diff --git a/.gitignore b/.gitignore
--- a/.gitignore
+++ b/.gitignore
@@ -1,4 +1,5 @@
 dist
+dist-newstyle
 docs
 wiki
 TAGS
@@ -12,3 +13,21 @@
 *~
 *#
 *.imports
+.stack-work/
+cabal-dev
+*.chi
+*.chs.h
+*.dyn_o
+*.dyn_hi
+.hpc
+.hsenv
+.cabal-sandbox/
+cabal.sandbox.config
+*.prof
+*.aux
+*.hp
+*.eventlog
+cabal.project.local
+cabal.project.local~
+.HTF/
+.ghc.environment.*
diff --git a/.travis.yml b/.travis.yml
--- a/.travis.yml
+++ b/.travis.yml
@@ -1,100 +1,146 @@
-# This file has been generated -- see https://github.com/hvr/multi-ghc-travis
+# This Travis job script has been generated by a script via
 #
-# Modifications: benchmarks disabled, cabal check disabled, IRC notifications
+#   runghc make_travis_yml_2.hs '-o' '.travis.yml' '--irc-channel=irc.freenode.org#haskell-lens' '--no-no-tests-no-bench' '--no-installed' 'cabal.project'
+#
+# For more information, see https://github.com/hvr/multi-ghc-travis
+#
 language: c
 sudo: false
 
+git:
+  submodules: false  # whether to recursively clone submodules
+
+notifications:
+  irc:
+    channels:
+      - "irc.freenode.org#haskell-lens"
+    skip_join: true
+    template:
+      - "\x0313ad\x03/\x0306%{branch}\x03 \x0314%{commit}\x03 %{build_url} %{message}"
+
 cache:
   directories:
-    - $HOME/.cabsnap
     - $HOME/.cabal/packages
+    - $HOME/.cabal/store
 
 before_cache:
   - rm -fv $HOME/.cabal/packages/hackage.haskell.org/build-reports.log
-  - rm -fv $HOME/.cabal/packages/hackage.haskell.org/00-index.tar
+  # remove files that are regenerated by 'cabal update'
+  - rm -fv $HOME/.cabal/packages/hackage.haskell.org/00-index.*
+  - rm -fv $HOME/.cabal/packages/hackage.haskell.org/*.json
+  - rm -fv $HOME/.cabal/packages/hackage.haskell.org/01-index.cache
+  - rm -fv $HOME/.cabal/packages/hackage.haskell.org/01-index.tar
+  - rm -fv $HOME/.cabal/packages/hackage.haskell.org/01-index.tar.idx
 
+  - rm -rfv $HOME/.cabal/packages/head.hackage
+
+addons:
+  apt:
+    packages: &apt_packages
+      - ghc-ppa-tools
+      - alex-3.1.7
+      - happy-1.19.5
+
 matrix:
   include:
-    - env: CABALVER=1.24 GHCVER=7.4.2 ALEXVER=3.1.7 HAPPYVER=1.19.5
-      compiler: ": #GHC 7.4.2"
-      addons: {apt: {packages: [cabal-install-1.24,ghc-7.4.2,alex-3.1.7,happy-1.19.5], sources: [hvr-ghc]}}
-    - env: CABALVER=1.24 GHCVER=7.6.3 ALEXVER=3.1.7 HAPPYVER=1.19.5
-      compiler: ": #GHC 7.6.3"
-      addons: {apt: {packages: [cabal-install-1.24,ghc-7.6.3,alex-3.1.7,happy-1.19.5], sources: [hvr-ghc]}}
-    - env: CABALVER=1.24 GHCVER=7.8.4 ALEXVER=3.1.7 HAPPYVER=1.19.5
-      compiler: ": #GHC 7.8.4"
-      addons: {apt: {packages: [cabal-install-1.24,ghc-7.8.4,alex-3.1.7,happy-1.19.5], sources: [hvr-ghc]}}
-    - env: CABALVER=1.24 GHCVER=7.10.3 ALEXVER=3.1.7 HAPPYVER=1.19.5
-      compiler: ": #GHC 7.10.3"
-      addons: {apt: {packages: [cabal-install-1.24,ghc-7.10.3,alex-3.1.7,happy-1.19.5], sources: [hvr-ghc]}}
-    - env: CABALVER=1.24 GHCVER=8.0.2 ALEXVER=3.1.7 HAPPYVER=1.19.5
-      compiler: ": #GHC 8.0.2"
-      addons: {apt: {packages: [cabal-install-1.24,ghc-8.0.2,alex-3.1.7,happy-1.19.5], sources: [hvr-ghc]}}
-    - env: CABALVER=2.0 GHCVER=8.2.1 ALEXVER=3.1.7 HAPPYVER=1.19.5
-      compiler: ": #GHC 8.2.1"
-      addons: {apt: {packages: [cabal-install-2.0,ghc-8.2.1,alex-3.1.7,happy-1.19.5], sources: [hvr-ghc]}}
+    - compiler: "ghc-7.4.2"
+    # env: TEST=--disable-tests BENCH=--disable-benchmarks
+      addons: {apt: {packages: [*apt_packages,cabal-install-2.0,ghc-7.4.2], sources: [hvr-ghc]}}
+    - compiler: "ghc-7.6.3"
+    # env: TEST=--disable-tests BENCH=--disable-benchmarks
+      addons: {apt: {packages: [*apt_packages,cabal-install-2.0,ghc-7.6.3], sources: [hvr-ghc]}}
+    - compiler: "ghc-7.8.4"
+    # env: TEST=--disable-tests BENCH=--disable-benchmarks
+      addons: {apt: {packages: [*apt_packages,cabal-install-2.0,ghc-7.8.4], sources: [hvr-ghc]}}
+    - compiler: "ghc-7.10.3"
+    # env: TEST=--disable-tests BENCH=--disable-benchmarks
+      addons: {apt: {packages: [*apt_packages,cabal-install-2.0,ghc-7.10.3], sources: [hvr-ghc]}}
+    - compiler: "ghc-8.0.2"
+    # env: TEST=--disable-tests BENCH=--disable-benchmarks
+      addons: {apt: {packages: [*apt_packages,cabal-install-2.0,ghc-8.0.2], sources: [hvr-ghc]}}
+    - compiler: "ghc-8.2.2"
+    # env: TEST=--disable-tests BENCH=--disable-benchmarks
+      addons: {apt: {packages: [*apt_packages,cabal-install-2.0,ghc-8.2.2], sources: [hvr-ghc]}}
+    - compiler: "ghc-8.4.1"
+      env: GHCHEAD=true
+      addons: {apt: {packages: [*apt_packages,cabal-install-head,ghc-8.4.1], sources: [hvr-ghc]}}
+    - compiler: "ghc-head"
+      env: GHCHEAD=true
+      addons: {apt: {packages: [*apt_packages,cabal-install-head,ghc-head], sources: [hvr-ghc]}}
 
+  allow_failures:
+    - compiler: "ghc-8.4.1"
+    - compiler: "ghc-head"
+
 before_install:
- - unset CC
- - export PATH=/opt/ghc/$GHCVER/bin:/opt/cabal/$CABALVER/bin:/opt/alex/$ALEXVER/bin:/opt/happy/$HAPPYVER/bin:$PATH
+  - HC=${CC}
+  - HCPKG=${HC/ghc/ghc-pkg}
+  - unset CC
+  - ROOTDIR=$(pwd)
+  - mkdir -p $HOME/.local/bin
+  - "PATH=/opt/ghc/bin:/opt/ghc-ppa-tools/bin:$HOME/local/bin:$PATH"
+  - HCNUMVER=$(( $(${HC} --numeric-version|sed -E 's/([0-9]+)\.([0-9]+)\.([0-9]+).*/\1 * 10000 + \2 * 100 + \3/') ))
+  - echo $HCNUMVER
 
 install:
- - cabal --version
- - echo "$(ghc --version) [$(ghc --print-project-git-commit-id 2> /dev/null || echo '?')]"
- - if [ -f $HOME/.cabal/packages/hackage.haskell.org/00-index.tar.gz ];
-   then
-     zcat $HOME/.cabal/packages/hackage.haskell.org/00-index.tar.gz >
-          $HOME/.cabal/packages/hackage.haskell.org/00-index.tar;
-   fi
- - travis_retry cabal update -v
- - sed -i 's/^jobs:/-- jobs:/' ${HOME}/.cabal/config
- - cabal install --only-dependencies --enable-tests --dry -v > installplan.txt
- - sed -i -e '1,/^Resolving /d' installplan.txt; cat installplan.txt
+  - cabal --version
+  - echo "$(${HC} --version) [$(${HC} --print-project-git-commit-id 2> /dev/null || echo '?')]"
+  - BENCH=${BENCH---enable-benchmarks}
+  - TEST=${TEST---enable-tests}
+  - HADDOCK=${HADDOCK-true}
+  - INSTALLED=${INSTALLED-true}
+  - GHCHEAD=${GHCHEAD-false}
+  - travis_retry cabal update -v
+  - "sed -i.bak 's/^jobs:/-- jobs:/' ${HOME}/.cabal/config"
+  - rm -fv cabal.project cabal.project.local
+  # Overlay Hackage Package Index for GHC HEAD: https://github.com/hvr/head.hackage
+  - |
+    if $GHCHEAD; then
+      sed -i.bak 's/-- allow-newer:.*/allow-newer: *:base, *:template-haskell, *:ghc, *:Cabal/' ${HOME}/.cabal/config
 
-# check whether current requested install-plan matches cached package-db snapshot
- - if diff -u installplan.txt $HOME/.cabsnap/installplan.txt;
-   then
-     echo "cabal build-cache HIT";
-     rm -rfv .ghc;
-     cp -a $HOME/.cabsnap/ghc $HOME/.ghc;
-     cp -a $HOME/.cabsnap/lib $HOME/.cabsnap/share $HOME/.cabsnap/bin $HOME/.cabal/;
-   else
-     echo "cabal build-cache MISS";
-     rm -rf $HOME/.cabsnap;
-     mkdir -p $HOME/.ghc $HOME/.cabal/lib $HOME/.cabal/share $HOME/.cabal/bin;
-     cabal install -j --only-dependencies --enable-tests;
-   fi
+      echo 'repository head.hackage'                                                        >> ${HOME}/.cabal/config
+      echo '   url: http://head.hackage.haskell.org/'                                       >> ${HOME}/.cabal/config
+      echo '   secure: True'                                                                >> ${HOME}/.cabal/config
+      echo '   root-keys: 07c59cb65787dedfaef5bd5f987ceb5f7e5ebf88b904bbd4c5cbdeb2ff71b740' >> ${HOME}/.cabal/config
+      echo '              2e8555dde16ebd8df076f1a8ef13b8f14c66bad8eafefd7d9e37d0ed711821fb' >> ${HOME}/.cabal/config
+      echo '              8f79fd2389ab2967354407ec852cbe73f2e8635793ac446d09461ffb99527f6e' >> ${HOME}/.cabal/config
+      echo '   key-threshold: 3'                                                            >> ${HOME}/.cabal.config
 
-# snapshot package-db on cache miss
- - if [ ! -d $HOME/.cabsnap ];
-   then
-      echo "snapshotting package-db to build-cache";
-      mkdir $HOME/.cabsnap;
-      cp -a $HOME/.ghc $HOME/.cabsnap/ghc;
-      cp -a $HOME/.cabal/lib $HOME/.cabal/share $HOME/.cabal/bin installplan.txt $HOME/.cabsnap/;
-   fi
+      cabal new-update head.hackage -v
+    fi
+  - grep -Ev -- '^\s*--' ${HOME}/.cabal/config | grep -Ev '^\s*$'
+  - "printf 'packages: \".\"\\n' > cabal.project"
+  - cat cabal.project
+  - if [ -f "./configure.ac" ]; then
+      (cd "." && autoreconf -i);
+    fi
+  - rm -f cabal.project.freeze
+  - cabal new-build -w ${HC} ${TEST} ${BENCH} --project-file="cabal.project" --dep -j2 all
+  - rm -rf "."/.ghc.environment.* "."/dist
+  - DISTDIR=$(mktemp -d /tmp/dist-test.XXXX)
 
 # Here starts the actual work to be performed for the package under test;
 # any command which exits with a non-zero exit code causes the build to fail.
 script:
- - if [ -f configure.ac ]; then autoreconf -i; fi
- - cabal configure --enable-tests -v2  # -v2 provides useful information for debugging
- - cabal build   # this builds all libraries and executables (including tests)
- - cabal test
- - cabal sdist   # tests that a source-distribution can be generated
-                 # - cabal check disabled due to spurious -O2 warning
+  # test that source-distributions can be generated
+  - (cd "." && cabal sdist)
+  - mv "."/dist/ad-*.tar.gz ${DISTDIR}/
+  - cd ${DISTDIR} || false
+  - find . -maxdepth 1 -name '*.tar.gz' -exec tar -xvf '{}' \;
+  - "printf 'packages: ad-*/*.cabal\\n' > cabal.project"
+  - cat cabal.project
 
-# Check that the resulting source distribution can be built & installed.
-# If there are no other `.tar.gz` files in `dist`, this can be even simpler:
-# `cabal install --force-reinstalls dist/*-*.tar.gz`
- - SRC_TGZ=$(cabal info . | awk '{print $2;exit}').tar.gz &&
-   (cd dist && cabal install --force-reinstalls "$SRC_TGZ")
 
-notifications:
-  irc:
-    channels:
-      - "irc.freenode.org#haskell-lens"
-    skip_join: true
-    template:
-      - "\x0313ad\x0f/\x0306%{branch}\x0f \x0314%{commit}\x0f %{message} \x0302\x1f%{build_url}\x0f"
+  # build & run tests, build benchmarks
+  - cabal new-build -w ${HC} ${TEST} ${BENCH} all
+  - if [ "x$TEST" = "x--enable-tests" ]; then cabal new-test -w ${HC} ${TEST} ${BENCH} all; fi
+
+  # cabal check
+  - (cd ad-* && cabal check)
+
+  # haddock
+  - rm -rf ./dist-newstyle
+  - if $HADDOCK; then cabal new-haddock -w ${HC} ${TEST} ${BENCH} all; else echo "Skipping haddock generation";fi
+
+# REGENDATA ["-o",".travis.yml","--irc-channel=irc.freenode.org#haskell-lens","--no-no-tests-no-bench","--no-installed","cabal.project"]
 # EOF
diff --git a/CHANGELOG.markdown b/CHANGELOG.markdown
--- a/CHANGELOG.markdown
+++ b/CHANGELOG.markdown
@@ -1,3 +1,7 @@
+4.3.5 [2018.01.18]
+------------------
+* Add `Semigroup` instance for `Id`.
+
 4.3.4
 -----
 * Support `doctest-0.12`
diff --git a/ad.cabal b/ad.cabal
--- a/ad.cabal
+++ b/ad.cabal
@@ -1,5 +1,5 @@
 name:          ad
-version:       4.3.4
+version:       4.3.5
 license:       BSD3
 license-File:  LICENSE
 copyright:     (c) Edward Kmett 2010-2015,
@@ -12,7 +12,7 @@
 bug-reports:   http://github.com/ekmett/ad/issues
 build-type:    Custom
 cabal-version: >= 1.10
-tested-with:   GHC == 7.4.2, GHC == 7.6.3, GHC == 7.8.4, GHC == 7.10.3, GHC == 8.0.2, GHC == 8.2.1
+tested-with:   GHC == 7.4.2, GHC == 7.6.3, GHC == 7.8.4, GHC == 7.10.3, GHC == 8.0.2, GHC == 8.2.2, GHC == 8.4.1
 synopsis:      Automatic Differentiation
 extra-source-files:
   .ghci
@@ -119,6 +119,7 @@
     free             >= 4.6.1 && < 5,
     nats             >= 0.1.2 && < 2,
     reflection       >= 1.4   && < 3,
+    semigroups       >= 0.16  && < 1,
     transformers     >= 0.3   && < 0.6
 
   if impl(ghc < 7.8)
@@ -179,7 +180,7 @@
     ad,
     base,
     directory,
-    doctest >= 0.9.0.1 && < 0.13,
+    doctest >= 0.9.0.1 && < 0.14,
     filepath
   ghc-options: -Wall -threaded
   hs-source-dirs: tests
diff --git a/src/Numeric/AD/Internal/Identity.hs b/src/Numeric/AD/Internal/Identity.hs
--- a/src/Numeric/AD/Internal/Identity.hs
+++ b/src/Numeric/AD/Internal/Identity.hs
@@ -25,15 +25,18 @@
   ) where
 
 import Data.Data (Data)
-#if __GLASGOW_HASKELL__ < 710
-import Data.Monoid
+#if !(MIN_VERSION_base(4,8,0))
+import Data.Monoid (Monoid(..))
 #endif
+#if !(MIN_VERSION_base(4,11,0))
+import Data.Semigroup (Semigroup(..))
+#endif
 import Data.Number.Erf
 import Data.Typeable (Typeable)
 import Numeric.AD.Mode
 
 newtype Id a = Id { runId :: a } deriving
-  (Eq, Ord, Show, Enum, Bounded, Num, Real, Fractional, Floating, RealFrac, RealFloat, Monoid, Data, Typeable, Erf, InvErf)
+  (Eq, Ord, Show, Enum, Bounded, Num, Real, Fractional, Floating, RealFrac, RealFloat, Semigroup, Monoid, Data, Typeable, Erf, InvErf)
 
 probe :: a -> Id a
 probe = Id
diff --git a/src/Numeric/AD/Internal/Kahn.hs b/src/Numeric/AD/Internal/Kahn.hs
--- a/src/Numeric/AD/Internal/Kahn.hs
+++ b/src/Numeric/AD/Internal/Kahn.hs
@@ -116,7 +116,7 @@
 Kahn Zero <**> y             = auto (0 ** primal y)
 _         <**> Kahn Zero     = auto 1
 x         <**> Kahn (Lift y) = lift1 (**y) (\z -> y *^ z ** Id (y-1)) x
-x         <**> y             = lift2_ (**) (\z xi yi -> (yi * z / xi, z * log xi)) x y
+x         <**> y             = lift2_ (**) (\z xi yi -> (yi * xi ** (yi - 1), z * log xi)) x y
 
 primal :: Num a => Kahn a -> a
 primal (Kahn Zero) = 0
diff --git a/src/Numeric/AD/Internal/Reverse.hs b/src/Numeric/AD/Internal/Reverse.hs
--- a/src/Numeric/AD/Internal/Reverse.hs
+++ b/src/Numeric/AD/Internal/Reverse.hs
@@ -158,7 +158,7 @@
 Zero <**> y      = auto (0 ** primal y)
 _    <**> Zero   = auto 1
 x    <**> Lift y = lift1 (**y) (\z -> y *^ z ** Id (y - 1)) x
-x    <**> y      = lift2_ (**) (\z xi yi -> (yi * z / xi, z * log xi)) x y
+x    <**> y      = lift2_ (**) (\z xi yi -> (yi * xi ** (yi - 1), z * log xi)) x y
 
 primal :: Num a => Reverse s a -> a
 primal Zero = 0
diff --git a/src/Numeric/AD/Internal/Sparse.hs b/src/Numeric/AD/Internal/Sparse.hs
--- a/src/Numeric/AD/Internal/Sparse.hs
+++ b/src/Numeric/AD/Internal/Sparse.hs
@@ -144,7 +144,7 @@
 _    <**> Zero = auto 1
 x    <**> y@(Sparse b bs)
   | IntMap.null bs = lift1 (**b) (\z -> b *^ z <**> Sparse (b-1) IntMap.empty) x
-  | otherwise      = lift2_ (**) (\z xi yi -> (yi * z / xi, z * log xi)) x y
+  | otherwise      = lift2_ (**) (\z xi yi -> (yi * xi ** (yi - 1), z * log xi)) x y
 
 instance Num a => Mode (Sparse a) where
   type Scalar (Sparse a) = a
diff --git a/src/Numeric/AD/Mode.hs b/src/Numeric/AD/Mode.hs
--- a/src/Numeric/AD/Mode.hs
+++ b/src/Numeric/AD/Mode.hs
@@ -40,7 +40,7 @@
   isKnownConstant :: t -> Bool
   isKnownConstant _ = False
 
-  -- | allowed to return False for zero, but we give more NaN's than strictly necessary then
+  -- | allowed to return False for zero, but we give more NaN's than strictly necessary
   isKnownZero :: t -> Bool
   isKnownZero _ = False
 
diff --git a/src/Numeric/AD/Mode/Kahn.hs b/src/Numeric/AD/Mode/Kahn.hs
--- a/src/Numeric/AD/Mode/Kahn.hs
+++ b/src/Numeric/AD/Mode/Kahn.hs
@@ -58,8 +58,12 @@
 
 -- | The 'grad' function calculates the gradient of a non-scalar-to-scalar function with kahn-mode AD in a single pass.
 --
+--
 -- >>> grad (\[x,y,z] -> x*y+z) [1,2,3]
 -- [2,1,1]
+--
+-- >>> grad (\[x,y] -> x**y) [0,2]
+-- [0.0,NaN]
 grad :: (Traversable f, Num a) => (forall s. f (AD s (Kahn a)) -> AD s (Kahn a)) -> f a -> f a
 grad f = Rank1.grad (runAD.f.fmap AD)
 {-# INLINE grad #-}
diff --git a/src/Numeric/AD/Mode/Reverse.hs b/src/Numeric/AD/Mode/Reverse.hs
--- a/src/Numeric/AD/Mode/Reverse.hs
+++ b/src/Numeric/AD/Mode/Reverse.hs
@@ -61,6 +61,9 @@
 --
 -- >>> grad (\[x,y,z] -> x*y+z) [1,2,3]
 -- [2,1,1]
+--
+-- >>> grad (\[x,y] -> x**y) [0,2]
+-- [0.0,NaN]
 grad :: (Traversable f, Num a) => (forall s. Reifies s Tape => f (Reverse s a) -> Reverse s a) -> f a -> f a
 grad f as = reifyTape (snd bds) $ \p -> unbind vs $! partialArrayOf p bds $! f vs where
   (vs, bds) = bind as
diff --git a/src/Numeric/AD/Mode/Sparse.hs b/src/Numeric/AD/Mode/Sparse.hs
--- a/src/Numeric/AD/Mode/Sparse.hs
+++ b/src/Numeric/AD/Mode/Sparse.hs
@@ -45,7 +45,14 @@
 import Numeric.AD.Internal.Type
 import Numeric.AD.Mode
 
-
+-- | The 'grad' function calculates the gradient of a non-scalar-to-scalar function with sparse-mode AD in a single pass.
+--
+--
+-- >>> grad (\[x,y,z] -> x*y+z) [1,2,3]
+-- [2,1,1]
+--
+-- >>> grad (\[x,y] -> x**y) [0,2]
+-- [0.0,NaN]
 grad :: (Traversable f, Num a) => (forall s. f (AD s (Sparse a)) -> AD s (Sparse a)) -> f a -> f a
 grad f = Rank1.grad (runAD.f.fmap AD)
 {-# INLINE grad #-}