NAG Library Manual, Mark 27.3
```!   D01FDF Example Program Text
!   Mark 27.3 Release. NAG Copyright 2021.

Module d01fdfe_mod

!     D01FDF Example Program Module:
!            Parameters and User-defined Routines

!     .. Use Statements ..
Use nag_library, Only: nag_wp
!     .. Implicit None Statement ..
Implicit None
!     .. Accessibility Statements ..
Private
Public                           :: f, region
!     .. Parameters ..
Integer, Parameter, Public       :: nout = 6
Contains
Function f(ndim,x)

!       .. Function Return Value ..
Real (Kind=nag_wp)             :: f
!       .. Scalar Arguments ..
Integer, Intent (In)           :: ndim
!       .. Array Arguments ..
Real (Kind=nag_wp), Intent (In) :: x(ndim)
!       .. Intrinsic Procedures ..
Intrinsic                      :: abs, sqrt, sum
!       .. Executable Statements ..
f = 1.0E0_nag_wp/sqrt(abs(2.25E0_nag_wp-sum(x(1:ndim)**2)))

Return

End Function f
Subroutine region(ndim,x,j,c,d)

!       .. Scalar Arguments ..
Real (Kind=nag_wp), Intent (Out) :: c, d
Integer, Intent (In)           :: j, ndim
!       .. Array Arguments ..
Real (Kind=nag_wp), Intent (In) :: x(ndim)
!       .. Intrinsic Procedures ..
Intrinsic                      :: abs, sqrt, sum
!       .. Executable Statements ..
If (j>1) Then
d = sqrt(abs(2.25E0_nag_wp-sum(x(1:(j-1))**2)))
c = -d
Else
c = -1.5E0_nag_wp
d = 1.5E0_nag_wp
End If

Return

End Subroutine region
End Module d01fdfe_mod
Program d01fdfe

!     D01FDF Example Main Program

!     .. Use Statements ..
Use d01fdfe_mod, Only: f, nout, region
Use nag_library, Only: d01fdf, d01fdv, nag_wp
!     .. Implicit None Statement ..
Implicit None
!     .. Local Scalars ..
Real (Kind=nag_wp)               :: r0, result, sigma, u
Integer                          :: ifail, limit, ncalls, ndim
!     .. Executable Statements ..
Write (nout,*) 'D01FDF Example Program Results'

ndim = 3
limit = 8000
u = 1.5E0_nag_wp
sigma = 1.5E0_nag_wp
r0 = 0.9E0_nag_wp

ifail = 0
Call d01fdf(ndim,f,sigma,d01fdv,limit,r0,u,result,ncalls,ifail)

Write (nout,*)
Write (nout,*) 'Sphere-to-sphere transformation'
Write (nout,*)
Write (nout,99999) 'Estimated value of the integral = ', result
Write (nout,99998) 'Number of integrand evaluations = ', ncalls
Write (nout,*)
Write (nout,*) 'Product region transformation'

sigma = -1.0E0_nag_wp
r0 = 0.8E0_nag_wp

ifail = 0
Call d01fdf(ndim,f,sigma,region,limit,r0,u,result,ncalls,ifail)

Write (nout,*)
Write (nout,99999) 'Estimated value of the integral = ', result
Write (nout,99998) 'Number of integrand evaluations = ', ncalls

99999 Format (1X,A,F9.3)
99998 Format (1X,A,I5)
End Program d01fdfe
```