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authorBjörn Gustavsson <[email protected]>2014-12-08 12:33:12 +0100
committerBjörn Gustavsson <[email protected]>2015-01-09 13:18:44 +0100
commit7b10ff77235532923558a30759ed9b5fe6d994a5 (patch)
tree98bdf359c9df1f077335dfe16ca7c621cc5a3777
parent4f7edc376ee61238699f68c8721ab23ee56eafee (diff)
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beam_dead: Optimize branches from relational conditionals
The BEAM compiler translates code such as: is_hex_digit(D) when $0 =< D, D =< $9 -> true; is_hex_digit(D) when $a =< D, D =< $z -> true; is_hex_digit(D) when $A =< D, D =< $Z -> true; is_hex_digit(_) -> false. to something like this: L0: test is_ge L1 {x,0} 48 test is_ge L1 57 {x,0} move true {x,0} return. L1: test is_ge L2 {x,0} 97 test is_ge L2 122 {x,0} move true {x,0} return L2: test is_ge L3 {x,0} 65 test is_ge L3 90 {x,0} move true {x,0} return L3: move false {x,0} return We can see that tests will be repeated even if they cannot possibly succeed. For instance, if we pass in {x,0} equal to 32, the first test that {x,0} is greater than or equal to 48 at L0 will fail. The control will transfer to L1, where it will be tested whether {x,0} is greater than 97. That test will fail and control will pass to L2, where again the test will fail. The compiler can do better by short-circuiting repeating tests: L0: test is_ge L3 {x,0} 48 test is_ge L1 57 {x,0} move true {x,0} return. L1: test is_ge L2 {x,0} 97 test is_ge L3 122 {x,0} move true {x,0} return L2: test is_ge L3 {x,0} 65 test is_ge L3 90 {x,0} move true {x,0} return L3: move false {x,0} return
-rw-r--r--lib/compiler/src/beam_dead.erl349
-rw-r--r--lib/compiler/test/guard_SUITE.erl230
2 files changed, 563 insertions, 16 deletions
diff --git a/lib/compiler/src/beam_dead.erl b/lib/compiler/src/beam_dead.erl
index b15adfa889..9703b1a847 100644
--- a/lib/compiler/src/beam_dead.erl
+++ b/lib/compiler/src/beam_dead.erl
@@ -215,15 +215,13 @@ forward([{test,is_eq_exact,_,[Dst,Src]}=I,{move,Src,Dst}|Is], D, Lc, Acc) ->
forward([I|Is], D, Lc, Acc);
forward([{test,is_nil,_,[Dst]}=I,{move,nil,Dst}|Is], D, Lc, Acc) ->
forward([I|Is], D, Lc, Acc);
-forward([{test,is_eq_exact,_,_}=I|Is], D, Lc, Acc) ->
- case Is of
- [{label,_}|_] -> forward(Is, D, Lc, [I|Acc]);
- _ -> forward(Is, D, Lc+1, [{label,Lc},I|Acc])
- end;
-forward([{test,is_ne_exact,_,_}=I|Is], D, Lc, Acc) ->
- case Is of
- [{label,_}|_] -> forward(Is, D, Lc, [I|Acc]);
- _ -> forward(Is, D, Lc+1, [{label,Lc},I|Acc])
+forward([{test,_,_,_}=I|Is]=Is0, D, Lc, Acc) ->
+ %% Help the second, backward pass to by inserting labels after
+ %% relational operators so that they can be skipped if they are
+ %% known to be true.
+ case useful_to_insert_label(Is0) of
+ false -> forward(Is, D, Lc, [I|Acc]);
+ true -> forward(Is, D, Lc+1, [{label,Lc},I|Acc])
end;
forward([I|Is], D, Lc, Acc) ->
forward(Is, D, Lc, [I|Acc]);
@@ -239,6 +237,17 @@ update_value_dict([Lit,{f,Lbl}|T], Reg, D0) ->
update_value_dict(T, Reg, D);
update_value_dict([], _, D) -> D.
+useful_to_insert_label([_,{label,_}|_]) ->
+ false;
+useful_to_insert_label([{test,Op,_,_}|_]) ->
+ case Op of
+ is_lt -> true;
+ is_ge -> true;
+ is_eq_exact -> true;
+ is_ne_exact -> true;
+ _ -> false
+ end.
+
%%%
%%% Scan instructions in reverse execution order and remove dead code.
%%%
@@ -309,20 +318,22 @@ backward([{test,is_eq_exact,{f,To0},[Reg,{atom,Val}]=Ops}|Is], D, Acc) ->
backward([{test,Op,{f,To0},Ops0}|Is], D, Acc) ->
To1 = shortcut_bs_test(To0, Is, D),
To2 = shortcut_label(To1, D),
+ To3 = shortcut_rel_op(To2, Op, Ops0, D),
+
%% Try to shortcut a repeated test:
%%
%% test Op {f,Fail1} Operands test Op {f,Fail2} Operands
%% . . . ==> ...
%% Fail1: test Op {f,Fail2} Operands Fail1: test Op {f,Fail2} Operands
%%
- To = case beam_utils:code_at(To2, D) of
- [{test,Op,{f,To3},Ops}|_] ->
+ To = case beam_utils:code_at(To3, D) of
+ [{test,Op,{f,To4},Ops}|_] ->
case equal_ops(Ops0, Ops) of
- true -> To3;
- false -> To2
+ true -> To4;
+ false -> To3
end;
_Code ->
- To2
+ To3
end,
I = case Op of
is_eq_exact -> combine_eqs(To, Ops0, D, Acc);
@@ -562,3 +573,313 @@ shortcut_bs_start_match_2([{test,bs_start_match2,{f,To},_,[Reg|_],_}|_], Reg, _)
To;
shortcut_bs_start_match_2(_Is, _Reg, To) ->
To.
+
+%% shortcut_rel_op(FailLabel, Operator, [Operand], D) -> FailLabel'
+%% Try to shortcut the given test instruction. Example:
+%%
+%% is_ge L1 {x,0} 48
+%% .
+%% .
+%% .
+%% L1: is_ge L2 {x,0} 65
+%%
+%% The first test instruction can be rewritten to "is_ge L2 {x,0} 48"
+%% since the instruction at L1 will also fail.
+%%
+%% If there are instructions between L1 and the other test instruction
+%% it may still be possible to do the shortcut. For example:
+%%
+%% L1: is_eq_exact L3 {x,0} 92
+%% is_ge L2 {x,0} 65
+%%
+%% Since the first test instruction failed, we know that {x,0} must
+%% be less than 48; therefore, we know that {x,0} cannot be equal to
+%% 92 and the jump to L3 cannot happen.
+
+shortcut_rel_op(To, Op, Ops, D) ->
+ case normalize_op({test,Op,{f,To},Ops}) of
+ {{NormOp,A,B},_} ->
+ Normalized = {negate_op(NormOp),A,B},
+ shortcut_rel_op_fp(To, Normalized, D);
+ {_,_} ->
+ To;
+ error ->
+ To
+ end.
+
+shortcut_rel_op_fp(To0, Normalized, D) ->
+ Code = beam_utils:code_at(To0, D),
+ case shortcut_any_label(Code, Normalized) of
+ error ->
+ To0;
+ To ->
+ shortcut_rel_op_fp(To, Normalized, D)
+ end.
+
+%% shortcut_any_label([Instruction], PrevCondition) -> FailLabel | error
+%% Using PrevCondition (a previous condition known to be true),
+%% try to shortcut to another failure label.
+
+shortcut_any_label([{jump,{f,Lbl}}|_], _Prev) ->
+ Lbl;
+shortcut_any_label([{label,Lbl}|_], _Prev) ->
+ Lbl;
+shortcut_any_label([{select,select_val,R,{f,Fail},L}|_], Prev) ->
+ shortcut_selectval(L, R, Fail, Prev);
+shortcut_any_label([I|Is], Prev) ->
+ case normalize_op(I) of
+ error ->
+ error;
+ {Normalized,Fail} ->
+ %% We have a relational operator.
+ case will_succeed(Prev, Normalized) of
+ no ->
+ %% This test instruction will always branch
+ %% to Fail.
+ Fail;
+ yes ->
+ %% This test instruction will never branch,
+ %% so we will look at the next instruction.
+ shortcut_any_label(Is, Prev);
+ maybe ->
+ %% May or may not branch. From now on, we can only
+ %% shortcut to the this specific failure label
+ %% Fail.
+ shortcut_specific_label(Is, Fail, Prev)
+ end
+ end.
+
+%% shortcut_specific_label([Instruction], FailLabel, PrevCondition) ->
+%% FailLabel | error
+%% We have previously encountered a test instruction that may or
+%% may not branch to FailLabel. Therefore we are only allowed
+%% to do the shortcut to the same fail label (FailLabel).
+
+shortcut_specific_label([{label,_}|Is], Fail, Prev) ->
+ shortcut_specific_label(Is, Fail, Prev);
+shortcut_specific_label([{select,select_val,R,{f,F},L}|_], Fail, Prev) ->
+ case shortcut_selectval(L, R, F, Prev) of
+ Fail -> Fail;
+ _ -> error
+ end;
+shortcut_specific_label([I|Is], Fail, Prev) ->
+ case normalize_op(I) of
+ error ->
+ error;
+ {Normalized,Fail} ->
+ case will_succeed(Prev, Normalized) of
+ no ->
+ %% Will branch to FailLabel.
+ Fail;
+ yes ->
+ %% Will definitely never branch.
+ shortcut_specific_label(Is, Fail, Prev);
+ maybe ->
+ %% May branch, but still OK since it will branch
+ %% to FailLabel.
+ shortcut_specific_label(Is, Fail, Prev)
+ end;
+ {Normalized,_} ->
+ %% This test instruction will branch to a different
+ %% fail label, if it branches at all.
+ case will_succeed(Prev, Normalized) of
+ yes ->
+ %% Still OK, since the branch will never be
+ %% taken.
+ shortcut_specific_label(Is, Fail, Prev);
+ no ->
+ %% Give up. The branch will definitely be taken
+ %% to a different fail label.
+ error;
+ maybe ->
+ %% Give up. If the branch is taken, it will be
+ %% to a different fail label.
+ error
+ end
+ end.
+
+
+%% shortcut_selectval(List, Reg, Fail, PrevCond) -> FailLabel | error
+%% Try to shortcut a selectval instruction. A selectval instruction
+%% is equivalent to the following instruction sequence:
+%%
+%% is_ne_exact L1 Reg Value1
+%% .
+%% .
+%% .
+%% is_ne_exact LN Reg ValueN
+%% jump DefaultFailLabel
+%%
+shortcut_selectval([Val,{f,Lbl}|T], R, Fail, Prev) ->
+ case will_succeed(Prev, {'=/=',R,get_literal(Val)}) of
+ yes -> shortcut_selectval(T, R, Fail, Prev);
+ no -> Lbl;
+ maybe -> error
+ end;
+shortcut_selectval([], _, Fail, _) -> Fail.
+
+%% will_succeed(PrevCondition, Condition) -> yes | no | maybe
+%% PrevCondition is a condition known to be true. This function
+%% will tell whether Condition will succeed.
+
+will_succeed({Op1,Reg,A}, {Op2,Reg,B}) ->
+ will_succeed_1(Op1, A, Op2, B);
+will_succeed({'=:=',Reg,{literal,A}}, {TypeTest,Reg}) ->
+ case erlang:TypeTest(A) of
+ false -> no;
+ true -> yes
+ end;
+will_succeed({_,_,_}, maybe) ->
+ maybe;
+will_succeed({_,_,_}, Test) when is_tuple(Test) ->
+ maybe.
+
+will_succeed_1('=:=', A, '<', B) ->
+ if
+ B =< A -> no;
+ true -> yes
+ end;
+will_succeed_1('=:=', A, '=<', B) ->
+ if
+ B < A -> no;
+ true -> yes
+ end;
+will_succeed_1('=:=', A, '=:=', B) ->
+ if
+ A =:= B -> yes;
+ true -> no
+ end;
+will_succeed_1('=:=', A, '=/=', B) ->
+ if
+ A =:= B -> no;
+ true -> yes
+ end;
+will_succeed_1('=:=', A, '>=', B) ->
+ if
+ B > A -> no;
+ true -> yes
+ end;
+will_succeed_1('=:=', A, '>', B) ->
+ if
+ B >= A -> no;
+ true -> yes
+ end;
+
+will_succeed_1('=/=', A, '=/=', B) when A =:= B -> yes;
+will_succeed_1('=/=', A, '=:=', B) when A =:= B -> no;
+
+will_succeed_1('<', A, '=:=', B) when B >= A -> no;
+will_succeed_1('<', A, '=/=', B) when B >= A -> yes;
+will_succeed_1('<', A, '<', B) when B >= A -> yes;
+will_succeed_1('<', A, '=<', B) when B > A -> yes;
+will_succeed_1('<', A, '>=', B) when B > A -> no;
+will_succeed_1('<', A, '>', B) when B >= A -> no;
+
+will_succeed_1('=<', A, '=:=', B) when B > A -> no;
+will_succeed_1('=<', A, '=/=', B) when B > A -> yes;
+will_succeed_1('=<', A, '<', B) when B > A -> yes;
+will_succeed_1('=<', A, '=<', B) when B >= A -> yes;
+will_succeed_1('=<', A, '>=', B) when B > A -> no;
+will_succeed_1('=<', A, '>', B) when B >= A -> no;
+
+will_succeed_1('>=', A, '=:=', B) when B < A -> no;
+will_succeed_1('>=', A, '=/=', B) when B < A -> yes;
+will_succeed_1('>=', A, '<', B) when B =< A -> no;
+will_succeed_1('>=', A, '=<', B) when B < A -> no;
+will_succeed_1('>=', A, '>=', B) when B =< A -> yes;
+will_succeed_1('>=', A, '>', B) when B < A -> yes;
+
+will_succeed_1('>', A, '=:=', B) when B =< A -> no;
+will_succeed_1('>', A, '=/=', B) when B =< A -> yes;
+will_succeed_1('>', A, '<', B) when B =< A -> no;
+will_succeed_1('>', A, '=<', B) when B < A -> no;
+will_succeed_1('>', A, '>=', B) when B =< A -> yes;
+will_succeed_1('>', A, '>', B) when B < A -> yes;
+
+will_succeed_1(_, _, _, _) -> maybe.
+
+%% normalize_op(Instruction) -> {Normalized,FailLabel} | error
+%% Normalized = {Operator,Register,Literal} |
+%% {TypeTest,Register} |
+%% maybe
+%% Operation = '<' | '=<' | '=:=' | '=/=' | '>=' | '>'
+%% TypeTest = is_atom | is_integer ...
+%% Literal = {literal,Term}
+%%
+%% Normalize a relational operator to facilitate further
+%% comparisons between operators. Always make the register
+%% operand the first operand. Thus the following instruction:
+%%
+%% {test,is_ge,{f,99},{integer,13},{x,0}}
+%%
+%% will be normalized to:
+%%
+%% {'=<',{x,0},{literal,13}}
+%%
+%% NOTE: Bit syntax test instructions are scary. They may change the
+%% state of match contexts and update registers, so we don't dare
+%% mess with them.
+
+normalize_op({test,is_ge,{f,Fail},Ops}) ->
+ normalize_op_1('>=', Ops, Fail);
+normalize_op({test,is_lt,{f,Fail},Ops}) ->
+ normalize_op_1('<', Ops, Fail);
+normalize_op({test,is_eq_exact,{f,Fail},Ops}) ->
+ normalize_op_1('=:=', Ops, Fail);
+normalize_op({test,is_ne_exact,{f,Fail},Ops}) ->
+ normalize_op_1('=/=', Ops, Fail);
+normalize_op({test,is_nil,{f,Fail},[R]}) ->
+ normalize_op_1('=:=', [R,nil], Fail);
+normalize_op({test,Op,{f,Fail},[R]}) ->
+ case erl_internal:new_type_test(Op, 1) of
+ true -> {{Op,R},Fail};
+ false -> {maybe,Fail}
+ end;
+normalize_op({test,_,{f,Fail},_}=I) ->
+ case beam_utils:is_pure_test(I) of
+ true -> {maybe,Fail};
+ false -> error
+ end;
+normalize_op(_) ->
+ error.
+
+normalize_op_1(Op, [Op1,Op2], Fail) ->
+ case {get_literal(Op1),get_literal(Op2)} of
+ {error,error} ->
+ %% Both operands are registers.
+ {maybe,Fail};
+ {error,Lit} ->
+ {{Op,Op1,Lit},Fail};
+ {Lit,error} ->
+ {{turn_op(Op),Op2,Lit},Fail};
+ {_,_} ->
+ %% Both operands are literals. Can probably only
+ %% happen if the Core Erlang optimizations passes were
+ %% turned off, so don't bother trying to do something
+ %% smart here.
+ {maybe,Fail}
+ end.
+
+turn_op('<') -> '>';
+turn_op('>=') -> '=<';
+turn_op('=:='=Op) -> Op;
+turn_op('=/='=Op) -> Op.
+
+negate_op('>=') -> '<';
+negate_op('<') -> '>=';
+negate_op('=<') -> '>';
+negate_op('>') -> '=<';
+negate_op('=:=') -> '=/=';
+negate_op('=/=') -> '=:='.
+
+get_literal({atom,Val}) ->
+ {literal,Val};
+get_literal({integer,Val}) ->
+ {literal,Val};
+get_literal({float,Val}) ->
+ {literal,Val};
+get_literal(nil) ->
+ {literal,[]};
+get_literal({literal,_}=Lit) ->
+ Lit;
+get_literal({_,_}) -> error.
diff --git a/lib/compiler/test/guard_SUITE.erl b/lib/compiler/test/guard_SUITE.erl
index eb205d09a7..689c65f537 100644
--- a/lib/compiler/test/guard_SUITE.erl
+++ b/lib/compiler/test/guard_SUITE.erl
@@ -30,7 +30,7 @@
old_guard_tests/1,
build_in_guard/1,gbif/1,
t_is_boolean/1,is_function_2/1,
- tricky/1,rel_ops/1,literal_type_tests/1,
+ tricky/1,rel_ops/1,rel_op_combinations/1,literal_type_tests/1,
basic_andalso_orelse/1,traverse_dcd/1,
check_qlc_hrl/1,andalso_semi/1,t_tuple_size/1,binary_part/1,
bad_constants/1,bad_guards/1]).
@@ -47,7 +47,8 @@ groups() ->
semicolon,complex_semicolon,comma,or_guard,
more_or_guards,complex_or_guards,and_guard,xor_guard,
more_xor_guards,build_in_guard,old_guard_tests,gbif,
- t_is_boolean,is_function_2,tricky,rel_ops,
+ t_is_boolean,is_function_2,tricky,
+ rel_ops,rel_op_combinations,
literal_type_tests,basic_andalso_orelse,traverse_dcd,
check_qlc_hrl,andalso_semi,t_tuple_size,binary_part,
bad_constants,bad_guards]}].
@@ -1122,6 +1123,231 @@ rel_ops(Config) when is_list(Config) ->
-undef(TestOp).
+rel_op_combinations(Config) when is_list(Config) ->
+ Digits0 = lists:seq(16#0030, 16#0039) ++
+ lists:seq(16#0660, 16#0669) ++
+ lists:seq(16#06F0, 16#06F9),
+ Digits = gb_sets:from_list(Digits0),
+ rel_op_combinations_1(16#0700, Digits),
+
+ BrokenRange0 = lists:seq(3, 5) ++
+ lists:seq(10, 12) ++ lists:seq(14, 20),
+ BrokenRange = gb_sets:from_list(BrokenRange0),
+ rel_op_combinations_2(30, BrokenRange),
+
+ Red0 = [{I,2*I} || I <- lists:seq(0, 50)] ++
+ [{I,5*I} || I <- lists:seq(51, 80)],
+ Red = gb_trees:from_orddict(Red0),
+ rel_op_combinations_3(100, Red).
+
+rel_op_combinations_1(0, _) ->
+ ok;
+rel_op_combinations_1(N, Digits) ->
+ Bool = gb_sets:is_member(N, Digits),
+ Bool = is_digit_1(N),
+ Bool = is_digit_2(N),
+ Bool = is_digit_3(N),
+ Bool = is_digit_4(N),
+ Bool = is_digit_5(N),
+ Bool = is_digit_6(N),
+ Bool = is_digit_7(N),
+ Bool = is_digit_8(N),
+ rel_op_combinations_1(N-1, Digits).
+
+is_digit_1(X) when 16#0660 =< X, X =< 16#0669 -> true;
+is_digit_1(X) when 16#0030 =< X, X =< 16#0039 -> true;
+is_digit_1(X) when 16#06F0 =< X, X =< 16#06F9 -> true;
+is_digit_1(_) -> false.
+
+is_digit_2(X) when (16#0030-1) < X, X =< 16#0039 -> true;
+is_digit_2(X) when (16#0660-1) < X, X =< 16#0669 -> true;
+is_digit_2(X) when (16#06F0-1) < X, X =< 16#06F9 -> true;
+is_digit_2(_) -> false.
+
+is_digit_3(X) when 16#0660 =< X, X < (16#0669+1) -> true;
+is_digit_3(X) when 16#0030 =< X, X < (16#0039+1) -> true;
+is_digit_3(X) when 16#06F0 =< X, X < (16#06F9+1) -> true;
+is_digit_3(_) -> false.
+
+is_digit_4(X) when (16#0660-1) < X, X < (16#0669+1) -> true;
+is_digit_4(X) when (16#0030-1) < X, X < (16#0039+1) -> true;
+is_digit_4(X) when (16#06F0-1) < X, X < (16#06F9+1) -> true;
+is_digit_4(_) -> false.
+
+is_digit_5(X) when X >= 16#0660, X =< 16#0669 -> true;
+is_digit_5(X) when X >= 16#0030, X =< 16#0039 -> true;
+is_digit_5(X) when X >= 16#06F0, X =< 16#06F9 -> true;
+is_digit_5(_) -> false.
+
+is_digit_6(X) when X > (16#0660-1), X =< 16#0669 -> true;
+is_digit_6(X) when X > (16#0030-1), X =< 16#0039 -> true;
+is_digit_6(X) when X > (16#06F0-1), X =< 16#06F9 -> true;
+is_digit_6(_) -> false.
+
+is_digit_7(X) when 16#0660 =< X, X =< 16#0669 -> true;
+is_digit_7(X) when 16#0030 =< X, X =< 16#003A, X =/= 16#003A -> true;
+is_digit_7(X) when 16#06F0 =< X, X =< 16#06F9 -> true;
+is_digit_7(_) -> false.
+
+is_digit_8(X) when X =< 16#0039, X > (16#0030-1) -> true;
+is_digit_8(X) when X =< 16#06F9, X > (16#06F0-1) -> true;
+is_digit_8(X) when X =< 16#0669, X > (16#0660-1) -> true;
+is_digit_8(16#0670) -> false;
+is_digit_8(_) -> false.
+
+rel_op_combinations_2(0, _) ->
+ ok;
+rel_op_combinations_2(N, Range) ->
+ Bool = gb_sets:is_member(N, Range),
+ Bool = broken_range_1(N),
+ Bool = broken_range_2(N),
+ Bool = broken_range_3(N),
+ Bool = broken_range_4(N),
+ Bool = broken_range_5(N),
+ Bool = broken_range_6(N),
+ Bool = broken_range_7(N),
+ Bool = broken_range_8(N),
+ Bool = broken_range_9(N),
+ Bool = broken_range_10(N),
+ Bool = broken_range_11(N),
+ Bool = broken_range_12(N),
+ Bool = broken_range_13(N),
+ rel_op_combinations_2(N-1, Range).
+
+broken_range_1(X) when X >= 10, X =< 20, X =/= 13 -> true;
+broken_range_1(X) when X >= 3, X =< 5 -> true;
+broken_range_1(_) -> false.
+
+broken_range_2(X) when X >= 10, X =< 12 -> true;
+broken_range_2(X) when X >= 14, X =< 20 -> true;
+broken_range_2(X) when X >= 3, X =< 5 -> true;
+broken_range_2(_) -> false.
+
+broken_range_3(X) when X >= 10, X =< 12 -> true;
+broken_range_3(X) when X >= 14, X < 21 -> true;
+broken_range_3(3) -> true;
+broken_range_3(4) -> true;
+broken_range_3(5) -> true;
+broken_range_3(_) -> false.
+
+broken_range_4(X) when X =< 5, X >= 3 -> true;
+broken_range_4(X) when X >= 10, X =< 20, X =/= 13 -> true;
+broken_range_4(X) when X =< 100 -> false;
+broken_range_4(_) -> false.
+
+broken_range_5(X) when X >= 10, X =< 20, X =/= 13 -> true;
+broken_range_5(X) when X > 2, X =< 5 -> true;
+broken_range_5(_) -> false.
+
+broken_range_6(X) when X >= 10, X =< 20, X =/= 13 -> true;
+broken_range_6(X) when X > 2, X < 6 -> true;
+broken_range_6(_) -> false.
+
+broken_range_7(X) when X > 2, X < 6 -> true;
+broken_range_7(X) when X >= 10, X =< 20, X =/= 13 -> true;
+broken_range_7(X) when X > 30 -> false;
+broken_range_7(_) -> false.
+
+broken_range_8(X) when X >= 10, X =< 20, X =/= 13 -> true;
+broken_range_8(X) when X =:= 3 -> true;
+broken_range_8(X) when X >= 3, X =< 5 -> true;
+broken_range_8(_) -> false.
+
+broken_range_9(X) when X >= 10, X =< 20, X =/= 13 -> true;
+broken_range_9(X) when X =:= 13 -> false;
+broken_range_9(X) when X >= 3, X =< 5 -> true;
+broken_range_9(_) -> false.
+
+broken_range_10(X) when X >= 3, X =< 5 -> true;
+broken_range_10(X) when X >= 10, X =< 20, X =/= 13 -> true;
+broken_range_10(X) when X =/= 13 -> false;
+broken_range_10(_) -> false.
+
+broken_range_11(X) when X >= 10, X =< 20, X =/= 13 -> true;
+broken_range_11(X) when is_tuple(X), X =:= 10 -> true;
+broken_range_11(X) when X >= 3, X =< 5 -> true;
+broken_range_11(_) -> false.
+
+broken_range_12(X) when X >= 3, X =< 5 -> true;
+broken_range_12(X) when X >= 10, X =< 20, X =/= 13 -> true;
+broken_range_12(X) when X < 30, X > 20 -> false;
+broken_range_12(_) -> false.
+
+broken_range_13(X) when X >= 10, X =< 20, 13 =/= X -> true;
+broken_range_13(X) when X >= 3, X =< 5 -> true;
+broken_range_13(_) -> false.
+
+rel_op_combinations_3(0, _) ->
+ ok;
+rel_op_combinations_3(N, Red) ->
+ Val = case gb_trees:lookup(N, Red) of
+ none -> none;
+ {value,V} -> V
+ end,
+ Val = redundant_1(N),
+ Val = redundant_2(N),
+ Val = redundant_3(N),
+ Val = redundant_4(N),
+ Val = redundant_5(N),
+ Val = redundant_6(N),
+ Val = redundant_7(N),
+ Val = redundant_8(N),
+ Val = redundant_9(N),
+ Val = redundant_10(N),
+ Val = redundant_11(N),
+ rel_op_combinations_3(N-1, Red).
+
+redundant_1(X) when X >= 51, X =< 80 -> 5*X;
+redundant_1(X) when X < 51 -> 2*X;
+redundant_1(_) -> none.
+
+redundant_2(X) when X < 51 -> 2*X;
+redundant_2(X) when X >= 51, X =< 80 -> 5*X;
+redundant_2(_) -> none.
+
+redundant_3(X) when X < 51 -> 2*X;
+redundant_3(X) when X =< 80, X >= 51 -> 5*X;
+redundant_3(X) when X =/= 100 -> none;
+redundant_3(_) -> none.
+
+redundant_4(X) when X < 51 -> 2*X;
+redundant_4(X) when X =< 80, X > 50 -> 5*X;
+redundant_4(X) when X =/= 100 -> none;
+redundant_4(_) -> none.
+
+redundant_5(X) when X < 51 -> 2*X;
+redundant_5(X) when X > 50, X < 81 -> 5*X;
+redundant_5(X) when X =< 10 -> none;
+redundant_5(_) -> none.
+
+redundant_6(X) when X > 50, X =< 80 -> 5*X;
+redundant_6(X) when X < 51 -> 2*X;
+redundant_6(_) -> none.
+
+redundant_7(X) when is_integer(X), X >= 51, X =< 80 -> 5*X;
+redundant_7(X) when is_integer(X), X < 51 -> 2*X;
+redundant_7(_) -> none.
+
+redundant_8(X) when X >= 51, X =< 80 -> 5*X;
+redundant_8(X) when X < 51 -> 2*X;
+redundant_8(_) -> none.
+
+redundant_9(X) when X >= 51, X =< 80 -> 5*X;
+redundant_9(X) when X < 51 -> 2*X;
+redundant_9(90) -> none;
+redundant_9(X) when X =/= 90 -> none;
+redundant_9(_) -> none.
+
+redundant_10(X) when X >= 51, X =< 80 -> 5*X;
+redundant_10(X) when X < 51 -> 2*X;
+redundant_10(90) -> none;
+redundant_10(X) when X =:= 90 -> none;
+redundant_10(_) -> none.
+
+redundant_11(X) when X < 51 -> 2*X;
+redundant_11(X) when X =:= 10 -> 2*X;
+redundant_11(X) when X >= 51, X =< 80 -> 5*X;
+redundant_11(_) -> none.
%% Test type tests on literal values. (From emulator test suites.)
literal_type_tests(Config) when is_list(Config) ->