CSE P 501 – Compilers Dataflow Analysis Hal Perkins Spring 2018 UW CSE P 501 Spring 2018 T-1 Agenda • Dataflow analysis: a framework and algorithm for many common compiler analyses • Initial example: dataflow analysis for common subexpression elimination • Other analysis problems that work in the same framework • Some of these are optimizations we’ve seen, but now more formally and with details UW CSE P 501 Spring 2018 T-2 The Story So Far… • Redundant expression elimination – Local Value Numbering – Superlocal Value Numbering • Extends VN to EBBs • SSA-like namespace – Dominator VN Technique (DVNT) • All oF these propagate along Forward edges • None are global – In particular, can’t handle back edges (loops) UW CSE P 501 Spring 2018 T-3 Dominator Value Numbering A m = a + b • 0 0 0 Most sophisticated n = a + b algorithm so far 0 0 0 B C • Still misses some p = c + d 0 0 0 q0 = a0 + b0 opportunities r = c + d 0 0 0 r1 = c0 + d0 • Can’t handle loops D E e0 = b0 + 18 e1 = a0 + 17 s0 = a0 + b0 t0 = c0 + d0 u0 = e0 + f0 u1 = e1 + f0 F e2 = Φ(e0,e1) u2 = Φ(u0,u1) v0 = a0 + b0 G w = c + d r2 = Φ(r0,r1) 0 0 0 x = e + f y0 = a0 + b0 0 2 0 z0 = c0 + d0 UW CSE P 501 Spring 2018 T-4 Available Expressions • Goal: use dataflow analysis to find common subexpressions whose range spans basic blocks • Idea: calculate available expressions at beginning of each basic block • Avoid re-evaluation of an available expression – use a copy operation UW CSE P 501 Spring 2018 T-5 “Available” and Other Terms • An expression e is defined at point p in the CFG if its value a+b is computed at p defined t1 = a + b – Sometimes called definition site ... • An expression e is killed at point p if one of its operands a+b is defined at p available t10 = a + b – Sometimes called kill site … • An expression e is available at point p if every path a+b leading to p contains a prior killed b = 7 definition of e and e is not … killed between that definition and p UW CSE P 501 Spring 2018 T-6 Available Expression Sets • To compute available expressions, for each block b, define – AVAIL(b) – the set of expressions available on entry to b – NKILL(b) – the set of expressions not killed in b • i.e., all expressions in the program except for those killed in b – DEF(b) – the set of expressions defined in b and not subsequently killed in b UW CSE P 501 Spring 2018 T-7 Computing Available Expressions • AVAIL(b) is the set AVAIL(b) = ÇxÎpreds(b) (DEF(x) È (AVAIL(x) Ç NKILL(x))) – preds(b) is the set of b’s predecessors in the CFG – The set of expressions available on entry to b is the set of expressions that were available at the end of every predeCessor basiC bloCk x – The expressions available on exit from bloCk b are those defined in b or available on entry to b and not killed in b • This gives a system of simultaneous equations – a dataflow problem UW CSE P 501 Spring 2018 T-8 Name Space Issues • In previous value-numbering algorithms, we used a SSA-like renaming to keep track of versions • In global dataflow problems, we use the original namespace – we require a+b have the same value along all paths to its use – If a or b is updated along any path to its use, then a+b has the “wrong” value – so original names are exactly what we want • The KILL information captures when a value is no longer available UW CSE P 501 Spring 2018 T-9 Computing Available Expressions • Big Picture – Build control-flow graph – Calculate initial local data – DEF(b) and NKILL(b) • This only needs to be done once for each block b and depends only on the statements in b – Iteratively calculate AVAIL(b) by repeatedly evaluating equations until nothing changes • Another fixed-point algorithm UW CSE P 501 Spring 2018 T-10 Computing DEF and NKILL (1) • For each block b with operations o1, o2, …, ok KILLED = Æ // killed variables, not expressions DEF(b) = Æ for i = k to 1 // note: working back to Front assume oi is “x = y + z” add x to KILLED if (y Ï KILLED and z Ï KILLED) add “y + z” to DEF(b) … UW CSE P 501 Spring 2018 T-11 Computing DEF and NKILL (2) • After computing DEF and KILLED for a block b, compute set of all expressions in the program not killed in b NKILL(b) = { all expressions } for each expression e for each variable v Î e if v Î KILLED then NKILL(b) = NKILL(b) - e UW CSE P 501 Spring 2018 T-12 Example: Compute DEF and NKILL j = 2 * a k = 2 * b x = a + b c = 5 * n b = c + d m = 5 * n h = 2 * a UW CSE P 501 Spring 2018 T-14 Example: Compute DEF and NKILL j = 2 * a DEF = { 2*a, 2*b } k = 2 * b NKILL = exprs w/o j or k x = a + b c = 5 * n b = c + d m = 5 * n h = 2 * a UW CSE P 501 Spring 2018 T-15 Example: Compute DEF and NKILL j = 2 * a DEF = { 2*a, 2*b } k = 2 * b NKILL = exprs w/o j or k x = a + b c = 5 * n DEF = { 5*n } b = c + d NKILL = exprs w/o c m = 5 * n h = 2 * a UW CSE P 501 Spring 2018 T-16 Example: Compute DEF and NKILL j = 2 * a DEF = { 2*a, 2*b } k = 2 * b NKILL = exprs w/o j or k DEF = { 5*n, c+d } x = a + b c = 5 * n DEF = { 5*n } NKILL = exprs w/o b = c + d NKILL = exprs w/o c m, x, b m = 5 * n h = 2 * a UW CSE P 501 Spring 2018 T-17 Example: Compute DEF and NKILL j = 2 * a DEF = { 2*a, 2*b } k = 2 * b NKILL = exprs w/o j or k DEF = { 5*n, c+d } x = a + b c = 5 * n DEF = { 5*n } NKILL = exprs w/o b = c + d NKILL = exprs w/o c m, x, b m = 5 * n DEF = { 2*a } h = 2 * a NKILL = exprs w/o h UW CSE P 501 Spring 2018 T-18 Computing Available Expressions Once DEF(b) and NKILL(b) are computed for all blocks b Worklist = { all blocks bi } while (Worklist ¹ Æ) remove a block b from Worklist recompute AVAIL(b) if AVAIL(b) changed Worklist = Worklist È successors(b) UW CSE P 501 Spring 2018 T-19 Example: Find Available Expressions AVAIL(b) = ÇxÎpreds(b) (DEF(x) È (AVAIL(x) Ç NKILL(x))) j = 2 * a DEF = { 2*a, 2*b } k = 2 * b NKILL = exprs w/o j or k DEF = { 5*n, c+d } x = a + b c = 5 * n DEF = { 5*n } NKILL = exprs w/o b = c + d NKILL = exprs w/o c m, x, b m = 5 * n DEF = { 2*a } h = 2 * a NKILL = exprs w/o h = in worklist = processing UW CSE P 501 Spring 2018 T-20 Example: Find Available Expressions AVAIL(b) = ÇxÎpreds(b) (DEF(x) È (AVAIL(x) Ç NKILL(x))) AVAIL = { } j = 2 * a DEF = { 2*a, 2*b } k = 2 * b NKILL = exprs w/o j or k DEF = { 5*n, c+d } x = a + b c = 5 * n DEF = { 5*n } NKILL = exprs w/o b = c + d NKILL = exprs w/o c m, x, b m = 5 * n DEF = { 2*a } h = 2 * a NKILL = exprs w/o h = in worklist = processing UW CSE P 501 Spring 2018 T-21 Example: Find Available Expressions AVAIL(b) = ÇxÎpreds(b) (DEF(x) È (AVAIL(x) Ç NKILL(x))) AVAIL = { } j = 2 * a DEF = { 2*a, 2*b } k = 2 * b NKILL = exprs w/o j or k DEF = { 5*n, c+d } x = a + b c = 5 * n DEF = { 5*n } NKILL = exprs w/o b = c + d NKILL = exprs w/o c m, x, b m = 5 * n AVAIL = { 5*n } h = 2 * a DEF = { 2*a } NKILL = exprs w/o h = in worklist = processing UW CSE P 501 Spring 2018 T-22 Example: Find Available Expressions AVAIL(b) = ÇxÎpreds(b) (DEF(x) È (AVAIL(x) Ç NKILL(x))) AVAIL = { } j = 2 * a DEF = { 2*a, 2*b } k = 2 * b NKILL = exprs w/o j or k AVAIL = { 2*a, 2*b } x = a + b c = 5 * n DEF = { 5*n } DEF = { 5*n, c+d } NKILL = exprs w/o c NKILL = exprs w/o b = c + d m, x, b m = 5 * n AVAIL = { 5*n } h = 2 * a DEF = { 2*a } NKILL = exprs w/o h = in worklist = processing UW CSE P 501 Spring 2018 T-23 Example: Find Available Expressions AVAIL(b) = ÇxÎpreds(b) (DEF(x) È (AVAIL(x) Ç NKILL(x))) AVAIL = { } j = 2 * a DEF = { 2*a, 2*b } k = 2 * b NKILL = exprs w/o j or k AVAIL = { 2*a, 2*b } AVAIL = { 2*a, 2*b } DEF = { 5*n, c+d } x = a + b c = 5 * n DEF = { 5*n } NKILL = exprs w/o b = c + d NKILL = exprs w/o c m, x, b m = 5 * n AVAIL = { 5*n } h = 2 * a DEF = { 2*a } NKILL = exprs w/o h = in worklist = processing UW CSE P 501 Spring 2018 T-24 Example: Find Available Expressions AVAIL(b) = ÇxÎpreds(b) (DEF(x) È (AVAIL(x) Ç NKILL(x))) AVAIL = { } j = 2 * a DEF = { 2*a, 2*b } k = 2 * b NKILL = exprs w/o j or k AVAIL = { 2*a, 2*b } AVAIL = { 2*a, 2*b } DEF = { 5*n, c+d } x = a + b c = 5 * n DEF = { 5*n } NKILL = exprs w/o b = c + d NKILL = exprs w/o c m, x, b m = 5 * n AVAIL = { 5*n, 2*a } h = 2 * a DEF = { 2*a } NKILL = exprs w/o h = in worklist = processing UW CSE P 501 Spring 2018 T-25 Example: Find Available Expressions AVAIL(b) = ÇxÎpreds(b) (DEF(x) È (AVAIL(x) Ç NKILL(x))) AVAIL = { } j = 2 * a DEF = { 2*a, 2*b } k = 2 * b NKILL = exprs w/o j or k AVAIL = { 2*a, 2*b } AVAIL = { 2*a, 2*b } DEF = { 5*n, c+d } x = a + b c = 5 * n DEF = { 5*n } NKILL = exprs w/o b = c + d NKILL = exprs w/o c m, x, b m = 5 * n AVAIL = { 5*n, 2*a } h = 2 * a DEF = { 2*a } NKILL = exprs w/o h = in worklist = processing And The common subexpression is??? UW CSE P 501 Spring 2018 T-26 Example: Find Available Expressions AVAIL(b) = ÇxÎpreds(b) (DEF(x) È (AVAIL(x) Ç NKILL(x))) AVAIL = { } j = 2 * a DEF = { 2*a, 2*b } k = 2 * b NKILL = exprs w/o j or k AVAIL = { 2*a, 2*b } AVAIL = { 2*a, 2*b } DEF = { 5*n, c+d } x = a + b c = 5 * n DEF = { 5*n } NKILL = exprs w/o b = c + d NKILL = exprs w/o c m, x, b m = 5 * n AVAIL = { 5*n, 2*a } h = 2 * a DEF = { 2*a } NKILL = exprs w/o h = in worklist = processing UW CSE P 501 Spring 2018 T-27 Comparing Algorithms A • m = a + b LVN – Local Value n = a + b Numbering • SVN – Superlocal Value B C p = c + d q = a + b Numbering r = c + d r = c + d • DVN – DominatoT-based D E Value Numbering e = b + 18 e = a + 17 • GRE – Global Redundancy s = a + b t = c + d Elimination u = e + f u = e + f F v = a + b w = c + d x = e + f G y = a + b z = c + d UW CSE P 501 Spring 2018 T-28 Comparing Algorithms (2) • LVN => SVN => DVN form a strict hierarchy – later algorithms find a superset of previous information • Global RE finds a somewhat