diff --git a/org/qmckl_sherman_morrison_woodbury.org b/org/qmckl_sherman_morrison_woodbury.org
index a10cdc6..37785f8 100644
--- a/org/qmckl_sherman_morrison_woodbury.org
+++ b/org/qmckl_sherman_morrison_woodbury.org
@@ -1423,7 +1423,7 @@ with Sherman-Morrison and update splitting. Please look at the performance recco
 
 *** Introduction
 The Woodbury 2x2 kernel. It is used to apply two rank-1 updates at once. The formula used in
-this algorithm is called the Woodbury Matrix Identity
+this algorithm is called the Woodbury Matrix Id
 \[
 (S + U V)^{-1}  =  S^{-1} - C B^{-1} D
 \]
@@ -1485,12 +1485,15 @@ integer function qmckl_woodbury_2x2_doc_f(&
   real*8    , intent(inout)         :: s_inv(dim * lds)
   real*8    , intent(inout)         :: determinant
 
-  real*8 , dimension(lds, 2)        :: Updates
-  real*8 , dimension(dim, lds)      :: Inverse
-  real*8 , dimension(dim, 2)        :: C
-  real*8 , dimension(2, dim)        :: D
-  real*8                            :: denominator, idenominator, update
-  integer*8                         :: i, j, l, row
+  integer*8 , dimension(2, dim)     :: V
+  integer*8 , dimension(2, 2)       :: Id
+  real*8    , dimension(dim, dim)   :: Inverse
+  real*8    , dimension(dim, 2)     :: Updates, C
+  real*8    , dimension(2, 2)       :: D, invD
+  real*8    , dimension(2, dim)     :: E, F
+  
+  real*8                            :: detD, idenominator, update
+  integer*8                         :: i, j, k, l
 
   info = QMCKL_FAILURE
 
@@ -1499,19 +1502,86 @@ integer function qmckl_woodbury_2x2_doc_f(&
     return
   endif
 
+  ! Construct V(2, dim) matrix
+  V = 0
+  V(1, updates_index(1)) = 1
+  V(2, updates_index(2)) = 1
+
+  ! Construct Id(2, 2) matrix
+  Id = 0
+  Id(1, 1) = 1
+  Id(2, 2) = 1
+
   ! Convert 'upds' and 's_inv' into the more easily readable Fortran
   ! matrices 'Updates' and 'Inverse'.
   call convert(upds, s_inv, Updates, Inverse, int(2,8), lds, dim)
 
-  ! Compute C(dim,2) = Inverse(dim,dim) x Updates(dim,2) 
+  ! Compute C(dim, 2) = Inverse(dim, dim) x Updates(dim, 2)
+  C = 0
   do i = 1, dim
-    do j = 1, dim
-      C(i,1) = C(i,1) + Inverse(1,j) * Updates(j,1)
-      C(i,2) = C(i,1) + Inverse(1,j) * Updates(j,2)
+    do j = 1, 2
+      do k = 1, dim
+        C(i, j) = C(i, j) + Inverse(i, k) * Updates(k, j)
+      end do
     end do
   end do
 
+  ! Construct matrix D(2, 2) := I(2, 2) + V(2, dim) x C(dim, 2)
+  D = 0
+  do i = 1, 2 
+     do j = 1, 2
+        do k = 2, dim
+           D(i, j) = D(i, j) + V(i, k) * C(k, j)
+        end do
+      end do
+  end do
+  D = Id + D
   
+  ! Compute determinant := det(D) explicitly
+  detD = D(1,1) * D(2,2) - D(1,2) * D(2,1)
+
+    ! Return early if det(D) is too small
+  if (abs(detD) < breakdown) return
+
+  ! Update det(S)
+  determinant = determinant * detD
+
+  ! Compute inv(D) explicitly
+  invD(1,1) =   D(2,2)
+  invD(1,2) = - D(1,2)
+  invD(2,1) = - D(2,1)
+  invD(2,2) =   D(1,1)
+  invD = invD / detD
+
+  ! Compute E(2, dim) := V(2, dim) x Inverse(dim, dim)
+  E = 0
+  do i = 1, 2 
+    do j = 1, dim
+      do k = 1, dim
+        E(i, j) = E(i, j) + V(i, k) * Inverse(k, j)
+      end do
+    end do
+  end do
+
+  ! Compute F(2, dim) := invD(2, 2) x E(2, dim)
+  F = 0
+  do i = 1, 2
+    do j = 1, dim
+      do k = 1, 2
+        F(i, j) = F(i, j) + invD(i, k) * E(k, j)
+      end do
+    end do
+  end do
+
+  ! Compute Inverse(dim, dim) := Inverse(dim, dim) - C(dim, 2) x F(2, dim)
+  do i = 1, dim
+    do j = 1, dim
+      do k = 1, 2
+        Inverse(i, j) = Inverse(i, j) - C(i, k) * F(k, j)
+      end do
+    end do
+  end do
+
   ! Copy updated inverse and later updates
   ! back to s_inv and later_upds
   call copy_back_inv(Inverse, s_inv, lds, dim)
@@ -1830,16 +1900,7 @@ qmckl_exit_code qmckl_woodbury_2x2(const qmckl_context context,
         determinant);
   }
   #else
-  // return qmckl_woodbury_2x2_doc(
-  //     context,
-  //     LDS,
-  //     Dim,
-  //     Updates,
-  //     Updates_index,
-  //     breakdown,
-  //     Slater_inv,
-  //     determinant);
-  return qmckl_woodbury_2x2_hpc(
+  return qmckl_woodbury_2x2_doc(
       context,
       LDS,
       Dim,
@@ -1848,6 +1909,15 @@ qmckl_exit_code qmckl_woodbury_2x2(const qmckl_context context,
       breakdown,
       Slater_inv,
       determinant);
+  // return qmckl_woodbury_2x2_hpc(
+  //     context,
+  //     LDS,
+  //     Dim,
+  //     Updates,
+  //     Updates_index,
+  //     breakdown,
+  //     Slater_inv,
+  //     determinant);
   #endif
 
   return QMCKL_FAILURE;