text: /*
  ** NARROW: Narrowing of numbers to integers (double to int32_t).
  ** STRIPOV: Stripping of overflow checks.
  ** Copyright (C) 2005-2015 Mike Pall. See Copyright Notice in luajit.h
  */
  
  #define lj_opt_narrow_c
  #define LUA_CORE
  
  #include "lj_obj.h"
  
  #if LJ_HASJIT
  
  #include "lj_bc.h"
  #include "lj_ir.h"
  #include "lj_jit.h"
  #include "lj_iropt.h"
  #include "lj_trace.h"
  #include "lj_vm.h"
  #include "lj_strscan.h"
  
  /* Rationale for narrowing optimizations:
  **
  ** Lua has only a single number type and this is a FP double by default.
  ** Narrowing doubles to integers does not pay off for the interpreter on a
  ** current-generation x86/x64 machine. Most FP operations need the same
  ** amount of execution resources as their integer counterparts, except
  ** with slightly longer latencies. Longer latencies are a non-issue for
  ** the interpreter, since they are usually hidden by other overhead.
  **
  ** The total CPU execution bandwidth is the sum of the bandwidth of the FP
  ** and the integer units, because they execute in parallel. The FP units
  ** have an equal or higher bandwidth than the integer units. Not using
  ** them means losing execution bandwidth. Moving work away from them to
  ** the already quite busy integer units is a losing proposition.
  **
  ** The situation for JIT-compiled code is a bit different: the higher code
  ** density makes the extra latencies much more visible. Tight loops expose
  ** the latencies for updating the induction variables. Array indexing
  ** requires narrowing conversions with high latencies and additional
  ** guards (to check that the index is really an integer). And many common
  ** optimizations only work on integers.
  **
  ** One solution would be speculative, eager narrowing of all number loads.
  ** This causes many problems, like losing -0 or the need to resolve type
  ** mismatches between traces. It also effectively forces the integer type
  ** to have overflow-checking semantics. This impedes many basic
  ** optimizations and requires adding overflow checks to all integer
  ** arithmetic operations (whereas FP arithmetics can do without).
  **
  ** Always replacing an FP op with an integer op plus an overflow check is
  ** counter-productive on a current-generation super-scalar CPU. Although
  ** the overflow check branches are highly predictable, they will clog the
  ** execution port for the branch unit and tie up reorder buffers. This is
  ** turning a pure data-flow dependency into a different data-flow
  ** dependency (with slightly lower latency) *plus* a control dependency.
  ** In general, you don't want to do this since latencies due to data-flow
  ** dependencies can be well hidden by out-of-order execution.
  **
  ** A better solution is to keep all numbers as FP values and only narrow
  ** when it's beneficial to do so. LuaJIT uses predictive narrowing for
  ** induction variables and demand-driven narrowing for index expressions,
  ** integer arguments and bit operations. Additionally it can eliminate or
  ** hoist most of the resulting overflow checks. Regular arithmetic
  ** computations are never narrowed to integers.
  **
  ** The integer type in the IR has convenient wrap-around semantics and
  ** ignores overflow. Extra operations have been added for
  ** overflow-checking arithmetic (ADDOV/SUBOV) instead of an extra type.
  ** Apart from reducing overall complexity of the compiler, this also
  ** nicely solves the problem where you want to apply algebraic
  ** simplifications to ADD, but not to ADDOV. And the x86/x64 assembler can
  ** use lea instead of an add for integer ADD, but not for ADDOV (lea does
  ** not affect the flags, but it helps to avoid register moves).
  **
  **
  ** All of the above has to be reconsidered for architectures with slow FP
  ** operations or without a hardware FPU. The dual-number mode of LuaJIT
  ** addresses this issue. Arithmetic operations are performed on integers
  ** as far as possible and overflow checks are added as needed.
  **
  ** This implies that narrowing for integer arguments and bit operations
  ** should also strip overflow checks, e.g. replace ADDOV with ADD. The
  ** original overflow guards are weak and can be eliminated by DCE, if
  ** there's no other use.
  **
  ** A slight twist is that it's usually beneficial to use overflow-checked
  ** integer arithmetics if all inputs are already integers. This is the only
  ** change that affects the single-number mode, too.
  */
  
  /* Some local macros to save typing. Undef'd at the end. */
  #define IR(ref)			(&J->cur.ir[(ref)])
  #define fins			(&J->fold.ins)
  
  /* Pass IR on to next optimization in chain (FOLD). */
  #define emitir(ot, a, b)	(lj_ir_set(J, (ot), (a), (b)), lj_opt_fold(J))
  
  #define emitir_raw(ot, a, b)	(lj_ir_set(J, (ot), (a), (b)), lj_ir_emit(J))
  
  /* -- Elimination of narrowing type conversions --------------------------- */
  
  /* Narrowing of index expressions and bit operations is demand-driven. The
  ** trace recorder emits a narrowing type conversion (CONV.int.num or TOBIT)
  ** in all of these cases (e.g. array indexing or string indexing). FOLD
  ** already takes care of eliminating simple redundant conversions like
  ** CONV.int.num(CONV.num.int(x)) ==> x.
  **
  ** But the surrounding code is FP-heavy and arithmetic operations are
  ** performed on FP numbers (for the single-number mode). Consider a common
  ** example such as 'x=t[i+1]', with 'i' already an integer (due to induction
  ** variable narrowing). The index expression would be recorded as
  **   CONV.int.num(ADD(CONV.num.int(i), 1))
  ** which is clearly suboptimal.
  **
  ** One can do better by recursively backpropagating the narrowing type
  ** conversion across FP arithmetic operations. This turns FP ops into
  ** their corresponding integer counterparts. Depending on the semantics of
  ** the conversion they also need to check for overflow. Currently only ADD
  ** and SUB are supported.
  **
  ** The above example can be rewritten as
  **   ADDOV(CONV.int.num(CONV.num.int(i)), 1)
  ** and then into ADDOV(i, 1) after folding of the conversions. The original
  ** FP ops remain in the IR and are eliminated by DCE since all references to
  ** them are gone.
  **
  ** [In dual-number mode the trace recorder already emits ADDOV etc., but
  ** this can be further reduced. See below.]
  **
  ** Special care has to be taken to avoid narrowing across an operation
  ** which is potentially operating on non-integral operands. One obvious
  ** case is when an expression contains a non-integral constant, but ends
  ** up as an integer index at runtime (like t[x+1.5] with x=0.5).
  **
  ** Operations with two non-constant operands illustrate a similar problem
  ** (like t[a+b] with a=1.5 and b=2.5). Backpropagation has to stop there,
  ** unless it can be proven that either operand is integral (e.g. by CSEing
  ** a previous conversion). As a not-so-obvious corollary this logic also
  ** applies for a whole expression tree (e.g. t[(a+1)+(b+1)]).
  **
  ** Correctness of the transformation is guaranteed by avoiding to expand
  ** the tree by adding more conversions than the one we would need to emit
  ** if not backpropagating. TOBIT employs a more optimistic rule, because
  ** the conversion has special semantics, designed to make the life of the
  ** compiler writer easier. ;-)
  **
  ** Using on-the-fly backpropagation of an expression tree doesn't work
  ** because it's unknown whether the transform is correct until the end.
  ** This either requires IR rollback and cache invalidation for every
  ** subtree or a two-pass algorithm. The former didn't work out too well,
  ** so the code now combines a recursive collector with a stack-based
  ** emitter.
  **
  ** [A recursive backpropagation algorithm with backtracking, employing
  ** skip-list lookup and round-robin caching, emitting stack operations
  ** on-the-fly for a stack-based interpreter -- and all of that in a meager
  ** kilobyte? Yep, compilers are a great treasure chest. Throw away your
  ** textbooks and read the codebase of a compiler today!]
  **
  ** There's another optimization opportunity for array indexing: it's
  ** always accompanied by an array bounds-check. The outermost overflow
  ** check may be delegated to the ABC operation. This works because ABC is
  ** an unsigned comparison and wrap-around due to overflow creates negative
  ** numbers.
  **
  ** But this optimization is only valid for constants that cannot overflow
  ** an int32_t into the range of valid array indexes [0..2^27+1). A check
  ** for +-2^30 is safe since -2^31 - 2^30 wraps to 2^30 and 2^31-1 + 2^30
  ** wraps to -2^30-1.
  **
  ** It's also good enough in practice, since e.g. t[i+1] or t[i-10] are
  ** quite common. So the above example finally ends up as ADD(i, 1)!
  **
  ** Later on, the assembler is able to fuse the whole array reference and
  ** the ADD into the memory operands of loads and other instructions. This
  ** is why LuaJIT is able to generate very pretty (and fast) machine code
  ** for array indexing. And that, my dear, concludes another story about
  ** one of the hidden secrets of LuaJIT ...
  */
  
  /* Maximum backpropagation depth and maximum stack size. */
  #define NARROW_MAX_BACKPROP	100
  #define NARROW_MAX_STACK	256
  
  /* The stack machine has a 32 bit instruction format: [IROpT | IRRef1]
  ** The lower 16 bits hold a reference (or 0). The upper 16 bits hold
  ** the IR opcode + type or one of the following special opcodes:
  */
  enum {
    NARROW_REF,		/* Push ref. */
    NARROW_CONV,		/* Push conversion of ref. */
    NARROW_SEXT,		/* Push sign-extension of ref. */
    NARROW_INT		/* Push KINT ref. The next code holds an int32_t. */
  };
  
  typedef uint32_t NarrowIns;
  
  #define NARROWINS(op, ref)	(((op) << 16) + (ref))
  #define narrow_op(ins)		((IROpT)((ins) >> 16))
  #define narrow_ref(ins)		((IRRef1)(ins))
  
  /* Context used for narrowing of type conversions. */
  typedef struct NarrowConv {
    jit_State *J;		/* JIT compiler state. */
    NarrowIns *sp;	/* Current stack pointer. */
    NarrowIns *maxsp;	/* Maximum stack pointer minus redzone. */
    IRRef mode;		/* Conversion mode (IRCONV_*). */
    IRType t;		/* Destination type: IRT_INT or IRT_I64. */
    NarrowIns stack[NARROW_MAX_STACK];  /* Stack holding stack-machine code. */
  } NarrowConv;
  
  /* Lookup a reference in the backpropagation cache. */
  static BPropEntry *narrow_bpc_get(jit_State *J, IRRef1 key, IRRef mode)
  {
    ptrdiff_t i;
    for (i = 0; i < BPROP_SLOTS; i++) {
      BPropEntry *bp = &J->bpropcache[i];
      /* Stronger checks are ok, too. */
      if (bp->key == key && bp->mode >= mode &&
  	((bp->mode ^ mode) & IRCONV_MODEMASK) == 0)
        return bp;
    }
    return NULL;
  }
  
  /* Add an entry to the backpropagation cache. */
  static void narrow_bpc_set(jit_State *J, IRRef1 key, IRRef1 val, IRRef mode)
  {
    uint32_t slot = J->bpropslot;
    BPropEntry *bp = &J->bpropcache[slot];
    J->bpropslot = (slot + 1) & (BPROP_SLOTS-1);
    bp->key = key;
    bp->val = val;
    bp->mode = mode;
  }
  
  /* Backpropagate overflow stripping. */
  static void narrow_stripov_backprop(NarrowConv *nc, IRRef ref, int depth)
  {
    jit_State *J = nc->J;
    IRIns *ir = IR(ref);
    if (ir->o == IR_ADDOV || ir->o == IR_SUBOV ||
        (ir->o == IR_MULOV && (nc->mode & IRCONV_CONVMASK) == IRCONV_ANY)) {
      BPropEntry *bp = narrow_bpc_get(nc->J, ref, IRCONV_TOBIT);
      if (bp) {
        ref = bp->val;
      } else if (++depth < NARROW_MAX_BACKPROP && nc->sp < nc->maxsp) {
        NarrowIns *savesp = nc->sp;
        narrow_stripov_backprop(nc, ir->op1, depth);
        if (nc->sp < nc->maxsp) {
  	narrow_stripov_backprop(nc, ir->op2, depth);
  	if (nc->sp < nc->maxsp) {
  	  *nc->sp++ = NARROWINS(IRT(ir->o - IR_ADDOV + IR_ADD, IRT_INT), ref);
  	  return;
  	}
        }
        nc->sp = savesp;  /* Path too deep, need to backtrack. */
      }
    }
    *nc->sp++ = NARROWINS(NARROW_REF, ref);
  }
  
  /* Backpropagate narrowing conversion. Return number of needed conversions. */
  static int narrow_conv_backprop(NarrowConv *nc, IRRef ref, int depth)
  {
    jit_State *J = nc->J;
    IRIns *ir = IR(ref);
    IRRef cref;
  
    if (nc->sp >= nc->maxsp) return 10;  /* Path too deep. */
  
    /* Check the easy cases first. */
    if (ir->o == IR_CONV && (ir->op2 & IRCONV_SRCMASK) == IRT_INT) {
      if ((nc->mode & IRCONV_CONVMASK) <= IRCONV_ANY)
        narrow_stripov_backprop(nc, ir->op1, depth+1);
      else
        *nc->sp++ = NARROWINS(NARROW_REF, ir->op1);  /* Undo conversion. */
      if (nc->t == IRT_I64)
        *nc->sp++ = NARROWINS(NARROW_SEXT, 0);  /* Sign-extend integer. */
      return 0;
    } else if (ir->o == IR_KNUM) {  /* Narrow FP constant. */
      lua_Number n = ir_knum(ir)->n;
      if ((nc->mode & IRCONV_CONVMASK) == IRCONV_TOBIT) {
        /* Allows a wider range of constants. */
        int64_t k64 = (int64_t)n;
        if (n == (lua_Number)k64) {  /* Only if const doesn't lose precision. */
  	*nc->sp++ = NARROWINS(NARROW_INT, 0);
  	*nc->sp++ = (NarrowIns)k64;  /* But always truncate to 32 bits. */
  	return 0;
        }
      } else {
        int32_t k = lj_num2int(n);
        /* Only if constant is a small integer. */
        if (checki16(k) && n == (lua_Number)k) {
  	*nc->sp++ = NARROWINS(NARROW_INT, 0);
  	*nc->sp++ = (NarrowIns)k;
  	return 0;
        }
      }
      return 10;  /* Never narrow other FP constants (this is rare). */
    }
  
    /* Try to CSE the conversion. Stronger checks are ok, too. */
    cref = J->chain[fins->o];
    while (cref > ref) {
      IRIns *cr = IR(cref);
      if (cr->op1 == ref &&
  	(fins->o == IR_TOBIT ||
  	 ((cr->op2 & IRCONV_MODEMASK) == (nc->mode & IRCONV_MODEMASK) &&
  	  irt_isguard(cr->t) >= irt_isguard(fins->t)))) {
        *nc->sp++ = NARROWINS(NARROW_REF, cref);
        return 0;  /* Already there, no additional conversion needed. */
      }
      cref = cr->prev;
    }
  
    /* Backpropagate across ADD/SUB. */
    if (ir->o == IR_ADD || ir->o == IR_SUB) {
      /* Try cache lookup first. */
      IRRef mode = nc->mode;
      BPropEntry *bp;
      /* Inner conversions need a stronger check. */
      if ((mode & IRCONV_CONVMASK) == IRCONV_INDEX && depth > 0)
        mode += IRCONV_CHECK-IRCONV_INDEX;
      bp = narrow_bpc_get(nc->J, (IRRef1)ref, mode);
      if (bp) {
        *nc->sp++ = NARROWINS(NARROW_REF, bp->val);
        return 0;
      } else if (nc->t == IRT_I64) {
        /* Try sign-extending from an existing (checked) conversion to int. */
        mode = (IRT_INT<<5)|IRT_NUM|IRCONV_INDEX;
        bp = narrow_bpc_get(nc->J, (IRRef1)ref, mode);
        if (bp) {
  	*nc->sp++ = NARROWINS(NARROW_REF, bp->val);
  	*nc->sp++ = NARROWINS(NARROW_SEXT, 0);
  	return 0;
        }
      }
      if (++depth < NARROW_MAX_BACKPROP && nc->sp < nc->maxsp) {
        NarrowIns *savesp = nc->sp;
        int count = narrow_conv_backprop(nc, ir->op1, depth);
        count += narrow_conv_backprop(nc, ir->op2, depth);
        if (count <= 1) {  /* Limit total number of conversions. */
  	*nc->sp++ = NARROWINS(IRT(ir->o, nc->t), ref);
  	return count;
        }
        nc->sp = savesp;  /* Too many conversions, need to backtrack. */
      }
    }
  
    /* Otherwise add a conversion. */
    *nc->sp++ = NARROWINS(NARROW_CONV, ref);
    return 1;
  }
  
  /* Emit the conversions collected during backpropagation. */
  static IRRef narrow_conv_emit(jit_State *J, NarrowConv *nc)
  {
    /* The fins fields must be saved now -- emitir() overwrites them. */
    IROpT guardot = irt_isguard(fins->t) ? IRTG(IR_ADDOV-IR_ADD, 0) : 0;
    IROpT convot = fins->ot;
    IRRef1 convop2 = fins->op2;
    NarrowIns *next = nc->stack;  /* List of instructions from backpropagation. */
    NarrowIns *last = nc->sp;
    NarrowIns *sp = nc->stack;  /* Recycle the stack to store operands. */
    while (next < last) {  /* Simple stack machine to process the ins. list. */
      NarrowIns ref = *next++;
      IROpT op = narrow_op(ref);
      if (op == NARROW_REF) {
        *sp++ = ref;
      } else if (op == NARROW_CONV) {
        *sp++ = emitir_raw(convot, ref, convop2);  /* Raw emit avoids a loop. */
      } else if (op == NARROW_SEXT) {
        lua_assert(sp >= nc->stack+1);
        sp[-1] = emitir(IRT(IR_CONV, IRT_I64), sp[-1],
  		      (IRT_I64<<5)|IRT_INT|IRCONV_SEXT);
      } else if (op == NARROW_INT) {
        lua_assert(next < last);
        *sp++ = nc->t == IRT_I64 ?
  	      lj_ir_kint64(J, (int64_t)(int32_t)*next++) :
  	      lj_ir_kint(J, *next++);
      } else {  /* Regular IROpT. Pops two operands and pushes one result. */
        IRRef mode = nc->mode;
        lua_assert(sp >= nc->stack+2);
        sp--;
        /* Omit some overflow checks for array indexing. See comments above. */
        if ((mode & IRCONV_CONVMASK) == IRCONV_INDEX) {
  	if (next == last && irref_isk(narrow_ref(sp[0])) &&
  	  (uint32_t)IR(narrow_ref(sp[0]))->i + 0x40000000u < 0x80000000u)
  	  guardot = 0;
  	else  /* Otherwise cache a stronger check. */
  	  mode += IRCONV_CHECK-IRCONV_INDEX;
        }
        sp[-1] = emitir(op+guardot, sp[-1], sp[0]);
        /* Add to cache. */
        if (narrow_ref(ref))
  	narrow_bpc_set(J, narrow_ref(ref), narrow_ref(sp[-1]), mode);
      }
    }
    lua_assert(sp == nc->stack+1);
    return nc->stack[0];
  }
  
  /* Narrow a type conversion of an arithmetic operation. */
  TRef LJ_FASTCALL lj_opt_narrow_convert(jit_State *J)
  {
    if ((J->flags & JIT_F_OPT_NARROW)) {
      NarrowConv nc;
      nc.J = J;
      nc.sp = nc.stack;
      nc.maxsp = &nc.stack[NARROW_MAX_STACK-4];
      nc.t = irt_type(fins->t);
      if (fins->o == IR_TOBIT) {
        nc.mode = IRCONV_TOBIT;  /* Used only in the backpropagation cache. */
      } else {
        nc.mode = fins->op2;
      }
      if (narrow_conv_backprop(&nc, fins->op1, 0) <= 1)
        return narrow_conv_emit(J, &nc);
    }
    return NEXTFOLD;
  }
  
  /* -- Narrowing of implicit conversions ----------------------------------- */
  
  /* Recursively strip overflow checks. */
  static TRef narrow_stripov(jit_State *J, TRef tr, int lastop, IRRef mode)
  {
    IRRef ref = tref_ref(tr);
    IRIns *ir = IR(ref);
    int op = ir->o;
    if (op >= IR_ADDOV && op <= lastop) {
      BPropEntry *bp = narrow_bpc_get(J, ref, mode);
      if (bp) {
        return TREF(bp->val, irt_t(IR(bp->val)->t));
      } else {
        IRRef op1 = ir->op1, op2 = ir->op2;  /* The IR may be reallocated. */
        op1 = narrow_stripov(J, op1, lastop, mode);
        op2 = narrow_stripov(J, op2, lastop, mode);
        tr = emitir(IRT(op - IR_ADDOV + IR_ADD,
  		      ((mode & IRCONV_DSTMASK) >> IRCONV_DSH)), op1, op2);
        narrow_bpc_set(J, ref, tref_ref(tr), mode);
      }
    } else if (LJ_64 && (mode & IRCONV_SEXT) && !irt_is64(ir->t)) {
      tr = emitir(IRT(IR_CONV, IRT_INTP), tr, mode);
    }
    return tr;
  }
  
  /* Narrow array index. */
  TRef LJ_FASTCALL lj_opt_narrow_index(jit_State *J, TRef tr)
  {
    IRIns *ir;
    lua_assert(tref_isnumber(tr));
    if (tref_isnum(tr))  /* Conversion may be narrowed, too. See above. */
      return emitir(IRTGI(IR_CONV), tr, IRCONV_INT_NUM|IRCONV_INDEX);
    /* Omit some overflow checks for array indexing. See comments above. */
    ir = IR(tref_ref(tr));
    if ((ir->o == IR_ADDOV || ir->o == IR_SUBOV) && irref_isk(ir->op2) &&
        (uint32_t)IR(ir->op2)->i + 0x40000000u < 0x80000000u)
      return emitir(IRTI(ir->o - IR_ADDOV + IR_ADD), ir->op1, ir->op2);
    return tr;
  }
  
  /* Narrow conversion to integer operand (overflow undefined). */
  TRef LJ_FASTCALL lj_opt_narrow_toint(jit_State *J, TRef tr)
  {
    if (tref_isstr(tr))
      tr = emitir(IRTG(IR_STRTO, IRT_NUM), tr, 0);
    if (tref_isnum(tr))  /* Conversion may be narrowed, too. See above. */
      return emitir(IRTI(IR_CONV), tr, IRCONV_INT_NUM|IRCONV_ANY);
    if (!tref_isinteger(tr))
      lj_trace_err(J, LJ_TRERR_BADTYPE);
    /*
    ** Undefined overflow semantics allow stripping of ADDOV, SUBOV and MULOV.
    ** Use IRCONV_TOBIT for the cache entries, since the semantics are the same.
    */
    return narrow_stripov(J, tr, IR_MULOV, (IRT_INT<<5)|IRT_INT|IRCONV_TOBIT);
  }
  
  /* Narrow conversion to bitop operand (overflow wrapped). */
  TRef LJ_FASTCALL lj_opt_narrow_tobit(jit_State *J, TRef tr)
  {
    if (tref_isstr(tr))
      tr = emitir(IRTG(IR_STRTO, IRT_NUM), tr, 0);
    if (tref_isnum(tr))  /* Conversion may be narrowed, too. See above. */
      return emitir(IRTI(IR_TOBIT), tr, lj_ir_knum_tobit(J));
    if (!tref_isinteger(tr))
      lj_trace_err(J, LJ_TRERR_BADTYPE);
    /*
    ** Wrapped overflow semantics allow stripping of ADDOV and SUBOV.
    ** MULOV cannot be stripped due to precision widening.
    */
    return narrow_stripov(J, tr, IR_SUBOV, (IRT_INT<<5)|IRT_INT|IRCONV_TOBIT);
  }
  
  #if LJ_HASFFI
  /* Narrow C array index (overflow undefined). */
  TRef LJ_FASTCALL lj_opt_narrow_cindex(jit_State *J, TRef tr)
  {
    lua_assert(tref_isnumber(tr));
    if (tref_isnum(tr))
      return emitir(IRT(IR_CONV, IRT_INTP), tr, (IRT_INTP<<5)|IRT_NUM|IRCONV_ANY);
    /* Undefined overflow semantics allow stripping of ADDOV, SUBOV and MULOV. */
    return narrow_stripov(J, tr, IR_MULOV,
  			LJ_64 ? ((IRT_INTP<<5)|IRT_INT|IRCONV_SEXT) :
  				((IRT_INTP<<5)|IRT_INT|IRCONV_TOBIT));
  }
  #endif
  
  /* -- Narrowing of arithmetic operators ----------------------------------- */
  
  /* Check whether a number fits into an int32_t (-0 is ok, too). */
  static int numisint(lua_Number n)
  {
    return (n == (lua_Number)lj_num2int(n));
  }
  
  /* Narrowing of arithmetic operations. */
  TRef lj_opt_narrow_arith(jit_State *J, TRef rb, TRef rc,
  			 TValue *vb, TValue *vc, IROp op)
  {
    if (tref_isstr(rb)) {
      rb = emitir(IRTG(IR_STRTO, IRT_NUM), rb, 0);
      lj_strscan_num(strV(vb), vb);
    }
    if (tref_isstr(rc)) {
      rc = emitir(IRTG(IR_STRTO, IRT_NUM), rc, 0);
      lj_strscan_num(strV(vc), vc);
    }
    /* Must not narrow MUL in non-DUALNUM variant, because it loses -0. */
    if ((op >= IR_ADD && op <= (LJ_DUALNUM ? IR_MUL : IR_SUB)) &&
        tref_isinteger(rb) && tref_isinteger(rc) &&
        numisint(lj_vm_foldarith(numberVnum(vb), numberVnum(vc),
  			       (int)op - (int)IR_ADD)))
      return emitir(IRTGI((int)op - (int)IR_ADD + (int)IR_ADDOV), rb, rc);
    if (!tref_isnum(rb)) rb = emitir(IRTN(IR_CONV), rb, IRCONV_NUM_INT);
    if (!tref_isnum(rc)) rc = emitir(IRTN(IR_CONV), rc, IRCONV_NUM_INT);
    return emitir(IRTN(op), rb, rc);
  }
  
  /* Narrowing of unary minus operator. */
  TRef lj_opt_narrow_unm(jit_State *J, TRef rc, TValue *vc)
  {
    if (tref_isstr(rc)) {
      rc = emitir(IRTG(IR_STRTO, IRT_NUM), rc, 0);
      lj_strscan_num(strV(vc), vc);
    }
    if (tref_isinteger(rc)) {
      if ((uint32_t)numberVint(vc) != 0x80000000u)
        return emitir(IRTGI(IR_SUBOV), lj_ir_kint(J, 0), rc);
      rc = emitir(IRTN(IR_CONV), rc, IRCONV_NUM_INT);
    }
    return emitir(IRTN(IR_NEG), rc, lj_ir_knum_neg(J));
  }
  
  /* Narrowing of modulo operator. */
  TRef lj_opt_narrow_mod(jit_State *J, TRef rb, TRef rc, TValue *vc)
  {
    TRef tmp;
    if (tvisstr(vc) && !lj_strscan_num(strV(vc), vc))
      lj_trace_err(J, LJ_TRERR_BADTYPE);
    if ((LJ_DUALNUM || (J->flags & JIT_F_OPT_NARROW)) &&
        tref_isinteger(rb) && tref_isinteger(rc) &&
        (tvisint(vc) ? intV(vc) != 0 : !tviszero(vc))) {
      emitir(IRTGI(IR_NE), rc, lj_ir_kint(J, 0));
      return emitir(IRTI(IR_MOD), rb, rc);
    }
    /* b % c ==> b - floor(b/c)*c */
    rb = lj_ir_tonum(J, rb);
    rc = lj_ir_tonum(J, rc);
    tmp = emitir(IRTN(IR_DIV), rb, rc);
    tmp = emitir(IRTN(IR_FPMATH), tmp, IRFPM_FLOOR);
    tmp = emitir(IRTN(IR_MUL), tmp, rc);
    return emitir(IRTN(IR_SUB), rb, tmp);
  }
  
  /* Narrowing of power operator or math.pow. */
  TRef lj_opt_narrow_pow(jit_State *J, TRef rb, TRef rc, TValue *vc)
  {
    if (tvisstr(vc) && !lj_strscan_num(strV(vc), vc))
      lj_trace_err(J, LJ_TRERR_BADTYPE);
    /* Narrowing must be unconditional to preserve (-x)^i semantics. */
    if (tvisint(vc) || numisint(numV(vc))) {
      int checkrange = 0;
      /* Split pow is faster for bigger exponents. But do this only for (+k)^i. */
      if (tref_isk(rb) && (int32_t)ir_knum(IR(tref_ref(rb)))->u32.hi >= 0) {
        int32_t k = numberVint(vc);
        if (!(k >= -65536 && k <= 65536)) goto split_pow;
        checkrange = 1;
      }
      if (!tref_isinteger(rc)) {
        if (tref_isstr(rc))
  	rc = emitir(IRTG(IR_STRTO, IRT_NUM), rc, 0);
        /* Guarded conversion to integer! */
        rc = emitir(IRTGI(IR_CONV), rc, IRCONV_INT_NUM|IRCONV_CHECK);
      }
      if (checkrange && !tref_isk(rc)) {  /* Range guard: -65536 <= i <= 65536 */
        TRef tmp = emitir(IRTI(IR_ADD), rc, lj_ir_kint(J, 65536));
        emitir(IRTGI(IR_ULE), tmp, lj_ir_kint(J, 2*65536));
      }
      return emitir(IRTN(IR_POW), rb, rc);
    }
  split_pow:
    /* FOLD covers most cases, but some are easier to do here. */
    if (tref_isk(rb) && tvispone(ir_knum(IR(tref_ref(rb)))))
      return rb;  /* 1 ^ x ==> 1 */
    rc = lj_ir_tonum(J, rc);
    if (tref_isk(rc) && ir_knum(IR(tref_ref(rc)))->n == 0.5)
      return emitir(IRTN(IR_FPMATH), rb, IRFPM_SQRT);  /* x ^ 0.5 ==> sqrt(x) */
    /* Split up b^c into exp2(c*log2(b)). Assembler may rejoin later. */
    rb = emitir(IRTN(IR_FPMATH), rb, IRFPM_LOG2);
    rc = emitir(IRTN(IR_MUL), rb, rc);
    return emitir(IRTN(IR_FPMATH), rc, IRFPM_EXP2);
  }
  
  /* -- Predictive narrowing of induction variables ------------------------- */
  
  /* Narrow a single runtime value. */
  static int narrow_forl(jit_State *J, cTValue *o)
  {
    if (tvisint(o)) return 1;
    if (LJ_DUALNUM || (J->flags & JIT_F_OPT_NARROW)) return numisint(numV(o));
    return 0;
  }
  
  /* Narrow the FORL index type by looking at the runtime values. */
  IRType lj_opt_narrow_forl(jit_State *J, cTValue *tv)
  {
    lua_assert(tvisnumber(&tv[FORL_IDX]) &&
  	     tvisnumber(&tv[FORL_STOP]) &&
  	     tvisnumber(&tv[FORL_STEP]));
    /* Narrow only if the runtime values of start/stop/step are all integers. */
    if (narrow_forl(J, &tv[FORL_IDX]) &&
        narrow_forl(J, &tv[FORL_STOP]) &&
        narrow_forl(J, &tv[FORL_STEP])) {
      /* And if the loop index can't possibly overflow. */
      lua_Number step = numberVnum(&tv[FORL_STEP]);
      lua_Number sum = numberVnum(&tv[FORL_STOP]) + step;
      if (0 <= step ? (sum <= 2147483647.0) : (sum >= -2147483648.0))
        return IRT_INT;
    }
    return IRT_NUM;
  }
  
  #undef IR
  #undef fins
  #undef emitir
  #undef emitir_raw
  
  #endif
  