Is it possible to overwrite intrinsics?

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Yes, in principle though technically in modern Fortran, a practitioner will be advised to

  1. furnish an explicit interface instead of using EXTERNAL, and
  2. employ the INTRINSIC attribute/statement to indicate explicitly a reference to an intrinsic procedure

With other readers in mind, an example will be as follows:

   real :: sinx
   sinx = sin(0.5)
   print *, "sin(0.5) = ", sinx
   block
      intrinsic :: sin
      print *, "Using intrinsic implementation:"
      print *, "sin(0.5) = ", sin(0.5)
   end block
contains
   impure elemental function sin(x) result(r)
      real, intent(in) :: x
      real :: r
      real, parameter :: PI = 3.1415926535
      real, parameter :: PI2 = PI*PI
      print *, "Using Bhaskara approximation:"
      r = 16*x*(PI-x)/(5*PI2-4*x*(PI-x))
   end function
end 

C:\temp>ifort /standard-semantics p.f90
Intel(R) Fortran Intel(R) 64 Compiler Classic for applications running on Intel(R) 64, Version 2021.5.0 Build 20211109_000000
Copyright (C) 1985-2021 Intel Corporation.  All rights reserved.

Microsoft (R) Incremental Linker Version 14.31.31105.0
Copyright (C) Microsoft Corporation.  All rights reserved.

-out:p.exe
-subsystem:console
p.obj

C:\temp>p.exe
 Using Bhaskara approximation:
 sin(0.5) =  0.4795828
 Using intrinsic implementation:
 sin(0.5) =  0.4794255
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