diff --git a/packages/compiler/src/backend/llvm/emitter.ts b/packages/compiler/src/backend/llvm/emitter.ts
index f5fccab80..1aa1e4197 100644
--- a/packages/compiler/src/backend/llvm/emitter.ts
+++ b/packages/compiler/src/backend/llvm/emitter.ts
@@ -56,6 +56,14 @@ import {
   matchIntegerCountedForLoop,
 } from "../../ir/integer-loops.js";
 import { emitCountedLoopLimit } from "./counted-loops.js";
+import {
+  emitSafeIntegerCheck,
+  emitSpeculativeInteger,
+  planSpeculativeLoop,
+  speculativeReads,
+  speculativeWriteTarget,
+  SPECULATIVE_RANGE,
+} from "./speculative-integers.js";
 import { emitLiteralSwitch } from "./switch-dispatch.js";
 import { findBytesBounds } from "./bytes-bounds.js";
 import {
@@ -956,6 +964,22 @@ export class LlEmitter {
     { slot: string; type: "i32" | "i64"; signed?: boolean; range?: { min: number; max: number } }
   >();
   private countedLoopsEnabled = false;
+  /** Speculative integer loops (speculative-integers.ts) run in this function. */
+  private speculativeLoopsEnabled = false;
+  /** The fast copy of a speculative integer loop being emitted: its
+   * integer-computed writes and where each resumes in the ordinary copy. */
+  private speculation: {
+    writes: Set<IrStmt>;
+    resume: Map<IrStmt, string>;
+    /** Globals whose double is written back only when the copy is left. */
+    deferred: Set<string>;
+  } | null = null;
+  /** The ordinary copy being emitted: resume labels still to place. */
+  private speculationResume: Map<IrStmt, { label: string; placed: boolean }> | null = null;
+  /** The function being emitted, for speculative globals. */
+  private speculativeFunction = "";
+  /** Every function that writes each number global (computed on demand). */
+  private globalWriters: Map<string, Set<string>> | null = null;
   /** Whole-program final reference fields (ir/final-fields.ts). */
   private finalFields!: FinalFields;
   /** The current function may borrow final fields of borrowable objects:
@@ -4920,6 +4944,266 @@ export class LlEmitter {
     return slot;
   }
 
+  /** Emit a loop's condition, body and update through `emitCopy`, twice
+   * when speculative-integers.ts plans a fast integer copy. Both copies
+   * share the loop's setup and exit; the ordinary copy comes first so the
+   * fast copy is only emitted when every resume point exists. */
+  private emitLoopCopies(
+    loop: Extract<IrStmt, { kind: "while" | "doWhile" | "for" }>,
+    allowed: boolean,
+    exit: string,
+    emitCopy: (exit: string) => void,
+  ): void {
+    const plan =
+      allowed &&
+      this.speculativeLoopsEnabled &&
+      this.speculation === null &&
+      this.speculationResume === null
+        ? planSpeculativeLoop(
+            loop,
+            (id) => this.speculativeLocal(id),
+            (id) => this.integerLoopBindings.has(id) || this.int32Locals.has(id),
+            (e) => {
+              const range = this.integerRanges.get(e);
+              return (
+                range !== undefined &&
+                range !== null &&
+                range.min >= -Number.MAX_SAFE_INTEGER &&
+                range.max <= Number.MAX_SAFE_INTEGER
+              );
+            },
+            (id) => this.speculativeLocal(id) || this.speculativeInvariantGlobal(id),
+          )
+        : null;
+    if (!plan) {
+      emitCopy(exit);
+      return;
+    }
+    const B = this.B;
+    const entries = new Map<string, string>();
+    let ok = "true";
+    for (const [id, live] of plan.locals) {
+      if (!live) continue;
+      const value = B.tmp();
+      B.line(`${value} = load double, ptr ${this.binding(id).slot}`);
+      const [integer, exact] = emitSafeIntegerCheck(this, value);
+      entries.set(id, integer);
+      if (ok === "true") ok = exact;
+      else {
+        const both = B.tmp();
+        B.line(`${both} = and i1 ${ok}, ${exact}`);
+        ok = both;
+      }
+    }
+    const fast = B.newLabel("spec.fast"),
+      slow = B.newLabel("spec.slow");
+    B.condBr(ok, fast, slow);
+    const resume = new Map<IrStmt, { label: string; placed: boolean }>();
+    for (const s of plan.writes) resume.set(s, { label: B.newLabel("spec.resume"), placed: false });
+    B.startBlock(slow);
+    this.speculationResume = resume;
+    try {
+      emitCopy(exit);
+    } finally {
+      this.speculationResume = null;
+    }
+    B.startBlock(fast);
+    if (![...resume.values()].every((r) => r.placed)) {
+      B.br(slow);
+      return;
+    }
+    for (const id of plan.locals.keys()) {
+      const shadow = B.slot();
+      B.entryAllocas.push(
+        `${shadow} = alloca i64 ; speculative integer ${this.currentLocals.get(id)?.name ?? id}`,
+      );
+      const entry = entries.get(id);
+      if (entry !== undefined) {
+        // The same double, now visibly an integer conversion: the double
+        // phis of the fast copy fold into integer phis.
+        const exact = B.tmp();
+        B.line(`store i64 ${entry}, ptr ${shadow}`);
+        B.line(`${exact} = sitofp i64 ${entry} to double`);
+        B.line(`store double ${exact}, ptr ${this.binding(id).slot}`);
+      } else B.line(`store i64 0, ptr ${shadow}`);
+      this.integerLoopBindings.set(id, {
+        slot: shadow,
+        type: "i64",
+        signed: true,
+        range: SPECULATIVE_RANGE,
+      });
+    }
+    // A global's store stays in the loop (calls may read it). When nothing
+    // in the loop can observe it, its double is written once on leaving the
+    // copy instead, so the loop keeps it in a register as the double copy
+    // would.
+    const deferred = new Set<string>();
+    // Worker loops also leave through their cancellation poll.
+    if (plan.observerFree && !this.mod.workers)
+      for (const s of plan.writes) {
+        const id = speculativeWriteTarget(s)!.localId;
+        if (!this.currentLocals.has(id)) deferred.add(id);
+      }
+    this.speculation = {
+      writes: plan.writes,
+      resume: new Map([...resume].map(([s, r]) => [s, r.label])),
+      deferred,
+    };
+    try {
+      if (deferred.size === 0) emitCopy(exit);
+      else {
+        const leave = B.newLabel("spec.leave");
+        emitCopy(leave);
+        B.startBlock(leave);
+        this.writeBackSpeculativeGlobals(deferred);
+        B.br(exit);
+      }
+    } finally {
+      this.speculation = null;
+      for (const id of plan.locals.keys()) this.integerLoopBindings.delete(id);
+    }
+  }
+
+  private writeBackSpeculativeGlobals(deferred: Set<string>): void {
+    const B = this.B;
+    for (const id of deferred) {
+      const integer = B.tmp(),
+        number = B.tmp();
+      B.line(`${integer} = load i64, ptr ${this.integerLoopBindings.get(id)!.slot}`);
+      B.line(`${number} = sitofp i64 ${integer} to double`);
+      B.line(`store double ${number}, ptr ${this.binding(id).slot}`);
+    }
+  }
+
+  /** A number local whose only storage is its own double slot, so an i64
+   * shadow beside it can stand for it within one loop. */
+  private speculativeLocal(id: string): boolean {
+    const local = this.currentLocals.get(id);
+    if (!local) return this.speculativeGlobal(id);
+    if (local.type.kind !== "f64" || local.boxed || local.tdz) return false;
+    return !(
+      this.captureIds.has(id) ||
+      this.int32Locals.has(id) ||
+      this.integerLoopBindings.has(id) ||
+      this.integerViews.has(id) ||
+      this.localUnionStorage.has(id) ||
+      this.reboundOwnerSlots.has(id) ||
+      this.lazyCaptureBoxes.has(id) ||
+      this.privateSplitLocals.has(id)
+    );
+  }
+
+  /** A module-level number written only by the file's top-level code that
+   * is emitting this loop. Calls may still read it: the fast copy keeps its
+   * double current. An init function runs once and is never re-entered, so
+   * no call from the loop can write the global behind the shadow. */
+  private speculativeGlobal(id: string): boolean {
+    if (!this.speculativeFunction.startsWith("%init.")) return false;
+    const writers = this.numberGlobalWriters(id);
+    return (
+      writers !== null &&
+      (writers.size === 0 || (writers.size === 1 && writers.has(this.speculativeFunction)))
+    );
+  }
+
+  /** A module-level number a loop of any function only reads, written by
+   * nothing but top-level code: an init function never runs during a loop
+   * of another function, so the entry value stays its value. */
+  private speculativeInvariantGlobal(id: string): boolean {
+    const writers = this.numberGlobalWriters(id);
+    return writers !== null && [...writers].every((name) => name.startsWith("%init."));
+  }
+
+  /** The functions writing a number global a shadow may stand for, or null
+   * when the global is not such a number. */
+  private numberGlobalWriters(id: string): ReadonlySet<string> | null {
+    if (this.globalTypes.get(id)?.kind !== "f64") return null;
+    if (this.tdzGlobals.has(id) || this.integerViews.has(id)) return null;
+    if (!this.globalWriters) {
+      const writers = new Map<string, Set<string>>();
+      for (const fn of this.mod.functions) {
+        const note = (localId: string): void => {
+          if (!localId.startsWith("%g.")) return;
+          const set = writers.get(localId) ?? new Set<string>();
+          set.add(fn.name);
+          writers.set(localId, set);
+        };
+        everyStmtList(fn.body, {
+          stmt: (s) => {
+            if (s.kind === "assign" || s.kind === "varDecl") note(s.localId);
+            return true;
+          },
+          expr: (e) => {
+            if (e.kind === "assignExpr" || e.kind === "incDec") note(e.localId);
+            return true;
+          },
+        });
+      }
+      this.globalWriters = writers;
+    }
+    return this.globalWriters.get(id) ?? new Set<string>();
+  }
+
+  /** A speculated write in the fast copy: compute the value as a safe
+   * integer, store it to the shadow and its exact double to the local.
+   * Any failed check re-executes the statement in the ordinary copy. */
+  private emitSpeculativeWrite(s: IrStmt, resume: string): void {
+    const B = this.B;
+    const target = speculativeWriteTarget(s)!;
+    const binding = this.integerLoopBindings.get(target.localId)!;
+    const failed = B.newLabel("spec.bail");
+    let bails = false;
+    this.declare("declare i1 @llvm.expect.i1(i1, i1)");
+    const bail = (condition: string): void => {
+      if (condition === "false") return;
+      bails = true;
+      const expected = B.tmp();
+      B.line(`${expected} = call i1 @llvm.expect.i1(i1 ${condition}, i1 false)`);
+      const next = B.newLabel("spec.ok");
+      B.condBr(expected, failed, next);
+      B.startBlock(next);
+    };
+    // Heap reads come first, before any check, and release their temporaries
+    // at once: a bailing path then owns nothing, and reads have no effect
+    // whose reordering before the arithmetic could be observed.
+    const reads = new Map<IrExpr, LlValue>();
+    if (target.value !== null) {
+      this.frames.push([]);
+      for (const read of speculativeReads(target.value)) reads.set(read, this.emitExpr(read));
+      this.releaseFrame(this.frames.pop()!);
+    }
+    this.frames.push([]);
+    let result: string;
+    if (target.value === null) {
+      const old = B.tmp();
+      result = B.tmp();
+      B.line(`${old} = load i64, ptr ${binding.slot}`);
+      B.line(`${result} = ${target.op === "+" ? "add" : "sub"} nsw i64 ${old}, 1`);
+      const biased = B.tmp(),
+        outside = B.tmp();
+      B.line(`${biased} = add i64 ${result}, ${Number.MAX_SAFE_INTEGER}`);
+      B.line(`${outside} = icmp ugt i64 ${biased}, ${2 * Number.MAX_SAFE_INTEGER}`);
+      bail(outside);
+    } else result = emitSpeculativeInteger(this, target.value, bail, reads);
+    // Speculable values read only numbers past this point.
+    if (this.frames.pop()!.length > 0)
+      throw new InternalCompilerError("llvm emitter bug: speculative write owns a temporary");
+    B.line(`store i64 ${result}, ptr ${binding.slot}`);
+    const deferred = this.speculation!.deferred;
+    if (!deferred.has(target.localId)) {
+      const number = B.tmp();
+      B.line(`${number} = sitofp i64 ${result} to double`);
+      B.line(`store double ${number}, ptr ${this.binding(target.localId).slot}`);
+    }
+    if (!bails) return;
+    const done = B.newLabel("spec.done");
+    B.br(done);
+    B.startBlock(failed);
+    this.writeBackSpeculativeGlobals(deferred);
+    B.br(resume);
+    B.startBlock(done);
+  }
+
   /** The amortized cancellation poll, emitted on the loop's back edge (the
    * continue target), after the body. A poll in the header would be a loop
    * exit ahead of the body; LLVM then cannot rotate the loop, so loads that
@@ -6033,6 +6317,10 @@ export class LlEmitter {
       fn.captures !== undefined ? this.lazyCaptures.stableCaptures(fn.name) : new Set();
     this.integerLoopBindings.clear();
     this.countedLoopsEnabled = this.debug === null && !fn.async && !fn.generator;
+    this.speculativeLoopsEnabled = this.countedLoopsEnabled;
+    this.speculativeFunction = fn.name;
+    this.speculation = null;
+    this.speculationResume = null;
     this.finalFieldBorrows =
       !fn.async && !fn.generator && constructorClass(fn.name) === null && this.finalFields.size > 0;
     this.byteWindowsEnabled = this.countedLoopsEnabled;
@@ -7056,6 +7344,17 @@ export class LlEmitter {
   }
 
   emitStmt(s: IrStmt): void {
+    const resume = this.speculationResume?.get(s);
+    if (resume && !resume.placed && !this.B.isTerminated()) {
+      // The fast copy's failed checks re-execute this statement here.
+      resume.placed = true;
+      this.B.br(resume.label);
+      this.B.startBlock(resume.label);
+    }
+    if (this.speculation?.writes.has(s)) {
+      this.emitSpeculativeWrite(s, this.speculation.resume.get(s)!);
+      return;
+    }
     const previous = this.B.debugLocation;
     this.B.debugLocation = this.debug?.location(s.loc, this.debugScope) ?? null;
     const loop =
@@ -7716,57 +8015,61 @@ export class LlEmitter {
       }
       case "while": {
         const workerBudget = this.workerLoopBudget(s);
-        const lc = B.newLabel("loop.c");
-        const lb = B.newLabel("loop.b");
         const le = B.newLabel("loop.e");
-        const lp = workerBudget !== null ? B.newLabel("loop.p") : lc;
-        B.br(lc);
-        B.startBlock(lc);
-        B.condBr(this.emitCondition(s.cond), lb, le);
-        B.startBlock(lb);
-        this.jumpTargets.push({
-          kind: "loop",
-          brkLabel: le,
-          contLabel: lp,
-          labels: s.labels,
-          frameDepth: this.frames.length,
-          scopeDepth: this.scopes.length,
-          finallyDepth: this.finallyStack.length,
-        });
-        this.emitBlock(s.body);
-        this.jumpTargets.pop();
-        B.br(lp);
-        if (lp !== lc) {
-          B.startBlock(lp);
-          this.emitWorkerLoopCheck(workerBudget);
+        this.emitLoopCopies(s, true, le, (exit) => {
+          const lc = B.newLabel("loop.c");
+          const lb = B.newLabel("loop.b");
+          const lp = workerBudget !== null ? B.newLabel("loop.p") : lc;
           B.br(lc);
-        }
+          B.startBlock(lc);
+          B.condBr(this.emitCondition(s.cond), lb, exit);
+          B.startBlock(lb);
+          this.jumpTargets.push({
+            kind: "loop",
+            brkLabel: exit,
+            contLabel: lp,
+            labels: s.labels,
+            frameDepth: this.frames.length,
+            scopeDepth: this.scopes.length,
+            finallyDepth: this.finallyStack.length,
+          });
+          this.emitBlock(s.body);
+          this.jumpTargets.pop();
+          B.br(lp);
+          if (lp !== lc) {
+            B.startBlock(lp);
+            this.emitWorkerLoopCheck(workerBudget);
+            B.br(lc);
+          }
+        });
         B.startBlock(le);
         break;
       }
       case "doWhile": {
         const workerBudget = this.workerLoopBudget(s);
         // Body first (runs at least once); continue jumps to the CONDITION.
-        const lb = B.newLabel("loop.b");
-        const lc = B.newLabel("loop.c");
         const le = B.newLabel("loop.e");
-        B.br(lb);
-        B.startBlock(lb);
-        this.jumpTargets.push({
-          kind: "loop",
-          brkLabel: le,
-          contLabel: lc,
-          labels: s.labels,
-          frameDepth: this.frames.length,
-          scopeDepth: this.scopes.length,
-          finallyDepth: this.finallyStack.length,
+        this.emitLoopCopies(s, true, le, (exit) => {
+          const lb = B.newLabel("loop.b");
+          const lc = B.newLabel("loop.c");
+          B.br(lb);
+          B.startBlock(lb);
+          this.jumpTargets.push({
+            kind: "loop",
+            brkLabel: exit,
+            contLabel: lc,
+            labels: s.labels,
+            frameDepth: this.frames.length,
+            scopeDepth: this.scopes.length,
+            finallyDepth: this.finallyStack.length,
+          });
+          this.emitBlock(s.body);
+          this.jumpTargets.pop();
+          B.br(lc);
+          B.startBlock(lc);
+          this.emitWorkerLoopCheck(workerBudget);
+          B.condBr(this.emitCondition(s.cond), lb, exit);
         });
-        this.emitBlock(s.body);
-        this.jumpTargets.pop();
-        B.br(lc);
-        B.startBlock(lc);
-        this.emitWorkerLoopCheck(workerBudget);
-        B.condBr(this.emitCondition(s.cond), lb, le);
         B.startBlock(le);
         break;
       }
@@ -7922,8 +8225,6 @@ export class LlEmitter {
           const initializers = s.init.kind === "block" ? s.init.body : [s.init];
           for (const initializer of initializers) this.emitStmt(initializer);
         }
-        const lc = B.newLabel("loop.c");
-        const lb = B.newLabel("loop.b");
         const le = B.newLabel("loop.e");
         // JS `for (let i ...)`: each iteration gets a FRESH binding holding
         // a copy of the previous one (closures made in iteration k keep
@@ -7952,69 +8253,75 @@ export class LlEmitter {
           }
         };
         const workerBudget = this.workerLoopBudget(s);
-        const lu =
-          s.update || capturedLets.length > 0 || workerBudget !== null ? B.newLabel("loop.u") : lc;
         freshenBindings();
-        B.br(lc);
-        B.startBlock(lc);
-        if (integerLoop && integerSlot) {
-          const receiver = this.emitStableReceiver(integerLoop.limitReceiver, []);
-          const lenPtr = B.tmp();
-          const len = B.tmp();
-          const index = B.tmp();
-          const inBounds = B.tmp();
-          B.line(
-            `${lenPtr} = getelementptr inbounds %${integerArrayLoop ? "ScrArr" : "ScrBytes"}, ptr ${receiver.name}, i64 0, i32 1`,
-          );
-          if (integerArrayLoop) this.markMemoryPointer(lenPtr, "array:header");
-          B.line(
-            `${len} = load ${this.sizeType}, ptr ${lenPtr}${this.fieldAliasAttachment(lenPtr)}`,
-          );
-          B.line(`${index} = load ${this.sizeType}, ptr ${integerSlot}`);
-          B.line(`${inBounds} = icmp ult ${this.sizeType} ${index}, ${len}`);
-          B.condBr(inBounds, lb, le);
-        } else if (countedLoop && integerSlot && countedLimit) {
-          const index = B.tmp();
-          const inBounds = B.tmp();
-          B.line(`${index} = load i64, ptr ${integerSlot}`);
-          B.line(
-            `${inBounds} = icmp ${countedLoop.step > 0 ? "slt" : "sgt"} i64 ${index}, ${countedLimit}`,
-          );
-          B.condBr(inBounds, lb, le);
-        } else if (s.cond) B.condBr(this.emitCondition(s.cond), lb, le);
-        else B.br(lb);
-        B.startBlock(lb);
-        this.jumpTargets.push({
-          kind: "loop",
-          brkLabel: le,
-          contLabel: lu,
-          labels: s.labels,
-          frameDepth: this.frames.length,
-          scopeDepth: this.scopes.length,
-          finallyDepth: this.finallyStack.length,
-        });
-        this.emitBlock(s.body);
-        this.jumpTargets.pop();
-        B.br(lu);
-        if (lu !== lc) {
-          B.startBlock(lu);
-          this.emitWorkerLoopCheck(workerBudget);
-          freshenBindings();
-          if (integerLoop && integerSlot) {
-            const old = B.tmp();
-            const next = B.tmp();
-            B.line(`${old} = load ${this.sizeType}, ptr ${integerSlot}`);
-            B.line(`${next} = add nuw ${this.sizeType} ${old}, 1`);
-            B.line(`store ${this.sizeType} ${next}, ptr ${integerSlot}`);
-          } else if (countedLoop && integerSlot) {
-            const old = B.tmp();
-            const next = B.tmp();
-            B.line(`${old} = load i64, ptr ${integerSlot}`);
-            B.line(`${next} = add nsw i64 ${old}, ${countedLoop.step}`);
-            B.line(`store i64 ${next}, ptr ${integerSlot}`);
-          } else if (s.update) this.emitStmt(s.update);
+        this.emitLoopCopies(s, capturedLets.length === 0, le, (exit) => {
+          const lc = B.newLabel("loop.c");
+          const lb = B.newLabel("loop.b");
+          const lu =
+            s.update || capturedLets.length > 0 || workerBudget !== null
+              ? B.newLabel("loop.u")
+              : lc;
           B.br(lc);
-        }
+          B.startBlock(lc);
+          if (integerLoop && integerSlot) {
+            const receiver = this.emitStableReceiver(integerLoop.limitReceiver, []);
+            const lenPtr = B.tmp();
+            const len = B.tmp();
+            const index = B.tmp();
+            const inBounds = B.tmp();
+            B.line(
+              `${lenPtr} = getelementptr inbounds %${integerArrayLoop ? "ScrArr" : "ScrBytes"}, ptr ${receiver.name}, i64 0, i32 1`,
+            );
+            if (integerArrayLoop) this.markMemoryPointer(lenPtr, "array:header");
+            B.line(
+              `${len} = load ${this.sizeType}, ptr ${lenPtr}${this.fieldAliasAttachment(lenPtr)}`,
+            );
+            B.line(`${index} = load ${this.sizeType}, ptr ${integerSlot}`);
+            B.line(`${inBounds} = icmp ult ${this.sizeType} ${index}, ${len}`);
+            B.condBr(inBounds, lb, exit);
+          } else if (countedLoop && integerSlot && countedLimit) {
+            const index = B.tmp();
+            const inBounds = B.tmp();
+            B.line(`${index} = load i64, ptr ${integerSlot}`);
+            B.line(
+              `${inBounds} = icmp ${countedLoop.step > 0 ? "slt" : "sgt"} i64 ${index}, ${countedLimit}`,
+            );
+            B.condBr(inBounds, lb, exit);
+          } else if (s.cond) B.condBr(this.emitCondition(s.cond), lb, exit);
+          else B.br(lb);
+          B.startBlock(lb);
+          this.jumpTargets.push({
+            kind: "loop",
+            brkLabel: exit,
+            contLabel: lu,
+            labels: s.labels,
+            frameDepth: this.frames.length,
+            scopeDepth: this.scopes.length,
+            finallyDepth: this.finallyStack.length,
+          });
+          this.emitBlock(s.body);
+          this.jumpTargets.pop();
+          B.br(lu);
+          if (lu !== lc) {
+            B.startBlock(lu);
+            this.emitWorkerLoopCheck(workerBudget);
+            freshenBindings();
+            if (integerLoop && integerSlot) {
+              const old = B.tmp();
+              const next = B.tmp();
+              B.line(`${old} = load ${this.sizeType}, ptr ${integerSlot}`);
+              B.line(`${next} = add nuw ${this.sizeType} ${old}, 1`);
+              B.line(`store ${this.sizeType} ${next}, ptr ${integerSlot}`);
+            } else if (countedLoop && integerSlot) {
+              const old = B.tmp();
+              const next = B.tmp();
+              B.line(`${old} = load i64, ptr ${integerSlot}`);
+              B.line(`${next} = add nsw i64 ${old}, ${countedLoop.step}`);
+              B.line(`store i64 ${next}, ptr ${integerSlot}`);
+            } else if (s.update) this.emitStmt(s.update);
+            B.br(lc);
+          }
+        });
         B.startBlock(le);
         this.releaseScope(this.scopes.pop()!);
         if (integerLoop) this.integerLoopBindings.delete(integerLoop.localId);
diff --git a/packages/compiler/src/backend/llvm/speculative-integers.test.ts b/packages/compiler/src/backend/llvm/speculative-integers.test.ts
new file mode 100644
index 000000000..ae9172298
--- /dev/null
+++ b/packages/compiler/src/backend/llvm/speculative-integers.test.ts
@@ -0,0 +1,196 @@
+import { expect, test } from "vitest";
+import { F64, type IrExpr, type IrStmt } from "../../ir/ir.js";
+import {
+  isSpeculableValue,
+  planSpeculativeLoop,
+  speculativeReads,
+} from "./speculative-integers.js";
+
+const loc = { file: "speculative.ts", start: 0, end: 0 };
+const num = (value: number): IrExpr => ({ kind: "numLit", value, type: F64, loc });
+const ref = (localId: string): IrExpr => ({ kind: "varRef", localId, type: F64, loc });
+const bin = (op: "+" | "-" | "*" | "/" | "%" | "<" | "|", left: IrExpr, right: IrExpr): IrExpr => ({
+  kind: "bin",
+  op,
+  left,
+  right,
+  type: op === "<" ? { kind: "bool" } : F64,
+  loc,
+});
+const assign = (localId: string, value: IrExpr): IrStmt => ({
+  kind: "assign",
+  localId,
+  value,
+  loc,
+});
+const increment = (localId: string): IrStmt => ({
+  kind: "exprStmt",
+  expr: { kind: "incDec", op: "+", prefix: false, localId, type: F64, loc },
+  loc,
+});
+const loop = (body: IrStmt[], cond: IrExpr = bin("<", ref("i"), ref("n"))): IrStmt => ({
+  kind: "while",
+  cond,
+  body,
+  loc,
+});
+const all = (): boolean => true;
+
+test("integer writes and their loop-invariant inputs are speculated", () => {
+  const write = assign(
+    "v",
+    bin("%", bin("+", bin("*", ref("v"), num(31)), ref("k")), num(1000003)),
+  );
+  const step = increment("i");
+  const plan = planSpeculativeLoop(loop([write, step]) as never, all)!;
+  expect(plan.writes).toEqual(new Set([write, step]));
+  // `n` (condition) and `k` (write input) are invariant and checked on entry.
+  expect([...plan.locals]).toEqual([
+    ["v", true],
+    ["i", true],
+    ["k", true],
+    ["n", true],
+  ]);
+});
+
+test("locals declared in the body need no entry check", () => {
+  const declare: IrStmt = { kind: "varDecl", localId: "t", init: bin("*", ref("i"), num(2)), loc };
+  const plan = planSpeculativeLoop(
+    loop([
+      declare,
+      assign("s", bin("%", bin("*", ref("s"), ref("t")), num(65521))),
+      increment("i"),
+    ]) as never,
+    all,
+  )!;
+  expect(plan.locals.get("t")).toBe(false);
+  expect(plan.locals.get("s")).toBe(true);
+});
+
+test("non-integer, expression-position and ineligible writes keep the double copy", () => {
+  expect(isSpeculableValue(bin("/", ref("a"), num(2)))).toBe(false);
+  expect(isSpeculableValue(num(0.5))).toBe(false);
+  expect(isSpeculableValue(num(-0))).toBe(false);
+  expect(isSpeculableValue(bin("|", ref("a"), num(0)))).toBe(true);
+  // A fractional write leaves only the counter, which alone is not worth a copy.
+  const halves = loop([assign("x", bin("/", ref("x"), num(2))), increment("i")]);
+  expect(planSpeculativeLoop(halves as never, all)?.locals.has("x")).toBeFalsy();
+  // A write inside a sequence expression cannot restart its statement.
+  const sequence: IrStmt = {
+    kind: "exprStmt",
+    expr: {
+      kind: "seqExpr",
+      stmts: [assign("v", bin("+", ref("v"), num(1)))],
+      result: num(0),
+      type: F64,
+      loc,
+    },
+    loc,
+  };
+  const plan = planSpeculativeLoop(
+    loop([sequence, assign("v", bin("+", ref("v"), num(2))), increment("i")]) as never,
+    all,
+  );
+  expect(plan?.locals.has("v")).toBeFalsy();
+  expect(
+    planSpeculativeLoop(
+      loop([assign("v", bin("+", ref("v"), num(2)))]) as never,
+      (id) => id !== "v",
+    ),
+  ).toBeNull();
+});
+
+test("accumulators that are only added to keep their double", () => {
+  const sum = loop([assign("s", bin("+", ref("s"), bin("%", ref("i"), num(7)))), increment("i")]);
+  expect(planSpeculativeLoop(sum as never, all)?.locals.has("s")).toBeFalsy();
+});
+
+test("nested loops are left to the inner loop", () => {
+  const inner = loop([assign("v", bin("+", ref("v"), num(1)))]);
+  expect(planSpeculativeLoop(loop([inner, increment("i")]) as never, all)).toBeNull();
+});
+
+test("statically proven locals are never shadowed and leave nothing to duplicate", () => {
+  const write = assign(
+    "v",
+    bin("%", bin("+", bin("*", ref("v"), num(31)), ref("k")), num(1000003)),
+  );
+  const body = [write, increment("i")];
+  // Every read proven: no plan at all.
+  expect(
+    planSpeculativeLoop(
+      loop(body) as never,
+      all,
+      () => false,
+      () => true,
+    ),
+  ).toBeNull();
+  // Only `k` proven: `v` and the counter are shadowed, `k` is not checked.
+  const plan = planSpeculativeLoop(
+    loop(body) as never,
+    all,
+    () => false,
+    (e) => e.kind === "varRef" && e.localId === "k",
+  )!;
+  expect(plan.locals.has("v")).toBe(true);
+  expect(plan.locals.has("k")).toBe(false);
+});
+
+test("heap reads are speculative leaves, read before any check", () => {
+  const rows: IrExpr = {
+    kind: "varRef",
+    localId: "rows",
+    type: { kind: "array", elem: F64 },
+    loc,
+  } as never;
+  const element: IrExpr = { kind: "arrayGet", arr: rows, index: ref("i"), type: F64, loc };
+  const text: IrExpr = { kind: "varRef", localId: "s", type: { kind: "string" }, loc } as never;
+  const code: IrExpr = {
+    kind: "strIntrinsic",
+    method: "charCodeAt",
+    receiver: text,
+    args: [ref("i")],
+    type: F64,
+    loc,
+  } as never;
+  const value = bin("%", bin("+", bin("*", ref("h"), num(33)), code), num(2147483647));
+  expect(isSpeculableValue(value)).toBe(true);
+  expect(speculativeReads(bin("+", element, code))).toEqual([element, code]);
+  // The ordinary emitter evaluates bitwise operators; heap reads stay out.
+  expect(isSpeculableValue(bin("|", code, num(0)))).toBe(false);
+  const eligible = (id: string): boolean => id !== "rows" && id !== "s";
+  const plan = planSpeculativeLoop(loop([assign("h", value), increment("i")]) as never, eligible)!;
+  expect(plan.locals.has("h")).toBe(true);
+});
+
+test("invariants may come from bindings no shadow could be written to", () => {
+  const write = assign("h", bin("%", bin("+", bin("*", ref("h"), num(33)), ref("i")), ref("m")));
+  const body = [write, increment("i")];
+  const locals = (id: string): boolean => id !== "m";
+  expect(planSpeculativeLoop(loop(body) as never, locals)?.locals.has("h")).toBeFalsy();
+  const plan = planSpeculativeLoop(
+    loop(body) as never,
+    locals,
+    () => false,
+    () => false,
+    all,
+  )!;
+  expect(plan.locals.get("m")).toBe(true);
+  expect(plan.locals.has("h")).toBe(true);
+});
+
+test("loops that only count and compare keep one copy", () => {
+  const counter = loop([increment("i")]);
+  expect(planSpeculativeLoop(counter as never, all)).toBeNull();
+  // A product in the condition alone is not.
+  const bounded = loop([increment("i")], bin("<", ref("i"), bin("*", ref("n"), ref("scale"))));
+  expect(planSpeculativeLoop(bounded as never, all)).toBeNull();
+  // A variable step (a sieve's `j += i`) is worth a copy.
+  const sieve = loop([assign("i", bin("+", ref("i"), ref("step")))]);
+  expect(planSpeculativeLoop(sieve as never, all)?.locals.has("i")).toBe(true);
+  const scaled = loop([
+    assign("j", bin("+", ref("j"), bin("*", ref("i"), ref("n")))),
+    increment("i"),
+  ]);
+  expect(planSpeculativeLoop(scaled as never, all)?.locals.has("n")).toBe(true);
+});
diff --git a/packages/compiler/src/backend/llvm/speculative-integers.ts b/packages/compiler/src/backend/llvm/speculative-integers.ts
new file mode 100644
index 000000000..cf1295fa2
--- /dev/null
+++ b/packages/compiler/src/backend/llvm/speculative-integers.ts
@@ -0,0 +1,634 @@
+import type { IrExpr, IrStmt } from "../../ir/ir.js";
+import { everyExprChild, everyStmtChild } from "../../ir/traverse.js";
+import { f64Lit } from "./common.js";
+import type { LlValue, LlvmEmitterContext } from "./expr-context.js";
+import { exactInteger, widenInteger } from "./integer-values.js";
+import { withinInt32 } from "../../ir/integer-ranges.js";
+
+/** Speculative integer loops.
+ *
+ * An innermost loop whose number locals are only written by statement-level
+ * integer arithmetic (`x = x * 31 + n`, `x += y % m`, `x++`) runs as two
+ * copies. The fast copy keeps those locals in i64 shadow slots holding safe
+ * integers and computes the writes with checked integer arithmetic; every
+ * write also stores the exact double, so the ordinary slot is always current
+ * and loop exits, calls and jumps need no write-back. A check that fails
+ * (overflow past 2^53 - 1, a result of -0, a fractional or non-finite leaf, a
+ * zero divisor) branches to the same statement in the ordinary double copy,
+ * which re-executes it and finishes the loop there. Re-execution is exact
+ * because a speculated statement evaluates nothing but locals, literals and
+ * arithmetic before it writes.
+ *
+ * The loop is entered through the fast copy only when every speculated local
+ * that is live on entry already holds a safe integer other than -0. */
+export interface SpeculativeLoopPlan {
+  /** Speculated locals; `true` when the entry value is live and checked. */
+  locals: Map<string, boolean>;
+  /** Statement-level writes computed as integers in the fast copy. */
+  writes: Set<IrStmt>;
+  /** Nothing in the loop can observe a module global or leave it other
+   * than through its condition, `break` or a failed check: no calls, no
+   * throwing operations, no labeled jumps and no `return`. */
+  observerFree: boolean;
+}
+
+type SpeculativeLoop = Extract<IrStmt, { kind: "for" | "while" | "doWhile" }>;
+
+/** The fast copy duplicates the loop; keep the duplicated code small. */
+const MAX_LOOP_NODES = 400;
+
+/** Statements whose emission is plain control flow or a single write.
+ * Nested loops, switches and try statements keep the loop on the ordinary
+ * path (their emission keeps state that the two copies would not share). */
+const SIMPLE_STATEMENTS = new Set<string>([
+  "varDecl",
+  "assign",
+  "exprStmt",
+  "if",
+  "block",
+  "arraySet",
+  "bytesSet",
+  "fieldSet",
+  "recordSet",
+  "break",
+  "continue",
+  "return",
+]);
+
+/** Statements and expressions that run no other code and cannot throw. */
+const OBSERVER_FREE_STATEMENTS = new Set<string>([
+  "varDecl",
+  "assign",
+  "exprStmt",
+  "if",
+  "block",
+  "break",
+  "continue",
+]);
+const OBSERVER_FREE_EXPRESSIONS = new Set<string>([
+  "numLit",
+  "boolLit",
+  "varRef",
+  "bin",
+  "unary",
+  "incDec",
+  "assignExpr",
+  "toBool",
+  "logical",
+  "ternary",
+]);
+
+/** Operators whose double form costs conversions or long latency, and that
+ * take an integer operand directly. */
+const CONSUMING_OPS = new Set<string>([
+  "*",
+  "/",
+  "%",
+  "**",
+  "&",
+  "|",
+  "^",
+  "<<",
+  ">>",
+  ">>>",
+  "<",
+  "<=",
+  ">",
+  ">=",
+  "===",
+  "!==",
+]);
+
+const INTEGER_OPS = new Set<string>(["+", "-", "*", "%", "&", "|", "^", "<<", ">>", ">>>"]);
+const ARITHMETIC_OPS = new Set<string>(["+", "-", "*", "%"]);
+
+/** A number expression that reads only locals and literals through integer
+ * operators: evaluating it has no effects, so a failed check may restart its
+ * statement. Division and exponentiation usually leave the integers and
+ * would only bail. */
+export function isSpeculableValue(e: IrExpr, reads = true): boolean {
+  if (e.type.kind !== "f64") return false;
+  switch (e.kind) {
+    case "numLit":
+      return Number.isSafeInteger(e.value) && !Object.is(e.value, -0);
+    case "varRef":
+      return true;
+    case "bin": {
+      if (!INTEGER_OPS.has(e.op)) return false;
+      // Bitwise operators evaluate through the ordinary emitter, which
+      // must not read the heap after a check (see speculativeReads).
+      const inner = reads && ARITHMETIC_OPS.has(e.op);
+      return isSpeculableValue(e.left, inner) && isSpeculableValue(e.right, inner);
+    }
+    case "unary":
+      return (e.op === "-" || e.op === "~") && isSpeculableValue(e.operand, reads && e.op === "-");
+    default:
+      return reads && isSpeculativeRead(e);
+  }
+}
+
+/** A number read from a string, an array, a record or an object field that
+ * has no effect: it may throw (a missing element), but it changes nothing,
+ * so a failed check may restart its statement. Its value carries no range,
+ * so it is checked like any other unproven integer. */
+export function isSpeculativeRead(e: IrExpr): boolean {
+  if (e.type.kind !== "f64") return false;
+  switch (e.kind) {
+    case "strIntrinsic":
+      return (
+        e.method === "charCodeAt" &&
+        isEffectFree(e.receiver) &&
+        e.args.length === 1 &&
+        isEffectFree(e.args[0]!)
+      );
+    case "arrayGet":
+      return isEffectFree(e.arr) && isEffectFree(e.index);
+    case "recordGet":
+    case "fieldGet":
+      return isEffectFree(e.obj);
+    default:
+      return false;
+  }
+}
+
+/** Receivers and indexes of speculative reads: locals, literals, number
+ * arithmetic and nested reads. */
+function isEffectFree(e: IrExpr): boolean {
+  switch (e.kind) {
+    case "numLit":
+    case "varRef":
+      return true;
+    case "bin":
+      return e.left.type.kind === "f64" && isEffectFree(e.left) && isEffectFree(e.right);
+    case "arrayGet":
+      return isEffectFree(e.arr) && isEffectFree(e.index);
+    case "recordGet":
+    case "fieldGet":
+      return isEffectFree(e.obj);
+    default:
+      return false;
+  }
+}
+
+/** The speculative reads of a speculable value in evaluation order. */
+export function speculativeReads(e: IrExpr, out: IrExpr[] = []): IrExpr[] {
+  if (e.kind === "bin" && ARITHMETIC_OPS.has(e.op)) {
+    speculativeReads(e.left, out);
+    speculativeReads(e.right, out);
+  } else if (e.kind === "unary" && e.op === "-") speculativeReads(e.operand, out);
+  else if (isSpeculativeRead(e)) out.push(e);
+  return out;
+}
+
+/** `x = x + y` or `x = x - y` with a step that is not a literal, as in a
+ * sieve's `j += i`: the double copy pays a conversion of every index and
+ * the step's own latency, while counters stepping by a literal usually
+ * have a static proof or too little work to repay a second copy. */
+function isVariableStep(localId: string, e: IrExpr): boolean {
+  if (e.kind !== "bin" || (e.op !== "+" && e.op !== "-")) return false;
+  const self = (x: IrExpr): boolean => x.kind === "varRef" && x.localId === localId;
+  const step = self(e.left) ? e.right : e.op === "+" && self(e.right) ? e.left : null;
+  return step !== null && step.kind !== "numLit";
+}
+
+function hasProduct(e: IrExpr): boolean {
+  if (e.kind === "bin")
+    return e.op === "*" || e.op === "%" || hasProduct(e.left) || hasProduct(e.right);
+  if (e.kind === "unary") return hasProduct(e.operand);
+  return false;
+}
+
+function hasArithmetic(e: IrExpr): boolean {
+  if (e.kind === "bin")
+    return ARITHMETIC_OPS.has(e.op) || hasArithmetic(e.left) || hasArithmetic(e.right);
+  if (e.kind === "unary") return e.op === "-" || hasArithmetic(e.operand);
+  return false;
+}
+
+/** The written local and value of a statement-level write, or null. A
+ * value of null with a local means `x++`/`x--`. */
+export function speculativeWriteTarget(
+  s: IrStmt,
+): { localId: string; value: IrExpr | null; op: "+" | "-" } | null {
+  switch (s.kind) {
+    case "assign":
+      return { localId: s.localId, value: s.value, op: "+" };
+    case "varDecl":
+      return s.init ? { localId: s.localId, value: s.init, op: "+" } : null;
+    case "exprStmt":
+      if (s.expr.kind === "incDec") return { localId: s.expr.localId, value: null, op: s.expr.op };
+      if (s.expr.kind === "assignExpr")
+        return { localId: s.expr.localId, value: s.expr.value, op: "+" };
+      return null;
+    default:
+      return null;
+  }
+}
+
+/** `eligible` names the locals a shadow may stand for; `integral` those the
+ * emitter already holds as integers (counters, int32 locals); `proven` the
+ * expressions the static range analysis already proves integral. Static
+ * proofs come first: a local whose reads are all proven is never shadowed
+ * or checked, and a loop with nothing unproven is never duplicated. */
+export function planSpeculativeLoop(
+  loop: SpeculativeLoop,
+  eligible: (localId: string) => boolean,
+  integral: (localId: string) => boolean = () => false,
+  proven: (e: IrExpr) => boolean = () => false,
+  invariantEligible: (localId: string) => boolean = eligible,
+): SpeculativeLoopPlan | null {
+  const writes = new Map<string, IrStmt[]>();
+  const unspeculable = new Set<string>();
+  const reads = new Set<string>();
+  // Locals with at least one read the static analysis does not prove.
+  const unproven = new Set<string>();
+  const declared = new Set<string>();
+  let nodes = 0;
+  let rejected = false;
+  let arithmetic = false;
+  let observerFree = true;
+  // Locals the loop reads as integers: operands of multiplication,
+  // remainder, bitwise operators and comparisons, and indexes. An
+  // accumulator only ever added to gains little from an integer shadow
+  // (the double add is cheap), so it keeps its double.
+  const consumed = new Set<string>();
+  const consume = (e: IrExpr): void => {
+    if (e.kind === "varRef") consumed.add(e.localId);
+  };
+  // Operands of multiplication, division and remainder: the operations whose
+  // double forms cost the most. A loop whose integers only count and compare
+  // gains nothing measurable from a second copy, so it keeps one.
+  const multiplied = new Set<string>();
+  // A product in the condition alone (`i < Math.floor(n * scale)`) does not
+  // make the body's work integer.
+  let inCondition = false;
+
+  const expr = (e: IrExpr): boolean => {
+    if (rejected) return false;
+    nodes++;
+    if (!OBSERVER_FREE_EXPRESSIONS.has(e.kind) || (e.kind === "bin" && e.left.type.kind !== "f64"))
+      observerFree = false;
+    if (e.kind === "varRef") {
+      reads.add(e.localId);
+      if (!proven(e)) unproven.add(e.localId);
+    } else if (e.kind === "assignExpr" || e.kind === "incDec") unspeculable.add(e.localId);
+    else if (e.kind === "bin" && CONSUMING_OPS.has(e.op)) {
+      consume(e.left);
+      consume(e.right);
+      if (!inCondition && (e.op === "*" || e.op === "/" || e.op === "%"))
+        for (const operand of [e.left, e.right])
+          if (operand.kind === "varRef") multiplied.add(operand.localId);
+    } else if (e.kind === "arrayGet" || e.kind === "arrayHas" || e.kind === "arrayState")
+      consume(e.index);
+    else if (e.kind === "bytesIntrinsic" || e.kind === "strIntrinsic") e.args.forEach(consume);
+    return everyExprChild(e, expr, nested);
+  };
+  // Statements inside expressions (sequence expressions) run mid-expression,
+  // where no statement can restart: their writes stay unspeculated.
+  const nested = (s: IrStmt): boolean => {
+    nodes++;
+    observerFree = false;
+    if (s.kind === "assign" || s.kind === "varDecl") unspeculable.add(s.localId);
+    return everyStmtChild(s, expr, nested);
+  };
+  const stmt = (s: IrStmt): boolean => {
+    if (rejected) return false;
+    nodes++;
+    if (!SIMPLE_STATEMENTS.has(s.kind)) {
+      rejected = true;
+      return false;
+    }
+    if (
+      !OBSERVER_FREE_STATEMENTS.has(s.kind) ||
+      ((s.kind === "break" || s.kind === "continue") && s.label !== undefined)
+    )
+      observerFree = false;
+    if (s.kind === "varDecl") declared.add(s.localId);
+    if (s.kind === "arraySet" || s.kind === "bytesSet") consume(s.index);
+    const target = speculativeWriteTarget(s);
+    if (target) {
+      if (target.value === null || isSpeculableValue(target.value)) {
+        const list = writes.get(target.localId) ?? [];
+        list.push(s);
+        writes.set(target.localId, list);
+        if (target.value === null) {
+          reads.add(target.localId);
+          if (s.kind !== "exprStmt" || !proven(s.expr)) unproven.add(target.localId);
+        } else expr(target.value);
+        return !rejected;
+      }
+      unspeculable.add(target.localId);
+      if (target.value) expr(target.value);
+      return !rejected;
+    }
+    return everyStmtChild(s, expr, stmt);
+  };
+
+  if (loop.cond) {
+    inCondition = true;
+    expr(loop.cond);
+    inCondition = false;
+  }
+  if (loop.kind === "for" && loop.update) stmt(loop.update);
+  for (const s of loop.body) stmt(s);
+  if (rejected || nodes > MAX_LOOP_NODES) return null;
+
+  const locals = new Map<string, boolean>();
+  for (const id of writes.keys())
+    if (!unspeculable.has(id) && consumed.has(id) && unproven.has(id) && eligible(id))
+      locals.set(id, !declared.has(id));
+  // Every local a speculated write reads must be an integer whenever the
+  // write runs: speculated itself, already an integer, or loop-invariant
+  // (checked on entry). A value computed from doubles the loop writes
+  // would only fail its check on every entry.
+  const grounded = (e: IrExpr): boolean => {
+    if (e.kind === "varRef") {
+      const id = e.localId;
+      return (
+        proven(e) ||
+        locals.has(id) ||
+        integral(id) ||
+        (!writes.has(id) && !unspeculable.has(id) && invariantEligible(id))
+      );
+    }
+    if (isSpeculativeRead(e)) return true;
+    return everyExprChild(e, grounded, () => false);
+  };
+  for (let changed = true; changed;) {
+    changed = false;
+    for (const id of [...locals.keys()]) {
+      for (const s of writes.get(id)!) {
+        const target = speculativeWriteTarget(s)!;
+        if (target.value !== null && !grounded(target.value)) {
+          locals.delete(id);
+          changed = true;
+          break;
+        }
+      }
+    }
+  }
+  const speculated = new Set<IrStmt>();
+  for (const id of locals.keys()) for (const s of writes.get(id)!) speculated.add(s);
+  if (speculated.size === 0) return null;
+  // Loop-invariant integer inputs of the writes and the condition join the
+  // shadows, so their reads carry integer facts too.
+  const invariant = (e: IrExpr): boolean => {
+    if (e.kind === "varRef") {
+      const id = e.localId;
+      if (
+        !proven(e) &&
+        !locals.has(id) &&
+        !writes.has(id) &&
+        !unspeculable.has(id) &&
+        invariantEligible(id)
+      )
+        locals.set(id, true);
+      return true;
+    }
+    return everyExprChild(e, invariant, () => true);
+  };
+  for (const s of speculated) {
+    const target = speculativeWriteTarget(s)!;
+    if (target.value === null) arithmetic = true;
+    else {
+      if (hasArithmetic(target.value)) arithmetic = true;
+      invariant(target.value);
+    }
+  }
+  if (!arithmetic) return null;
+  if (loop.cond) invariant(loop.cond);
+  const heavy =
+    [...locals.keys()].some((id) => multiplied.has(id)) ||
+    [...speculated].some((s) => {
+      const target = speculativeWriteTarget(s)!;
+      return (
+        target.value !== null &&
+        (hasProduct(target.value) || isVariableStep(target.localId, target.value))
+      );
+    });
+  if (!heavy) return null;
+  return { locals, writes: speculated, observerFree };
+}
+
+const SAFE = Number.MAX_SAFE_INTEGER;
+
+/** The range every speculative shadow holds. */
+export const SPECULATIVE_RANGE = { min: -SAFE, max: SAFE };
+
+/** Convert a double to i64 and test that it is a safe integer other than
+ * -0: the ordered range tests reject NaN and infinities, and the bit
+ * patterns of the value and its round trip agree only for integers that are
+ * not -0. `freeze` keeps out-of-range conversions from producing poison. */
+export function emitSafeIntegerCheck(host: LlvmEmitterContext, value: string): [string, string] {
+  const B = host.B;
+  const low = B.tmp(),
+    high = B.tmp(),
+    inRange = B.tmp(),
+    raw = B.tmp(),
+    integer = B.tmp(),
+    back = B.tmp(),
+    bits = B.tmp(),
+    backBits = B.tmp(),
+    exact = B.tmp(),
+    ok = B.tmp();
+  B.line(`${low} = fcmp oge double ${value}, ${f64Lit(-SAFE)}`);
+  B.line(`${high} = fcmp ole double ${value}, ${f64Lit(SAFE)}`);
+  B.line(`${inRange} = and i1 ${low}, ${high}`);
+  B.line(`${raw} = fptosi double ${value} to i64`);
+  B.line(`${integer} = freeze i64 ${raw}`);
+  B.line(`${back} = sitofp i64 ${integer} to double`);
+  B.line(`${bits} = bitcast double ${value} to i64`);
+  B.line(`${backBits} = bitcast double ${back} to i64`);
+  B.line(`${exact} = icmp eq i64 ${bits}, ${backBits}`);
+  B.line(`${ok} = and i1 ${inRange}, ${exact}`);
+  return [integer, ok];
+}
+
+/** A condition that is true when an i64 lies outside the safe range. */
+function unsafeInteger(host: LlvmEmitterContext, value: string): string {
+  const B = host.B;
+  const biased = B.tmp(),
+    outside = B.tmp();
+  B.line(`${biased} = add i64 ${value}, ${SAFE}`);
+  B.line(`${outside} = icmp ugt i64 ${biased}, ${2 * SAFE}`);
+  return outside;
+}
+
+/** A condition that is true when a safe i64 lies outside [low, high]. Every
+ * speculative value is within ±(2^53 - 1), so bounds at or past those
+ * limits need no test, and the remaining bounds are exact integers. */
+function outsideRange(host: LlvmEmitterContext, value: string, low: number, high: number): string {
+  const B = host.B;
+  let outside = "false";
+  if (low > -SAFE) {
+    outside = B.tmp();
+    B.line(`${outside} = icmp slt i64 ${value}, ${low}`);
+  }
+  if (high < SAFE) {
+    const above = B.tmp();
+    B.line(`${above} = icmp sgt i64 ${value}, ${high}`);
+    outside = outside === "false" ? above : either(host, outside, above);
+  }
+  return outside;
+}
+
+/** A literal's value or the range analysis' proof for an expression whose
+ * range lies well inside the safe integers. */
+function staticRange(
+  host: LlvmEmitterContext,
+  e: IrExpr,
+): { min: number; max: number } | undefined {
+  const range = e.kind === "numLit" ? { min: e.value, max: e.value } : host.integerRanges.get(e);
+  if (!range) return undefined;
+  const limit = 2 ** 52;
+  return range.min >= -limit && range.max <= limit ? range : undefined;
+}
+
+function either(host: LlvmEmitterContext, a: string, b: string): string {
+  const t = host.B.tmp();
+  host.B.line(`${t} = or i1 ${a}, ${b}`);
+  return t;
+}
+
+/** Evaluate a speculable value (isSpeculableValue) as a safe i64. `bail`
+ * branches away when its condition holds, so the integer result equals the
+ * JS number exactly whenever control continues: sums and products stay
+ * within 2^53 - 1, no result is -0, and `%` never divides by zero. Other
+ * operators evaluate through the ordinary emitter and must yield a safe
+ * integer. */
+export function emitSpeculativeInteger(
+  host: LlvmEmitterContext,
+  e: IrExpr,
+  bail: (condition: string) => void,
+  reads: ReadonlyMap<IrExpr, LlValue> = new Map(),
+): string {
+  const B = host.B;
+  // A value the range analysis already proves takes the ordinary integer
+  // emission, which needs no checks.
+  const proven = e.kind === "bin" ? host.integerRanges.get(e) : undefined;
+  if (proven && proven.min >= -SAFE && proven.max <= SAFE && speculativeReads(e).length === 0)
+    return speculativeLeaf(host, host.emitExpr(e), e, bail);
+  switch (e.kind) {
+    case "numLit":
+      return String(e.value);
+    case "varRef": {
+      const slot = host.integerLoopBindings.get(e.localId);
+      if (slot !== undefined) {
+        const value = B.tmp();
+        B.line(`${value} = load ${slot.type}, ptr ${slot.slot}`);
+        if (slot.type === "i64") return value;
+        const wide = B.tmp();
+        B.line(`${wide} = ${slot.signed ? "sext" : "zext"} i32 ${value} to i64`);
+        return wide;
+      }
+      const int32 = host.int32Locals.get(e.localId);
+      if (int32 !== undefined) {
+        const bits = B.tmp(),
+          wide = B.tmp();
+        B.line(`${bits} = load i32, ptr ${host.binding(e.localId).slot}`);
+        B.line(`${wide} = ${int32.signed ? "sext" : "zext"} i32 ${bits} to i64`);
+        return wide;
+      }
+      break;
+    }
+    case "bin": {
+      if (e.op !== "+" && e.op !== "-" && e.op !== "*" && e.op !== "%") break;
+      const a = emitSpeculativeInteger(host, e.left, bail, reads);
+      const b = emitSpeculativeInteger(host, e.right, bail, reads);
+      const result = B.tmp();
+      if (e.op === "+" || e.op === "-") {
+        // With one operand's range known, the other is checked against the
+        // narrowed bounds before the operation: the check then depends on
+        // no new value, and LLVM keeps the operation itself simple.
+        const right = staticRange(host, e.right);
+        const left = e.op === "+" ? staticRange(host, e.left) : undefined;
+        if (right) {
+          // a + b with b in [lo, hi] is safe iff a in [-SAFE - lo, SAFE - hi];
+          // a - b iff a in [-SAFE + hi, SAFE + lo].
+          const low = e.op === "+" ? -SAFE - right.min : -SAFE + right.max;
+          const high = e.op === "+" ? SAFE - right.max : SAFE + right.min;
+          bail(outsideRange(host, a, low, high));
+        } else if (left) bail(outsideRange(host, b, -SAFE - left.min, SAFE - left.max));
+        // Safe operands cannot overflow i64; an exact zero sum is +0.
+        B.line(`${result} = ${e.op === "+" ? "add" : "sub"} nsw i64 ${a}, ${b}`);
+        if (!right && !left) bail(unsafeInteger(host, result));
+        return result;
+      }
+      if (e.op === "*") {
+        host.declare("declare { i64, i1 } @llvm.smul.with.overflow.i64(i64, i64)");
+        const pair = B.tmp(),
+          overflow = B.tmp(),
+          zero = B.tmp(),
+          signs = B.tmp(),
+          negative = B.tmp(),
+          negativeZero = B.tmp();
+        B.line(`${pair} = call { i64, i1 } @llvm.smul.with.overflow.i64(i64 ${a}, i64 ${b})`);
+        B.line(`${result} = extractvalue { i64, i1 } ${pair}, 0`);
+        B.line(`${overflow} = extractvalue { i64, i1 } ${pair}, 1`);
+        // A zero product with a negative factor is -0.
+        B.line(`${zero} = icmp eq i64 ${result}, 0`);
+        B.line(`${signs} = or i64 ${a}, ${b}`);
+        B.line(`${negative} = icmp slt i64 ${signs}, 0`);
+        B.line(`${negativeZero} = and i1 ${zero}, ${negative}`);
+        bail(either(host, either(host, overflow, unsafeInteger(host, result)), negativeZero));
+        return result;
+      }
+      // `%`: a zero divisor yields NaN, and a zero remainder of a negative
+      // dividend is -0. Safe operands cannot hit INT64_MIN % -1.
+      const zeroDivisor = B.tmp();
+      B.line(`${zeroDivisor} = icmp eq i64 ${b}, 0`);
+      bail(zeroDivisor);
+      const zero = B.tmp(),
+        negative = B.tmp(),
+        negativeZero = B.tmp();
+      B.line(`${result} = srem i64 ${a}, ${b}`);
+      B.line(`${zero} = icmp eq i64 ${result}, 0`);
+      B.line(`${negative} = icmp slt i64 ${a}, 0`);
+      B.line(`${negativeZero} = and i1 ${zero}, ${negative}`);
+      bail(negativeZero);
+      return result;
+    }
+    case "unary": {
+      if (e.op !== "-") break;
+      const a = emitSpeculativeInteger(host, e.operand, bail, reads);
+      const zero = B.tmp(),
+        result = B.tmp();
+      // -0 is not an integer result.
+      B.line(`${zero} = icmp eq i64 ${a}, 0`);
+      bail(zero);
+      B.line(`${result} = sub nsw i64 0, ${a}`);
+      return result;
+    }
+    default:
+      break;
+  }
+  return speculativeLeaf(host, reads.get(e) ?? host.emitExpr(e), e, bail);
+}
+
+/** The safe i64 of an evaluated number: its proven integer, or a checked
+ * conversion. */
+function speculativeLeaf(
+  host: LlvmEmitterContext,
+  value: LlValue,
+  e: IrExpr,
+  bail: (condition: string) => void,
+): string {
+  const integer = exactInteger(host, value, e);
+  if (integer && integer.type === "i64" && withinInt32(integer.range)) {
+    // Only the low half is used, as by a double conversion of an int32:
+    // LLVM then narrows the operation that produced it (a 64-bit remainder
+    // by a constant costs a wide multiply).
+    const narrow = host.B.tmp(),
+      wide = host.B.tmp();
+    host.B.line(`${narrow} = trunc i64 ${integer.name} to i32`);
+    host.B.line(`${wide} = sext i32 ${narrow} to i64`);
+    return wide;
+  }
+  if (integer && integer.range.min >= -SAFE && integer.range.max <= SAFE)
+    return widenInteger(host, integer);
+  const [wide, ok] = emitSafeIntegerCheck(host, value.name);
+  const fail = host.B.tmp();
+  host.B.line(`${fail} = xor i1 ${ok}, true`);
+  bail(fail);
+  return wide;
+}
diff --git a/packages/compiler/test/ts7/baselines/order-parity/tests/corpus/speculative-integer-globals.ts.json b/packages/compiler/test/ts7/baselines/order-parity/tests/corpus/speculative-integer-globals.ts.json
new file mode 100644
index 000000000..f09db682d
--- /dev/null
+++ b/packages/compiler/test/ts7/baselines/order-parity/tests/corpus/speculative-integer-globals.ts.json
@@ -0,0 +1,6 @@
+{
+ "order": [
+  "<repo>/tests/corpus/speculative-integer-globals.ts"
+ ],
+ "diags": []
+}
diff --git a/packages/compiler/test/ts7/baselines/order-parity/tests/corpus/speculative-integer-loops.ts.json b/packages/compiler/test/ts7/baselines/order-parity/tests/corpus/speculative-integer-loops.ts.json
new file mode 100644
index 000000000..f79d2bfa2
--- /dev/null
+++ b/packages/compiler/test/ts7/baselines/order-parity/tests/corpus/speculative-integer-loops.ts.json
@@ -0,0 +1,6 @@
+{
+ "order": [
+  "<repo>/tests/corpus/speculative-integer-loops.ts"
+ ],
+ "diags": []
+}
diff --git a/packages/compiler/test/ts7/baselines/order-parity/tests/corpus/speculative-integer-reads.ts.json b/packages/compiler/test/ts7/baselines/order-parity/tests/corpus/speculative-integer-reads.ts.json
new file mode 100644
index 000000000..b05283134
--- /dev/null
+++ b/packages/compiler/test/ts7/baselines/order-parity/tests/corpus/speculative-integer-reads.ts.json
@@ -0,0 +1,6 @@
+{
+ "order": [
+  "<repo>/tests/corpus/speculative-integer-reads.ts"
+ ],
+ "diags": []
+}
diff --git a/tests/corpus/speculative-integer-globals.ts b/tests/corpus/speculative-integer-globals.ts
new file mode 100644
index 000000000..fcce1ee5e
--- /dev/null
+++ b/tests/corpus/speculative-integer-globals.ts
@@ -0,0 +1,66 @@
+// Module-level numbers accumulated by top-level loops: calls in the loop
+// read them while the loop keeps them as integers, and sums that leave
+// the safe integers continue as doubles.
+
+let total: number = 0;
+let calls: number = 0;
+let big: number = 9007199254740000;
+
+function peek(): number {
+  calls++;
+  return total;
+}
+
+for (let i: number = 0; i < 1000; i++) {
+  total = total + (i % 7);
+  if (total < 0) break;
+  if (i % 250 === 0) console.log("peek", peek());
+}
+console.log("total", total, "calls", calls);
+
+for (let i: number = 0; i < 3000; i++) {
+  big += i;
+  if (big % 7 === -1) break;
+}
+console.log("big", big, Number.isSafeInteger(big));
+
+let product: number = 1;
+let rounds: number = 0;
+while (rounds < 70) {
+  product = product * 3;
+  rounds++;
+}
+console.log("product", product);
+
+let zero: number = 1;
+for (let i: number = 0; i < 2; i++) zero = zero * -0;
+console.log("zero", Object.is(zero, -0));
+
+let mixed: number = 10;
+for (let i: number = 0; i < 6; i++) {
+  mixed = mixed - 3;
+  mixed = mixed * 2;
+}
+console.log("mixed", mixed);
+
+let found: number = 0;
+for (let i: number = 0; i < 100; i++) {
+  found = found + i * i;
+  if (found > 1000) break;
+}
+console.log("found", found);
+
+let acc: number = 1;
+let steps: number = 0;
+while (true) {
+  acc = acc * 7;
+  steps++;
+  if (steps > 30) break;
+}
+console.log("acc", acc, steps);
+
+let countdown: number = 5;
+do {
+  countdown = countdown - 2;
+} while (countdown > -3);
+console.log("countdown", countdown);
diff --git a/tests/corpus/speculative-integer-loops.ts b/tests/corpus/speculative-integer-loops.ts
new file mode 100644
index 000000000..1397a3357
--- /dev/null
+++ b/tests/corpus/speculative-integer-loops.ts
@@ -0,0 +1,253 @@
+// Loops whose number locals hold integers until they do not: every check
+// that leaves the integer copy (overflow past 2^53 - 1, -0, fractions,
+// NaN, zero divisors) must continue with exactly Node's doubles.
+
+function show(label: string, value: number): void {
+  console.log(label, Object.is(value, -0) ? "-0" : String(value));
+}
+
+// A remainder chain that stays small.
+function chain(start: number, n: number): number {
+  let v: number = start;
+  for (let i: number = 0; i < 100000; i++) v = (v * 31 + n) % 1000003;
+  return v;
+}
+show("chain", chain(7, 3));
+show("chain negative", chain(-7, -3));
+show("chain fraction start", chain(7.5, 3));
+show("chain fraction step", chain(7, 0.25));
+show("chain NaN", chain(NaN, 1));
+show("chain -0", chain(-0, 0));
+
+// Products that leave the safe integers part-way and keep growing.
+function grow(start: number, factor: number): number {
+  let x: number = start;
+  let steps: number = 0;
+  while (steps < 80) {
+    x = x * factor + 1;
+    steps++;
+  }
+  return x;
+}
+show("grow 3", grow(1, 3));
+show("grow -3", grow(1, -3));
+show("grow 2", grow(0, 2));
+
+// Sums that cross 2^53 - 1 on the way.
+function crossing(): number {
+  let s: number = 9007199254740900;
+  for (let i: number = 0; i < 200; i++) {
+    s += i;
+    if (s % 1000 === -1) break;
+  }
+  return s;
+}
+show("crossing", crossing());
+
+// -0 from a product, a remainder and a negation, each in its own loop.
+function negativeZeroProduct(k: number): number {
+  let p: number = 5;
+  for (let i: number = 0; i < 3; i++) p = p * k;
+  return p;
+}
+show("product -0", negativeZeroProduct(0) * -1);
+show("product zero", negativeZeroProduct(0));
+function negativeZeroTimes(): number {
+  let p: number = 0;
+  let q: number = 1;
+  for (let i: number = 0; i < 4; i++) {
+    p = q * (i - 2);
+    q = p * 0;
+  }
+  return q;
+}
+show("times -0", negativeZeroTimes());
+function negativeZeroRemainder(): number {
+  let r: number = -9;
+  let out: number = 1;
+  for (let i: number = 1; i < 5; i++) {
+    out = r % 3;
+    r = r + i;
+    if (out === 7) break;
+  }
+  return out;
+}
+show("remainder", negativeZeroRemainder());
+function remainderNegZero(): number {
+  let r: number = 0;
+  for (let i: number = 0; i < 2; i++) {
+    r = -6 % (3 + i * 0);
+    if (r > 100) break;
+  }
+  return r;
+}
+show("remainder -0", remainderNegZero());
+function negation(): number {
+  let a: number = 3;
+  let b: number = 1;
+  for (let i: number = 0; i < 4; i++) {
+    a = a - 1;
+    b = -a;
+    if (a * b === 999) break;
+  }
+  return b;
+}
+show("negation", negation());
+function negationZero(): number {
+  let a: number = 2;
+  let b: number = 1;
+  for (let i: number = 0; i < 2; i++) {
+    a = a - 1;
+    b = -a;
+    if (a * b === 999) break;
+  }
+  return b;
+}
+show("negation -0", negationZero());
+
+// A zero divisor part-way through.
+function zeroDivisor(): number {
+  let d: number = 3;
+  let acc: number = 100;
+  for (let i: number = 0; i < 6; i++) {
+    acc = acc % d;
+    d = d - 1;
+    acc = acc + 50;
+  }
+  return acc;
+}
+show("zero divisor", zeroDivisor());
+
+// Jumps out of the integer copy keep the locals current.
+function jumps(limit: number): number {
+  let total: number = 0;
+  let count: number = 0;
+  for (let i: number = 0; i < 1000; i++) {
+    if (i % 3 === 0) continue;
+    total = total + i * i;
+    count++;
+    if (total > limit) break;
+  }
+  return total * 1000 + count;
+}
+show("jumps", jumps(5000));
+show("jumps huge", jumps(1e300));
+function early(values: number[]): number {
+  let sum: number = 0;
+  let i: number = 0;
+  while (i < values.length) {
+    sum = sum + i;
+    if (sum > 10) return sum;
+    i++;
+  }
+  return -sum;
+}
+show("early", early([1, 2, 3, 4, 5, 6, 7]));
+show("early short", early([1, 2]));
+function labeled(): number {
+  let hits: number = 0;
+  outer: for (let a: number = 0; a < 5; a++) {
+    let b: number = 0;
+    while (b < 5) {
+      b = b + 1;
+      hits = hits + a * b;
+      if (b === a) continue outer;
+    }
+  }
+  return hits;
+}
+show("labeled", labeled());
+
+// do-while with a bail in the condition-adjacent update.
+function collatz(start: number): number {
+  let n: number = start;
+  let steps: number = 0;
+  do {
+    n = n % 2 === 0 ? n / 2 : 3 * n + 1;
+    steps = steps + 1;
+  } while (n !== 1 && steps < 1000);
+  return steps;
+}
+show("collatz", collatz(27));
+function doubling(): number {
+  let x: number = 1;
+  let n: number = 0;
+  do {
+    x = x * 2;
+    n++;
+  } while (n < 60);
+  return x;
+}
+show("doubling", doubling());
+
+// The update statement itself bails.
+function bigStep(): number {
+  let last: number = 0;
+  for (let j: number = 1; j < 2e16; j = j * 1000 + 1) last = j;
+  return last;
+}
+show("big step", bigStep());
+
+// Speculated indexes into typed arrays and arrays.
+const bytes: Uint8Array = new Uint8Array(64);
+let k: number = 0;
+for (let j: number = 3; j < 64; j += 3) {
+  bytes[j] = j;
+  k = k + j;
+}
+show("bytes", bytes[63] + bytes[3] + k);
+const squares: number[] = [];
+for (let j: number = 0; j < 10; j++) {
+  let sq: number = j * j;
+  sq = sq - j;
+  squares.push(sq);
+}
+console.log(squares.join(","));
+
+// Bitwise leaves and conversions.
+function lcg(seed: number): number {
+  let s: number = seed;
+  let mixed: number = 0;
+  for (let i: number = 0; i < 1000; i++) {
+    s = (s * 1103515245 + 12345) | 0;
+    mixed = mixed + (s >>> 16);
+    mixed = mixed % 4294967296;
+  }
+  return mixed;
+}
+show("lcg", lcg(42));
+function unsigned(): number {
+  let u: number = 0;
+  for (let i: number = 0; i < 40; i++) u = (u * 2 + 1) >>> 0;
+  return u;
+}
+show("unsigned", unsigned());
+function complement(): number {
+  let c: number = 5;
+  for (let i: number = 0; i < 3; i++) c = ~c + c * 2;
+  return c;
+}
+show("complement", complement());
+
+// Inputs that are not integers on entry, and loop-invariant inputs.
+function scaled(step: number): number {
+  let x: number = 0;
+  for (let i: number = 0; i < 10; i++) {
+    x = x + step;
+    if (x % 3 === -1) break;
+  }
+  return x;
+}
+show("scaled int", scaled(3));
+show("scaled fraction", scaled(0.1));
+show("scaled huge", scaled(1e300));
+show("scaled Infinity", scaled(Infinity));
+function invariantNegZero(z: number): number {
+  let x: number = 0;
+  for (let i: number = 0; i < 3; i++) {
+    x = z * 1;
+    if (x === 5) break;
+  }
+  return x;
+}
+show("invariant -0", invariantNegZero(-0));
diff --git a/tests/corpus/speculative-integer-reads.ts b/tests/corpus/speculative-integer-reads.ts
new file mode 100644
index 000000000..86b1eec9f
--- /dev/null
+++ b/tests/corpus/speculative-integer-reads.ts
@@ -0,0 +1,90 @@
+// Speculative integer loops over values read from strings, arrays, records
+// and module constants: reads that are not integers (NaN past a string's
+// end, fractional elements and fields, a fractional divisor) must continue
+// with exactly Node's doubles.
+
+function show(label: string, value: number): void {
+  console.log(label, Object.is(value, -0) ? "-0" : String(value));
+}
+
+function hashText(s: string, extra: number): number {
+  let h: number = 0;
+  for (let i: number = 0; i < s.length + extra; i++) h = (h * 33 + s.charCodeAt(i)) % 2147483647;
+  return h;
+}
+show("hash", hashText("speculative integer loops", 0));
+show("hash past end", hashText("speculative integer loops", 1));
+show("hash empty", hashText("", 0));
+show("hash unicode", hashText("héllo wörld \u{1F600}", 0));
+
+function mixElements(xs: number[]): number {
+  let a: number = 1;
+  for (let i: number = 0; i < xs.length; i++) a = (a * 7 + (xs[i]! % 100)) % 65521;
+  return a;
+}
+show("elements", mixElements([1, 2, 3, 400, 5000, 60000]));
+show("elements fraction", mixElements([1, 2, 3.5, 4, 5]));
+show("elements negative", mixElements([1, -2, -300, 4]));
+show("elements huge", mixElements([1, 2 ** 60, 3]));
+show("elements NaN", mixElements([1, NaN, 3]));
+
+interface Item {
+  id: number;
+  weight: number;
+}
+function checksum(items: Item[]): number {
+  let c: number = 0;
+  for (let i: number = 0; i < items.length; i++) c = (c * 31 + items[i]!.id) % 1000000007;
+  return c;
+}
+const items: Item[] = [];
+for (let i: number = 0; i < 50; i++) items.push({ id: (i * 7919) % 1000003, weight: i / 4 });
+show("checksum", checksum(items));
+items[25] = { id: 12.25, weight: 0 };
+show("checksum fraction", checksum(items));
+items[25] = { id: -0, weight: 0 };
+show("checksum -0", checksum(items));
+
+class Cell {
+  value: number;
+  constructor(value: number) {
+    this.value = value;
+  }
+}
+function sumCells(cells: Cell[]): number {
+  let s: number = 0;
+  for (let i: number = 0; i < cells.length; i++) s = (s * 3 + cells[i]!.value) % 1000003;
+  return s;
+}
+show("cells", sumCells([new Cell(1), new Cell(2), new Cell(3)]));
+show("cells fraction", sumCells([new Cell(1), new Cell(2.5), new Cell(3)]));
+
+// Module constants written only by top-level code act as loop invariants.
+const modulus: number = Number("1048573");
+const fractional: number = Number("2.5");
+const zero: number = Number("0");
+function hashWith(n: number, m: number): number {
+  let h: number = 7;
+  for (let i: number = 0; i < n; i++) h = (h * 33 + i) % m;
+  return h;
+}
+function hashModule(n: number): number {
+  let h: number = 7;
+  for (let i: number = 0; i < n; i++) h = (h * 33 + i) % modulus;
+  return h;
+}
+function hashFractional(n: number): number {
+  let h: number = 7;
+  for (let i: number = 0; i < n; i++) h = (h * 33 + i) % fractional;
+  return h;
+}
+function hashZero(n: number): number {
+  let h: number = 7;
+  for (let i: number = 0; i < n; i++) h = (h * 33 + i) % zero;
+  return h;
+}
+show("module modulus", hashModule(10000));
+show("module fractional", hashFractional(1000));
+show("module zero", hashZero(10));
+show("parameter modulus", hashWith(10000, modulus));
+show("parameter fractional", hashWith(100, fractional));