
.value
}, {
.expr2 <- log(u1 * u2)
.expr4 <- log(u2)/.expr2
.expr5 <- -alpha
.value <- exp(.expr2 * (1 - (.expr4^.expr5 + (1 - .expr4)^.expr5)^(-1/alpha)))
.grad <- array(0, c(length(.value), 1L), list(NULL, c("s")))
.grad[, "s"] <- 0
attr(.value, "gradient") <- .grad
.value
})
`galambosCopula.cdfDerWrtArg.expr` <-
expression(0, exp(log(u1 * u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha))) * (u2/(u1 *
u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(-1/alpha)) - log(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)))))))
`galambosCopula.cdfDerWrtArg.algr` <-
expression({
.value <- 0
.grad <- array(0, c(length(.value), 1L), list(NULL, c("s")))
.grad[, "s"] <- 0
attr(.value, "gradient") <- .grad
.value
}, {
.expr1 <- u1 * u2
.expr2 <- log(.expr1)
.expr3 <- log(u2)
.expr4 <- .expr3/.expr2
.expr5 <- -alpha
.expr7 <- 1 - .expr4
.expr9 <- .expr4^.expr5 + .expr7^.expr5
.expr11 <- -1/alpha
.expr13 <- 1 - .expr9^.expr11
.expr16 <- u2/.expr1
.expr20 <- .expr5 - 1
.expr25 <- .expr5 * (.expr3 * .expr16/.expr2^2)
.value <- exp(.expr2 * .expr13) * (.expr16 * .expr13 - .expr2 *
(.expr9^(.expr11 - 1) * (.expr11 * (.expr7^.expr20 *
.expr25 - .expr4^.expr20 * .expr25))))
.grad <- array(0, c(length(.value), 1L), list(NULL, c("s")))
.grad[, "s"] <- 0
attr(.value, "gradient") <- .grad
.value
})
`galambosCopula.cdfDerWrtPar.expr` <-
expression(0, -(exp(log(u1 * u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha))) * (log(u1 *
u2) * (((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(-1/alpha) * (log(((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))) * (1/alpha^2)) -
((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * ((-1/alpha) * ((1 -
(log(u2)/log(u1 * u2)))^(-alpha) * log((1 - (log(u2)/log(u1 *
u2)))) + (log(u2)/log(u1 * u2))^(-alpha) * log((log(u2)/log(u1 *
u2)))))))))
`galambosCopula.cdfDerWrtPar.algr` <-
expression({
.value <- 0
.grad <- array(0, c(length(.value), 1L), list(NULL, c("s")))
.grad[, "s"] <- 0
attr(.value, "gradient") <- .grad
.value
}, {
.expr2 <- log(u1 * u2)
.expr4 <- log(u2)/.expr2
.expr5 <- -alpha
.expr6 <- .expr4^.expr5
.expr7 <- 1 - .expr4
.expr8 <- .expr7^.expr5
.expr9 <- .expr6 + .expr8
.expr11 <- -1/alpha
.expr12 <- .expr9^.expr11
.value <- -(exp(.expr2 * (1 - .expr12)) * (.expr2 * (.expr12 *
(log(.expr9) * (1/alpha^2)) - .expr9^(.expr11 - 1) *
(.expr11 * (.expr8 * log(.expr7) + .expr6 * log(.expr4))))))
.grad <- array(0, c(length(.value), 1L), list(NULL, c("s")))
.grad[, "s"] <- 0
attr(.value, "gradient") <- .grad
.value
})
`galambosCopula.pdf.expr` <-
expression(0, exp(log(u1 * u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha))) * (u1/(u1 *
u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(-1/alpha)) - log(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) - 1) *
((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) *
((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)))))) * (u2/(u1 * u2) * (1 - ((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha)) -
log(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * ((-1/alpha) * ((1 -
(log(u2)/log(u1 * u2)))^((-alpha) - 1) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) - (log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2)))))) + exp(log(u1 * u2) * (1 - ((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha))) *
((1/(u1 * u2) - u2 * u1/(u1 * u2)^2) * (1 - ((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha)) -
u2/(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) + (1 -
(log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) - 1) *
((-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2))))) -
(u1/(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) + (1 -
(log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) - 1) *
((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2))))) + log(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(((-1/alpha) -
1) - 1) * (((-1/alpha) - 1) * ((log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)) -
(1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) * ((-alpha) *
(1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)))) * ((-1/alpha) * ((1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)))) + ((log(u2)/log(u1 * u2))^(-alpha) + (1 -
(log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) - 1) *
((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * ((1/u2 * (u2/(u1 * u2)) + log(u2) *
(1/(u1 * u2) - u2 * u1/(u1 * u2)^2))/log(u1 *
u2)^2 - log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 *
u2) * log(u1 * u2)))/(log(u1 * u2)^2)^2)) - (1 -
(log(u2)/log(u1 * u2)))^(((-alpha) - 1) - 1) *
(((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
((log(u2)/log(u1 * u2))^(((-alpha) - 1) - 1) *
(((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) +
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
((1/u2 * (u2/(u1 * u2)) + log(u2) * (1/(u1 *
u2) - u2 * u1/(u1 * u2)^2))/log(u1 * u2)^2 -
log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 *
u2) * log(u1 * u2)))/(log(u1 * u2)^2)^2)))))))))
`galambosCopula.pdf.algr` <-
expression({
.value <- 0
.grad <- array(0, c(length(.value), 1L), list(NULL, c("s")))
.grad[, "s"] <- 0
attr(.value, "gradient") <- .grad
.value
}, {
.expr1 <- u1 * u2
.expr2 <- log(.expr1)
.expr3 <- log(u2)
.expr4 <- .expr3/.expr2
.expr5 <- -alpha
.expr7 <- 1 - .expr4
.expr9 <- .expr4^.expr5 + .expr7^.expr5
.expr11 <- -1/alpha
.expr13 <- 1 - .expr9^.expr11
.expr15 <- exp(.expr2 * .expr13)
.expr16 <- u1/.expr1
.expr18 <- .expr11 - 1
.expr19 <- .expr9^.expr18
.expr20 <- .expr5 - 1
.expr21 <- .expr4^.expr20
.expr22 <- 1/u2
.expr25 <- .expr2^2
.expr27 <- .expr22/.expr2 - .expr3 * .expr16/.expr25
.expr28 <- .expr5 * .expr27
.expr30 <- .expr7^.expr20
.expr32 <- .expr21 * .expr28 - .expr30 * .expr28
.expr34 <- .expr19 * (.expr11 * .expr32)
.expr38 <- u2/.expr1
.expr40 <- .expr3 * .expr38
.expr42 <- .expr5 * (.expr40/.expr25)
.expr46 <- .expr11 * (.expr30 * .expr42 - .expr21 * .expr42)
.expr47 <- .expr19 * .expr46
.expr55 <- 1/.expr1 - u2 * u1/.expr1^2
.expr75 <- .expr5 * ((.expr22 * .expr38 + .expr3 * .expr55)/.expr25 -
.expr40 * (2 * (.expr16 * .expr2))/.expr25^2)
.expr77 <- .expr20 - 1
.expr79 <- .expr20 * .expr27
.value <- .expr15 * (.expr16 * .expr13 - .expr2 * .expr34) *
(.expr38 * .expr13 - .expr2 * .expr47) + .expr15 * (.expr55 *
.expr13 - .expr38 * .expr34 - (.expr16 * .expr47 + .expr2 *
(.expr9^(.expr18 - 1) * (.expr18 * .expr32) * .expr46 +
.expr19 * (.expr11 * (.expr30 * .expr75 - .expr7^.expr77 *
.expr79 * .expr42 - (.expr4^.expr77 * .expr79 *
.expr42 + .expr21 * .expr75))))))
.grad <- array(0, c(length(.value), 1L), list(NULL, c("s")))
.grad[, "s"] <- 0
attr(.value, "gradient") <- .grad
.value
})
`galambosCopula.pdfDerWrtArg.expr` <-
expression(0, (exp(log(u1 * u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha))) * (u2/(u1 *
u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(-1/alpha)) - log(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)))))) * (u1/(u1 * u2) *
(1 - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(-1/alpha)) - log(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) - 1) *
((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) *
((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)))))) + exp(log(u1 * u2) * (1 - ((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha))) *
((1/(u1 * u2) - u1 * u2/(u1 * u2)^2) * (1 - ((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha)) -
u1/(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) + (1 -
(log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) - 1) *
((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2))))) - (u2/(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2))))) + log(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(((-1/alpha) -
1) - 1) * (((-1/alpha) - 1) * ((1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)))) *
((-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) - 1) *
((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)))) -
((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * ((-1/alpha) *
((log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
(1/u2 * (u2/(u1 * u2))/log(u1 * u2)^2 + (log(u2) *
(1/(u1 * u2) - u1 * u2/(u1 * u2)^2)/log(u1 *
u2)^2 - log(u2) * (u1/(u1 * u2)) * (2 * (u2/(u1 *
u2) * log(u1 * u2)))/(log(u1 * u2)^2)^2))) +
(log(u2)/log(u1 * u2))^(((-alpha) - 1) - 1) *
(((-alpha) - 1) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) * ((-alpha) * (1/u2/log(u1 * u2) -
log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)) +
((1 - (log(u2)/log(u1 * u2)))^(((-alpha) - 1) -
1) * (((-alpha) - 1) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2)) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)) -
(1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) *
((-alpha) * (1/u2 * (u2/(u1 * u2))/log(u1 *
u2)^2 + (log(u2) * (1/(u1 * u2) - u1 *
u2/(u1 * u2)^2)/log(u1 * u2)^2 - log(u2) *
(u1/(u1 * u2)) * (2 * (u2/(u1 * u2) * log(u1 *
u2)))/(log(u1 * u2)^2)^2)))))))))) * (u2/(u1 *
u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(-1/alpha)) - log(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)))))) - exp(log(u1 * u2) *
(1 - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(-1/alpha))) * (u1/(u1 * u2) * (1 - ((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha)) -
log(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * ((-1/alpha) * ((log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 * u2) -
log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 * u2) -
log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)))))) * (u2/(u1 *
u2) * (((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2))))) + u2 * u2/(u1 *
u2)^2 * (1 - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(-1/alpha)) + (u2/(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2))))) + log(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(((-1/alpha) -
1) - 1) * (((-1/alpha) - 1) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)))) * ((-1/alpha) * ((1 -
(log(u2)/log(u1 * u2)))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)))) +
((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * ((-1/alpha) * ((1 -
(log(u2)/log(u1 * u2)))^(((-alpha) - 1) - 1) * (((-alpha) -
1) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (log(u2) * (u2 * u2/(u1 *
u2)^2)/log(u1 * u2)^2 + log(u2) * (u2/(u1 * u2)) * (2 *
(u2/(u1 * u2) * log(u1 * u2)))/(log(u1 * u2)^2)^2)) +
((log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
(log(u2) * (u2 * u2/(u1 * u2)^2)/log(u1 * u2)^2 +
log(u2) * (u2/(u1 * u2)) * (2 * (u2/(u1 * u2) *
log(u1 * u2)))/(log(u1 * u2)^2)^2)) + (log(u2)/log(u1 *
u2))^(((-alpha) - 1) - 1) * (((-alpha) - 1) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)))))))) + (exp(log(u1 *
u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(-1/alpha))) * (u2/(u1 * u2) * (1 - ((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha)) -
log(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * ((-1/alpha) * ((1 -
(log(u2)/log(u1 * u2)))^((-alpha) - 1) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) - (log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2)))))) * ((1/(u1 * u2) - u2 * u1/(u1 *
u2)^2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(-1/alpha)) - u2/(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) - 1) *
((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) *
((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2))))) - (u1/(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) - 1) *
((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) *
((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2))))) + log(u1 * u2) *
(((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(((-1/alpha) - 1) - 1) * (((-1/alpha) -
1) * ((log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
(1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2)))) * ((-1/alpha) * ((1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)))) +
((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * ((-1/alpha) *
((1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) * ((-alpha) *
((1/u2 * (u2/(u1 * u2)) + log(u2) * (1/(u1 *
u2) - u2 * u1/(u1 * u2)^2))/log(u1 * u2)^2 -
log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 * u2) *
log(u1 * u2)))/(log(u1 * u2)^2)^2)) - (1 -
(log(u2)/log(u1 * u2)))^(((-alpha) - 1) - 1) *
(((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
((log(u2)/log(u1 * u2))^(((-alpha) - 1) - 1) *
(((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) +
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
((1/u2 * (u2/(u1 * u2)) + log(u2) * (1/(u1 *
u2) - u2 * u1/(u1 * u2)^2))/log(u1 * u2)^2 -
log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 *
u2) * log(u1 * u2)))/(log(u1 * u2)^2)^2)))))))) -
exp(log(u1 * u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha))) *
((1/(u1 * u2) - u2 * u1/(u1 * u2)^2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) - 1) *
((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2))))) +
(u2/(u1 * u2)^2 + (u2/(u1 * u2)^2 - u2 * u1 * (2 *
(u2 * (u1 * u2)))/((u1 * u2)^2)^2)) * (1 - ((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha)) +
(u2/(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(((-1/alpha) -
1) - 1) * (((-1/alpha) - 1) * ((1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) - (log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)))) * ((-1/alpha) *
((log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
(1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)))) - ((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (1/u2 * (u2/(u1 * u2))/log(u1 *
u2)^2 + (log(u2) * (1/(u1 * u2) - u1 * u2/(u1 *
u2)^2)/log(u1 * u2)^2 - log(u2) * (u1/(u1 * u2)) *
(2 * (u2/(u1 * u2) * log(u1 * u2)))/(log(u1 *
u2)^2)^2))) + (log(u2)/log(u1 * u2))^(((-alpha) -
1) - 1) * (((-alpha) - 1) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2)) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)) +
((1 - (log(u2)/log(u1 * u2)))^(((-alpha) - 1) -
1) * (((-alpha) - 1) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2)) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)) -
(1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) *
((-alpha) * (1/u2 * (u2/(u1 * u2))/log(u1 *
u2)^2 + (log(u2) * (1/(u1 * u2) - u1 *
u2/(u1 * u2)^2)/log(u1 * u2)^2 - log(u2) *
(u1/(u1 * u2)) * (2 * (u2/(u1 * u2) * log(u1 *
u2)))/(log(u1 * u2)^2)^2))))))) - u2 *
u2/(u1 * u2)^2 * (((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)))))) +
((1/(u1 * u2) - u1 * u2/(u1 * u2)^2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2))))) + u1/(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(((-1/alpha) -
1) - 1) * (((-1/alpha) - 1) * ((1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) - (log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)))) * ((-1/alpha) *
((1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) *
((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)))) + ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^(((-alpha) -
1) - 1) * (((-alpha) - 1) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2)) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2 * u2/(u1 * u2)^2)/log(u1 * u2)^2 + log(u2) *
(u2/(u1 * u2)) * (2 * (u2/(u1 * u2) * log(u1 *
u2)))/(log(u1 * u2)^2)^2)) + ((log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2 * u2/(u1 * u2)^2)/log(u1 * u2)^2 + log(u2) *
(u2/(u1 * u2)) * (2 * (u2/(u1 * u2) * log(u1 *
u2)))/(log(u1 * u2)^2)^2)) + (log(u2)/log(u1 *
u2))^(((-alpha) - 1) - 1) * (((-alpha) - 1) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) *
((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)))))) + (u2/(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(((-1/alpha) -
1) - 1) * (((-1/alpha) - 1) * ((log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)) -
(1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) *
((-alpha) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)))) * ((-1/alpha) *
((1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) *
((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)))) + ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * ((1/u2 * (u2/(u1 * u2)) + log(u2) *
(1/(u1 * u2) - u2 * u1/(u1 * u2)^2))/log(u1 *
u2)^2 - log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 *
u2) * log(u1 * u2)))/(log(u1 * u2)^2)^2)) - (1 -
(log(u2)/log(u1 * u2)))^(((-alpha) - 1) - 1) *
(((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
((log(u2)/log(u1 * u2))^(((-alpha) - 1) - 1) *
(((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) +
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
((1/u2 * (u2/(u1 * u2)) + log(u2) * (1/(u1 *
u2) - u2 * u1/(u1 * u2)^2))/log(u1 * u2)^2 -
log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 *
u2) * log(u1 * u2)))/(log(u1 * u2)^2)^2)))))) +
log(u1 * u2) * ((((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((((-1/alpha) -
1) - 1) - 1) * ((((-1/alpha) - 1) - 1) * ((1 -
(log(u2)/log(u1 * u2)))^((-alpha) - 1) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)))) *
(((-1/alpha) - 1) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)))) -
((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(((-1/alpha) - 1) - 1) *
(((-1/alpha) - 1) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (1/u2 * (u2/(u1 * u2))/log(u1 *
u2)^2 + (log(u2) * (1/(u1 * u2) - u1 *
u2/(u1 * u2)^2)/log(u1 * u2)^2 - log(u2) *
(u1/(u1 * u2)) * (2 * (u2/(u1 * u2) * log(u1 *
u2)))/(log(u1 * u2)^2)^2))) + (log(u2)/log(u1 *
u2))^(((-alpha) - 1) - 1) * (((-alpha) -
1) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) * ((-alpha) * (1/u2/log(u1 * u2) -
log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)) +
((1 - (log(u2)/log(u1 * u2)))^(((-alpha) -
1) - 1) * (((-alpha) - 1) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) *
(1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (1/u2 *
(u2/(u1 * u2))/log(u1 * u2)^2 + (log(u2) *
(1/(u1 * u2) - u1 * u2/(u1 * u2)^2)/log(u1 *
u2)^2 - log(u2) * (u1/(u1 * u2)) * (2 *
(u2/(u1 * u2) * log(u1 * u2)))/(log(u1 *
u2)^2)^2))))))) * ((-1/alpha) * ((1 -
(log(u2)/log(u1 * u2)))^((-alpha) - 1) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)))) +
((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(((-1/alpha) - 1) - 1) *
(((-1/alpha) - 1) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) - (1 -
(log(u2)/log(u1 * u2)))^((-alpha) - 1) *
((-alpha) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)))) * ((-1/alpha) *
((1 - (log(u2)/log(u1 * u2)))^(((-alpha) -
1) - 1) * (((-alpha) - 1) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
(1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2 * u2/(u1 *
u2)^2)/log(u1 * u2)^2 + log(u2) * (u2/(u1 *
u2)) * (2 * (u2/(u1 * u2) * log(u1 *
u2)))/(log(u1 * u2)^2)^2)) + ((log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2 * u2/(u1 * u2)^2)/log(u1 * u2)^2 +
log(u2) * (u2/(u1 * u2)) * (2 * (u2/(u1 *
u2) * log(u1 * u2)))/(log(u1 * u2)^2)^2)) +
(log(u2)/log(u1 * u2))^(((-alpha) - 1) -
1) * (((-alpha) - 1) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2)) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2))))) +
(((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(((-1/alpha) - 1) - 1) *
(((-1/alpha) - 1) * ((1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) - (log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)))) * ((-1/alpha) *
((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * ((1/u2 * (u2/(u1 * u2)) +
log(u2) * (1/(u1 * u2) - u2 * u1/(u1 *
u2)^2))/log(u1 * u2)^2 - log(u2) * (u2/(u1 *
u2)) * (2 * (u1/(u1 * u2) * log(u1 * u2)))/(log(u1 *
u2)^2)^2)) - (1 - (log(u2)/log(u1 * u2)))^(((-alpha) -
1) - 1) * (((-alpha) - 1) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)) * ((-alpha) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2)) - ((log(u2)/log(u1 *
u2))^(((-alpha) - 1) - 1) * (((-alpha) -
1) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2)) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) + (log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * ((1/u2 *
(u2/(u1 * u2)) + log(u2) * (1/(u1 * u2) -
u2 * u1/(u1 * u2)^2))/log(u1 * u2)^2 -
log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 *
u2) * log(u1 * u2)))/(log(u1 * u2)^2)^2))))) +
((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * ((-1/alpha) *
((1 - (log(u2)/log(u1 * u2)))^(((-alpha) -
1) - 1) * (((-alpha) - 1) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) *
((1/u2 * (u2/(u1 * u2)) + log(u2) * (1/(u1 *
u2) - u2 * u1/(u1 * u2)^2))/log(u1 *
u2)^2 - log(u2) * (u2/(u1 * u2)) *
(2 * (u1/(u1 * u2) * log(u1 * u2)))/(log(u1 *
u2)^2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * ((log(u2) *
(u2/(u1 * u2)^2 + (u2/(u1 * u2)^2 - u2 *
u1 * (2 * (u2 * (u1 * u2)))/((u1 *
u2)^2)^2)) + 1/u2 * (u2 * u2/(u1 *
u2)^2))/log(u1 * u2)^2 + (1/u2 * (u2/(u1 *
u2)) + log(u2) * (1/(u1 * u2) - u2 *
u1/(u1 * u2)^2)) * (2 * (u2/(u1 * u2) *
log(u1 * u2)))/(log(u1 * u2)^2)^2 + ((log(u2) *
(u2/(u1 * u2)) * (2 * ((1/(u1 * u2) -
u1 * u2/(u1 * u2)^2) * log(u1 * u2) +
u1/(u1 * u2) * (u2/(u1 * u2)))) - log(u2) *
(u2 * u2/(u1 * u2)^2) * (2 * (u1/(u1 *
u2) * log(u1 * u2))))/(log(u1 * u2)^2)^2 -
log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 *
u2) * log(u1 * u2))) * (2 * (2 * (u2/(u1 *
u2) * log(u1 * u2)) * (log(u1 * u2)^2)))/((log(u1 *
u2)^2)^2)^2))) - (((1 - (log(u2)/log(u1 *
u2)))^((((-alpha) - 1) - 1) - 1) * ((((-alpha) -
1) - 1) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) * (((-alpha) - 1) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^(((-alpha) -
1) - 1) * (((-alpha) - 1) * (1/u2 * (u2/(u1 *
u2))/log(u1 * u2)^2 + (log(u2) * (1/(u1 *
u2) - u1 * u2/(u1 * u2)^2)/log(u1 * u2)^2 -
log(u2) * (u1/(u1 * u2)) * (2 * (u2/(u1 *
u2) * log(u1 * u2)))/(log(u1 * u2)^2)^2)))) *
((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^(((-alpha) -
1) - 1) * (((-alpha) - 1) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)) * ((-alpha) * (log(u2) * (u2 *
u2/(u1 * u2)^2)/log(u1 * u2)^2 + log(u2) *
(u2/(u1 * u2)) * (2 * (u2/(u1 * u2) *
log(u1 * u2)))/(log(u1 * u2)^2)^2))) +
((log(u2)/log(u1 * u2))^((-alpha) - 1) *
((-alpha) * ((log(u2) * (u2/(u1 * u2)^2 +
(u2/(u1 * u2)^2 - u2 * u1 * (2 *
(u2 * (u1 * u2)))/((u1 * u2)^2)^2)) +
1/u2 * (u2 * u2/(u1 * u2)^2))/log(u1 *
u2)^2 + (1/u2 * (u2/(u1 * u2)) +
log(u2) * (1/(u1 * u2) - u2 * u1/(u1 *
u2)^2)) * (2 * (u2/(u1 * u2) *
log(u1 * u2)))/(log(u1 * u2)^2)^2 +
((log(u2) * (u2/(u1 * u2)) * (2 *
((1/(u1 * u2) - u1 * u2/(u1 * u2)^2) *
log(u1 * u2) + u1/(u1 * u2) *
(u2/(u1 * u2)))) - log(u2) *
(u2 * u2/(u1 * u2)^2) * (2 * (u1/(u1 *
u2) * log(u1 * u2))))/(log(u1 *
u2)^2)^2 - log(u2) * (u2/(u1 *
u2)) * (2 * (u1/(u1 * u2) * log(u1 *
u2))) * (2 * (2 * (u2/(u1 * u2) *
log(u1 * u2)) * (log(u1 * u2)^2)))/((log(u1 *
u2)^2)^2)^2))) + (log(u2)/log(u1 *
u2))^(((-alpha) - 1) - 1) * (((-alpha) -
1) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) * ((-alpha) * ((1/u2 * (u2/(u1 *
u2)) + log(u2) * (1/(u1 * u2) - u2 *
u1/(u1 * u2)^2))/log(u1 * u2)^2 - log(u2) *
(u2/(u1 * u2)) * (2 * (u1/(u1 * u2) *
log(u1 * u2)))/(log(u1 * u2)^2)^2)) +
((log(u2)/log(u1 * u2))^(((-alpha) -
1) - 1) * (((-alpha) - 1) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)) * ((-alpha) * (log(u2) *
(u2 * u2/(u1 * u2)^2)/log(u1 * u2)^2 +
log(u2) * (u2/(u1 * u2)) * (2 * (u2/(u1 *
u2) * log(u1 * u2)))/(log(u1 *
u2)^2)^2)) + ((log(u2)/log(u1 *
u2))^(((-alpha) - 1) - 1) * (((-alpha) -
1) * (1/u2 * (u2/(u1 * u2))/log(u1 *
u2)^2 + (log(u2) * (1/(u1 * u2) -
u1 * u2/(u1 * u2)^2)/log(u1 * u2)^2 -
log(u2) * (u1/(u1 * u2)) * (2 * (u2/(u1 *
u2) * log(u1 * u2)))/(log(u1 *
u2)^2)^2))) + (log(u2)/log(u1 *
u2))^((((-alpha) - 1) - 1) - 1) *
((((-alpha) - 1) - 1) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) *
(((-alpha) - 1) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2))) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)))))))))))))
`galambosCopula.pdfDerWrtArg.algr` <-
expression({
.value <- 0
.grad <- array(0, c(length(.value), 1L), list(NULL, c("s")))
.grad[, "s"] <- 0
attr(.value, "gradient") <- .grad
.value
}, {
.expr1 <- u1 * u2
.expr2 <- log(.expr1)
.expr3 <- log(u2)
.expr4 <- .expr3/.expr2
.expr5 <- -alpha
.expr7 <- 1 - .expr4
.expr9 <- .expr4^.expr5 + .expr7^.expr5
.expr11 <- -1/alpha
.expr13 <- 1 - .expr9^.expr11
.expr15 <- exp(.expr2 * .expr13)
.expr16 <- u2/.expr1
.expr18 <- .expr11 - 1
.expr19 <- .expr9^.expr18
.expr20 <- .expr5 - 1
.expr21 <- .expr7^.expr20
.expr22 <- .expr3 * .expr16
.expr23 <- .expr2^2
.expr24 <- .expr22/.expr23
.expr25 <- .expr5 * .expr24
.expr27 <- .expr4^.expr20
.expr29 <- .expr21 * .expr25 - .expr27 * .expr25
.expr30 <- .expr11 * .expr29
.expr31 <- .expr19 * .expr30
.expr33 <- .expr16 * .expr13 - .expr2 * .expr31
.expr34 <- .expr15 * .expr33
.expr35 <- u1/.expr1
.expr37 <- 1/u2
.expr39 <- .expr3 * .expr35
.expr41 <- .expr37/.expr2 - .expr39/.expr23
.expr42 <- .expr5 * .expr41
.expr45 <- .expr27 * .expr42 - .expr21 * .expr42
.expr46 <- .expr11 * .expr45
.expr47 <- .expr19 * .expr46
.expr49 <- .expr35 * .expr13 - .expr2 * .expr47
.expr51 <- 1/.expr1
.expr52 <- .expr1^2
.expr54 <- .expr51 - .expr1/.expr52
.expr56 <- .expr35 * .expr31
.expr58 <- .expr16 * .expr47
.expr59 <- .expr18 - 1
.expr60 <- .expr9^.expr59
.expr62 <- .expr60 * (.expr18 * .expr29)
.expr64 <- .expr37 * .expr16
.expr69 <- 2 * (.expr16 * .expr2)
.expr71 <- .expr23^2
.expr74 <- .expr64/.expr23 + (.expr3 * .expr54/.expr23 -
.expr39 * .expr69/.expr71)
.expr75 <- .expr5 * .expr74
.expr77 <- .expr20 - 1
.expr78 <- .expr4^.expr77
.expr79 <- .expr20 * .expr24
.expr80 <- .expr78 * .expr79
.expr83 <- .expr7^.expr77
.expr84 <- .expr83 * .expr79
.expr88 <- .expr27 * .expr75 + .expr80 * .expr42 + (.expr84 *
.expr42 - .expr21 * .expr75)
.expr91 <- .expr62 * .expr46 - .expr19 * (.expr11 * .expr88)
.expr99 <- .expr16 * .expr31
.expr101 <- u2 * u2/.expr52
.expr106 <- .expr3 * .expr101
.expr111 <- .expr5 * (.expr106/.expr23 + .expr22 * .expr69/.expr71)
.expr118 <- .expr11 * (.expr84 * .expr25 - .expr21 * .expr111 +
(.expr27 * .expr111 + .expr80 * .expr25))
.expr120 <- .expr62 * .expr30 + .expr19 * .expr118
.expr126 <- u2 * u1
.expr128 <- .expr51 - .expr126/.expr52
.expr131 <- .expr18 * .expr45
.expr132 <- .expr60 * .expr131
.expr135 <- .expr64 + .expr3 * .expr128
.expr138 <- 2 * (.expr35 * .expr2)
.expr139 <- .expr22 * .expr138
.expr142 <- .expr5 * (.expr135/.expr23 - .expr139/.expr71)
.expr144 <- .expr20 * .expr41
.expr145 <- .expr83 * .expr144
.expr148 <- .expr78 * .expr144
.expr153 <- .expr11 * (.expr21 * .expr142 - .expr145 * .expr25 -
(.expr148 * .expr25 + .expr27 * .expr142))
.expr155 <- .expr132 * .expr30 + .expr19 * .expr153
.expr161 <- u2/.expr52
.expr168 <- .expr161 + (.expr161 - .expr126 * (2 * (u2 *
.expr1))/.expr52^2)
.expr214 <- .expr5 * ((.expr3 * .expr168 + .expr37 * .expr101)/.expr23 +
.expr135 * .expr69/.expr71 + ((.expr22 * (2 * (.expr54 *
.expr2 + .expr35 * .expr16)) - .expr106 * .expr138)/.expr71 -
.expr139 * (2 * (.expr69 * .expr23))/.expr71^2))
.expr217 <- .expr77 - 1
.expr219 <- .expr77 * .expr24
.expr222 <- .expr20 * .expr74
.value <- (.expr34 * .expr49 + .expr15 * (.expr54 * .expr13 -
.expr56 - (.expr58 + .expr2 * .expr91))) * .expr33 -
.expr15 * .expr49 * (.expr99 + .expr101 * .expr13 + (.expr99 +
.expr2 * .expr120)) + (.expr34 * (.expr128 * .expr13 -
.expr58 - (.expr56 + .expr2 * .expr155)) - .expr15 *
(.expr128 * .expr31 + .expr168 * .expr13 + (.expr16 *
.expr91 - .expr101 * .expr47) + (.expr54 * .expr31 +
.expr35 * .expr120 + (.expr16 * .expr155 + .expr2 *
((.expr9^(.expr59 - 1) * (.expr59 * .expr29) * .expr131 -
.expr60 * (.expr18 * .expr88)) * .expr30 + .expr132 *
.expr118 + (.expr62 * .expr153 + .expr19 * (.expr11 *
(.expr84 * .expr142 - .expr21 * .expr214 - ((.expr7^.expr217 *
.expr219 * .expr144 - .expr83 * .expr222) *
.expr25 - .expr145 * .expr111) + (.expr27 *
.expr214 + .expr80 * .expr142 + (.expr148 *
.expr111 + (.expr78 * .expr222 + .expr4^.expr217 *
.expr219 * .expr144) * .expr25))))))))))
.grad <- array(0, c(length(.value), 1L), list(NULL, c("s")))
.grad[, "s"] <- 0
attr(.value, "gradient") <- .grad
.value
})
`galambosCopula.pdfDerWrtPar.expr` <-
expression(0, -(exp(log(u1 * u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha))) * ((1/(u1 *
u2) - u2 * u1/(u1 * u2)^2) * (((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha) * (log(((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))) *
(1/alpha^2)) - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 *
u2)))^(-alpha) * log((1 - (log(u2)/log(u1 * u2)))) + (log(u2)/log(u1 *
u2))^(-alpha) * log((log(u2)/log(u1 * u2)))))) + u2/(u1 *
u2) * ((((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * (log(((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))) *
(1/alpha^2)) - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(((-1/alpha) - 1) - 1) * (((-1/alpha) - 1) *
((1 - (log(u2)/log(u1 * u2)))^(-alpha) * log((1 - (log(u2)/log(u1 *
u2)))) + (log(u2)/log(u1 * u2))^(-alpha) * log((log(u2)/log(u1 *
u2)))))) * ((-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2)))) + ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) - 1) *
(1/alpha^2 * ((log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
(1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2))) - (-1/alpha) * ((log(u2)/log(u1 *
u2))^((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2) + (log(u2)/log(u1 * u2))^((-alpha) -
1) * log((log(u2)/log(u1 * u2))) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)) - ((1 -
(log(u2)/log(u1 * u2)))^((-alpha) - 1) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2) + (1 -
(log(u2)/log(u1 * u2)))^((-alpha) - 1) * log((1 - (log(u2)/log(u1 *
u2)))) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2)))))) + (u1/(u1 * u2) * ((((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * (log(((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))) * (1/alpha^2)) - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(((-1/alpha) - 1) -
1) * (((-1/alpha) - 1) * ((1 - (log(u2)/log(u1 * u2)))^(-alpha) *
log((1 - (log(u2)/log(u1 * u2)))) + (log(u2)/log(u1 * u2))^(-alpha) *
log((log(u2)/log(u1 * u2)))))) * ((-1/alpha) * ((1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)))) + ((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * (1/alpha^2 * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2))) - (-1/alpha) * ((1 -
(log(u2)/log(u1 * u2)))^((-alpha) - 1) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2) + (1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * log((1 - (log(u2)/log(u1 * u2)))) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) - ((log(u2)/log(u1 * u2))^((-alpha) -
1) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2) + (log(u2)/log(u1 *
u2))^((-alpha) - 1) * log((log(u2)/log(u1 * u2))) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)))))) + log(u1 *
u2) * (((((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(((-1/alpha) - 1) - 1) * (log(((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))) *
(1/alpha^2)) - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((((-1/alpha) - 1) - 1) - 1) * ((((-1/alpha) -
1) - 1) * ((1 - (log(u2)/log(u1 * u2)))^(-alpha) * log((1 -
(log(u2)/log(u1 * u2)))) + (log(u2)/log(u1 * u2))^(-alpha) *
log((log(u2)/log(u1 * u2)))))) * (((-1/alpha) - 1) * ((log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)))) + ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(((-1/alpha) - 1) -
1) * (1/alpha^2 * ((log(u2)/log(u1 * u2))^((-alpha) - 1) *
((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) *
((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2))) - ((-1/alpha) - 1) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2) + (log(u2)/log(u1 * u2))^((-alpha) - 1) * log((log(u2)/log(u1 *
u2))) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2)) - ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2) + (1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) * log((1 -
(log(u2)/log(u1 * u2)))) * ((-alpha) * (1/u2/log(u1 * u2) -
log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)))))) * ((-1/alpha) *
((1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) - (log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2)))) + ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(((-1/alpha) - 1) -
1) * (((-1/alpha) - 1) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2)))) * (1/alpha^2 * ((1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2))) - (-1/alpha) *
((1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2) + (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * log((1 - (log(u2)/log(u1 * u2)))) *
((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
((log(u2)/log(u1 * u2))^((-alpha) - 1) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2) + (log(u2)/log(u1 * u2))^((-alpha) -
1) * log((log(u2)/log(u1 * u2))) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2))))) + ((((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * (log(((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))) * (1/alpha^2)) - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(((-1/alpha) - 1) -
1) * (((-1/alpha) - 1) * ((1 - (log(u2)/log(u1 * u2)))^(-alpha) *
log((1 - (log(u2)/log(u1 * u2)))) + (log(u2)/log(u1 * u2))^(-alpha) *
log((log(u2)/log(u1 * u2)))))) * ((-1/alpha) * ((1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * ((1/u2 * (u2/(u1 * u2)) +
log(u2) * (1/(u1 * u2) - u2 * u1/(u1 * u2)^2))/log(u1 * u2)^2 -
log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 * u2) * log(u1 *
u2)))/(log(u1 * u2)^2)^2)) - (1 - (log(u2)/log(u1 * u2)))^(((-alpha) -
1) - 1) * (((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) - ((log(u2)/log(u1 * u2))^(((-alpha) -
1) - 1) * (((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) + (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * ((1/u2 * (u2/(u1 * u2)) + log(u2) * (1/(u1 *
u2) - u2 * u1/(u1 * u2)^2))/log(u1 * u2)^2 - log(u2) * (u2/(u1 *
u2)) * (2 * (u1/(u1 * u2) * log(u1 * u2)))/(log(u1 * u2)^2)^2))))) +
((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * (1/alpha^2 * ((1 -
(log(u2)/log(u1 * u2)))^((-alpha) - 1) * ((-alpha) *
((1/u2 * (u2/(u1 * u2)) + log(u2) * (1/(u1 * u2) - u2 *
u1/(u1 * u2)^2))/log(u1 * u2)^2 - log(u2) * (u2/(u1 *
u2)) * (2 * (u1/(u1 * u2) * log(u1 * u2)))/(log(u1 *
u2)^2)^2)) - (1 - (log(u2)/log(u1 * u2)))^(((-alpha) -
1) - 1) * (((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)) - ((log(u2)/log(u1 *
u2))^(((-alpha) - 1) - 1) * (((-alpha) - 1) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) + (log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((-alpha) * ((1/u2 * (u2/(u1 *
u2)) + log(u2) * (1/(u1 * u2) - u2 * u1/(u1 * u2)^2))/log(u1 *
u2)^2 - log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 * u2) *
log(u1 * u2)))/(log(u1 * u2)^2)^2)))) - (-1/alpha) *
((1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) * ((1/u2 *
(u2/(u1 * u2)) + log(u2) * (1/(u1 * u2) - u2 * u1/(u1 *
u2)^2))/log(u1 * u2)^2 - log(u2) * (u2/(u1 * u2)) *
(2 * (u1/(u1 * u2) * log(u1 * u2)))/(log(u1 * u2)^2)^2) +
(1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) * log((1 -
(log(u2)/log(u1 * u2)))) * ((-alpha) * ((1/u2 *
(u2/(u1 * u2)) + log(u2) * (1/(u1 * u2) - u2 *
u1/(u1 * u2)^2))/log(u1 * u2)^2 - log(u2) * (u2/(u1 *
u2)) * (2 * (u1/(u1 * u2) * log(u1 * u2)))/(log(u1 *
u2)^2)^2)) - ((1 - (log(u2)/log(u1 * u2)))^(((-alpha) -
1) - 1) * (((-alpha) - 1) * (1/u2/log(u1 * u2) -
log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2) + ((1 - (log(u2)/log(u1 *
u2)))^(((-alpha) - 1) - 1) * (1/u2/log(u1 * u2) -
log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2) + (1 - (log(u2)/log(u1 *
u2)))^(((-alpha) - 1) - 1) * log((1 - (log(u2)/log(u1 *
u2)))) * (((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2))) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2))) - ((log(u2)/log(u1 *
u2))^((-alpha) - 1) * ((1/u2 * (u2/(u1 * u2)) + log(u2) *
(1/(u1 * u2) - u2 * u1/(u1 * u2)^2))/log(u1 * u2)^2 -
log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 * u2) * log(u1 *
u2)))/(log(u1 * u2)^2)^2) + (log(u2)/log(u1 *
u2))^((-alpha) - 1) * log((log(u2)/log(u1 * u2))) *
((-alpha) * ((1/u2 * (u2/(u1 * u2)) + log(u2) * (1/(u1 *
u2) - u2 * u1/(u1 * u2)^2))/log(u1 * u2)^2 -
log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 * u2) *
log(u1 * u2)))/(log(u1 * u2)^2)^2)) + ((log(u2)/log(u1 *
u2))^(((-alpha) - 1) - 1) * (((-alpha) - 1) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2) + ((log(u2)/log(u1 *
u2))^(((-alpha) - 1) - 1) * (1/u2/log(u1 * u2) -
log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2) + (log(u2)/log(u1 *
u2))^(((-alpha) - 1) - 1) * log((log(u2)/log(u1 *
u2))) * (((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2))) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2)))))))))) + exp(log(u1 *
u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(-1/alpha))) * (log(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha) *
(log(((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))) * (1/alpha^2)) - ((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^(-alpha) *
log((1 - (log(u2)/log(u1 * u2)))) + (log(u2)/log(u1 * u2))^(-alpha) *
log((log(u2)/log(u1 * u2))))))) * ((1/(u1 * u2) - u2 * u1/(u1 *
u2)^2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(-1/alpha)) - u2/(u1 * u2) * (((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) - 1) *
((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) *
((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2))))) - (u1/(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) - 1) *
((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) *
((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2))))) + log(u1 * u2) *
(((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(((-1/alpha) - 1) - 1) * (((-1/alpha) -
1) * ((log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
(1/u2/log(u1 * u2) - log(u2) * (u1/(u1 * u2))/log(u1 *
u2)^2)) - (1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2)))) * ((-1/alpha) * ((1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (log(u2) * (u2/(u1 *
u2))/log(u1 * u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)))) +
((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * ((-1/alpha) *
((1 - (log(u2)/log(u1 * u2)))^((-alpha) - 1) * ((-alpha) *
((1/u2 * (u2/(u1 * u2)) + log(u2) * (1/(u1 *
u2) - u2 * u1/(u1 * u2)^2))/log(u1 * u2)^2 -
log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 * u2) *
log(u1 * u2)))/(log(u1 * u2)^2)^2)) - (1 -
(log(u2)/log(u1 * u2)))^(((-alpha) - 1) - 1) *
(((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
((log(u2)/log(u1 * u2))^(((-alpha) - 1) - 1) *
(((-alpha) - 1) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) +
(log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
((1/u2 * (u2/(u1 * u2)) + log(u2) * (1/(u1 *
u2) - u2 * u1/(u1 * u2)^2))/log(u1 * u2)^2 -
log(u2) * (u2/(u1 * u2)) * (2 * (u1/(u1 *
u2) * log(u1 * u2)))/(log(u1 * u2)^2)^2)))))))) +
(exp(log(u1 * u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha))) *
(u1/(u1 * u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha)) -
log(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)))))) *
(u2/(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) + (1 -
(log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha) * (log(((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))) *
(1/alpha^2)) - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^(-alpha) *
log((1 - (log(u2)/log(u1 * u2)))) + (log(u2)/log(u1 *
u2))^(-alpha) * log((log(u2)/log(u1 * u2)))))) +
log(u1 * u2) * ((((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * (log(((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))) * (1/alpha^2)) -
((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(((-1/alpha) - 1) - 1) * (((-1/alpha) -
1) * ((1 - (log(u2)/log(u1 * u2)))^(-alpha) *
log((1 - (log(u2)/log(u1 * u2)))) + (log(u2)/log(u1 *
u2))^(-alpha) * log((log(u2)/log(u1 * u2)))))) *
((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)))) + ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * (1/alpha^2 * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2))) - (-1/alpha) * ((1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2) + (1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * log((1 - (log(u2)/log(u1 * u2)))) * ((-alpha) *
(log(u2) * (u2/(u1 * u2))/log(u1 * u2)^2)) -
((log(u2)/log(u1 * u2))^((-alpha) - 1) * (log(u2) *
(u2/(u1 * u2))/log(u1 * u2)^2) + (log(u2)/log(u1 *
u2))^((-alpha) - 1) * log((log(u2)/log(u1 *
u2))) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2))))))) + (exp(log(u1 * u2) * (1 - ((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha))) *
(u1/(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) + (1 -
(log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha) * (log(((log(u2)/log(u1 *
u2))^(-alpha) + (1 - (log(u2)/log(u1 * u2)))^(-alpha))) *
(1/alpha^2)) - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^(-alpha) *
log((1 - (log(u2)/log(u1 * u2)))) + (log(u2)/log(u1 *
u2))^(-alpha) * log((log(u2)/log(u1 * u2)))))) +
log(u1 * u2) * ((((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * (log(((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))) * (1/alpha^2)) -
((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^(((-1/alpha) - 1) - 1) * (((-1/alpha) -
1) * ((1 - (log(u2)/log(u1 * u2)))^(-alpha) *
log((1 - (log(u2)/log(u1 * u2)))) + (log(u2)/log(u1 *
u2))^(-alpha) * log((log(u2)/log(u1 * u2)))))) *
((-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)))) +
((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * (1/alpha^2 *
((log(u2)/log(u1 * u2))^((-alpha) - 1) * ((-alpha) *
(1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2))) -
(-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2) + (log(u2)/log(u1 *
u2))^((-alpha) - 1) * log((log(u2)/log(u1 *
u2))) * ((-alpha) * (1/u2/log(u1 * u2) -
log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2)) -
((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * (1/u2/log(u1 * u2) - log(u2) * (u1/(u1 *
u2))/log(u1 * u2)^2) + (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * log((1 - (log(u2)/log(u1 *
u2)))) * ((-alpha) * (1/u2/log(u1 * u2) -
log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2))))))) +
exp(log(u1 * u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha))) *
(log(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha) *
(log(((log(u2)/log(u1 * u2))^(-alpha) + (1 -
(log(u2)/log(u1 * u2)))^(-alpha))) * (1/alpha^2)) -
((log(u2)/log(u1 * u2))^(-alpha) + (1 - (log(u2)/log(u1 *
u2)))^(-alpha))^((-1/alpha) - 1) * ((-1/alpha) *
((1 - (log(u2)/log(u1 * u2)))^(-alpha) * log((1 -
(log(u2)/log(u1 * u2)))) + (log(u2)/log(u1 *
u2))^(-alpha) * log((log(u2)/log(u1 * u2))))))) *
(u1/(u1 * u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha)) -
log(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (1/u2/log(u1 * u2) - log(u2) *
(u1/(u1 * u2))/log(u1 * u2)^2)) - (1 - (log(u2)/log(u1 *
u2)))^((-alpha) - 1) * ((-alpha) * (1/u2/log(u1 *
u2) - log(u2) * (u1/(u1 * u2))/log(u1 * u2)^2))))))) *
(u2/(u1 * u2) * (1 - ((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^(-1/alpha)) -
log(u1 * u2) * (((log(u2)/log(u1 * u2))^(-alpha) +
(1 - (log(u2)/log(u1 * u2)))^(-alpha))^((-1/alpha) -
1) * ((-1/alpha) * ((1 - (log(u2)/log(u1 * u2)))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)) - (log(u2)/log(u1 * u2))^((-alpha) -
1) * ((-alpha) * (log(u2) * (u2/(u1 * u2))/log(u1 *
u2)^2)))))))))
`galambosCopula.pdfDerWrtPar.algr` <-
expression({
.value <- 0
.grad <- array(0, c(length(.value), 1L), list(NULL, c("s")))
.grad[, "s"] <- 0
attr(.value, "gradient") <- .grad
.value
}, {
.expr1 <- u1 * u2
.expr2 <- log(.expr1)
.expr3 <- log(u2)
.expr4 <- .expr3/.expr2
.expr5 <- -alpha
.expr6 <- .expr4^.expr5
.expr7 <- 1 - .expr4
.expr8 <- .expr7^.expr5
.expr9 <- .expr6 + .expr8
.expr11 <- -1/alpha
.expr12 <- .expr9^.expr11
.expr13 <- 1 - .expr12
.expr15 <- exp(.expr2 * .expr13)
.expr20 <- 1/.expr1 - u2 * u1/.expr1^2
.expr23 <- 1/alpha^2
.expr24 <- log(.expr9) * .expr23
.expr26 <- .expr11 - 1
.expr27 <- .expr9^.expr26
.expr28 <- log(.expr7)
.expr30 <- log(.expr4)
.expr32 <- .expr8 * .expr28 + .expr6 * .expr30
.expr35 <- .expr12 * .expr24 - .expr27 * (.expr11 * .expr32)
.expr37 <- u2/.expr1
.expr39 <- .expr26 - 1
.expr40 <- .expr9^.expr39
.expr43 <- .expr27 * .expr24 - .expr40 * (.expr26 * .expr32)
.expr44 <- .expr5 - 1
.expr45 <- .expr4^.expr44
.expr46 <- 1/u2
.expr48 <- u1/.expr1
.expr50 <- .expr2^2
.expr52 <- .expr46/.expr2 - .expr3 * .expr48/.expr50
.expr53 <- .expr5 * .expr52
.expr55 <- .expr7^.expr44
.expr57 <- .expr45 * .expr53 - .expr55 * .expr53
.expr58 <- .expr11 * .expr57
.expr60 <- .expr23 * .expr57
.expr62 <- .expr45 * .expr30
.expr66 <- .expr55 * .expr28
.expr69 <- .expr45 * .expr52 + .expr62 * .expr53 - (.expr55 *
.expr52 + .expr66 * .expr53)
.expr73 <- .expr43 * .expr58 + .expr27 * (.expr60 - .expr11 *
.expr69)
.expr76 <- .expr3 * .expr37
.expr77 <- .expr76/.expr50
.expr78 <- .expr5 * .expr77
.expr81 <- .expr55 * .expr78 - .expr45 * .expr78
.expr82 <- .expr11 * .expr81
.expr93 <- .expr23 * .expr81 - .expr11 * (.expr55 * .expr77 +
.expr66 * .expr78 - (.expr45 * .expr77 + .expr62 * .expr78))
.expr95 <- .expr43 * .expr82 + .expr27 * .expr93
.expr103 <- .expr26 * .expr57
.expr110 <- .expr40 * .expr103
.expr122 <- (.expr46 * .expr37 + .expr3 * .expr20)/.expr50 -
.expr76 * (2 * (.expr48 * .expr2))/.expr50^2
.expr123 <- .expr5 * .expr122
.expr125 <- .expr44 - 1
.expr126 <- .expr7^.expr125
.expr127 <- .expr44 * .expr52
.expr128 <- .expr126 * .expr127
.expr131 <- .expr4^.expr125
.expr132 <- .expr131 * .expr127
.expr136 <- .expr55 * .expr123 - .expr128 * .expr78 - (.expr132 *
.expr78 + .expr45 * .expr123)
.expr137 <- .expr11 * .expr136
.expr173 <- .expr15 * (.expr2 * .expr35)
.expr175 <- .expr27 * .expr58
.expr178 <- .expr27 * .expr82
.expr190 <- .expr48 * .expr13 - .expr2 * .expr175
.value <- -(.expr15 * (.expr20 * .expr35 + .expr37 * .expr73 +
(.expr48 * .expr95 + .expr2 * (((.expr40 * .expr24 -
.expr9^(.expr39 - 1) * (.expr39 * .expr32)) * .expr103 +
.expr40 * (.expr60 - .expr26 * .expr69)) * .expr82 +
.expr110 * .expr93 + (.expr43 * .expr137 + .expr27 *
(.expr23 * .expr136 - .expr11 * (.expr55 * .expr122 +
.expr66 * .expr123 - (.expr128 * .expr77 + (.expr126 *
.expr52 + .expr126 * .expr28 * .expr127) * .expr78) -
(.expr45 * .expr122 + .expr62 * .expr123 + (.expr132 *
.expr77 + (.expr131 * .expr52 + .expr131 *
.expr30 * .expr127) * .expr78)))))))) + .expr173 *
(.expr20 * .expr13 - .expr37 * .expr175 - (.expr48 *
.expr178 + .expr2 * (.expr110 * .expr82 + .expr27 *
.expr137))) + (.expr15 * .expr190 * (.expr37 * .expr35 +
.expr2 * .expr95) + (.expr15 * (.expr48 * .expr35 + .expr2 *
.expr73) + .expr173 * .expr190) * (.expr37 * .expr13 -
.expr2 * .expr178)))
.grad <- array(0, c(length(.value), 1L), list(NULL, c("s")))
.grad[, "s"] <- 0
attr(.value, "gradient") <- .grad
.value
})
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