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Simplify.lhs 65.4 KB
 keithw committed May 15, 2000 1 %  simonm committed Dec 02, 1998 2 % (c) The AQUA Project, Glasgow University, 1993-1998  partain committed Jan 08, 1996 3 4 5 6 % \section[Simplify]{The main module of the simplifier} \begin{code}  simonpj committed May 18, 1999 7 module Simplify ( simplTopBinds, simplExpr ) where  partain committed Jan 08, 1996 8   simonm committed Jan 08, 1998 9 #include "HsVersions.h"  partain committed Jan 08, 1996 10   simonmar committed Mar 18, 2005 11 import DynFlags ( dopt, DynFlag(Opt_D_dump_inlinings),  simonm committed Dec 02, 1998 12  SimplifierSwitch(..)  sof committed Sep 09, 1997 13  )  simonm committed Dec 02, 1998 14 import SimplMonad  simonpj@microsoft.com committed Nov 01, 2006 15 import Type hiding ( substTy, extendTvSubst )  simonpj committed Dec 24, 2004 16 import SimplEnv  simonpj@microsoft.com committed Nov 01, 2006 17 18 19 20 import SimplUtils import Id import IdInfo import Coercion  chak@cse.unsw.edu.au. committed Sep 15, 2006 21 import TcGadt ( dataConCanMatch )  simonpj@microsoft.com committed Sep 23, 2006 22 import DataCon ( dataConTyCon, dataConRepStrictness )  simonpj@microsoft.com committed Apr 12, 2006 23 import TyCon ( tyConArity, isAlgTyCon, isNewTyCon, tyConDataCons_maybe )  simonm committed Dec 02, 1998 24 import CoreSyn  simonpj committed Jan 11, 2001 25 import PprCore ( pprParendExpr, pprCoreExpr )  simonpj committed Aug 18, 2005 26 import CoreUnfold ( mkUnfolding, callSiteInline )  simonpj@microsoft.com committed Nov 01, 2006 27 import CoreUtils  simonpj committed May 18, 1999 28 import Rules ( lookupRule )  simonpj committed Sep 14, 2001 29 import BasicTypes ( isMarkedStrict )  simonmar committed Jul 11, 2000 30 import CostCentre ( currentCCS )  simonm committed Dec 02, 1998 31 import TysPrim ( realWorldStatePrimTy )  simonpj committed May 18, 1999 32 import PrelInfo ( realWorldPrimId )  simonpj committed Oct 18, 2001 33 import BasicTypes ( TopLevelFlag(..), isTopLevel,  simonpj@microsoft.com committed Nov 01, 2006 34  RecFlag(..), isNonRuleLoopBreaker )  simonpj@microsoft.com committed Apr 12, 2006 35 import List ( nub )  simonpj committed Sep 23, 2003 36 import Maybes ( orElse )  simonm committed Dec 02, 1998 37 import Outputable  simonpj@microsoft.com committed Nov 01, 2006 38 import Util  partain committed Jan 08, 1996 39 40 41 \end{code}  simonpj committed Sep 26, 2001 42 43 The guts of the simplifier is in this module, but the driver loop for the simplifier is in SimplCore.lhs.  simonpj committed May 18, 1999 44 45   simonpj committed Aug 01, 2000 46 47 48 49 50 51 52 53 ----------------------------------------- *** IMPORTANT NOTE *** ----------------------------------------- The simplifier used to guarantee that the output had no shadowing, but it does not do so any more. (Actually, it never did!) The reason is documented with simplifyArgs.  simonpj committed Sep 26, 2001 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 ----------------------------------------- *** IMPORTANT NOTE *** ----------------------------------------- Many parts of the simplifier return a bunch of "floats" as well as an expression. This is wrapped as a datatype SimplUtils.FloatsWith. All "floats" are let-binds, not case-binds, but some non-rec lets may be unlifted (with RHS ok-for-speculation). ----------------------------------------- ORGANISATION OF FUNCTIONS ----------------------------------------- simplTopBinds - simplify all top-level binders - for NonRec, call simplRecOrTopPair - for Rec, call simplRecBind ------------------------------ simplExpr (applied lambda) ==> simplNonRecBind simplExpr (Let (NonRec ...) ..) ==> simplNonRecBind simplExpr (Let (Rec ...) ..) ==> simplify binders; simplRecBind ------------------------------ simplRecBind [binders already simplfied] - use simplRecOrTopPair on each pair in turn simplRecOrTopPair [binder already simplified] Used for: recursive bindings (top level and nested) top-level non-recursive bindings Returns: - check for PreInlineUnconditionally - simplLazyBind simplNonRecBind Used for: non-top-level non-recursive bindings beta reductions (which amount to the same thing) Because it can deal with strict arts, it takes a "thing-inside" and returns an expression - check for PreInlineUnconditionally - simplify binder, including its IdInfo - if strict binding simplStrictArg mkAtomicArgs completeNonRecX else simplLazyBind addFloats simplNonRecX: [given a *simplified* RHS, but an *unsimplified* binder] Used for: binding case-binder and constr args in a known-constructor case - check for PreInLineUnconditionally - simplify binder - completeNonRecX ------------------------------ simplLazyBind: [binder already simplified, RHS not] Used for: recursive bindings (top level and nested) top-level non-recursive bindings non-top-level, but *lazy* non-recursive bindings [must not be strict or unboxed] Returns floats + an augmented environment, not an expression - substituteIdInfo and add result to in-scope [so that rules are available in rec rhs] - simplify rhs - mkAtomicArgs - float if exposes constructor or PAP  simonpj@microsoft.com committed Nov 01, 2006 124  - completeBind  simonpj committed Sep 26, 2001 125 126 127 128 129 130  completeNonRecX: [binder and rhs both simplified] - if the the thing needs case binding (unlifted and not ok-for-spec) build a Case else  simonpj@microsoft.com committed Nov 01, 2006 131  completeBind  simonpj committed Sep 26, 2001 132 133  addFloats  simonpj@microsoft.com committed Nov 01, 2006 134 completeBind: [given a simplified RHS]  simonpj committed Sep 26, 2001 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192  [used for both rec and non-rec bindings, top level and not] - try PostInlineUnconditionally - add unfolding [this is the only place we add an unfolding] - add arity Right hand sides and arguments ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ In many ways we want to treat (a) the right hand side of a let(rec), and (b) a function argument in the same way. But not always! In particular, we would like to leave these arguments exactly as they are, so they will match a RULE more easily. f (g x, h x) g (+ x) It's harder to make the rule match if we ANF-ise the constructor, or eta-expand the PAP: f (let { a = g x; b = h x } in (a,b)) g (\y. + x y) On the other hand if we see the let-defns p = (g x, h x) q = + x then we *do* want to ANF-ise and eta-expand, so that p and q can be safely inlined. Even floating lets out is a bit dubious. For let RHS's we float lets out if that exposes a value, so that the value can be inlined more vigorously. For example r = let x = e in (x,x) Here, if we float the let out we'll expose a nice constructor. We did experiments that showed this to be a generally good thing. But it was a bad thing to float lets out unconditionally, because that meant they got allocated more often. For function arguments, there's less reason to expose a constructor (it won't get inlined). Just possibly it might make a rule match, but I'm pretty skeptical. So for the moment we don't float lets out of function arguments either. Eta expansion ~~~~~~~~~~~~~~ For eta expansion, we want to catch things like case e of (a,b) -> \x -> case a of (p,q) -> \y -> r If the \x was on the RHS of a let, we'd eta expand to bring the two lambdas together. And in general that's a good thing to do. Perhaps we should eta expand wherever we find a (value) lambda? Then the eta expansion at a let RHS can concentrate solely on the PAP case.  simonpj committed Aug 01, 2000 193 194   simonpj committed May 18, 1999 195 196 197 198 199 200 201 %************************************************************************ %* * \subsection{Bindings} %* * %************************************************************************ \begin{code}  simonpj committed Sep 26, 2001 202 simplTopBinds :: SimplEnv -> [InBind] -> SimplM [OutBind]  simonpj committed May 18, 1999 203   simonpj committed Sep 26, 2001 204 simplTopBinds env binds  simonpj@microsoft.com committed Nov 01, 2006 205 206 207 208 209 210 211 212 213 214  = do { -- Put all the top-level binders into scope at the start -- so that if a transformation rule has unexpectedly brought -- anything into scope, then we don't get a complaint about that. -- It's rather as if the top-level binders were imported. ; env <- simplRecBndrs env (bindersOfBinds binds) ; dflags <- getDOptsSmpl ; let dump_flag = dopt Opt_D_dump_inlinings dflags ; env' <- simpl_binds dump_flag env binds ; freeTick SimplifierDone ; return (getFloats env') }  simonpj committed May 18, 1999 215  where  simonpj committed Nov 01, 1999 216 217  -- We need to track the zapped top-level binders, because -- they should have their fragile IdInfo zapped (notably occurrence info)  simonpj committed Sep 26, 2001 218  -- That's why we run down binds and bndrs' simultaneously.  simonpj@microsoft.com committed Nov 01, 2006 219 220 221 222 223  simpl_binds :: Bool -> SimplEnv -> [InBind] -> SimplM SimplEnv simpl_binds dump env [] = return env simpl_binds dump env (bind:binds) = do { env' <- trace dump bind $simpl_bind env bind ; simpl_binds dump env' binds }  simonpj committed Sep 26, 2001 224   simonpj@microsoft.com committed Nov 01, 2006 225 226  trace True bind = pprTrace "SimplBind" (ppr (bindersOf bind)) trace False bind = \x -> x  simonpj@microsoft.com committed Aug 16, 2006 227   simonpj@microsoft.com committed Nov 01, 2006 228 229  simpl_bind env (NonRec b r) = simplRecOrTopPair env TopLevel b r simpl_bind env (Rec pairs) = simplRecBind env TopLevel pairs  simonpj committed Sep 26, 2001 230 231 232 233 234 235 236 237 238 239 240 241 242 243 \end{code} %************************************************************************ %* * \subsection{Lazy bindings} %* * %************************************************************************ simplRecBind is used for * recursive bindings only \begin{code} simplRecBind :: SimplEnv -> TopLevelFlag  simonpj@microsoft.com committed Nov 01, 2006 244 245 246 247 248 249 250  -> [(InId, InExpr)] -> SimplM SimplEnv simplRecBind env top_lvl pairs = do { env' <- go (zapFloats env) pairs ; return (env addRecFloats env') } -- addFloats adds the floats from env', -- *and* updates env with the in-scope set from env'  simonpj committed May 18, 1999 251  where  simonpj@microsoft.com committed Nov 01, 2006 252  go env [] = return env  simonpj committed May 18, 1999 253   simonpj@microsoft.com committed Nov 01, 2006 254 255 256  go env ((bndr, rhs) : pairs) = do { env <- simplRecOrTopPair env top_lvl bndr rhs ; go env pairs }  simonpj committed Sep 26, 2001 257 258 \end{code}  simonpj@microsoft.com committed Nov 01, 2006 259 simplOrTopPair is used for  simonpj committed Sep 26, 2001 260 261 262 263 264 265 266 267  * recursive bindings (whether top level or not) * top-level non-recursive bindings It assumes the binder has already been simplified, but not its IdInfo. \begin{code} simplRecOrTopPair :: SimplEnv -> TopLevelFlag  simonpj@microsoft.com committed Nov 01, 2006 268 269  -> InId -> InExpr -- Binder and rhs -> SimplM SimplEnv -- Returns an env that includes the binding  simonpj committed Sep 26, 2001 270   simonpj@microsoft.com committed Nov 01, 2006 271 simplRecOrTopPair env top_lvl bndr rhs  simonmar committed Aug 03, 2005 272  | preInlineUnconditionally env top_lvl bndr rhs -- Check for unconditional inline  simonpj@microsoft.com committed Nov 01, 2006 273 274  = do { tick (PreInlineUnconditionally bndr) ; return (extendIdSubst env bndr (mkContEx env rhs)) }  simonpj committed Sep 26, 2001 275 276  | otherwise  simonpj@microsoft.com committed Nov 01, 2006 277 278 279  = do { let bndr' = lookupRecBndr env bndr (env', bndr'') = addLetIdInfo env bndr bndr' ; simplLazyBind env' top_lvl Recursive bndr bndr'' rhs env' }  simonpj committed Sep 26, 2001 280 281 282 283 284  -- May not actually be recursive, but it doesn't matter \end{code} simplLazyBind is used for  simonpj@microsoft.com committed Nov 01, 2006 285 286 287  * [simplRecOrTopPair] recursive bindings (whether top level or not) * [simplRecOrTopPair] top-level non-recursive bindings * [simplNonRecE] non-top-level *lazy* non-recursive bindings  simonpj committed Sep 26, 2001 288 289 290  Nota bene: 1. It assumes that the binder is *already* simplified,  simonpj@microsoft.com committed Nov 01, 2006 291  and is in scope, and its IdInfo too, except unfolding  simonpj committed Sep 26, 2001 292 293 294 295 296 297 298 299 300 301  2. It assumes that the binder type is lifted. 3. It does not check for pre-inline-unconditionallly; that should have been done already. \begin{code} simplLazyBind :: SimplEnv -> TopLevelFlag -> RecFlag -> InId -> OutId -- Binder, both pre-and post simpl  simonpj@microsoft.com committed Nov 01, 2006 302  -- The OutId has IdInfo, except arity, unfolding  simonpj committed Sep 26, 2001 303  -> InExpr -> SimplEnv -- The RHS and its environment  simonpj@microsoft.com committed Nov 01, 2006 304  -> SimplM SimplEnv  simonpj committed Sep 26, 2001 305   simonpj@microsoft.com committed Mar 01, 2006 306 simplLazyBind env top_lvl is_rec bndr bndr1 rhs rhs_se  simonpj@microsoft.com committed Nov 01, 2006 307 308 309  = do { let rhs_env = rhs_se setInScope env rhs_cont = mkRhsStop (idType bndr1)  simonpj committed Sep 30, 2004 310  -- Simplify the RHS; note the mkRhsStop, which tells  simonpj committed Sep 26, 2001 311  -- the simplifier that this is the RHS of a let.  simonpj@microsoft.com committed Nov 01, 2006 312  ; (rhs_env1, rhs1) <- simplExprF rhs_env rhs rhs_cont  simonpj committed Sep 26, 2001 313 314  -- If any of the floats can't be floated, give up now  simonpj@microsoft.com committed Nov 01, 2006 315 316 317 318 319  -- (The canFloat predicate says True for empty floats.) ; if (not (canFloat top_lvl is_rec False rhs_env1)) then completeBind env top_lvl bndr bndr1 (wrapFloats rhs_env1 rhs1) else do  simonpj committed Sep 26, 2001 320  -- ANF-ise a constructor or PAP rhs  simonpj@microsoft.com committed Nov 01, 2006 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382  { (rhs_env2, rhs2) <- prepareRhs rhs_env1 rhs1 ; (env', rhs3) <- chooseRhsFloats top_lvl is_rec False env rhs_env2 rhs2 ; completeBind env' top_lvl bndr bndr1 rhs3 } } chooseRhsFloats :: TopLevelFlag -> RecFlag -> Bool -> SimplEnv -- Env for the let -> SimplEnv -- Env for the RHS, with RHS floats in it -> OutExpr -- ..and the RHS itself -> SimplM (SimplEnv, OutExpr) -- New env for let, and RHS chooseRhsFloats top_lvl is_rec is_strict env rhs_env rhs | not (isEmptyFloats rhs_env) -- Something to float , canFloat top_lvl is_rec is_strict rhs_env -- ...that can float , (isTopLevel top_lvl || exprIsCheap rhs) -- ...and we want to float = do { tick LetFloatFromLet -- Float ; return (addFloats env rhs_env, rhs) } -- Add the floats to the main env | otherwise -- Don't float = return (env, wrapFloats rhs_env rhs) -- Wrap the floats around the RHS \end{code} %************************************************************************ %* * \subsection{simplNonRec} %* * %************************************************************************ A specialised variant of simplNonRec used when the RHS is already simplified, notably in knownCon. It uses case-binding where necessary. \begin{code} simplNonRecX :: SimplEnv -> InId -- Old binder -> OutExpr -- Simplified RHS -> SimplM SimplEnv simplNonRecX env bndr new_rhs = do { (env, bndr') <- simplBinder env bndr ; completeNonRecX env NotTopLevel NonRecursive (isStrictBndr bndr) bndr bndr' new_rhs } completeNonRecX :: SimplEnv -> TopLevelFlag -> RecFlag -> Bool -> InId -- Old binder -> OutId -- New binder -> OutExpr -- Simplified RHS -> SimplM SimplEnv completeNonRecX env top_lvl is_rec is_strict old_bndr new_bndr new_rhs = do { (env1, rhs1) <- prepareRhs (zapFloats env) new_rhs ; (env2, rhs2) <- chooseRhsFloats top_lvl is_rec is_strict env env1 rhs1 ; completeBind env2 NotTopLevel old_bndr new_bndr rhs2 } \end{code} {- No, no, no! Do not try preInlineUnconditionally in completeNonRecX Doing so risks exponential behaviour, because new_rhs has been simplified once already In the cases described by the folowing commment, postInlineUnconditionally will catch many of the relevant cases. -- This happens; for example, the case_bndr during case of -- known constructor: case (a,b) of x { (p,q) -> ... } -- Here x isn't mentioned in the RHS, so we don't want to -- create the (dead) let-binding let x = (a,b) in ...  simonpj committed Sep 11, 2003 383  --  simonpj@microsoft.com committed Nov 01, 2006 384 385 386  -- Similarly, single occurrences can be inlined vigourously -- e.g. case (f x, g y) of (a,b) -> .... -- If a,b occur once we can avoid constructing the let binding for them.  simonpj committed Sep 26, 2001 387   simonpj@microsoft.com committed Nov 01, 2006 388 389 390 391 392 393 394  Furthermore in the case-binding case preInlineUnconditionally risks extra thunks -- Consider case I# (quotInt# x y) of -- I# v -> let w = J# v in ... -- If we gaily inline (quotInt# x y) for v, we end up building an -- extra thunk: -- let w = J# (quotInt# x y) in ... -- because quotInt# can fail.  simonpj committed Sep 26, 2001 395   simonpj@microsoft.com committed Nov 01, 2006 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412  | preInlineUnconditionally env NotTopLevel bndr new_rhs = thing_inside (extendIdSubst env bndr (DoneEx new_rhs)) -} prepareRhs takes a putative RHS, checks whether it's a PAP or constructor application and, if so, converts it to ANF, so that the resulting thing can be inlined more easily. Thus x = (f a, g b) becomes t1 = f a t2 = g b x = (t1,t2) \begin{code} prepareRhs :: SimplEnv -> OutExpr -> SimplM (SimplEnv, OutExpr) -- Adds new floats to the env iff that allows us to return a good RHS  simonpj@microsoft.com committed Jan 10, 2007 413 prepareRhs env (Cast rhs co) -- Note [Float coercions]  simonpj@microsoft.com committed Nov 01, 2006 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465  = do { (env', rhs') <- makeTrivial env rhs ; return (env', Cast rhs' co) } prepareRhs env rhs | (Var fun, args) <- collectArgs rhs -- It's an application , let n_args = valArgCount args , n_args > 0 -- ...but not a trivial one , isDataConWorkId fun || n_args < idArity fun -- ...and it's a constructor or PAP = go env (Var fun) args where go env fun [] = return (env, fun) go env fun (arg : args) = do { (env', arg') <- makeTrivial env arg ; go env' (App fun arg') args } prepareRhs env rhs -- The default case = return (env, rhs) \end{code} Note [Float coercions] ~~~~~~~~~~~~~~~~~~~~~~ When we find the binding x = e cast co we'd like to transform it to x' = e x = x cast co -- A trivial binding There's a chance that e will be a constructor application or function, or something like that, so moving the coerion to the usage site may well cancel the coersions and lead to further optimisation. Example: data family T a :: * data instance T Int = T Int foo :: Int -> Int -> Int foo m n = ... where x = T m go 0 = 0 go n = case x of { T m -> go (n-m) } -- This case should optimise \begin{code} makeTrivial :: SimplEnv -> OutExpr -> SimplM (SimplEnv, OutExpr) -- Binds the expression to a variable, if it's not trivial, returning the variable makeTrivial env expr | exprIsTrivial expr = return (env, expr) | otherwise -- See Note [Take care] below = do { var <- newId FSLIT("a") (exprType expr) ; env <- completeNonRecX env NotTopLevel NonRecursive False var var expr ; return (env, substExpr env (Var var)) }  simonpj committed May 18, 1999 466 \end{code}  partain committed Jan 08, 1996 467 468   simonpj committed Sep 26, 2001 469 470 471 472 473 474 %************************************************************************ %* * \subsection{Completing a lazy binding} %* * %************************************************************************  simonpj@microsoft.com committed Nov 01, 2006 475 476 477 478 479 completeBind * deals only with Ids, not TyVars * takes an already-simplified binder and RHS * is used for both recursive and non-recursive bindings * is used for both top-level and non-top-level bindings  simonpj committed Sep 26, 2001 480 481 482 483 484 485 486 487  It does the following: - tries discarding a dead binding - tries PostInlineUnconditionally - add unfolding [this is the only place we add an unfolding] - add arity It does *not* attempt to do let-to-case. Why? Because it is used for  simonpj@microsoft.com committed Nov 01, 2006 488 489  - top-level bindings (when let-to-case is impossible) - many situations where the "rhs" is known to be a WHNF  simonpj committed Sep 26, 2001 490 491  (so let-to-case is inappropriate).  simonpj@microsoft.com committed Nov 01, 2006 492 493 Nor does it do the atomic-argument thing  simonpj committed Sep 26, 2001 494 \begin{code}  simonpj@microsoft.com committed Nov 01, 2006 495 496 497 498 499 500 501 502 503 504 completeBind :: SimplEnv -> TopLevelFlag -- Flag stuck into unfolding -> InId -- Old binder -> OutId -> OutExpr -- New binder and RHS -> SimplM SimplEnv -- completeBind may choose to do its work -- * by extending the substitution (e.g. let x = y in ...) -- * or by adding to the floats in the envt completeBind env top_lvl old_bndr new_bndr new_rhs  simonmar committed Aug 12, 2005 505  | postInlineUnconditionally env top_lvl new_bndr occ_info new_rhs unfolding  simonpj@microsoft.com committed Nov 01, 2006 506 507 508 509 510 511  -- Inline and discard the binding = do { tick (PostInlineUnconditionally old_bndr) ; -- pprTrace "postInlineUnconditionally" (ppr old_bndr <+> ppr new_bndr <+> ppr new_rhs)$ return (extendIdSubst env old_bndr (DoneEx new_rhs)) } -- Use the substitution to make quite, quite sure that the -- substitution will happen, since we are going to discard the binding  simonpj committed Sep 26, 2001 512 513 514  | otherwise = let  simonpj@microsoft.com committed Oct 04, 2006 515  -- Arity info  simonpj committed Sep 26, 2001 516 517  new_bndr_info = idInfo new_bndr setArityInfo exprArity new_rhs  simonpj@microsoft.com committed Oct 04, 2006 518  -- Unfolding info  simonpj committed Sep 23, 2003 519 520 521 522 523  -- Add the unfolding *only* for non-loop-breakers -- Making loop breakers not have an unfolding at all -- means that we can avoid tests in exprIsConApp, for example. -- This is important: if exprIsConApp says 'yes' for a recursive -- thing, then we can get into an infinite loop  simonpj@microsoft.com committed Oct 04, 2006 524 525  -- Demand info  simonpj committed Sep 23, 2003 526 527 528 529 530 531 532 533 534 535  -- If the unfolding is a value, the demand info may -- go pear-shaped, so we nuke it. Example: -- let x = (a,b) in -- case x of (p,q) -> h p q x -- Here x is certainly demanded. But after we've nuked -- the case, we'll get just -- let x = (a,b) in h a b x -- and now x is not demanded (I'm assuming h is lazy) -- This really happens. Similarly -- let f = \x -> e in ...f..f...  simonpj@microsoft.com committed Oct 05, 2006 536  -- After inlining f at some of its call sites the original binding may  simonpj committed Sep 23, 2003 537 538 539 540 541 542 543 544  -- (for example) be no longer strictly demanded. -- The solution here is a bit ad hoc... info_w_unf = new_bndr_info setUnfoldingInfo unfolding final_info | loop_breaker = new_bndr_info | isEvaldUnfolding unfolding = zapDemandInfo info_w_unf orElse info_w_unf | otherwise = info_w_unf final_id = new_bndr setIdInfo final_info  simonpj committed Sep 26, 2001 545 546 547 548  in -- These seqs forces the Id, and hence its IdInfo, -- and hence any inner substitutions final_id seq  simonpj@microsoft.com committed Aug 16, 2006 549  -- pprTrace "Binding" (ppr final_id <+> ppr unfolding) $ simonpj@microsoft.com committed Nov 01, 2006 550  return (addNonRec env final_id new_rhs)  simonpj committed Sep 26, 2001 551  where  simonmar committed Aug 03, 2005 552  unfolding = mkUnfolding (isTopLevel top_lvl) new_rhs  simonpj@microsoft.com committed Oct 04, 2006 553  loop_breaker = isNonRuleLoopBreaker occ_info  simonpj committed Sep 26, 2001 554 555  old_info = idInfo old_bndr occ_info = occInfo old_info  SamB committed Nov 10, 2006 556 \end{code}  simonpj committed Sep 26, 2001 557 558 559   partain committed Jan 08, 1996 560 561 562 563 564 565 %************************************************************************ %* * \subsection[Simplify-simplExpr]{The main function: simplExpr} %* * %************************************************************************  simonpj committed Dec 18, 1998 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 The reason for this OutExprStuff stuff is that we want to float *after* simplifying a RHS, not before. If we do so naively we get quadratic behaviour as things float out. To see why it's important to do it after, consider this (real) example: let t = f x in fst t ==> let t = let a = e1 b = e2 in (a,b) in fst t ==> let a = e1 b = e2 t = (a,b) in a -- Can't inline a this round, cos it appears twice ==> e1 Each of the ==> steps is a round of simplification. We'd save a whole round if we float first. This can cascade. Consider let f = g d in \x -> ...f... ==> let f = let d1 = ..d.. in \y -> e in \x -> ...f... ==> let d1 = ..d.. in \x -> ...(\y ->e)... Only in this second round can the \y be applied, and it might do the same again.  partain committed Jan 08, 1996 604 \begin{code}  simonpj committed Sep 26, 2001 605 simplExpr :: SimplEnv -> CoreExpr -> SimplM CoreExpr  simonpj committed Sep 30, 2004 606 simplExpr env expr = simplExprC env expr (mkBoringStop expr_ty')  simonpj committed Sep 26, 2001 607  where  simonpj committed Dec 24, 2004 608  expr_ty' = substTy env (exprType expr)  simonpj committed Sep 26, 2001 609  -- The type in the Stop continuation, expr_ty', is usually not used  simonpj committed May 18, 1999 610  -- It's only needed when discarding continuations after finding  simonpj committed Jul 14, 1999 611 612  -- a function that returns bottom. -- Hence the lazy substitution  partain committed Jan 08, 1996 613   simonpj committed Dec 18, 1998 614   simonpj committed Sep 26, 2001 615 616 617 simplExprC :: SimplEnv -> CoreExpr -> SimplCont -> SimplM CoreExpr -- Simplify an expression, given a continuation simplExprC env expr cont  simonpj@microsoft.com committed Nov 01, 2006 618 619 620 621 622 623 624 625 626 627 628  = -- pprTrace "simplExprC" (ppr expr $$ppr cont {-$$ ppr (seIdSubst env) -} $$ppr (seFloats env) ) do { (env', expr') <- simplExprF (zapFloats env) expr cont ; -- pprTrace "simplExprC ret" (ppr expr$$ ppr expr')$ -- pprTrace "simplExprC ret3" (ppr (seInScope env')) $-- pprTrace "simplExprC ret4" (ppr (seFloats env'))$ return (wrapFloats env' expr') } -------------------------------------------------- simplExprF :: SimplEnv -> InExpr -> SimplCont -> SimplM (SimplEnv, OutExpr)  simonpj@microsoft.com committed Nov 06, 2006 629 630 631 simplExprF env e cont = -- pprTrace "simplExprF" (ppr e $$ppr cont$$ ppr (seTvSubst env) $$ppr (seIdSubst env) {-$$ ppr (seFloats env) -} ) $simplExprF' env e cont  simonpj@microsoft.com committed Nov 01, 2006 632   simonpj@microsoft.com committed Jan 03, 2007 633 simplExprF' env (Var v) cont = simplVar env v cont  simonpj@microsoft.com committed Nov 01, 2006 634 635 636 637 simplExprF' env (Lit lit) cont = rebuild env (Lit lit) cont simplExprF' env (Note n expr) cont = simplNote env n expr cont simplExprF' env (Cast body co) cont = simplCast env body co cont simplExprF' env (App fun arg) cont = simplExprF env fun$  simonpj@microsoft.com committed Jan 03, 2007 638  ApplyTo NoDup arg env cont  simonpj@microsoft.com committed Nov 01, 2006 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656  simplExprF' env expr@(Lam _ _) cont = simplLam env (map zap bndrs) body cont -- The main issue here is under-saturated lambdas -- (\x1. \x2. e) arg1 -- Here x1 might have "occurs-once" occ-info, because occ-info -- is computed assuming that a group of lambdas is applied -- all at once. If there are too few args, we must zap the -- occ-info. where n_args = countArgs cont n_params = length bndrs (bndrs, body) = collectBinders expr zap | n_args >= n_params = \b -> b | otherwise = \b -> if isTyVar b then b else zapLamIdInfo b -- NB: we count all the args incl type args -- so we must count all the binders (incl type lambdas)  simonm committed Dec 02, 1998 657   simonpj@microsoft.com committed Nov 01, 2006 658 simplExprF' env (Type ty) cont  simonpj committed Sep 26, 2001 659  = ASSERT( contIsRhsOrArg cont )  simonpj@microsoft.com committed Nov 01, 2006 660 661  do { ty' <- simplType env ty ; rebuild env (Type ty') cont }  simonpj committed Mar 08, 2001 662   simonpj@microsoft.com committed Nov 01, 2006 663 simplExprF' env (Case scrut bndr case_ty alts) cont  simonpj committed Sep 26, 2001 664 665 666  | not (switchIsOn (getSwitchChecker env) NoCaseOfCase) = -- Simplify the scrutinee with a Select continuation simplExprF env scrut (Select NoDup bndr alts env cont)  simonpj committed Mar 24, 2000 667   simonpj committed Sep 26, 2001 668 669  | otherwise = -- If case-of-case is off, simply simplify the case expression  simonpj committed Mar 24, 2000 670  -- in a vanilla Stop context, and rebuild the result around it  simonpj@microsoft.com committed Nov 01, 2006 671 672  do { case_expr' <- simplExprC env scrut case_cont ; rebuild env case_expr' cont }  simonpj committed Sep 26, 2001 673  where  simonpj committed Sep 30, 2004 674  case_cont = Select NoDup bndr alts env (mkBoringStop case_ty')  simonpj committed Dec 24, 2004 675  case_ty' = substTy env case_ty -- c.f. defn of simplExpr  simonpj committed May 18, 1999 676   simonpj@microsoft.com committed Nov 01, 2006 677 678 679 680 simplExprF' env (Let (Rec pairs) body) cont = do { env <- simplRecBndrs env (map fst pairs) -- NB: bndrs' don't have unfoldings or rules -- We add them as we go down  simonm committed Dec 02, 1998 681   simonpj@microsoft.com committed Nov 01, 2006 682 683  ; env <- simplRecBind env NotTopLevel pairs ; simplExprF env body cont }  simonpj committed May 18, 1999 684   simonpj@microsoft.com committed Nov 01, 2006 685 686 simplExprF' env (Let (NonRec bndr rhs) body) cont = simplNonRecE env bndr (rhs, env) ([], body) cont  simonpj committed May 18, 1999 687 688  ---------------------------------  simonpj committed Sep 26, 2001 689 690 691 simplType :: SimplEnv -> InType -> SimplM OutType -- Kept monadic just so we can do the seqType simplType env ty  simonpj@microsoft.com committed Nov 01, 2006 692 693  = -- pprTrace "simplType" (ppr ty $$ppr (seTvSubst env))  seqType new_ty seq returnSmpl new_ty  simonpj committed Sep 26, 2001 694  where  simonpj committed Dec 24, 2004 695  new_ty = substTy env ty  simonpj committed Mar 08, 2001 696 697 698 \end{code}  simonpj@microsoft.com committed Nov 01, 2006 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 %************************************************************************ %* * \subsection{The main rebuilder} %* * %************************************************************************ \begin{code} rebuild :: SimplEnv -> OutExpr -> SimplCont -> SimplM (SimplEnv, OutExpr) -- At this point the substitution in the SimplEnv should be irrelevant -- only the in-scope set and floats should matter rebuild env expr cont = -- pprTrace "rebuild" (ppr expr$$ ppr cont $$ppr (seFloats env))  case cont of Stop {} -> return (env, expr) CoerceIt co cont -> rebuild env (mkCoerce co expr) cont Select _ bndr alts se cont -> rebuildCase (se setFloats env) expr bndr alts cont StrictArg fun ty info cont -> rebuildCall env (fun App expr) (funResultTy ty) info cont StrictBind b bs body se cont -> do { env' <- simplNonRecX (se setFloats env) b expr ; simplLam env' bs body cont } ApplyTo _ arg se cont -> do { arg' <- simplExpr (se setInScope env) arg ; rebuild env (App expr arg') cont } \end{code}  chak@cse.unsw.edu.au. committed Sep 15, 2006 723 724 725 726 727 728 729 %************************************************************************ %* * \subsection{Lambdas} %* * %************************************************************************ \begin{code}  simonpj@microsoft.com committed Nov 01, 2006 730 731 simplCast :: SimplEnv -> InExpr -> Coercion -> SimplCont -> SimplM (SimplEnv, OutExpr)  chak@cse.unsw.edu.au. committed Sep 15, 2006 732 simplCast env body co cont  simonpj@microsoft.com committed Nov 01, 2006 733 734 735  = do { co' <- simplType env co ; simplExprF env body (addCoerce co' cont) } where  simonpj@microsoft.com committed Jan 03, 2007 736 737 738 739 740 741  addCoerce co cont = add_coerce co (coercionKind co) cont add_coerce co (s1, k1) cont | s1 coreEqType k1 = cont add_coerce co1 (s1, k2) (CoerceIt co2 cont) | (l1, t1) <- coercionKind co2  chak@cse.unsw.edu.au. committed Sep 15, 2006 742 743 744 745 746 747 748 749 750 751 752 753  -- coerce T1 S1 (coerce S1 K1 e) -- ==> -- e, if T1=K1 -- coerce T1 K1 e, otherwise -- -- For example, in the initial form of a worker -- we may find (coerce T (coerce S (\x.e))) y -- and we'd like it to simplify to e[y/x] in one round -- of simplification , s1 coreEqType t1 = cont -- The coerces cancel out | otherwise = CoerceIt (mkTransCoercion co1 co2) cont  simonpj@microsoft.com committed Jan 03, 2007 754 755  add_coerce co (s1s2, t1t2) (ApplyTo dup arg arg_se cont) | not (isTypeArg arg) -- This whole case only works for value args  chak@cse.unsw.edu.au. committed Sep 15, 2006 756  -- Could upgrade to have equiv thing for type apps too  simonpj@microsoft.com committed Jan 03, 2007 757  , isFunTy s1s2 -- t1t2 must be a function type, becuase it's applied  chak@cse.unsw.edu.au. committed Sep 15, 2006 758 759 760 761 762 763 764 765 766 767 768 769  -- co : s1s2 :=: t1t2 -- (coerce (T1->T2) (S1->S2) F) E -- ===> -- coerce T2 S2 (F (coerce S1 T1 E)) -- -- t1t2 must be a function type, T1->T2, because it's applied -- to something but s1s2 might conceivably not be -- -- When we build the ApplyTo we can't mix the out-types -- with the InExpr in the argument, so we simply substitute -- to make it all consistent. It's a bit messy. -- But it isn't a common case.  simonpj@microsoft.com committed Jan 03, 2007 770 771  -- -- Example of use: Trac #995  simonpj@microsoft.com committed Nov 01, 2006 772  = ApplyTo dup new_arg (zapSubstEnv env) (addCoerce co2 cont)  chak@cse.unsw.edu.au. committed Sep 15, 2006 773 774 775 776 777  where -- we split coercion t1->t2 :=: s1->s2 into t1 :=: s1 and -- t2 :=: s2 with left and right on the curried form: -- (->) t1 t2 :=: (->) s1 s2 [co1, co2] = decomposeCo 2 co  chak@cse.unsw.edu.au. committed Sep 19, 2006 778  new_arg = mkCoerce (mkSymCoercion co1) arg'  simonpj@microsoft.com committed Nov 01, 2006 779 780  arg' = substExpr arg_se arg  simonpj@microsoft.com committed Jan 03, 2007 781  add_coerce co _ cont = CoerceIt co cont  chak@cse.unsw.edu.au. committed Sep 15, 2006 782 783 \end{code}  simonpj@microsoft.com committed Nov 01, 2006 784   simonpj committed Mar 08, 2001 785 786 787 788 789 %************************************************************************ %* * \subsection{Lambdas} %* * %************************************************************************  simonpj committed May 18, 1999 790 791  \begin{code}  simonpj@microsoft.com committed Nov 01, 2006 792 793 794 795 simplLam :: SimplEnv -> [InId] -> InExpr -> SimplCont -> SimplM (SimplEnv, OutExpr) simplLam env [] body cont = simplExprF env body cont  simonpj committed May 18, 1999 796 797  -- Type-beta reduction  simonpj@microsoft.com committed Nov 01, 2006 798 799 800 801 802 simplLam env (bndr:bndrs) body (ApplyTo _ (Type ty_arg) arg_se cont) = ASSERT( isTyVar bndr ) do { tick (BetaReduction bndr) ; ty_arg' <- simplType (arg_se setInScope env) ty_arg ; simplLam (extendTvSubst env bndr ty_arg') bndrs body cont }  simonpj committed May 18, 1999 803 804  -- Ordinary beta reduction  simonpj@microsoft.com committed Nov 01, 2006 805 806 807 simplLam env (bndr:bndrs) body (ApplyTo _ arg arg_se cont) = do { tick (BetaReduction bndr) ; simplNonRecE env bndr (arg, arg_se) (bndrs, body) cont }  simonpj committed May 18, 1999 808   simonpj committed Sep 26, 2001 809  -- Not enough args, so there are real lambdas left to put in the result  simonpj@microsoft.com committed Nov 01, 2006 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 simplLam env bndrs body cont = do { (env, bndrs') <- simplLamBndrs env bndrs ; body' <- simplExpr env body ; new_lam <- mkLam bndrs' body' ; rebuild env new_lam cont } ------------------ simplNonRecE :: SimplEnv -> InId -- The binder -> (InExpr, SimplEnv) -- Rhs of binding (or arg of lambda) -> ([InId], InExpr) -- Body of the let/lambda -- \xs.e -> SimplCont -> SimplM (SimplEnv, OutExpr) -- simplNonRecE is used for -- * non-top-level non-recursive lets in expressions -- * beta reduction -- -- It deals with strict bindings, via the StrictBind continuation, -- which may abort the whole process -- -- The "body" of the binding comes as a pair of ([InId],InExpr) -- representing a lambda; so we recurse back to simplLam -- Why? Because of the binder-occ-info-zapping done before -- the call to simplLam in simplExprF (Lam ...) simplNonRecE env bndr (rhs, rhs_se) (bndrs, body) cont | preInlineUnconditionally env NotTopLevel bndr rhs = do { tick (PreInlineUnconditionally bndr) ; simplLam (extendIdSubst env bndr (mkContEx rhs_se rhs)) bndrs body cont } | isStrictBndr bndr = do { simplExprF (rhs_se setFloats env) rhs (StrictBind bndr bndrs body env cont) } | otherwise = do { (env, bndr') <- simplBinder env bndr ; env <- simplLazyBind env NotTopLevel NonRecursive bndr bndr' rhs rhs_se ; simplLam env bndrs body cont }  partain committed Jan 08, 1996 850 851 \end{code}  simonpj committed May 18, 1999 852   simonpj committed Mar 08, 2001 853 854 855 856 857 858 %************************************************************************ %* * \subsection{Notes} %* * %************************************************************************  sof committed May 18, 1997 859 \begin{code}  simonpj@microsoft.com committed Nov 01, 2006 860 861 -- Hack alert: we only distinguish subsumed cost centre stacks for the -- purposes of inlining. All other CCCSs are mapped to currentCCS.  simonpj committed Sep 26, 2001 862 simplNote env (SCC cc) e cont  simonpj@microsoft.com committed Nov 01, 2006 863 864  = do { e' <- simplExpr (setEnclosingCC env currentCCS) e ; rebuild env (mkSCC cc e') cont }  simonpj committed Sep 26, 2001 865 866 867 868  -- See notes with SimplMonad.inlineMode simplNote env InlineMe e cont | contIsRhsOrArg cont -- Totally boring continuation; see notes above  simonpj@microsoft.com committed Nov 01, 2006 869 870 871  = do { -- Don't inline inside an INLINE expression e' <- simplExpr (setMode inlineMode env) e ; rebuild env (mkInlineMe e') cont }  simonpj committed Mar 08, 2001 872 873 874 875  | otherwise -- Dissolve the InlineMe note if there's -- an interesting context of any kind to combine with -- (even a type application -- anything except Stop)  simonpj committed Sep 26, 2001 876  = simplExprF env e cont  simonpj committed Feb 20, 2003 877 878  simplNote env (CoreNote s) e cont  andy@galois.com committed Nov 29, 2006 879 880  = simplExpr env e thenSmpl \ e' -> rebuild env (Note (CoreNote s) e') cont  partain committed Jan 08, 1996 881 882 883 \end{code}  simonm committed Dec 02, 1998 884 885 %************************************************************************ %* *  simonpj committed Sep 26, 2001 886 \subsection{Dealing with calls}  simonm committed Dec 02, 1998 887 888 %* * %************************************************************************  partain committed Jan 08, 1996 889   simonpj committed May 18, 1999 890 \begin{code}  simonpj committed Sep 26, 2001 891 simplVar env var cont  simonpj committed Dec 24, 2004 892 893 894  = case substId env var of DoneEx e -> simplExprF (zapSubstEnv env) e cont ContEx tvs ids e -> simplExprF (setSubstEnv env tvs ids) e cont  simonpj@microsoft.com committed Oct 05, 2006 895  DoneId var1 -> completeCall (zapSubstEnv env) var1 cont  simonpj committed Sep 30, 2004 896  -- Note [zapSubstEnv]  simonpj committed Jun 22, 1999 897 898 899 900 901 902 903 904  -- The template is already simplified, so don't re-substitute. -- This is VITAL. Consider -- let x = e in -- let y = \z -> ...x... in -- \ x -> ...y... -- We'll clone the inner \x, adding x->x' in the id_subst -- Then when we inline y, we must *not* replace x by x' in -- the inlined copy!!  simonpj committed Sep 17, 1999 905   simonpj committed Mar 23, 2000 906 ---------------------------------------------------------  simonpj committed Oct 01, 2001 907 -- Dealing with a call site  simonpj committed Mar 23, 2000 908   simonpj@microsoft.com committed Oct 05, 2006 909 completeCall env var cont  simonpj@microsoft.com committed Nov 01, 2006 910 911 912 913 914 915 916 917 918 919 920 921  = do { dflags <- getDOptsSmpl ; let (args,call_cont) = contArgs cont -- The args are OutExprs, obtained by *lazily* substituting -- in the args found in cont. These args are only examined -- to limited depth (unless a rule fires). But we must do -- the substitution; rule matching on un-simplified args would -- be bogus ------------- First try rules ---------------- -- Do this before trying inlining. Some functions have -- rules *and* are strict; in this case, we don't want to -- inline the wrapper of the non-specialised thing; better  simonpj committed Sep 17, 1999 922  -- to call the specialised thing instead.  simonpj@microsoft.com committed Nov 01, 2006 923  --  simonpj committed Oct 01, 2001 924 925 926  -- We used to use the black-listing mechanism to ensure that inlining of -- the wrapper didn't occur for things that have specialisations till a -- later phase, so but now we just try RULES first  simonpj committed Jan 03, 2001 927  --  simonpj committed Jan 04, 2001 928 929 930 931 932 933 934 935 936 937 938  -- You might think that we shouldn't apply rules for a loop breaker: -- doing so might give rise to an infinite loop, because a RULE is -- rather like an extra equation for the function: -- RULE: f (g x) y = x+y -- Eqn: f a y = a-y -- -- But it's too drastic to disable rules for loop breakers. -- Even the foldr/build rule would be disabled, because foldr -- is recursive, and hence a loop breaker: -- foldr k z (build g) = g k z -- So it's up to the programmer: rules can cause divergence  simonpj@microsoft.com committed Nov 01, 2006 939 940 941 942 943 944 945 946 947  ; let in_scope = getInScope env rules = getRules env maybe_rule = case activeRule env of Nothing -> Nothing -- No rules apply Just act_fn -> lookupRule act_fn in_scope rules var args ; case maybe_rule of { Just (rule, rule_rhs) -> tick (RuleFired (ru_name rule)) thenSmpl_  simonpj committed Jan 11, 2001 948 949  (if dopt Opt_D_dump_inlinings dflags then pprTrace "Rule fired" (vcat [  simonpj@microsoft.com committed Nov 01, 2006 950  text "Rule:" <+> ftext (ru_name rule),  simonpj committed Oct 01, 2001 951  text "Before:" <+> ppr var <+> sep (map pprParendExpr args),  simonpj committed Nov 19, 2001 952 953  text "After: " <+> pprCoreExpr rule_rhs, text "Cont: " <+> ppr call_cont])  simonpj committed Jan 11, 2001 954 955  else id)   simonpj@microsoft.com committed Nov 01, 2006 956 957  simplExprF env rule_rhs (dropArgs (ruleArity rule) cont) -- The ruleArity says how many args the rule consumed  simonpj committed Sep 17, 1999 958   simonpj@microsoft.com committed Nov 01, 2006 959 960 961 962 963 964 965 966 967 968  ; Nothing -> do -- No rules ------------- Next try inlining ---------------- { let arg_infos = [interestingArg arg | arg <- args, isValArg arg] n_val_args = length arg_infos interesting_cont = interestingCallContext (notNull args) (notNull arg_infos) call_cont active_inline = activeInline env var maybe_inline = callSiteInline dflags active_inline  simonpj committed Oct 01, 2001 969  var arg_infos interesting_cont  simonpj@microsoft.com committed Nov 01, 2006 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989  ; case maybe_inline of { Just unfolding -- There is an inlining! -> do { tick (UnfoldingDone var) ; (if dopt Opt_D_dump_inlinings dflags then pprTrace "Inlining done" (vcat [ text "Before:" <+> ppr var <+> sep (map pprParendExpr args), text "Inlined fn: " <+> nest 2 (ppr unfolding), text "Cont: " <+> ppr call_cont]) else id) simplExprF env unfolding cont } ; Nothing -> -- No inlining! ------------- No inlining! ---------------- -- Next, look for rules or specialisations that match -- rebuildCall env (Var var) (idType var) (mkArgInfo var n_val_args call_cont) cont }}}}  simonpj committed Dec 22, 1998 990   simonpj@microsoft.com committed Nov 01, 2006 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 rebuildCall :: SimplEnv -> OutExpr -> OutType -- Function and its type -> (Bool, [Bool]) -- See SimplUtils.mkArgInfo -> SimplCont -> SimplM (SimplEnv, OutExpr) rebuildCall env fun fun_ty (has_rules, []) cont -- When we run out of strictness args, it means -- that the call is definitely bottom; see SimplUtils.mkArgInfo -- Then we want to discard the entire strict continuation. E.g. -- * case (error "hello") of { ... } -- * (error "Hello") arg -- * f (error "Hello") where f is strict -- etc -- Then, especially in the first of these cases, we'd like to discard -- the continuation, leaving just the bottoming expression. But the -- type might not be right, so we may have to add a coerce. | not (contIsTrivial cont) -- Only do thia if there is a non-trivial = return (env, mk_coerce fun) -- contination to discard, else we do it where -- again and again! cont_ty = contResultType cont co = mkUnsafeCoercion fun_ty cont_ty mk_coerce expr | cont_ty coreEqType fun_ty = fun | otherwise = mkCoerce co fun rebuildCall env fun fun_ty info (ApplyTo _ (Type arg_ty) se cont) = do { ty' <- simplType (se setInScope env) arg_ty ; rebuildCall env (fun App Type ty') (applyTy fun_ty ty') info cont } rebuildCall env fun fun_ty (has_rules, str:strs) (ApplyTo _ arg arg_se cont) | str || isStrictType arg_ty -- Strict argument = -- pprTrace "Strict Arg" (ppr arg$$ ppr (seIdSubst env) $$ppr (seInScope env))  simplExprF (arg_se setFloats env) arg (StrictArg fun fun_ty (has_rules, strs) cont) -- Note [Shadowing] | otherwise -- Lazy argument -- DO NOT float anything outside, hence simplExprC -- There is no benefit (unlike in a let-binding), and we'd -- have to be very careful about bogus strictness through -- floating a demanded let. = do { arg' <- simplExprC (arg_se setInScope env) arg (mkLazyArgStop arg_ty has_rules) ; rebuildCall env (fun App arg') res_ty (has_rules, strs) cont }  simonpj committed Sep 26, 2001 1034  where  simonpj@microsoft.com committed Nov 01, 2006 1035  (arg_ty, res_ty) = splitFunTy fun_ty  simonpj committed Dec 22, 1998 1036   simonpj@microsoft.com committed Nov 01, 2006 1037 1038 rebuildCall env fun fun_ty info cont = rebuild env fun cont  simonpj committed Sep 26, 2001 1039 \end{code}  simonpj committed Dec 18, 1998 1040   simonpj@microsoft.com committed Nov 01, 2006 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 Note [Shadowing] ~~~~~~~~~~~~~~~~ This part of the simplifier may break the no-shadowing invariant Consider f (...(\a -> e)...) (case y of (a,b) -> e') where f is strict in its second arg If we simplify the innermost one first we get (...(\a -> e)...) Simplifying the second arg makes us float the case out, so we end up with case y of (a,b) -> f (...(\a -> e)...) e' So the output does not have the no-shadowing invariant. However, there is no danger of getting name-capture, because when the first arg was simplified we used an in-scope set that at least mentioned all the variables free in its static environment, and that is enough. We can't just do innermost first, or we'd end up with a dual problem: case x of (a,b) -> f e (...(\a -> e')...) I spent hours trying to recover the no-shadowing invariant, but I just could not think of an elegant way to do it. The simplifier is already knee-deep in continuations. We have to keep the right in-scope set around; AND we have to get the effect that finding (error "foo") in a strict arg position will discard the entire application and replace it with (error "foo"). Getting all this at once is TOO HARD!  simonpj committed Dec 22, 1998 1064   simonpj committed Sep 26, 2001 1065 1066 %************************************************************************ %* *  simonpj@microsoft.com committed Nov 01, 2006 1067  Rebuilding a cse expression  simonpj committed Sep 26, 2001 1068 1069 %* * %************************************************************************  simonpj committed Dec 22, 1998 1070   simonpj committed Dec 18, 1998 1071 1072 1073 Blob of helper functions for the "case-of-something-else" situation. \begin{code}  simonpj committed Jan 04, 2000 1074 ---------------------------------------------------------  simonpj committed Mar 23, 2000 1075 -- Eliminate the case if possible  simonpj committed Jan 04, 2000 1076   simonpj committed Sep 26, 2001 1077 1078 1079 rebuildCase :: SimplEnv -> OutExpr -- Scrutinee -> InId -- Case binder  simonpj committed Dec 22, 2004 1080  -> [InAlt] -- Alternatives (inceasing order)  simonpj committed Sep 26, 2001 1081  -> SimplCont  simonpj@microsoft.com committed Nov 01, 2006 1082  -> SimplM (SimplEnv, OutExpr)  simonpj committed Mar 23, 2000 1083   simonpj committed Sep 26, 2001 1084 1085 1086 1087 rebuildCase env scrut case_bndr alts cont | Just (con,args) <- exprIsConApp_maybe scrut -- Works when the scrutinee is a variable with a known unfolding -- as well as when it's an explicit constructor application  simonpj@microsoft.com committed Aug 16, 2006 1088  = knownCon env scrut (DataAlt con) args case_bndr alts cont  simonpj committed Jan 04, 2000 1089   simonpj committed Sep 26, 2001 1090 1091  | Lit lit <- scrut -- No need for same treatment as constructors -- because literals are inlined more vigorously  simonpj@microsoft.com committed Aug 16, 2006 1092  = knownCon env scrut (LitAlt lit) [] case_bndr alts cont  simonpj committed Mar 23, 2000 1093   simonpj committed Sep 26, 2001 1094  | otherwise  simonpj@microsoft.com committed Nov 01, 2006 1095 1096 1097  = do { -- Prepare the continuation; -- The new subst_env is in place (env, dup_cont, nodup_cont) <- prepareCaseCont env alts cont  simonpj committed Sep 30, 2004 1098   simonpj@microsoft.com committed Nov 01, 2006 1099 1100 1101 1102  -- Simplify the alternatives ; (case_bndr', alts') <- simplAlts env scrut case_bndr alts dup_cont ; let res_ty' = contResultType dup_cont ; case_expr <- mkCase scrut case_bndr' res_ty' alts'  sof committed Sep 04, 1997 1103   simonpj committed Sep 26, 2001 1104 1105  -- Notice that rebuildDone returns the in-scope set from env, not alt_env -- The case binder *not* scope over the whole returned case-expression  simonpj@microsoft.com committed Nov 01, 2006 1106  ; rebuild env case_expr nodup_cont }  simonm committed Dec 02, 1998 1107 \end{code}  partain committed Jan 08, 1996 1108   simonpj committed Jun 18, 2000 1109 1110 1111 1112 1113 simplCaseBinder checks whether the scrutinee is a variable, v. If so, try to eliminate uses of v in the RHSs in favour of case_bndr; that way, there's a chance that v will now only be used once, and hence inlined.  simonpj@microsoft.com committed Oct 04, 2006 1114 1115 Note [no-case-of-case] ~~~~~~~~~~~~~~~~~~~~~~  simonpj committed Jun 18, 2000 1116 1117 1118 There is a time we *don't* want to do that, namely when -fno-case-of-case is on. This happens in the first simplifier pass, and enhances full laziness. Here's the bad case:  simonpj committed Mar 23, 2000 1119 1120 1121 1122 1123  f = \ y -> ...(case x of I# v -> ...(case x of ...) ... ) If we eliminate the inner case, we trap it inside the I# v -> arm, which might prevent some full laziness happening. I've seen this in action in spectral/cichelli/Prog.hs: [(m,n) | m <- [1..max], n <- [1..max]]  simonpj committed Sep 26, 2001 1124 1125 Hence the check for NoCaseOfCase.  simonpj@microsoft.com committed Jan 11, 2007 1126 1127 1128 1129 Note [Suppressing the case binder-swap] ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ There is another situation when it might make sense to suppress the case-expression binde-swap. If we have  simonpj committed Sep 26, 2001 1130 1131 1132 1133 1134 1135 1136 1137 1138  case x of w1 { DEFAULT -> case x of w2 { A -> e1; B -> e2 } ...other cases .... } We'll perform the binder-swap for the outer case, giving case x of w1 { DEFAULT -> case w1 of w2 { A -> e1; B -> e2 } ...other cases .... }  simonpj committed Dec 14, 2001 1139 1140 1141 1142 But there is no point in doing it for the inner case, because w1 can't be inlined anyway. Furthermore, doing the case-swapping involves zapping w2's occurrence info (see paragraphs that follow), and that forces us to bind w2 when doing case merging. So we get  simonpj committed Sep 26, 2001 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154  case x of w1 { A -> let w2 = w1 in e1 B -> let w2 = w1 in e2 ...other cases .... } This is plain silly in the common case where w2 is dead. Even so, I can't see a good way to implement this idea. I tried not doing the binder-swap if the scrutinee was already evaluated but that failed big-time: data T = MkT !Int  simonpj committed Mar 23, 2000 1155   simonpj committed Sep 26, 2001 1156 1157 1158  case v of w { MkT x -> case x of x1 { I# y1 -> case x of x2 { I# y2 -> ...  simonpj committed Mar 23, 2000 1159   simonpj committed Sep 26, 2001 1160 1161 1162 1163 1164 Notice that because MkT is strict, x is marked "evaluated". But to eliminate the last case, we must either make sure that x (as well as x1) has unfolding MkT y1. THe straightforward thing to do is to do the binder-swap. So this whole note is a no-op.  simonpj@microsoft.com committed Oct 04, 2006 1165 1166 Note [zapOccInfo] ~~~~~~~~~~~~~~~~~  simonpj committed Sep 26, 2001 1167 1168 1169 1170 If we replace the scrutinee, v, by tbe case binder, then we have to nuke any occurrence info (eg IAmDead) in the case binder, because the case-binder now effectively occurs whenever v does. AND we have to do the same for the pattern-bound variables! Example:  simonpj committed Sep 26, 1997 1171   simonm committed Dec 02, 1998 1172  (case x of { (a,b) -> a }) (case x of { (p,q) -> q })  simonpj committed Sep 26, 1997 1173   simonm committed Dec 02, 1998 1174 1175 Here, b and p are dead. But when we move the argment inside the first case RHS, and eliminate the second case, we get  simonpj committed Sep 26, 1997 1176   simonpj committed Oct 04, 2004 1177  case x of { (a,b) -> a b }  simonpj committed Sep 26, 1997 1178   simonm committed Dec 02, 1998 1179 Urk! b is alive! Reason: the scrutinee was a variable, and case elimination  simonpj committed Oct 04, 2004 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 happened. Indeed, this can happen anytime the case binder isn't dead: case of x { (a,b) -> case x of { (p,q) -> p } } Here (a,b) both look dead, but come alive after the inner case is eliminated. The point is that we bring into the envt a binding let x = (a,b) after the outer case, and that makes (a,b) alive. At least we do unless the case binder is guaranteed dead.  simonpj committed Sep 26, 1997 1190   simonpj@microsoft.com committed Jan 11, 2007 1191 1192 1193 1194 1195 1196 1197 1198 1199 Note [Case of cast] ~~~~~~~~~~~~~~~~~~~ Consider case (v cast co) of x { I# -> ... (case (v cast co) of {...}) ... We'd like to eliminate the inner case. We can get this neatly by arranging that inside the outer case we add the unfolding v |-> x cast (sym co) to v. Then we should inline v at the inner case, cancel the casts, and away we go  simonpj committed Sep 26, 1997 1200 \begin{code}  simonpj@microsoft.com committed Nov 01, 2006 1201 simplCaseBinder :: SimplEnv -> OutExpr -> InId -> SimplM (SimplEnv, OutId)  simonpj@microsoft.com committed Oct 04, 2006 1202 1203 1204 1205 1206 simplCaseBinder env scrut case_bndr | switchIsOn (getSwitchChecker env) NoCaseOfCase -- See Note [no-case-of-case] = do { (env, case_bndr') <- simplBinder env case_bndr ; return (env, case_bndr') }  simonpj committed Sep 26, 2001 1207   simonpj@microsoft.com committed Oct 04, 2006 1208 simplCaseBinder env (Var v) case_bndr  simonpj committed Sep 26, 2001 1209 1210 -- Failed try [see Note 2 above] -- not (isEvaldUnfolding (idUnfolding v))  simonpj@microsoft.com committed Oct 04, 2006 1211 1212  = do { (env, case_bndr') <- simplBinder env (zapOccInfo case_bndr) ; return (modifyInScope env v case_bndr', case_bndr') }  simonm committed Dec 02, 1998 1213 1214  -- We could extend the substitution instead, but it would be -- a hack because then the substitution wouldn't be idempotent  simonpj committed Oct 04, 2004 1215  -- any more (v is an OutId). And this does just as well.  simonm committed Dec 02, 1998 1216   simonpj@microsoft.com committed Oct 04, 2006 1217 1218 1219 1220 1221 simplCaseBinder env (Cast (Var v) co) case_bndr -- Note [Case of cast] = do { (env, case_bndr') <- simplBinder env (zapOccInfo case_bndr) ; let rhs = Cast (Var case_bndr') (mkSymCoercion co) ; return (addBinderUnfolding env v rhs, case_bndr') }  simonpj committed Sep 26, 2001 1222 simplCaseBinder env other_scrut case_bndr  simonpj@microsoft.com committed Oct 04, 2006 1223 1224  = do { (env, case_bndr') <- simplBinder env case_bndr ; return (env, case_bndr') }  simonpj@microsoft.com committed Aug 16, 2006 1225   simonpj@microsoft.com committed Oct 04, 2006 1226 zapOccInfo :: InId -> InId -- See Note [zapOccInfo]  simonpj@microsoft.com committed Aug 16, 2006 1227 zapOccInfo b = b setIdOccInfo NoOccInfo  simonm committed Dec 02, 1998 1228 \end{code}  simonpj committed Mar 09, 1998 1229 1230   simonpj@microsoft.com committed Apr 12, 2006 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 simplAlts does two things: 1. Eliminate alternatives that cannot match, including the DEFAULT alternative. 2. If the DEFAULT alternative can match only one possible constructor, then make that constructor explicit. e.g. case e of x { DEFAULT -> rhs } ===> case e of x { (a,b) -> rhs } where the type is a single constructor type. This gives better code when rhs also scrutinises x or e. Here "cannot match" includes knowledge from GADTs It's a good idea do do this stuff before simplifying the alternatives, to avoid simplifying alternatives we know can't happen, and to come up with the list of constructors that are handled, to put into the IdInfo of the case binder, for use when simplifying the alternatives. Eliminating the default alternative in (1) isn't so obvious, but it can happen: data Colour = Red | Green | Blue f x = case x of Red -> .. Green -> .. DEFAULT -> h x h y = case y of Blue -> .. DEFAULT -> [ case y of ... ] If we inline h into f, the default case of the inlined h can't happen. If we don't notice this, we may end up filtering out *all* the cases of the inner case y, which give us nowhere to go!  sof committed Apr 30, 1998 1270   simonm committed Dec 02, 1998 1271 \begin{code}  simonpj committed Sep 26, 2001 1272 simplAlts :: SimplEnv  simonpj@microsoft.com committed Apr 12, 2006 1273  -> OutExpr  simonpj@microsoft.com committed Nov 01, 2006 1274  -> InId -- Case binder  simonpj committed Sep 26, 2001 1275  -> [InAlt] -> SimplCont  simonpj@microsoft.com committed Nov 01, 2006 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294  -> SimplM (OutId, [OutAlt]) -- Includes the continuation -- Like simplExpr, this just returns the simplified alternatives; -- it not return an environment simplAlts env scrut case_bndr alts cont' = -- pprTrace "simplAlts" (ppr alts$$ ppr (seIdSubst env)) \$ do { let alt_env = zapFloats env ; (alt_env, case_bndr') <- simplCaseBinder alt_env scrut case_bndr ; default_alts <- prepareDefault alt_env case_bndr' imposs_deflt_cons cont' maybe_deflt ; let inst_tys = tyConAppArgs (idType case_bndr') trimmed_alts = filter (is_possible inst_tys) alts_wo_default in_alts = mergeAlts default_alts trimmed_alts -- We need the mergeAlts in case the new default_alt -- has turned into a constructor alternative. ; alts' <- mapM (simplAlt alt_env imposs_cons case_bndr' cont') in_alts ; return (case_bndr', alts') }  simonm committed Dec 02, 1998 1295  where  simonpj@microsoft.com committed Apr 12, 2006 1296 1297 1298 1299 1300 1301 1302 1303 1304  (alts_wo_default, maybe_deflt) = findDefault alts imposs_cons = case scrut of Var v -> otherCons (idUnfolding v) other -> [] -- "imposs_deflt_cons" are handled either by the context, -- OR by a branch in this case expression. (Don't include DEFAULT!!) imposs_deflt_cons = nub (imposs_cons ++ [con | (con,_,_) <- alts_wo_default])  simonpj@microsoft.com committed Nov 01, 2006 1305 1306 1307 1308 1309 1310 1311 1312  is_possible :: [Type] -> CoreAlt -> Bool is_possible tys (con, _, _) | con elem imposs_cons = False is_possible tys (DataAlt con, _, _) = dataConCanMatch tys con is_possible tys alt = True ------------------------------------ prepareDefault :: SimplEnv -> OutId -- Case binder; need just for its type. Note that as an  simonpj@microsoft.com committed Apr 12, 2006 1313 1314 1315 1316 1317  -- OutId, it has maximum information; this is important. -- Test simpl013 is an example -> [AltCon] -- These cons can't happen when matching the default -> SimplCont -> Maybe InExpr  simonpj@microsoft.com committed Nov 01, 2006 1318 1319  -> SimplM [InAlt] -- One branch or none; still unsimplified -- We use a list because it's what mergeAlts expects  simonpj@microsoft.com committed Apr 12, 2006 1320   simonpj@microsoft.com committed Nov 01, 2006 1321 prepareDefault env case_bndr' imposs_cons cont Nothing  simonpj@microsoft.com committed Apr 12, 2006 1322  = return [] -- No default branch  chak@cse.unsw.edu.au. committed Sep 20, 2006 1323   simonpj@microsoft.com committed Nov 01, 2006 1324 prepareDefault env case_bndr' imposs_cons cont (Just rhs)  simonpj@microsoft.com committed Apr 12, 2006 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344  | -- This branch handles the case where we are -- scrutinisng an algebraic data type Just (tycon, inst_tys) <- splitTyConApp_maybe (idType case_bndr'), isAlgTyCon tycon, -- It's a data type, tuple, or unboxed tuples. not (isNewTyCon tycon), -- We can have a newtype, if we are just doing an eval: -- case x of { DEFAULT -> e } -- and we don't want to fill in a default for them! Just all_cons <- tyConDataCons_maybe tycon, not (null all_cons), -- This is a tricky corner case. If the data type has no constructors, -- which GHC allows, then the case expression will have at most a default -- alternative. We don't want to eliminate that alternative, because the -- invariant is that there's always one alternative. It's more convenient -- to leave -- case x of { DEFAULT -> e } -- as it is, rather than transform it to -- error "case cant match" -- which would be quite legitmate. But it's a really obscure corner, and -- not worth wasting code on. let imposs_data_cons = [con | DataAlt con <- imposs_cons] -- We now know it's a data type  simonpj@microsoft.com committed Nov 01, 2006 1345 1346 1347 1348  is_possible con = not (con elem imposs_data_cons) && dataConCanMatch inst_tys con = case filter is_possible all_cons of [] -> return [] -- Eliminate the default alternative  simonpj@microsoft.com committed Apr 12, 2006 1349


text: Thank you! Come to Algebra-equation.com and discover rational expressions, math review and a great many other algebra subject areas Step 2: Simplify the radicals. I was struggling with fractions. Trigonometric Identities and Conditional Equations Identify and pull out powers of 4, using the fact that . Jeff Kasten, MI, I needed to test out of algebra to satisfy the entrance requirements for school. Multiplying Exponents. Simplify each radical, if possible, before multiplying. Homework problems on homogeneous linear equations m a √ = b if bm = a Radicals - Higher Roots Objective: Simplify radicals with an index greater than two. Choice of base Numbers. The "index" is the very small number written just to the left of the uppermost line in the radical symbol. Since all the radicals are fourth roots, you can use the rule to multiply the radicands. While we talk related with Simplifying Radicals Math Worksheets, we have collected particular similar images to give you more ideas. endstream endobj startxref 339yy2 ⋅. If there is no index number, the radical is understood to be a square root (index 2) and can be multiplied with other square roots. Linear Equations in Linear Algebra Let’s start with an example of multiplying roots with the different index The first step is to calculate the minimum common multiple of the indices: This will be the new common index, which we place already in the roots in the absence of the exponent of the radicando: Your software shows how the problems are solved and that was the answer for me. Algebrator's simple step by step solutions made him enjoy learning. Right from multiplying or divide radical terms with different indices to division, we have all of it included. 10 problems. Objective: Simplify radicals with an index greater than two. $�YAb�A�n �,�e "�A�bk��$�Z@�= "�čv&F�� .#E�Ƴ� + Solving Quadratic Equations by Factoring Choice of base Numbers. I note that 8 = 2 3 and 64 = 4 3, so I will actually be able to simplify the radicals completely. Linear Equations as Models I never used to take interest in math and always found algebra boring. Test Description for Quadratic Equations and Functions LinearEquations 1 Answer Jim H Mar 22, 2015 Make the indices the same (find a common index). My concepts are very clear and I love the step-by-step approach. Come to Sofsource.com and learn numbers, scientific notation and lots of other math subject areas ⋅ 14 21. b. We have $$\sqrt 3 \left( {2 - 3\sqrt 6 } \right) = 2\sqrt 3 - 3\sqrt {18}$$ Now since $$18 = 2 \cdot {3^2}$$, we can simplify the expression one more step. represent slope in a linear equation Be looking for powers of 4 in each radicand. Quadratic Equations Multiplying fractions worksheet, math tutoring software, Free Online Sats Papers, program to find slope calculator, free answers to algebra 2, graphs of common functions. J.V., Maryland, I started with this kind of programs as I am in an online class and there are times when "I have no clue". This is a differentiated worksheet on the topic of Index Laws. Two Equations containing Two Variables ©o 6KCuAtCav QSMoMfAtIw0akrLeD nLrLDCj.r m 0A0lsls 1r6i4gwh9tWsx 2rieAsKeLrFvpe9dc.c G 3Mfa0dZe7 UwBixtxhr AIunyfVi2nLimtqel bAmlCgQeNbarwaj w1Q.V-6-Worksheet by Kuta Software LLC Answers to Multiplying and Dividing Radicals 1) 3 2) −30 3) … Students will practice multiplying square roots (ie radicals). Quadratic Equations Simplify each radical, if possible, before multiplying. In this lesson, we are only going to deal with square roots only which is a specific type of radical expression with an index of \color{red}2.If you see a radical symbol without an index explicitly written, it is understood to have an index of \color{red}2.. Below are the basic rules in multiplying radical expressions. Note: When multiplying radicals with different indexes, change to rational exponents first, find a common denominator in order to add the exponents, then rewrite in radical notation as shown below: Example: 8 ˚ 2 Algebrator saved me. Students match the radicals to the simplified versions to reveal the answer to a riddle, so students will know right away if they've solved correctly! Fol-lowing is a deﬁnition of radicals. Multiplying radicals, though seemingly intimidating, is an incredibly simple process! This set of exponents worksheets provide practice multiplying simple exponential terms against numbers. h�bbdbZ$�� �� Number of problems 5 problems. 2 - focuses on finding non-square roots of numbers, and then investigates multiplying and dividing using indices. 12 lessons in Indices and standard form : Indices and Radicals; Indices and power of 0; Expressing powers of in different bases; Laws of indices; Fractional Indices (Part 1) Fractional Indices (Part 2) Negative indices; Powers of 10; Sorting large numbers; Multiplying and dividing in standard form; Addition and Subtraction in standard form 20 problems. Elementary Algebra Skill Multiplying Radicals of Index 2: No Variable Factors. Right now, they aren't. Home 20 problems. This worksheet covers the rules regarding multiplying and dividing indices with the same base, including when the variable has a coefficient. I had the questions and the answers but couldnt figure how to get from one to other. 33223 3 3 3 33 39 39 27 3 3 y yy y y y ⋅ =⋅y = = = b. Free math ratios answers, adding and subtracting fractions worksheet, 4th grade fraction worksheets, algebra ti-84 distance formula study cards, mixed numberto a decimal, multiplying radicals and integars. Solving Equations with Rational Expressions Created for high ability KS3 as an index laws introduction but could certainly be used for any class up to GCSE. A worksheet where you have to apply the law of multiplication with regards to indices. In order to simplify a radical, all we need to do is take the … 6 is the LCM of these two numbers because it is the smallest number that is evenly divisible by both 3 and 2. Solving Linear Equations Solving Multiple-Step Equations 6/3 = 2 and 6/2 = 3. Here we must remember to use the distributive property of multiplication, just like anytime. Solve Quadratic Equations by Completing the Square While square roots are the most common type of radical we work with, we can take higher roots of numbers as well: cube roots, fourth roots, ﬁfth roots, etc. endstream endobj 23 0 obj <> endobj 24 0 obj <> endobj 25 0 obj <>stream LinearEquations Algebra. Solving Quadratic Equations LinearEquations Multiplying Radicals Calculator is a free online tool that displays the product of two radicals. How to multiply and simplify radicals with different indices. Download math worksheet on finding square roots, cube roots and applying different operations on them to practice and score better in your classroom tests. Solving Exponential Equations Perform the indicated multiplication, and simplify completely. So whenever you are multiplying radicals with different indices, different roots, you always need to make your roots the same by doing and you do that by just changing your fraction to be a … Multiplying Radical Expressions. Identify and pull out powers of 4, using the fact that . Literal Equations 20 problems. Rewrite as the product of radicals. Right from multiplying or divide radical terms with different indices to division, we have all of it included. You can choose whether to have numbers or algebraic unknowns as the base. That is, numbers outside the radical multiply together, and numbers inside the radical multiply together. This worksheet has model problems worked out, step by step as well as 25 scaffolded questions that start out relatively easy and end with some real challenges. Linear Equations BYJU’S online multiplying radicals calculator tool performs the calculation faster and it displays the multiplication of two radical values in a fraction of seconds. You can choose whether to have numbers or algebraic unknowns as the base. 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Answers to Multiplying Radicals of Index 2: No Variable Factors 1) 6 2) 4 3) 10 problems. algebra 1 simplifying radical expressions worksheet, operations with scientific notation worksheet and multiplying radicals worksheet are three main things we want to show you based on the post title. Try the Free Math Solver or Scroll down to Tutorials! Step 3: Multiply the terms outside the radical. Functions, Equations, and Inequalities Number of problems 5 problems. Module 3: Multiplying Radical Expressions Recall the property of exponents that states that . This While square roots are the most common type of radical we work with, we can take higher roots of numbers as well: cube roots, fourth roots, ﬁfth roots, etc. 22 0 obj <> endobj Exponential and Logarithmic Equations SOLUTIONS: 2 14 21 14 21 2737 76 76 ⋅= ⋅ =⋅⋅⋅ =⋅ = a. Linear Equations and Matrices This worksheet covers the rules regarding multiplying and dividing indices with the same base, including when the variable has a coefficient. This worksheet is a fun way for your students to practice multiplying radicals! 10 problems. 2) To multiply radicals with different indices use fractional exponents and the laws of exponents. Access these printable radical worksheets, carefully designed and proposed for students of grade 8 and high school. How do you multiply radical expressions with different indices? How to multiply and simplify radicals with different indices. B.M., Vermont. Fol-lowing is a deﬁnition of radicals. SOLVING LINEAR AND QUADRATIC EQUATIONS 1) To multiply two or more radicals having the same index use . Division of radicals. %%EOF Parents can download these exponents worksheets for their kids to provide additional practice material at home while teachers can use it as an assignment in the classroom or as a test. Find the LCM of the indices for the following equation: 3 √(5) x 2 √(2) = ? SOLUTION: The key step when the indices of the radicals are different … Linear Equations and Inequalities ©o 6KCuAtCav QSMoMfAtIw0akrLeD nLrLDCj.r m 0A0lsls 1r6i4gwh9tWsx 2rieAsKeLrFvpe9dc.c G 3Mfa0dZe7 UwBixtxhr AIunyfVi2nLimtqel bAmlCgQeNbarwaj w1Q.V-6-Worksheet by Kuta Software LLC Answers to Multiplying and Dividing Radicals 1) 3 2) −30 3) … Examples. %PDF-1.5 %���� h޼Vmo�6�+��p�"R/@a�/u�mΌ�k��ج-@�IA��;R�$;�Z�'ޑw|��Q�F$'���+E�C�A�D@�"ʃ�%>s���R���� 2. Objective. Rewrite as the product of radicals. Solving Linear Equations Test Description for EXPRESSIONS AND EQUATIONS 54 0 obj <>stream Example. Equations Algebra Radicals and Geometry Connections Multiplication and Division of Radicals. Multiplying Radicals. Thanks. Step 1: If the radicals have the same index, multiply terms the outside the radical with terms outside the radical and terms inside the radical with terms inside the radical. These worksheets provide a gentle introduction into working with exponents in otherwise typical multiplication problems, and help reinforce the order of operation rules necessary to solve more complex problems later. We have So I'll simplify the radicals first, and then see if I can go any further. Answer sheet Include answer sheet. Dividing indices. Algebra. Simplifying Expressions & Solving Equations Like radicals can then be added or subtracted in the same way as other like terms. 0 You can choose whether to have numbers or algebraic unknowns as the base. Come to Sofsource.com and learn numbers, scientific notation and lots of other math subject areas Right from multiplying radicals with different indices to precalculus, we have got all the pieces included. Click here to review the steps for Simplifying Radicals. EXAMPLE: Perform the following multiplication: 3 x⋅4 x. But then we will use our property of multiplying radicals to handle the radical parts. If the radicals have the same index, multiply terms the outside the radical with terms outside the radical and terms inside the radical with terms inside the radical. Clay Timmons, TX, I really like your software. Right from multiplying radicals with different indices to precalculus, we have got all the pieces included. LinearEquations h�bf2ga��g�c@ >�r����!���vPXd�=b���!�$��P�����t�7s�䂲n�O�]�m���t�����a�4�f�v �e`�?�g�0 @�� Answers included! Multiplying Exponents Different Bases - Displaying top 8 worksheets found for this concept.. The pdf worksheets cover topics such as identifying the radicand and index in an expression, converting the radical form to exponential form and the other way around, reducing radicals to its simplest form, rationalizing the denominators, and simplifying the radical expressions. Time has changed since I bought this software. A worksheet where you have to apply the law of multiplication with regards to indices. 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Maximum norm error estimates for Div least-squares method for Darcy flows

• Least-squares finite element methods for second order elliptic partial differential equations such as Darcy flows are considered. While there has been a significant progress in terms of obtaining error estimates for the methods, the estimates are essentially based on $L_2$-norm of the error. In this paper, we provide maximum norm error estimates for the primary variable using a smoothed Green's function introduced in [33] and maximum norm error for the dual variables by taking advantage of the fact that least-squares solutions are higher-order perturbations of Galerkin solutions [8].
Mathematics Subject Classification: Primary: 65M60, 65M15.

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