path: root/lib/compiler/src/beam_utils.erl

%%
%% %CopyrightBegin%
%%
%% Copyright Ericsson AB 2007-2018. All Rights Reserved.
%%
%% Licensed under the Apache License, Version 2.0 (the "License");
%% you may not use this file except in compliance with the License.
%% You may obtain a copy of the License at
%%
%%     http://www.apache.org/licenses/LICENSE-2.0
%%
%% Unless required by applicable law or agreed to in writing, software
%% distributed under the License is distributed on an "AS IS" BASIS,
%% WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
%% See the License for the specific language governing permissions and
%% limitations under the License.
%%
%% %CopyrightEnd%
%%
%% Purpose : Common utilities used by several optimization passes.
%% 

-module(beam_utils).
-export([is_killed_block/2,is_killed/3,is_killed_at/3,
	 is_not_used/3,usage/3,
	 empty_label_index/0,index_label/3,index_labels/1,replace_labels/4,
	 code_at/2,bif_to_test/3,is_pure_test/1,
	 live_opt/1,delete_annos/1,combine_heap_needs/2,
         anno_defs/1,
	 split_even/1
        ]).

-export_type([code_index/0,module_code/0,instruction/0]).

-import(lists, [flatmap/2,map/2,member/2,sort/1,reverse/1,splitwith/2]).

-define(is_const(Val), (Val =:= nil orelse
                        element(1, Val) =:= integer orelse
                        element(1, Val) =:= float orelse
                        element(1, Val) =:= atom orelse
                        element(1, Val) =:= literal)).

%% instruction() describes all instructions that are used during optimzation
%% (from beam_a to beam_z).
-type instruction() :: atom() | tuple().

-type code_index() :: gb_trees:tree(beam_asm:label(), [instruction()]).

-type int_function() :: {'function',beam_asm:function_name(),arity(),
                         beam_asm:label(),[instruction()]}.

-type module_code() ::
        {module(),[_],[_],[int_function()],pos_integer()}.

%% Internal types.
-type fail() :: beam_asm:fail() | 'fail'.
-type test() :: {'test',atom(),fail(),[beam_asm:src()]} |
		{'test',atom(),fail(),integer(),list(),beam_asm:reg()}.
-type result_cache() :: gb_trees:tree(beam_asm:label(), 'killed' | 'used').

-record(live,
	{lbl :: code_index(),            %Label to code index.
	 res :: result_cache()}).        %Result cache for each label.

%% usage(Register, [Instruction], State) -> killed|not_used|used.
%%  Determine the usage of Register in the instruction sequence.
%%  The return value is one of:
%%
%%  killed   - The register is not used in any way.
%%  not_used - The register is referenced only by an allocating instruction
%%             (the actual value does not matter).
%%  used     - The register is used (its value do matter).

-spec usage(beam_asm:reg(), [instruction()], code_index()) ->
  'killed' | 'not_used' | 'used'.

usage(R, Is, D) ->
    St = #live{lbl=D,res=gb_trees:empty()},
    {Usage,_} = check_liveness(R, Is, St),
    Usage.


%% is_killed_block(Register, [Instruction]) -> true|false
%%  Determine whether a register is killed by the instruction sequence inside
%%  a block.
%%
%%  If true is returned, it means that the register will not be
%%  referenced in ANY way (not even indirectly by an allocate instruction);
%%  i.e. it is OK to enter the instruction sequence with Register
%%  containing garbage.

-spec is_killed_block(beam_asm:reg(), [instruction()]) -> boolean().

is_killed_block({x,X}, [{set,_,_,{alloc,Live,_}}|_]) ->
    X >= Live;
is_killed_block(R, [{set,Ds,Ss,_Op}|Is]) ->
    not member(R, Ss) andalso (member(R, Ds) orelse is_killed_block(R, Is));
is_killed_block(R, [{'%anno',{used,Regs}}|Is]) ->
    case R of
	{x,X} when (Regs bsr X) band 1 =:= 0 -> true;
	_ -> is_killed_block(R, Is)
    end;
is_killed_block(_, []) -> false.

%% is_killed(Register, [Instruction], State) -> true|false
%%  Determine whether a register is killed by the instruction sequence.
%%  If true is returned, it means that the register will not be
%%  referenced in ANY way (not even indirectly by an allocate instruction);
%%  i.e. it is OK to enter the instruction sequence with Register
%%  containing garbage.
%%
%%  The state (constructed by index_instructions/1) is used to allow us
%%  to determine the kill state across branches.

-spec is_killed(beam_asm:reg(), [instruction()], code_index()) -> boolean().

is_killed(R, Is, D) ->
    St = #live{lbl=D,res=gb_trees:empty()},
    case check_liveness(R, Is, St) of
	{killed,_} -> true;
	{exit_not_used,_} -> false;
	{_,_} -> false
    end.

%% is_killed_at(Reg, Lbl, State) -> true|false
%%  Determine whether Reg is killed at label Lbl.

-spec is_killed_at(beam_asm:reg(), beam_asm:label(), code_index()) -> boolean().

is_killed_at(R, Lbl, D) when is_integer(Lbl) ->
    St0 = #live{lbl=D,res=gb_trees:empty()},
    case check_liveness_at(R, Lbl, St0) of
	{killed,_} -> true;
	{exit_not_used,_} -> false;
	{_,_} -> false
    end.

%% is_not_used(Register, [Instruction], State) -> true|false
%%  Determine whether a register is never used in the instruction sequence
%%  (it could still be referenced by an allocate instruction, meaning that
%%  it MUST be initialized, but that its value does not matter).
%%    The state is used to allow us to determine the usage state
%%  across branches.

-spec is_not_used(beam_asm:reg(), [instruction()], code_index()) -> boolean().

is_not_used(R, Is, D) ->
    St = #live{lbl=D,res=gb_trees:empty()},
    case check_liveness(R, Is, St) of
	{used,_} -> false;
	{exit_not_used,_} -> true;
	{_,_} -> true
    end.

%% index_labels(FunctionIs) -> State
%%  Index the instruction sequence so that we can quickly
%%  look up the instruction following a specific label.

-spec index_labels([instruction()]) -> code_index().

index_labels(Is) ->
    index_labels_1(Is, []).

%% empty_label_index() -> State
%%  Create an empty label index.

-spec empty_label_index() -> code_index().

empty_label_index() ->
    gb_trees:empty().

%% index_label(Label, [Instruction], State) -> State
%%  Add an index for a label.

-spec index_label(beam_asm:label(), [instruction()], code_index()) ->
   code_index().

index_label(Lbl, Is0, Acc) ->
    Is = drop_labels(Is0),
    gb_trees:enter(Lbl, Is, Acc).


%% code_at(Label, State) -> [I].
%%  Retrieve the code at the given label.

-spec code_at(beam_asm:label(), code_index()) -> [instruction()].

code_at(L, Ll) ->
    gb_trees:get(L, Ll).

%% replace_labels(FunctionIs, Tail, ReplaceDb, Fallback) -> FunctionIs.
%%  Replace all labels in instructions according to the ReplaceDb.
%%  If label is not found the Fallback is called with the label to
%%  produce a new one.

-spec replace_labels([instruction()],
                     [instruction()],
                     #{beam_asm:label() => beam_asm:label()},
                     fun((beam_asm:label()) -> term())) -> [instruction()].
replace_labels(Is, Acc, D, Fb) ->
    replace_labels_1(Is, Acc, D, Fb).

%% bif_to_test(Bif, [Op], Fail) -> {test,Test,Fail,[Op]}
%%  Convert a BIF to a test. Fail if not possible.

-spec bif_to_test(atom(), list(), fail()) -> test().

bif_to_test(is_atom,     [_]=Ops, Fail) -> {test,is_atom,Fail,Ops};
bif_to_test(is_boolean,  [_]=Ops, Fail) -> {test,is_boolean,Fail,Ops};
bif_to_test(is_binary,   [_]=Ops, Fail) -> {test,is_binary,Fail,Ops};
bif_to_test(is_bitstring,[_]=Ops, Fail) -> {test,is_bitstr,Fail,Ops};
bif_to_test(is_float,    [_]=Ops, Fail) -> {test,is_float,Fail,Ops};
bif_to_test(is_function, [_]=Ops, Fail) -> {test,is_function,Fail,Ops};
bif_to_test(is_function, [_,_]=Ops, Fail) -> {test,is_function2,Fail,Ops};
bif_to_test(is_integer,  [_]=Ops, Fail) -> {test,is_integer,Fail,Ops};
bif_to_test(is_list,     [_]=Ops, Fail) -> {test,is_list,Fail,Ops};
bif_to_test(is_map,      [_]=Ops, Fail) -> {test,is_map,Fail,Ops};
bif_to_test(is_number,   [_]=Ops, Fail) -> {test,is_number,Fail,Ops};
bif_to_test(is_pid,      [_]=Ops, Fail) -> {test,is_pid,Fail,Ops};
bif_to_test(is_port,     [_]=Ops, Fail) -> {test,is_port,Fail,Ops};
bif_to_test(is_reference, [_]=Ops, Fail) -> {test,is_reference,Fail,Ops};
bif_to_test(is_tuple,    [_]=Ops, Fail)     -> {test,is_tuple,Fail,Ops};
bif_to_test('=<', [A,B], Fail) -> {test,is_ge,Fail,[B,A]};
bif_to_test('>', [A,B], Fail) -> {test,is_lt,Fail,[B,A]};
bif_to_test('<', [_,_]=Ops, Fail) -> {test,is_lt,Fail,Ops};
bif_to_test('>=', [_,_]=Ops, Fail) -> {test,is_ge,Fail,Ops};
bif_to_test('==', [A,nil], Fail) -> {test,is_nil,Fail,[A]};
bif_to_test('==', [nil,A], Fail) -> {test,is_nil,Fail,[A]};
bif_to_test('==', [C,A], Fail) when ?is_const(C) ->
    {test,is_eq,Fail,[A,C]};
bif_to_test('==', [_,_]=Ops, Fail) -> {test,is_eq,Fail,Ops};
bif_to_test('/=', [C,A], Fail) when ?is_const(C) ->
    {test,is_ne,Fail,[A,C]};
bif_to_test('/=', [_,_]=Ops, Fail) -> {test,is_ne,Fail,Ops};
bif_to_test('=:=', [A,nil], Fail) -> {test,is_nil,Fail,[A]};
bif_to_test('=:=', [nil,A], Fail) -> {test,is_nil,Fail,[A]};
bif_to_test('=:=', [C,A], Fail) when ?is_const(C) ->
    {test,is_eq_exact,Fail,[A,C]};
bif_to_test('=:=', [_,_]=Ops, Fail) -> {test,is_eq_exact,Fail,Ops};
bif_to_test('=/=', [C,A], Fail) when ?is_const(C) ->
    {test,is_ne_exact,Fail,[A,C]};
bif_to_test('=/=', [_,_]=Ops, Fail) -> {test,is_ne_exact,Fail,Ops};
bif_to_test(is_record, [_,_,_]=Ops, Fail) -> {test,is_record,Fail,Ops}.


%% is_pure_test({test,Op,Fail,Ops}) -> true|false.
%%  Return 'true' if the test instruction does not modify any
%%  registers and/or bit syntax matching state.
%%

-spec is_pure_test(test()) -> boolean().

is_pure_test({test,is_eq,_,[_,_]}) -> true;
is_pure_test({test,is_ne,_,[_,_]}) -> true;
is_pure_test({test,is_eq_exact,_,[_,_]}) -> true;
is_pure_test({test,is_ne_exact,_,[_,_]}) -> true;
is_pure_test({test,is_ge,_,[_,_]}) -> true;
is_pure_test({test,is_lt,_,[_,_]}) -> true;
is_pure_test({test,is_nil,_,[_]}) -> true;
is_pure_test({test,is_nonempty_list,_,[_]}) -> true;
is_pure_test({test,test_arity,_,[_,_]}) -> true;
is_pure_test({test,has_map_fields,_,[_|_]}) -> true;
is_pure_test({test,is_bitstr,_,[_]}) -> true;
is_pure_test({test,is_function2,_,[_,_]}) -> true;
is_pure_test({test,Op,_,Ops}) -> 
    erl_internal:new_type_test(Op, length(Ops)).

    
%% live_opt([Instruction]) -> [Instruction].
%%  Go through the instruction sequence in reverse execution
%%  order, keep track of liveness and remove 'move' instructions
%%  whose destination is a register that will not be used.
%%  Also insert {used,Regs} annotations at the beginning
%%  and end of each block.

-spec live_opt([instruction()]) -> [instruction()].

live_opt(Is0) ->
    {[{label,Fail}|_]=Bef,[Fi|Is]} =
	splitwith(fun({func_info,_,_,_}) -> false;
		     (_) -> true
		  end, Is0),
    {func_info,_,_,Live} = Fi,
    D = gb_trees:insert(Fail, live_call(Live), gb_trees:empty()),
    Bef ++ [Fi|live_opt(reverse(Is), 0, D, [])].


%% delete_annos([Instruction]) -> [Instruction].
%%  Delete all annotations.

-spec delete_annos([instruction()]) -> [instruction()].

delete_annos([{block,Bl0}|Is]) ->
    case delete_annos(Bl0) of
	[] -> delete_annos(Is);
	[_|_]=Bl -> [{block,Bl}|delete_annos(Is)]
    end;
delete_annos([{'%anno',_}|Is]) ->
    delete_annos(Is);
delete_annos([I|Is]) ->
    [I|delete_annos(Is)];
delete_annos([]) -> [].

%% combine_heap_needs(HeapNeed1, HeapNeed2) -> HeapNeed
%%  Combine the heap need for two allocation instructions.

-type heap_need_tag() :: 'floats' | 'words'.
-type heap_need() :: non_neg_integer() |
                     {'alloc',[{heap_need_tag(),non_neg_integer()}]}.
-spec combine_heap_needs(heap_need(), heap_need()) -> heap_need().

combine_heap_needs(H1, H2) when is_integer(H1), is_integer(H2) ->
    H1 + H2;
combine_heap_needs(H1, H2) ->
    {alloc,combine_alloc_lists([H1,H2])}.


%% anno_defs(Instructions) -> Instructions'
%%  Add {def,RegisterBitmap} annotations to the beginning of
%%  each block.  Iff bit X is set in the the bitmap, it means
%%  that {x,X} is defined when the block is entered.

-spec anno_defs([instruction()]) -> [instruction()].

anno_defs(Is0) ->
    {Bef,[Fi|Is1]} =
	splitwith(fun({func_info,_,_,_}) -> false;
		     (_) -> true
		  end, Is0),
    {func_info,_,_,Arity} = Fi,
    Regs = init_def_regs(Arity),
    Is = defs(Is1, Regs, #{}),
    Bef ++ [Fi|Is].

%% split_even/1
%% [1,2,3,4,5,6] -> {[1,3,5],[2,4,6]}

-spec split_even(list()) -> {list(),list()}.

split_even(Rs) -> split_even(Rs, [], []).

%%%
%%% Local functions.
%%%


%% check_liveness(Reg, [Instruction], #live{}) ->
%%                      {killed | not_used | used, #live{}}
%%  Find out whether Reg is used or killed in instruction sequence.
%%
%%    killed - Reg is assigned or killed by an allocation instruction.
%%    not_used - the value of Reg is not used, but Reg must not be garbage
%%    exit_not_used - the value of Reg is not used, but must not be garbage
%%                    because the stack will be scanned because an
%%                    exit BIF will raise an exception
%%    used - Reg is used

check_liveness({fr,_}, _, St) ->
    %% Conservatively always consider the floating point register used.
    {used,St};
check_liveness(R, [{block,Blk}|Is], St0) ->
    case check_liveness_block(R, Blk, St0) of
	{transparent,St1} ->
	    check_liveness(R, Is, St1);
	{alloc_used,St1} ->
            %% Used by an allocating instruction, but value not referenced.
            %% Must check the rest of the instructions.
	    not_used(check_liveness(R, Is, St1));
	{Other,_}=Res when is_atom(Other) ->
	    Res
    end;
check_liveness(R, [{label,_}|Is], St) ->
    check_liveness(R, Is, St);
check_liveness(R, [{test,_,{f,Fail},As}|Is], St0) ->
    case member(R, As) of
	true ->
	    {used,St0};
	false ->
	    case check_liveness_at(R, Fail, St0) of
		{killed,St1} ->
		    check_liveness(R, Is, St1);
		{exit_not_used,St1} ->
		    not_used(check_liveness(R, Is, St1));
		{not_used,St1} ->
		    not_used(check_liveness(R, Is, St1));
		{used,_}=Used ->
		    Used
	    end
    end;
check_liveness(R, [{test,Op,Fail,Live,Ss,Dst}|Is], St) ->
    %% Check this instruction as a block to get a less conservative
    %% result if the caller is is_not_used/3.
    Block = [{set,[Dst],Ss,{alloc,Live,{bif,Op,Fail}}}],
    check_liveness(R, [{block,Block}|Is], St);
check_liveness(R, [{select,_,R,_,_}|_], St) ->
    {used,St};
check_liveness(R, [{select,_,_,Fail,Branches}|_], St) ->
    check_liveness_everywhere(R, [Fail|Branches], St);
check_liveness(R, [{jump,{f,F}}|_], St) ->
    check_liveness_at(R, F, St);
check_liveness(R, [{case_end,Used}|_], St) ->
    check_liveness_exit(R, Used, St);
check_liveness(R, [{try_case_end,Used}|_], St) ->
    check_liveness_exit(R, Used, St);
check_liveness(R, [{badmatch,Used}|_], St) ->
    check_liveness_exit(R, Used, St);
check_liveness(R, [if_end|_], St) ->
    check_liveness_exit(R, ignore, St);
check_liveness(R, [{func_info,_,_,Ar}|_], St) ->
    case R of
	{x,X} when X < Ar -> {used,St};
	_ -> {killed,St}
    end;
check_liveness(R, [{kill,R}|_], St) ->
    {killed,St};
check_liveness(R, [{kill,_}|Is], St) ->
    check_liveness(R, Is, St);
check_liveness(R, [{bs_init,_,_,none,Ss,Dst}|Is], St) ->
    case member(R, Ss) of
	true ->
	    {used,St};
	false ->
	    if
		R =:= Dst -> {killed,St};
		true -> check_liveness(R, Is, St)
	    end
    end;
check_liveness(R, [{bs_init,_,_,Live,Ss,Dst}|Is], St) ->
    case R of
	{x,X} ->
            case member(R, Ss) of
                true ->
                    {used,St};
                false ->
                    if
                        X < Live ->
                            not_used(check_liveness(R, Is, St));
                        true ->
                            {killed,St}
                    end
	    end;
	{y,_} ->
	    case member(R, Ss) of
		true -> {used,St};
		false ->
                    %% If the exception is taken, the stack may
                    %% be scanned. Therefore the register is not
                    %% guaranteed to be killed.
		    if
                        R =:= Dst -> {not_used,St};
			true -> not_used(check_liveness(R, Is, St))
		    end
	    end
    end;
check_liveness(R, [{deallocate,_}|Is], St) ->
    case R of
	{y,_} -> {killed,St};
	_ -> check_liveness(R, Is, St)
    end;
check_liveness({x,_}=R, [return|_], St) ->
    case R of
	{x,0} -> {used,St};
	{x,_} -> {killed,St}
    end;
check_liveness(R, [{call,Live,_}|Is], St) ->
    case R of
	{x,X} when X < Live -> {used,St};
	{x,_} -> {killed,St};
	{y,_} -> not_used(check_liveness(R, Is, St))
    end;
check_liveness(R, [{call_ext,Live,_}=I|Is], St) ->
    case R of
	{x,X} when X < Live ->
	    {used,St};
	{x,_} ->
	    {killed,St};
	{y,_} ->
	    case beam_jump:is_exit_instruction(I) of
		false ->
		    not_used(check_liveness(R, Is, St));
		true ->
		    %% We must make sure we don't check beyond this
		    %% instruction or we will fall through into random
		    %% unrelated code and get stuck in a loop.
		    {exit_not_used,St}
	    end
    end;
check_liveness(R, [{call_fun,Live}|Is], St) ->
    case R of
	{x,X} when X =< Live -> {used,St};
	{x,_} -> {killed,St};
	{y,_} -> not_used(check_liveness(R, Is, St))
    end;
check_liveness(R, [{apply,Args}|Is], St) ->
    case R of
	{x,X} when X < Args+2 -> {used,St};
	{x,_} -> {killed,St};
	{y,_} -> not_used(check_liveness(R, Is, St))
    end;
check_liveness(R, [{bif,Op,Fail,Ss,D}|Is], St) ->
    Set = {set,[D],Ss,{bif,Op,Fail}},
    check_liveness(R, [{block,[Set]}|Is], St);
check_liveness(R, [{gc_bif,Op,{f,Fail},Live,Ss,D}|Is], St) ->
    Set = {set,[D],Ss,{alloc,Live,{gc_bif,Op,Fail}}},
    check_liveness(R, [{block,[Set]}|Is], St);
check_liveness(R, [{bs_put,{f,0},_,Ss}|Is], St) ->
    case member(R, Ss) of
	true -> {used,St};
	false -> check_liveness(R, Is, St)
    end;
check_liveness(R, [{bs_restore2,S,_}|Is], St) ->
    case R of
	S -> {used,St};
	_ -> check_liveness(R, Is, St)
    end;
check_liveness(R, [{bs_save2,S,_}|Is], St) ->
    case R of
	S -> {used,St};
	_ -> check_liveness(R, Is, St)
    end;
check_liveness(R, [{move,S,D}|Is], St) ->
    case R of
	S -> {used,St};
	D -> {killed,St};
	_ -> check_liveness(R, Is, St)
    end;
check_liveness(R, [{make_fun2,_,_,_,NumFree}|Is], St) ->
    case R of
	{x,X} when X < NumFree -> {used,St};
	{x,_} -> {killed,St};
	{y,_} -> not_used(check_liveness(R, Is, St))
    end;
check_liveness(R, [{'catch'=Op,Y,Fail}|Is], St) ->
    Set = {set,[Y],[],{try_catch,Op,Fail}},
    check_liveness(R, [{block,[Set]}|Is], St);
check_liveness(R, [{'try'=Op,Y,Fail}|Is], St) ->
    Set = {set,[Y],[],{try_catch,Op,Fail}},
    check_liveness(R, [{block,[Set]}|Is], St);
check_liveness(R, [{try_end,Y}|Is], St) ->
    case R of
	Y ->
	    {killed,St};
	{y,_} ->
	    %% y registers will be used if an exception occurs and
	    %% control transfers to the label given in the previous
	    %% try/2 instruction.
	    {used,St};
	_ ->
	    check_liveness(R, Is, St)
    end;
check_liveness(R, [{catch_end,Y}|Is], St) ->
    case R of
	Y -> {killed,St};
	_ -> check_liveness(R, Is, St)
    end;
check_liveness(R, [{get_tuple_element,S,_,D}|Is], St) ->
    case R of
	S -> {used,St};
	D -> {killed,St};
	_ -> check_liveness(R, Is, St)
    end;
check_liveness(R, [{bs_context_to_binary,S}|Is], St) ->
    case R of
	S -> {used,St};
	_ -> check_liveness(R, Is, St)
    end;
check_liveness(R, [{loop_rec,{f,_},{x,0}}|_], St) ->
    case R of
	{x,_} ->
	    {killed,St};
	_ ->
	    %% y register. Rarely happens. Be very conversative and
	    %% assume it's used.
	    {used,St}
    end;
check_liveness(R, [{loop_rec_end,{f,Fail}}|_], St) ->
    check_liveness_at(R, Fail, St);
check_liveness(R, [{line,_}|Is], St) ->
    check_liveness(R, Is, St);
check_liveness(R, [{get_map_elements,{f,Fail},S,{list,L}}|Is], St0) ->
    {Ss,Ds} = split_even(L),
    case member(R, [S|Ss]) of
	true ->
	    {used,St0};
	false ->
	    case check_liveness_at(R, Fail, St0) of
		{killed,St}=Killed ->
		    case member(R, Ds) of
			true -> Killed;
			false -> check_liveness(R, Is, St)
		    end;
		Other ->
		    Other
	    end
    end;
check_liveness(R, [{put_map,F,Op,S,D,Live,{list,Puts}}|Is], St) ->
    Set = {set,[D],[S|Puts],{alloc,Live,{put_map,Op,F}}},
    check_liveness(R, [{block,[Set]}||Is], St);
check_liveness(R, [{put_tuple,Ar,D}|Is], St) ->
    Set = {set,[D],[],{put_tuple,Ar}},
    check_liveness(R, [{block,[Set]}||Is], St);
check_liveness(R, [{put_list,S1,S2,D}|Is], St) ->
    Set = {set,[D],[S1,S2],put_list},
    check_liveness(R, [{block,[Set]}||Is], St);
check_liveness(R, [{test_heap,N,Live}|Is], St) ->
    I = {block,[{set,[],[],{alloc,Live,{nozero,nostack,N,[]}}}]},
    check_liveness(R, [I|Is], St);
check_liveness(R, [{allocate_zero,N,Live}|Is], St) ->
    I = {block,[{set,[],[],{alloc,Live,{zero,N,0,[]}}}]},
    check_liveness(R, [I|Is], St);
check_liveness(R, [{get_hd,S,D}|Is], St) ->
    I = {block,[{set,[D],[S],get_hd}]},
    check_liveness(R, [I|Is], St);
check_liveness(R, [{get_tl,S,D}|Is], St) ->
    I = {block,[{set,[D],[S],get_tl}]},
    check_liveness(R, [I|Is], St);
check_liveness(R, [remove_message|Is], St) ->
    check_liveness(R, Is, St);
check_liveness({x,X}, [build_stacktrace|_], St) when X > 0 ->
    {killed,St};
check_liveness(R, [{recv_mark,_}|Is], St) ->
    check_liveness(R, Is, St);
check_liveness(R, [{recv_set,_}|Is], St) ->
    check_liveness(R, Is, St);
check_liveness(R, [{'%',_}|Is], St) ->
    check_liveness(R, Is, St);
check_liveness(_R, Is, St) when is_list(Is) ->
    %% Not implemented. Conservatively assume that the register is used.
    {used,St}.

check_liveness_everywhere(R, Lbls, St0) ->
    check_liveness_everywhere_1(R, Lbls, killed, St0).

check_liveness_everywhere_1(R, [{f,Lbl}|T], Res0, St0) ->
    {Res1,St} = check_liveness_at(R, Lbl, St0),
    Res = case Res1 of
	      killed -> Res0;
	      _ -> Res1
	  end,
    case Res of
	used -> {used,St};
	_ -> check_liveness_everywhere_1(R, T, Res, St)
    end;
check_liveness_everywhere_1(R, [_|T], Res, St) ->
    check_liveness_everywhere_1(R, T, Res, St);
check_liveness_everywhere_1(_, [], Res, St) ->
    {Res,St}.

check_liveness_at(R, Lbl, #live{lbl=Ll,res=ResMemorized}=St0) ->
    case gb_trees:lookup(Lbl, ResMemorized) of
	{value,Res} ->
	    {Res,St0};
	none ->
	    {Res,St} = case gb_trees:lookup(Lbl, Ll) of
			   {value,Is} -> check_liveness(R, Is, St0);
			   none -> {used,St0}
		       end,
	    {Res,St#live{res=gb_trees:insert(Lbl, Res, St#live.res)}}
    end.

not_used({used,_}=Res) -> Res;
not_used({_,St}) -> {not_used,St}.

check_liveness_exit(R, R, St) -> {used,St};
check_liveness_exit({x,_}, _, St) -> {killed,St};
check_liveness_exit({y,_}, _, St) -> {exit_not_used,St}.

%% check_liveness_block(Reg, [Instruction], State) ->
%%     {killed | not_used | used | alloc_used | transparent,State'}
%%  Finds out how Reg is used in the instruction sequence inside a block.
%%  Returns one of:
%%    killed - Reg is assigned a new value or killed by an
%%       allocation instruction
%%    not_used - The value is not used, but the register is referenced
%%       e.g. by an allocation instruction
%%    transparent - Reg is neither used nor killed
%%    alloc_used - Used only in an allocate instruction
%%    used - Reg is explicitly used by an instruction
%%
%%  Annotations are not allowed.
%%
%%  (Unknown instructions will cause an exception.)

check_liveness_block({x,X}=R, [{set,Ds,Ss,{alloc,Live,Op}}|Is], St0) ->
    if 
	X >= Live ->
	    {killed,St0};
	true ->
	    case check_liveness_block_1(R, Ss, Ds, Op, Is, St0) of
                {transparent,St} -> {alloc_used,St};
		{_,_}=Res -> not_used(Res)
	    end
    end;
check_liveness_block({y,_}=R, [{set,Ds,Ss,{alloc,_Live,Op}}|Is], St0) ->
    case check_liveness_block_1(R, Ss, Ds, Op, Is, St0) of
        {transparent,St} -> {alloc_used,St};
        {_,_}=Res -> not_used(Res)
    end;
check_liveness_block({y,_}=R, [{set,Ds,Ss,{try_catch,_,Op}}|Is], St0) ->
    case Ds of
        [R] ->
            {killed,St0};
        _ ->
            case check_liveness_block_1(R, Ss, Ds, Op, Is, St0) of
                {exit_not_used,St} ->
                    {used,St};
                {transparent,St} ->
                    %% Conservatively assumed that it is used.
                    {used,St};
                {_,_}=Res ->
                    Res
            end
    end;
check_liveness_block(R, [{set,Ds,Ss,Op}|Is], St) ->
    check_liveness_block_1(R, Ss, Ds, Op, Is, St);
check_liveness_block(_, [], St) -> {transparent,St}.

check_liveness_block_1(R, Ss, Ds, Op, Is, St0) ->
    case member(R, Ss) of
	true ->
	    {used,St0};
	false ->
	    case check_liveness_block_2(R, Op, Ss, St0) of
		{killed,St} ->
		    case member(R, Ds) of
			true -> {killed,St};
			false -> check_liveness_block(R, Is, St)
		    end;
		{exit_not_used,St} ->
		    case member(R, Ds) of
			true -> {exit_not_used,St};
			false -> check_liveness_block(R, Is, St)
		    end;
		{not_used,St} ->
		    not_used(case member(R, Ds) of
				 true -> {killed,St};
				 false -> check_liveness_block(R, Is, St)
			     end);
		{used,St} ->
		    {used,St}
	    end
    end.

check_liveness_block_2(R, {gc_bif,Op,{f,Lbl}}, Ss, St) ->
    check_liveness_block_3(R, Lbl, {Op,length(Ss)}, St);
check_liveness_block_2(R, {bif,Op,{f,Lbl}}, Ss, St) ->
    Arity = length(Ss),
    case erl_internal:comp_op(Op, Arity) orelse
	erl_internal:new_type_test(Op, Arity) of
	true ->
	    {killed,St};
	false ->
	    check_liveness_block_3(R, Lbl, {Op,length(Ss)}, St)
    end;
check_liveness_block_2(R, {put_map,_Op,{f,Lbl}}, _Ss, St) ->
    check_liveness_block_3(R, Lbl, {unsafe,0}, St);
check_liveness_block_2(_, _, _, St) ->
    {killed,St}.

check_liveness_block_3({x,_}, 0, _FA, St) ->
    {killed,St};
check_liveness_block_3({y,_}, 0, {F,A}, St) ->
    %% If the exception is thrown, the stack may be scanned,
    %% thus implicitly using the y register.
    case erl_bifs:is_safe(erlang, F, A) of
        true -> {killed,St};
        false -> {used,St}
    end;
check_liveness_block_3(R, Lbl, _FA, St0) ->
    check_liveness_at(R, Lbl, St0).

index_labels_1([{label,Lbl}|Is0], Acc) ->
    Is = drop_labels(Is0),
    index_labels_1(Is0, [{Lbl,Is}|Acc]);
index_labels_1([_|Is], Acc) ->
    index_labels_1(Is, Acc);
index_labels_1([], Acc) -> gb_trees:from_orddict(sort(Acc)).

drop_labels([{label,_}|Is]) -> drop_labels(Is);
drop_labels(Is) -> Is.


replace_labels_1([{test,Test,{f,Lbl},Ops}|Is], Acc, D, Fb) ->
    replace_labels_1(Is, [{test,Test,{f,label(Lbl, D, Fb)},Ops}|Acc], D, Fb);
replace_labels_1([{test,Test,{f,Lbl},Live,Ops,Dst}|Is], Acc, D, Fb) ->
    replace_labels_1(Is, [{test,Test,{f,label(Lbl, D, Fb)},Live,Ops,Dst}|Acc], D, Fb);
replace_labels_1([{select,I,R,{f,Fail0},Vls0}|Is], Acc, D, Fb) ->
    Vls = map(fun ({f,L}) -> {f,label(L, D, Fb)};
		   (Other) -> Other
	      end, Vls0),
    Fail = label(Fail0, D, Fb),
    replace_labels_1(Is, [{select,I,R,{f,Fail},Vls}|Acc], D, Fb);
replace_labels_1([{'try',R,{f,Lbl}}|Is], Acc, D, Fb) ->
    replace_labels_1(Is, [{'try',R,{f,label(Lbl, D, Fb)}}|Acc], D, Fb);
replace_labels_1([{'catch',R,{f,Lbl}}|Is], Acc, D, Fb) ->
    replace_labels_1(Is, [{'catch',R,{f,label(Lbl, D, Fb)}}|Acc], D, Fb);
replace_labels_1([{jump,{f,Lbl}}|Is], Acc, D, Fb) ->
    replace_labels_1(Is, [{jump,{f,label(Lbl, D, Fb)}}|Acc], D, Fb);
replace_labels_1([{loop_rec,{f,Lbl},R}|Is], Acc, D, Fb) ->
    replace_labels_1(Is, [{loop_rec,{f,label(Lbl, D, Fb)},R}|Acc], D, Fb);
replace_labels_1([{loop_rec_end,{f,Lbl}}|Is], Acc, D, Fb) ->
    replace_labels_1(Is, [{loop_rec_end,{f,label(Lbl, D, Fb)}}|Acc], D, Fb);
replace_labels_1([{wait,{f,Lbl}}|Is], Acc, D, Fb) ->
    replace_labels_1(Is, [{wait,{f,label(Lbl, D, Fb)}}|Acc], D, Fb);
replace_labels_1([{wait_timeout,{f,Lbl},To}|Is], Acc, D, Fb) ->
    replace_labels_1(Is, [{wait_timeout,{f,label(Lbl, D, Fb)},To}|Acc], D, Fb);
replace_labels_1([{recv_mark=Op,{f,Lbl}}|Is], Acc, D, Fb) ->
    replace_labels_1(Is, [{Op,{f,label(Lbl, D, Fb)}}|Acc], D, Fb);
replace_labels_1([{recv_set=Op,{f,Lbl}}|Is], Acc, D, Fb) ->
    replace_labels_1(Is, [{Op,{f,label(Lbl, D, Fb)}}|Acc], D, Fb);
replace_labels_1([{bif,Name,{f,Lbl},As,R}|Is], Acc, D, Fb) when Lbl =/= 0 ->
    replace_labels_1(Is, [{bif,Name,{f,label(Lbl, D, Fb)},As,R}|Acc], D, Fb);
replace_labels_1([{gc_bif,Name,{f,Lbl},Live,As,R}|Is], Acc, D, Fb) when Lbl =/= 0 ->
    replace_labels_1(Is, [{gc_bif,Name,{f,label(Lbl, D, Fb)},Live,As,R}|Acc], D, Fb);
replace_labels_1([{call,Ar,{f,Lbl}}|Is], Acc, D, Fb) ->
    replace_labels_1(Is, [{call,Ar,{f,label(Lbl, D, Fb)}}|Acc], D, Fb);
replace_labels_1([{make_fun2,{f,Lbl},U1,U2,U3}|Is], Acc, D, Fb) ->
    replace_labels_1(Is, [{make_fun2,{f,label(Lbl, D, Fb)},U1,U2,U3}|Acc], D, Fb);
replace_labels_1([{bs_init,{f,Lbl},Info,Live,Ss,Dst}|Is], Acc, D, Fb) when Lbl =/= 0 ->
    replace_labels_1(Is, [{bs_init,{f,label(Lbl, D, Fb)},Info,Live,Ss,Dst}|Acc], D, Fb);
replace_labels_1([{bs_put,{f,Lbl},Info,Ss}|Is], Acc, D, Fb) when Lbl =/= 0 ->
    replace_labels_1(Is, [{bs_put,{f,label(Lbl, D, Fb)},Info,Ss}|Acc], D, Fb);
replace_labels_1([{put_map=I,{f,Lbl},Op,Src,Dst,Live,List}|Is], Acc, D, Fb)
  when Lbl =/= 0 ->
    replace_labels_1(Is, [{I,{f,label(Lbl, D, Fb)},Op,Src,Dst,Live,List}|Acc], D, Fb);
replace_labels_1([{get_map_elements=I,{f,Lbl},Src,List}|Is], Acc, D, Fb) when Lbl =/= 0 ->
    replace_labels_1(Is, [{I,{f,label(Lbl, D, Fb)},Src,List}|Acc], D, Fb);
replace_labels_1([I|Is], Acc, D, Fb) ->
    replace_labels_1(Is, [I|Acc], D, Fb);
replace_labels_1([], Acc, _, _) -> Acc.

label(Old, D, Fb) ->
    case D of
        #{Old := New} -> New;
        _ -> Fb(Old)
    end.

%% Help function for combine_heap_needs.

combine_alloc_lists(Al0) ->
    Al1 = flatmap(fun(Words) when is_integer(Words) ->
                         [{words,Words}];
                    ({alloc,List}) ->
                         List
                 end, Al0),
    Al2 = sofs:relation(Al1),
    Al3 = sofs:relation_to_family(Al2),
    Al4 = sofs:to_external(Al3),
    [{Tag,lists:sum(L)} || {Tag,L} <- Al4].

%% live_opt/4.

%% Bit syntax instructions.
live_opt([{bs_context_to_binary,Src}=I|Is], Regs0, D, Acc) ->
    Regs = x_live([Src], Regs0),
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{bs_init,Fail,_,none,Ss,Dst}=I|Is], Regs0, D, Acc) ->
    Regs1 = x_live(Ss, x_dead([Dst], Regs0)),
    Regs = live_join_label(Fail, D, Regs1),
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{bs_init,Fail,Info,Live0,Ss,Dst}|Is], Regs0, D, Acc) ->
    Regs1 = x_dead([Dst], Regs0),
    Live = live_regs(Regs1),
    true = Live =< Live0,	%Assertion.
    Regs2 = live_call(Live),
    Regs3 = x_live(Ss, Regs2),
    Regs = live_join_label(Fail, D, Regs3),
    I = {bs_init,Fail,Info,Live,Ss,Dst},
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{bs_put,Fail,_,Ss}=I|Is], Regs0, D, Acc) ->
    Regs1 = x_live(Ss, Regs0),
    Regs = live_join_label(Fail, D, Regs1),
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{bs_restore2,Src,_}=I|Is], Regs0, D, Acc) ->
    Regs = x_live([Src], Regs0),
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{bs_save2,Src,_}=I|Is], Regs0, D, Acc) ->
    Regs = x_live([Src], Regs0),
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{test,bs_start_match2,Fail,Live,[Src,_],_}=I|Is], _, D, Acc) ->
    Regs0 = live_call(Live),
    Regs1 = x_live([Src], Regs0),
    Regs = live_join_label(Fail, D, Regs1),
    live_opt(Is, Regs, D, [I|Acc]);

%% Other instructions.
live_opt([{block,Bl0}|Is], Regs0, D, Acc) ->
    Live0 = make_anno({used,Regs0}),
    {Bl,Regs} = live_opt_block(reverse(Bl0), Regs0, D, [Live0]),
    Live = make_anno({used,Regs}),
    live_opt(Is, Regs, D, [{block,[Live|Bl]}|Acc]);
live_opt([build_stacktrace=I|Is], _, D, Acc) ->
    live_opt(Is, live_call(1), D, [I|Acc]);
live_opt([raw_raise=I|Is], _, D, Acc) ->
    live_opt(Is, live_call(3), D, [I|Acc]);
live_opt([{label,L}=I|Is], Regs, D0, Acc) ->
    D = gb_trees:insert(L, Regs, D0),
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{jump,{f,L}}=I|Is], _, D, Acc) ->
    Regs = gb_trees:get(L, D),
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([return=I|Is], _, D, Acc) ->
    live_opt(Is, 1, D, [I|Acc]);
live_opt([{catch_end,_}=I|Is], _, D, Acc) ->
    live_opt(Is, live_call(1), D, [I|Acc]);
live_opt([{badmatch,Src}=I|Is], _, D, Acc) ->
    Regs = x_live([Src], 0),
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{case_end,Src}=I|Is], _, D, Acc) ->
    Regs = x_live([Src], 0),
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{try_case_end,Src}=I|Is], _, D, Acc) ->
    Regs = x_live([Src], 0),
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([if_end=I|Is], _, D, Acc) ->
    Regs = 0,
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{call,Arity,_}=I|Is], _, D, Acc) ->
    live_opt(Is, live_call(Arity), D, [I|Acc]);
live_opt([{call_ext,Arity,_}=I|Is], _, D, Acc) ->
    live_opt(Is, live_call(Arity), D, [I|Acc]);
live_opt([{call_fun,Arity}=I|Is], _, D, Acc) ->
    live_opt(Is, live_call(Arity+1), D, [I|Acc]);
live_opt([{apply,Arity}=I|Is], _, D, Acc) ->
    live_opt(Is, live_call(Arity+2), D, [I|Acc]);
live_opt([{make_fun2,_,_,_,Arity}=I|Is], _, D, Acc) ->
    live_opt(Is, live_call(Arity), D, [I|Acc]);
live_opt([{test,_,Fail,Ss}=I|Is], Regs0, D, Acc) ->
    Regs1 = x_live(Ss, Regs0),
    Regs = live_join_label(Fail, D, Regs1),
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{test,_,Fail,Live,Ss,_}=I|Is], _, D, Acc) ->
    Regs0 = live_call(Live),
    Regs1 = x_live(Ss, Regs0),
    Regs = live_join_label(Fail, D, Regs1),
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{select,_,Src,Fail,List}=I|Is], _, D, Acc) ->
    Regs0 = 0,
    Regs1 = x_live([Src], Regs0),
    Regs = live_join_labels([Fail|List], D, Regs1),
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{try_case,Y}=I|Is], Regs0, D, Acc) ->
    Regs = live_call(1),
    case Regs0 of
        0 ->
            live_opt(Is, Regs, D, [{try_end,Y}|Acc]);
        _ ->
            live_opt(Is, live_call(1), D, [I|Acc])
    end;
live_opt([{loop_rec,_Fail,_Dst}=I|Is], _, D, Acc) ->
    live_opt(Is, 0, D, [I|Acc]);
live_opt([timeout=I|Is], _, D, Acc) ->
    live_opt(Is, 0, D, [I|Acc]);
live_opt([{wait,_}=I|Is], _, D, Acc) ->
    live_opt(Is, 0, D, [I|Acc]);
live_opt([{get_map_elements,Fail,Src,{list,List}}=I|Is], Regs0, D, Acc) ->
    {Ss,Ds} = split_even(List),
    Regs1 = x_live([Src|Ss], x_dead(Ds, Regs0)),
    Regs = live_join_label(Fail, D, Regs1),
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{gc_bif,N,F,R,As,Dst}=I|Is], Regs0, D, Acc) ->
    Bl = [{set,[Dst],As,{alloc,R,{gc_bif,N,F}}}],
    {_,Regs} = live_opt_block(Bl, Regs0, D, []),
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{bif,N,F,As,Dst}=I|Is], Regs0, D, Acc) ->
    Bl = [{set,[Dst],As,{bif,N,F}}],
    {_,Regs} = live_opt_block(Bl, Regs0, D, []),
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{get_tuple_element,Src,Idx,Dst}=I|Is], Regs0, D, Acc) ->
    Bl = [{set,[Dst],[Src],{get_tuple_element,Idx}}],
    {_,Regs} = live_opt_block(Bl, Regs0, D, []),
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{move,Src,Dst}=I|Is], Regs0, D, Acc) ->
    Regs = x_live([Src], x_dead([Dst], Regs0)),
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{put_map,F,Op,S,Dst,R,{list,Puts}}=I|Is], Regs0, D, Acc) ->
    Bl = [{set,[Dst],[S|Puts],{alloc,R,{put_map,Op,F}}}],
    {_,Regs} = live_opt_block(Bl, Regs0, D, []),
    live_opt(Is, Regs, D, [I|Acc]);

%% Transparent instructions - they neither use nor modify x registers.
live_opt([{deallocate,_}=I|Is], Regs, D, Acc) ->
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{kill,_}=I|Is], Regs, D, Acc) ->
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{try_end,_}=I|Is], Regs, D, Acc) ->
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{loop_rec_end,_}=I|Is], Regs, D, Acc) ->
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{wait_timeout,_,nil}=I|Is], Regs, D, Acc) ->
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{wait_timeout,_,{Tag,_}}=I|Is], Regs, D, Acc) when Tag =/= x ->
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{line,_}=I|Is], Regs, D, Acc) ->
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{'catch',_,_}=I|Is], Regs, D, Acc) ->
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{'try',_,_}=I|Is], Regs, D, Acc) ->
    live_opt(Is, Regs, D, [I|Acc]);

%% The following instructions can occur if the "compilation" has been
%% started from a .S file using the 'from_asm' option.
live_opt([{trim,_,_}=I|Is], Regs, D, Acc) ->
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{'%',_}=I|Is], Regs, D, Acc) ->
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{recv_set,_}=I|Is], Regs, D, Acc) ->
    live_opt(Is, Regs, D, [I|Acc]);
live_opt([{recv_mark,_}=I|Is], Regs, D, Acc) ->
    live_opt(Is, Regs, D, [I|Acc]);

live_opt([], _, _, Acc) -> Acc.

live_opt_block([{set,[{x,X}]=Ds,Ss,move}=I|Is], Regs0, D, Acc) ->
    Regs = x_live(Ss, x_dead(Ds, Regs0)),
    case is_live(X, Regs0) of
        true ->
            live_opt_block(Is, Regs, D, [I|Acc]);
        false ->
            %% Useless move, will never be used.
            live_opt_block(Is, Regs, D, Acc)
    end;
live_opt_block([{set,Ds,Ss,{alloc,Live0,AllocOp}}|Is], Regs0, D, Acc) ->
    %% Calculate liveness from the point of view of the GC.
    %% There will never be a GC if the instruction fails, so we should
    %% ignore the failure branch.
    GcRegs1 = x_dead(Ds, Regs0),
    GcRegs = x_live(Ss, GcRegs1),
    Live = live_regs(GcRegs),

    %% The life-time analysis used by the code generator is sometimes too
    %% conservative, so it may be possible to lower the number of live
    %% registers based on the exact liveness information. The main benefit is
    %% that more optimizations that depend on liveness information (such as the
    %% beam_dead pass) may be applied.
    true = Live =< Live0,                       %Assertion.
    I = {set,Ds,Ss,{alloc,Live,AllocOp}},

    %% Calculate liveness from the point of view of the preceding instruction.
    %% The liveness is the union of live registers in the GC and the live
    %% registers at the failure label.
    Regs1 = live_call(Live),
    Regs = live_join_alloc(AllocOp, D, Regs1),
    live_opt_block(Is, Regs, D, [I|Acc]);
live_opt_block([{set,Ds,Ss,{bif,_,Fail}}=I|Is], Regs0, D, Acc) ->
    Regs1 = x_dead(Ds, Regs0),
    Regs2 = x_live(Ss, Regs1),
    Regs = live_join_label(Fail, D, Regs2),
    live_opt_block(Is, Regs, D, [I|Acc]);
live_opt_block([{set,Ds,Ss,_}=I|Is], Regs0, D, Acc) ->
    Regs = x_live(Ss, x_dead(Ds, Regs0)),
    live_opt_block(Is, Regs, D, [I|Acc]);
live_opt_block([{'%anno',_}|Is], Regs, D, Acc) ->
    live_opt_block(Is, Regs, D, Acc);
live_opt_block([], Regs, _, Acc) -> {Acc,Regs}.

live_join_alloc({Kind,_Name,Fail}, D, Regs) when Kind =:= gc_bif; Kind =:= put_map ->
    live_join_label(Fail, D, Regs);
live_join_alloc(_, _, Regs) -> Regs.

live_join_labels([{f,L}|T], D, Regs0) when L =/= 0 ->
    Regs = gb_trees:get(L, D) bor Regs0,
    live_join_labels(T, D, Regs);
live_join_labels([_|T], D, Regs) ->
    live_join_labels(T, D, Regs);
live_join_labels([], _, Regs) -> Regs.

live_join_label({f,0}, _, Regs) ->
    Regs;
live_join_label({f,L}, D, Regs) ->
    gb_trees:get(L, D) bor Regs.

live_call(Live) -> (1 bsl Live) - 1.

live_regs(Regs) ->
    live_regs_1(0, Regs).

live_regs_1(N, 0) -> N;
live_regs_1(N, Regs) -> live_regs_1(N+1, Regs bsr 1).

x_dead([{x,N}|Rs], Regs) -> x_dead(Rs, Regs band (bnot (1 bsl N)));
x_dead([_|Rs], Regs) -> x_dead(Rs, Regs);
x_dead([], Regs) -> Regs.

x_live([{x,N}|Rs], Regs) -> x_live(Rs, Regs bor (1 bsl N));
x_live([_|Rs], Regs) -> x_live(Rs, Regs);
x_live([], Regs) -> Regs.

is_live(X, Regs) -> ((Regs bsr X) band 1) =:= 1.

split_even([], Ss, Ds) ->
    {reverse(Ss),reverse(Ds)};
split_even([S,D|Rs], Ss, Ds) ->
    split_even(Rs, [S|Ss], [D|Ds]).

%%%
%%% Add annotations for defined registers.
%%%
%%% This analysis is done by scanning the instructions in
%%% execution order.
%%%

defs([{apply,_}=I|Is], _Regs, D) ->
    [I|defs(Is, 1, D)];
defs([{bif,_,{f,Fail},_Src,Dst}=I|Is], Regs0, D) ->
    Regs = def_regs([Dst], Regs0),
    [I|defs(Is, Regs, update_regs(Fail, Regs0, D))];
defs([{block,Block0}|Is], Regs0, D0) ->
    {Block,Regs,D} = defs_list(Block0, Regs0, D0),
    [{block,[make_anno({def,Regs0})|Block]}|defs(Is, Regs, D)];
defs([{bs_init,{f,L},_,_,_,Dst}=I|Is], Regs0, D) ->
    Regs = def_regs([Dst], Regs0),
    [I|defs(Is, Regs, update_regs(L, Regs, D))];
defs([{bs_put,{f,L},_,_}=I|Is], Regs, D) ->
    [I|defs(Is, Regs, update_regs(L, Regs, D))];
defs([build_stacktrace=I|Is], _Regs, D) ->
    [I|defs(Is, 1, D)];
defs([{call,_,_}=I|Is], _Regs, D) ->
    [I|defs(Is, 1, D)];
defs([{call_ext,_,{extfunc,M,F,A}}=I|Is], _Regs, D) ->
    case erl_bifs:is_exit_bif(M, F, A) of
        false ->
            [I|defs(Is, 1, D)];
        true ->
            [I|defs_unreachable(Is, D)]
    end;
defs([{call_ext,_,_}=I|Is], _Regs, D) ->
    [I|defs(Is, 1, D)];
defs([{call_fun,_}=I|Is], _Regs, D) ->
    [I|defs(Is, 1, D)];
defs([{'catch',_,{f,L}}=I|Is], Regs, D) ->
    RegsAtLabel = init_def_regs(1),
    [I|defs(Is, Regs, update_regs(L, RegsAtLabel, D))];
defs([{catch_end,_}=I|Is], _Regs, D) ->
    Regs = init_def_regs(1),
    [I|defs(Is, Regs, D)];
defs([{gc_bif,_,{f,Fail},Live,_Src,Dst}=I|Is], Regs0, D) ->
    true = all_defined(Live, Regs0),            %Assertion.
    Regs = def_regs([Dst], init_def_regs(Live)),
    [I|defs(Is, Regs, update_regs(Fail, Regs0, D))];
defs([{get_map_elements,{f,L},_Src,{list,DstList}}=I|Is], Regs0, D) ->
    {_,Ds} = beam_utils:split_even(DstList),
    Regs = def_regs(Ds, Regs0),
    [I|defs(Is, Regs, update_regs(L, Regs0, D))];
defs([{get_tuple_element,_,_,Dst}=I|Is], Regs0, D) ->
    Regs = def_regs([Dst], Regs0),
    [I|defs(Is, Regs, D)];
defs([{jump,{f,L}}=I|Is], Regs, D) ->
    [I|defs_unreachable(Is, update_regs(L, Regs, D))];
defs([{label,L}=I|Is], Regs0, D) ->
    case D of
        #{L:=Regs1} ->
            Regs = Regs0 band Regs1,
            [I|defs(Is, Regs, D)];
        #{} ->
            [I|defs(Is, Regs0, D)]
    end;
defs([{loop_rec,{f,L},{x,0}}=I|Is], _Regs, D0) ->
    RegsAtLabel = init_def_regs(0),
    D = update_regs(L, RegsAtLabel, D0),
    [I|defs(Is, init_def_regs(1), D)];
defs([{loop_rec_end,_}=I|Is], _Regs, D) ->
    [I|defs(Is, 0, D)];
defs([{make_fun2,_,_,_,_}=I|Is], _Regs, D) ->
    [I|defs(Is, 1, D)];
defs([{move,_,Dst}=I|Is], Regs0, D) ->
    Regs = def_regs([Dst], Regs0),
    [I|defs(Is, Regs, D)];
defs([{put_map,{f,Fail},_,_,Dst,_,_}=I|Is], Regs0, D) ->
    Regs = def_regs([Dst], Regs0),
    [I|defs(Is, Regs, update_regs(Fail, Regs0, D))];
defs([raw_raise=I|Is], _Regs, D) ->
    [I|defs(Is, 1, D)];
defs([return=I|Is], _Regs, D) ->
    [I|defs_unreachable(Is, D)];
defs([{select,_,_Src,Fail,List}=I|Is], Regs, D0) ->
    D = update_list([Fail|List], Regs, D0),
    [I|defs_unreachable(Is, D)];
defs([{test,_,{f,L},_}=I|Is], Regs, D) ->
    [I|defs(Is, Regs, update_regs(L, Regs, D))];
defs([{test,_,{f,L},Live,_,Dst}=I|Is], Regs0, D) ->
    true = all_defined(Live, Regs0),             %Assertion.
    Regs = def_regs([Dst], init_def_regs(Live)),
    [I|defs(Is, Regs, update_regs(L, Regs0, D))];
defs([{'try',_,{f,L}}=I|Is], Regs, D) ->
    RegsAtLabel = init_def_regs(3),
    [I|defs(Is, Regs, update_regs(L, RegsAtLabel, D))];
defs([{try_case,_}=I|Is], _Regs, D) ->
    [I|defs(Is, init_def_regs(3), D)];
defs([{wait,_}=I|Is], _Regs, D) ->
    [I|defs_unreachable(Is, D)];
defs([{wait_timeout,_,_}=I|Is], _Regs, D) ->
    [I|defs(Is, 0, D)];

%% Exceptions.
defs([{badmatch,_}=I|Is], _Regs, D) ->
    [I|defs_unreachable(Is, D)];
defs([{case_end,_}=I|Is], _Regs, D) ->
    [I|defs_unreachable(Is, D)];
defs([if_end=I|Is], _Regs, D) ->
    [I|defs_unreachable(Is, D)];
defs([{try_case_end,_}=I|Is], _Regs, D) ->
    [I|defs_unreachable(Is, D)];

%% Neutral instructions
defs([{bs_context_to_binary,_}=I|Is], Regs, D) ->
    [I|defs(Is, Regs, D)];
defs([{bs_restore2,_,_}=I|Is], Regs, D) ->
    [I|defs(Is, Regs, D)];
defs([{bs_save2,_,_}=I|Is], Regs, D) ->
    [I|defs(Is, Regs, D)];
defs([{deallocate,_}=I|Is], Regs, D) ->
    [I|defs(Is, Regs, D)];
defs([{kill,_}=I|Is], Regs, D) ->
    [I|defs(Is, Regs, D)];
defs([{line,_}=I|Is], Regs, D) ->
    [I|defs(Is, Regs, D)];
defs([{recv_mark,_}=I|Is], Regs, D) ->
    [I|defs(Is, Regs, D)];
defs([{recv_set,_}=I|Is], Regs, D) ->
    [I|defs(Is, Regs, D)];
defs([timeout=I|Is], Regs, D) ->
    [I|defs(Is, Regs, D)];
defs([{trim,_,_}=I|Is], Regs, D) ->
    [I|defs(Is, Regs, D)];
defs([{try_end,_}=I|Is], Regs, D) ->
    [I|defs(Is, Regs, D)];
defs([{'%',_}=I|Is], Regs, D) ->
    [I|defs(Is, Regs, D)];
defs([], _, _) -> [].

defs_unreachable([{label,L}=I|Is], D) ->
    case D of
        #{L:=Regs} ->
            [I|defs(Is, Regs, D)];
        #{} ->
            defs_unreachable(Is, D)
    end;
defs_unreachable([_|Is], D) ->
    defs_unreachable(Is, D);
defs_unreachable([], _D) -> [].

defs_list(Is, Regs, D) ->
    defs_list(Is, Regs, D, []).

defs_list([{set,Ds,_,{alloc,Live,Info}}=I|Is], Regs0, D0, Acc) ->
    true = all_defined(Live, Regs0),             %Assertion.
    D = case Info of
            {gc_bif,_,{f,Fail}} ->
                update_regs(Fail, Regs0, D0);
            {put_map,_,{f,Fail}} ->
                update_regs(Fail, Regs0, D0);
            _ ->
                D0
        end,
    Regs = def_regs(Ds, init_def_regs(Live)),
    defs_list(Is, Regs, D, [I|Acc]);
defs_list([{set,Ds,_,Info}=I|Is], Regs0, D0, Acc) ->
    D = case Info of
            {bif,_,{f,Fail}} ->
                update_regs(Fail, Regs0, D0);
            {try_catch,'catch',{f,Fail}} ->
                update_regs(Fail, init_def_regs(1), D0);
            {try_catch,'try',{f,Fail}} ->
                update_regs(Fail, init_def_regs(3), D0);
            _ ->
                D0
        end,
    Regs = def_regs(Ds, Regs0),
    defs_list(Is, Regs, D, [I|Acc]);
defs_list([], Regs, D, Acc) ->
    {reverse(Acc),Regs,D}.

init_def_regs(Arity) ->
    (1 bsl Arity) - 1.

def_regs([{x,X}|T], Regs) ->
    def_regs(T, Regs bor (1 bsl X));
def_regs([_|T], Regs) ->
    def_regs(T, Regs);
def_regs([], Regs) -> Regs.

update_list([{f,L}|T], Regs, D0) ->
    D = update_regs(L, Regs, D0),
    update_list(T, Regs, D);
update_list([_|T], Regs, D) ->
    update_list(T, Regs, D);
update_list([], _Regs, D) -> D.

update_regs(L, Regs0, D) ->
    case D of
        #{L:=Regs1} ->
            Regs = Regs0 band Regs1,
            D#{L:=Regs};
        #{} ->
            D#{L=>Regs0}
    end.

all_defined(Live, Regs) ->
    All = (1 bsl Live) - 1,
    Regs band All =:= All.

%%%
%%% Utilities.
%%%

%% make_anno(Anno) -> WrappedAnno.
%%  Wrap an annotation term.

make_anno(Anno) ->
    {'%anno',Anno}.