diff --git a/Data/Array/Accelerate/Trafo/Simplify.hs b/Data/Array/Accelerate/Trafo/Simplify.hs
index 617edf3ff..0022703c7 100644
--- a/Data/Array/Accelerate/Trafo/Simplify.hs
+++ b/Data/Array/Accelerate/Trafo/Simplify.hs
@@ -1,4 +1,3 @@
-{-# LANGUAGE BangPatterns         #-}
 {-# LANGUAGE FlexibleContexts     #-}
 {-# LANGUAGE FlexibleInstances    #-}
 {-# LANGUAGE GADTs                #-}
@@ -28,6 +27,7 @@ import Data.List                                        ( nubBy )
 import Data.Maybe
 import Data.Monoid
 import Data.Typeable
+import Text.Printf
 import Control.Applicative                              hiding ( Const )
 import Prelude                                          hiding ( exp, iterate )
 
@@ -301,15 +301,20 @@ simplifyOpenExp env = first getAny . cvtE
     -- If we are projecting elements from a tuple structure or tuple of constant
     -- valued tuple, pick out the appropriate component directly.
     --
+    -- Follow variable bindings, but only if they result in a simplification.
+    --
     prj :: forall s t. (Elt s, Elt t, IsTuple t)
         => TupleIdx (TupleRepr t) s
         -> (Any, PreOpenExp acc env aenv t)
         -> (Any, PreOpenExp acc env aenv s)
     prj ix exp@(_,exp')
-      | Tuple t <- exp' = Stats.inline "prj/Tuple" . yes $ prjT ix t
-      | Const c <- exp' = Stats.inline "prj/Const" . yes $ prjC ix (fromTuple (toElt c :: t))
-      | Let a b <- exp' = Stats.ruleFired "prj/Let"      $ cvtE (Let a (Prj ix b))
-      | otherwise       = Prj ix <$> exp
+      | Tuple t <- exp'                  = Stats.inline "prj/Tuple" . yes $ prjT ix t
+      | Const c <- exp'                  = Stats.inline "prj/Const" . yes $ prjC ix (fromTuple (toElt c :: t))
+      | Var v         <- exp'
+      , e             <- prjExp v env
+      , Nothing       <- match exp' e
+      , (Any True, c) <- prj ix (pure e) = Stats.inline "prj/Var"   . yes $ c
+      | otherwise                        = Prj ix <$> exp
       where
         prjT :: TupleIdx tup s -> Tuple (PreOpenExp acc env aenv) tup -> PreOpenExp acc env aenv s
         prjT ZeroTupIdx       (SnocTup _ e) = e
@@ -421,34 +426,35 @@ iterate
     -> (f a -> (Bool, f a))
     -> f a
     -> f a
-iterate ppr f = fix 0 . setup . simplify'
+iterate ppr f = fix 1 . setup
   where
     -- The maximum number of simplifier iterations. To be conservative and avoid
     -- excessive run times, we set this value very low.
     --
-    lIMIT       = 1
+    lIMIT       = 5
 
     simplify'   = Stats.simplifierDone . f
-    setup (_,x) = msg x x
+    setup x     = Stats.trace Stats.dump_simpl_iterations (printf "simplifier begin:\n%s\n" (ppr x))
+                $ snd (trace 0 "simplify" (simplify' x))
 
     fix :: Int -> f a -> f a
-    fix !i !x0
-      | i >= lIMIT      = $internalWarning "iterate" "iteration limit reached" (x0 ==^ f x0) x0
+    fix i x0
+      | i > lIMIT       = $internalWarning "iterate" "iteration limit reached" (x0 ==^ f x0) x0
       | not shrunk      = x1
       | not simplified  = x2
       | otherwise       = fix (i+1) x2
       where
-        (shrunk,     x1) = trace $ shrink' x0
-        (simplified, x2) = trace $ simplify' x1
+        (shrunk,     x1) = trace i "shrink"   $ shrink' x0
+        (simplified, x2) = trace i "simplify" $ simplify' x1
 
     -- debugging support
     --
     u ==^ (_,v)         = isJust (match u v)
 
-    trace v@(changed,x)
-      | changed         = msg x v
+    trace i s v@(changed,x)
+      | changed         = Stats.trace Stats.dump_simpl_iterations (msg i s x) v
       | otherwise       = v
 
-    msg :: f a -> x -> x
-    msg x next          = Stats.trace Stats.dump_simpl_iterations (unlines [ "simplifier done", ppr x ]) next
+    msg :: Int -> String -> f a -> String
+    msg i s x = printf "%s [%d/%d]:\n%s\n" s i lIMIT (ppr x)