New version of hash and ==

evaluation of Automorphism on the standard basis is expensive

in hash(g::Automorphism, h::UInt) we
1. compute and store savedhash (evaluating f, if necessary) with a fixed value HASHINGCONST
2. xor h with savedhash

in == we evaluate f only at hash-colision (or when
This way we evaluate f multiple times only in ~2% of cases (for SAut(F_4))

src/AutGroup.jl

@@ -218,20 +218,54 @@ end
 
 ###############################################################################
 #
-#   Basic manipulation
+#   Comparison
 #
 ###############################################################################
 
+const HASHINGCONST = 0x314a5215305f3ec1 # more or less random
+
 hash(s::AutSymbol, h::UInt) = hash(s.str, hash(s.pow, hash(:AutSymbol, h)))
 
 function hash(g::Automorphism, h::UInt)
-   if g.modified
-      g.savedhash = hash(g(domain(parent(g))), hash(typeof(g), hash(parent(g), h)))
-      g.modified = false
-   end
-   return g.savedhash
+    if g.modified
+        g.savedhash = hash(reduce!.(g(domain(parent(g)))),
+            hash(typeof(g), hash(parent(g), HASHINGCONST)))
+        g.modified = false
+        end
+    return xor(g.savedhash, h)
 end
 
+(==)(s::AutSymbol, t::AutSymbol) = s.str == t.str && s.pow == t.pow
+
+function (==)(g::Automorphism{N}, h::Automorphism{N}) where N
+    parent(g) == parent(h) || return false
+
+    if !g.modified && !h.modified
+        if g.savedhash != h.savedhash
+            return false
+        end
+    end
+
+    # expensive:
+    g_im = reduce!.(g(domain(parent(g))))
+    h_im = reduce!.(h(domain(parent(h))))
+    # cheap:
+    g.savedhash = hash(g_im,
+        hash(typeof(g), hash(parent(g), HASHINGCONST)))
+    g.modified = false
+    h.savedhash = hash(h_im,
+        hash(typeof(h), hash(parent(h), HASHINGCONST)))
+    h.modified = false
+
+    return g_im == h_im
+end
+
+###############################################################################
+#
+#   Basic manipulation
+#
+###############################################################################
+
 function change_pow(s::AutSymbol, n::Int)
     if n == 0
         return id_autsymbol()
@@ -266,19 +300,6 @@ function show(io::IO, G::AutGroup)
    print(io, "Generated by $(join(G.gens, ","))")
 end
 
-###############################################################################
-#
-#   Comparison
-#
-###############################################################################
-
-(==)(s::AutSymbol, t::AutSymbol) = s.str == t.str && s.pow == t.pow
-
-function (==)(g::Automorphism, h::Automorphism)
-   parent(g) == parent(h) || return false
-   return g(domain(parent(g))) == h(domain(parent(h)))
-end
-
 ###############################################################################
 #
 #   Binary operators