diff --git a/src/compat/tracker.jl b/src/compat/tracker.jl
index 6742ed16..8dab331a 100644
--- a/src/compat/tracker.jl
+++ b/src/compat/tracker.jl
@@ -477,8 +477,8 @@ _inv_link_chol_lkj(y::TrackedMatrix) = track(_inv_link_chol_lkj, y)
         @inbounds for j in 1:K
             Δtmp = Δw[j,j]
             for i in j:-1:2
-                Δz = Δw[i-1, j] * tmp_mat[i, j] + Δtmp * tmp_mat[i, j] * 1 / sqrt(1 - z_mat[i, j]^2) / 2 * (-2 * z_mat[i, j])
-                Δy[i-1, j] = Δz * 1 / cosh(y[i-1, j])^2
+                Δz = Δw[i-1, j] * tmp_mat[i, j] - Δtmp * tmp_mat[i, j] / sqrt(1 - z_mat[i, j]^2) * z_mat[i, j]
+                Δy[i-1, j] = Δz / cosh(y[i-1, j])^2
                 Δtmp = Δw[i-1, j] * z_mat[i, j] + Δtmp * sqrt(1 - z_mat[i, j]^2)
             end
         end
@@ -527,15 +527,14 @@ _link_chol_lkj(w::TrackedMatrix) = track(_link_chol_lkj, w)
             for i in (j-1):-1:2
                 p = w[i, j] / tmp_mat[i-1, j]
                 ftmp = sqrt(1 - p^2)
-                d_ftmp_p = 1 / ftmp / 2 * (-2 * p) 
+                d_ftmp_p = -p / ftmp
                 d_p_tmp = -w[i,j] / tmp_mat[i-1, j]^2
 
-                Δp = Δz[i,j] * 1/(1-p^2) + Δtmp * tmp_mat[i-1, j] * d_ftmp_p # TODO: simplify
+                Δp = Δz[i,j] / (1-p^2) + Δtmp * tmp_mat[i-1, j] * d_ftmp_p # TODO: simplify
                 Δw[i, j] = Δp / tmp_mat[i-1, j]
-                # Δtmp = Δp * d_p_tmp + Δtmp * (ftmp + tmp_mat[i-1, j] * d_ftmp_p * d_p_tmp) # update to "previous" Δtmp
-                Δtmp = Δp * d_p_tmp + Δtmp * (ftmp) # update to "previous" Δtmp
+                Δtmp = Δp * d_p_tmp + Δtmp * ftmp # update to "previous" Δtmp
             end
-            Δw[1, j] = Δz[1, j] * 1/(1-w[1,j]^2) + Δtmp / sqrt(1 - w[1,j]^2) / 2 * (-2 * w[1,j])
+            Δw[1, j] = Δz[1, j] / (1-w[1,j]^2) - Δtmp / sqrt(1 - w[1,j]^2) * w[1,j]
         end
 
         return (Δw,)
diff --git a/src/compat/zygote.jl b/src/compat/zygote.jl
index ed9d6a3e..48ff3076 100644
--- a/src/compat/zygote.jl
+++ b/src/compat/zygote.jl
@@ -227,8 +227,8 @@ end
         @inbounds for j in 1:K
             Δtmp = Δw[j,j]
             for i in j:-1:2
-                Δz = Δw[i-1, j] * tmp_mat[i, j] + Δtmp * tmp_mat[i, j] * 1 / sqrt(1 - z_mat[i, j]^2) / 2 * (-2 * z_mat[i, j])
-                Δy[i-1, j] = Δz * 1 / cosh(y[i-1, j])^2
+                Δz = Δw[i-1, j] * tmp_mat[i, j] - Δtmp * tmp_mat[i, j] / sqrt(1 - z_mat[i, j]^2) * z_mat[i, j]
+                Δy[i-1, j] = Δz / cosh(y[i-1, j])^2
                 Δtmp = Δw[i-1, j] * z_mat[i, j] + Δtmp * sqrt(1 - z_mat[i, j]^2)
             end
         end
@@ -274,15 +274,14 @@ end
             for i in (j-1):-1:2
                 p = w[i, j] / tmp_mat[i-1, j]
                 ftmp = sqrt(1 - p^2)
-                d_ftmp_p = 1 / ftmp / 2 * (-2 * p) 
+                d_ftmp_p = -p / ftmp
                 d_p_tmp = -w[i,j] / tmp_mat[i-1, j]^2
 
-                Δp = Δz[i,j] * 1/(1-p^2) + Δtmp * tmp_mat[i-1, j] * d_ftmp_p # TODO: simplify
+                Δp = Δz[i,j] / (1-p^2) + Δtmp * tmp_mat[i-1, j] * d_ftmp_p # TODO: simplify
                 Δw[i, j] = Δp / tmp_mat[i-1, j]
-                # Δtmp = Δp * d_p_tmp + Δtmp * (ftmp + tmp_mat[i-1, j] * d_ftmp_p * d_p_tmp) # update to "previous" Δtmp
-                Δtmp = Δp * d_p_tmp + Δtmp * (ftmp) # update to "previous" Δtmp
+                Δtmp = Δp * d_p_tmp + Δtmp * ftmp # update to "previous" Δtmp
             end
-            Δw[1, j] = Δz[1, j] * 1/(1-w[1,j]^2) + Δtmp / sqrt(1 - w[1,j]^2) / 2 * (-2 * w[1,j])
+            Δw[1, j] = Δz[1, j] / (1-w[1,j]^2) - Δtmp / sqrt(1 - w[1,j]^2) * w[1,j]
         end
 
         return (Δw,)