ljx

lj_opt_narrow.c (25734B)

---

 /*
 ** NARROW: Narrowing of numbers to integers (double to int32_t).
 ** STRIPOV: Stripping of overflow checks.
 ** Copyright (C) 2005-2016 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. */
 TRef LJ_FASTCALL lj_opt_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 = lj_opt_narrow_stripov(J, op1, lastop, mode);
       op2 = lj_opt_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 lj_opt_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 lj_opt_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 lj_opt_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));
 }
 
 /* Convert string to number. Error out for non-numeric string values. */
 static TRef conv_str_tonum(jit_State *J, TRef tr, TValue *o)
 {
   if (tref_isstr(tr)) {
     tr = emitir(IRTG(IR_STRTO, IRT_NUM), tr, 0);
     /* Would need an inverted STRTO for this rare and useless case. */
     if (!lj_strscan_num(strV(o), o))  /* Convert in-place. Value used below. */
       lj_trace_err(J, LJ_TRERR_BADTYPE);  /* Punt if non-numeric. */
   }
   return tr;
 }
 
 /* Narrowing of arithmetic operations. */
 TRef lj_opt_narrow_arith(jit_State *J, TRef rb, TRef rc,
 			 TValue *vb, TValue *vc, IROp op)
 {
   rb = conv_str_tonum(J, rb, vb);
   rc = conv_str_tonum(J, rc, 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)
 {
   rc = conv_str_tonum(J, rc, 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_ksimd(J, LJ_KSIMD_NEG));
 }
 
 /* Narrowing of modulo operator. */
 TRef lj_opt_narrow_mod(jit_State *J, TRef rb, TRef rc, TValue *vb, TValue *vc)
 {
   TRef tmp;
   rb = conv_str_tonum(J, rb, vb);
   rc = conv_str_tonum(J, rc, vc);
   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 *vb, TValue *vc)
 {
   rb = conv_str_tonum(J, rb, vb);
   rb = lj_ir_tonum(J, rb);  /* Left arg is always treated as an FP number. */
   rc = conv_str_tonum(J, rc, vc);
   /* 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)) {
       /* 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