diff --git a/src/emmy/calculus/derivative.cljc b/src/emmy/calculus/derivative.cljc
index f3a507dc..206767f6 100644
--- a/src/emmy/calculus/derivative.cljc
+++ b/src/emmy/calculus/derivative.cljc
@@ -256,10 +256,10 @@
   (fn [x]
     (let [tag    (d/fresh-tag)
           lifted
-          (d/bundle-element x 1 tag)
-
-          ;; TODO this is the only change we need to get this working. Also implement comparable.
-          #_(emmy.tape/->Dual tag x 1)]
+          ;; TODO if we want to do this, we can kill differentials and move the
+          ;; tests over...
+          #_(d/bundle-element x 1 tag)
+          (emmy.tape/->Dual tag x 1)]
       (-> (d/with-active-tag tag f [lifted])
           (d/extract-tangent tag))
       )))
@@ -548,9 +548,16 @@
      (letfn [(process-term [term]
                (g/simplify
                 (s/mapr (fn rec [x]
-                          (if (d/differential? x)
-                            (d/map-coefficients rec x)
-                            (-> (g/simplify x)
-                                (x/substitute replace-m))))
+                          (cond (d/differential? x) (d/map-coefficients rec x)
+                                (tape/dual? x)
+                                (tape/->Dual (tape/dual-tag x)
+                                             (rec (tape/dual-primal x))
+                                             (rec (tape/dual-tangent x)))
+
+                                (tape/tape? x)
+                                (u/illegal "TODO implement this using fmap style.")
+
+                                :else (-> (g/simplify x)
+                                          (x/substitute replace-m))))
                         term)))]
        (series/fmap process-term series)))))
diff --git a/src/emmy/series.cljc b/src/emmy/series.cljc
index 2d98fcd0..6c26c6ed 100644
--- a/src/emmy/series.cljc
+++ b/src/emmy/series.cljc
@@ -870,16 +870,18 @@
 (defmethod g/exact? [::series] [_] false)
 (defmethod g/exact? [::power-series] [_] false)
 (defmethod g/freeze [::power-series] [^PowerSeries s]
-        (let [prefix (->> (g/simplify (take 4 (.-xs s)))
-                          (g/freeze)
-                          (filter (complement g/zero?))
-                          (map-indexed
-                           (fn [n a]
-                             (if (g/one? a)
-                               `(~'expt ~'_ ~n)
-                               `(~'* ~a (~'expt ~'_ ~n))))))]
-          `(~'+ ~@prefix ~'...)))
+  (let [prefix (->> (g/simplify (take 4 (.-xs s)))
+                    (g/freeze)
+                    (into [] (comp
+                              (map-indexed
+                               (fn [n a]
+                                 (cond (g/zero? a) []
+                                       (g/one? a)  [(list 'expt '_ n)]
+                                       :else       [(list '* a (list 'expt '_ n))])))
+                              cat)))]
+    `(~'+ ~@prefix ~'...)))
+
 (defmethod g/freeze [::series] [^Series s]
-        (let [prefix (g/freeze
-                      (g/simplify (take 4 (.-xs s))))]
-          `(~'+ ~@prefix ~'...)))
+  (let [prefix (g/freeze
+                (g/simplify (take 4 (.-xs s))))]
+    `(~'+ ~@prefix ~'...)))
diff --git a/src/emmy/tape.cljc b/src/emmy/tape.cljc
index cf077481..e613c894 100644
--- a/src/emmy/tape.cljc
+++ b/src/emmy/tape.cljc
@@ -126,7 +126,7 @@
 
 ;; Here's the [[TapeCell]] type with the fields described above.
 
-(declare compare)
+(declare compare compare-dual dual-primal)
 
 (deftype Dual [tag primal tangent]
   v/IKind
@@ -147,6 +147,7 @@
 
   Object
   ;; TODO revisit all of this
+  #?(:clj (equals [_ b] (v/= primal (dual-primal b))))
   #?(:cljs (valueOf [_] (.valueOf primal)))
   (toString [_]
     (str "#emmy.tape.Dual"
@@ -154,8 +155,22 @@
           :primal primal
           :tangent tangent}))
 
-  ;; TODO add comparable block from below
-  )
+  #?@(:clj
+      ;; The motivation for this override is subtle. To participate in control
+      ;; flow operations, like comparison with both [[TapeCell]] and
+      ;; non-[[TapeCell]] instances, [[TapeCell]] instances should compare using
+      ;; ONLY their primal terms. This means that comparison will ignore any
+      ;; difference in `in->partial`.
+      [Comparable
+       (compareTo [a b] (compare-dual a b))]
+
+      :cljs
+      [IComparable
+       (-compare [a b]  (compare-dual a b))
+
+       IPrintWithWriter
+       (-pr-writer [x writer _]
+                   (write-all writer (.toString x)))]))
 
 
 (deftype TapeCell [tag id primal in->partial]
@@ -487,6 +502,12 @@
    (tape-primal a)
    (tape-primal b)))
 
+(defn compare-dual
+  [a b]
+  (v/compare
+   (dual-primal a)
+   (dual-primal b)))
+
 ;; ## Reverse-pass support
 
 (defn inner-tag
diff --git a/test/emmy/calculus/derivative_test.cljc b/test/emmy/calculus/derivative_test.cljc
index 4921f9e9..b2b79cbf 100644
--- a/test/emmy/calculus/derivative_test.cljc
+++ b/test/emmy/calculus/derivative_test.cljc
@@ -1569,8 +1569,7 @@
   (is (v/= [0 1 0 0]
            (take 4 ((D (fn [y]
                          (d/symbolic-taylor-series
-                          (fn [x] (g/* x y))
-                          0)))
+                          (fn [x] (g/* x y)))))
                     'a)))
       "proper function when symbolic-taylor-series is used INSIDE of a call to
       `D`; this shows that it can do proper symbolic replacement inside of
@@ -1578,8 +1577,8 @@
 
   (testing "compare, one stays symbolic:"
     (letfn [(f [[a b]]
-             (* (sin (* 3 a))
-                (cos (* 4 b))))]
+              (* (sin (* 3 a))
+                 (cos (* 4 b))))]
 
       (is (ish? [-0.020532965943782493
                  (s/down 0.4321318251769156 -0.558472974950351)]