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 Incoming links: Integer arithmetic

Constant folding 2026-10-13

Constant folding replaces a program expression whose value is known during translation by a literal value. For x=2+3*4, first replace the multiplication subtree by Constant(value=12), then replace the addition subtree by Constant(value=14). The variable assignment and its Store target survive: the result still assigns to x. This illustrates why a bottom-up abstract syntax tree traversal is convenient: replacing one child can make its parent eligible for the next replacement.

Figure 1.
Constant folding and postorder abstract syntax tree traversal for `2+3*4`: replace $3 \cdot 4$ by $12$ , then $2 + 12$ by $14$
. **Details**.

Here is a deliberately small Python transformer. Calling generic_visit first transforms the children; returning a node replaces the visited node. The location helpers retain or supply the metadata expected for compilation. ^[ref]

class FoldIntegers(ast.NodeTransformer):
    def visit_BinOp(self, node):
        self.generic_visit(node)
        if (isinstance(node.left, ast.Constant)
                and isinstance(node.right, ast.Constant)
                and type(node.left.value) is int
                and type(node.right.value) is int
                and type(node.op) in OPS):
            value = OPS[type(node.op)](node.left.value, node.right.value)
            return ast.copy_location(ast.Constant(value=value), node)
        return node

tree = ast.parse("x=2+3*4", optimize=0)
folded = ast.fix_missing_locations(FoldIntegers().visit(tree))
namespace = {}
exec(compile(folded, "<folded example>", "exec"), namespace)
assert namespace["x"] == 14
assert ast.unparse(folded) == "x = 14"

This transformer folds literal integer operands only. It leaves a+3*4 as a+12, without inventing a value for the program identifier a. It leaves calls intact: replacing 0*f() by 0 could discard a side effect or a failure. Nor does it reassociate floating-point arithmetic, where changing parentheses can change rounding. For the supported integer arithmetic, a short structural induction explains correctness: a retained literal value has the same value; at each folded binary node, the already-correct child values are combined by exactly the original operation in the original left/right order. This claim concerns the supported fragment's values, not unrestricted equivalence of arbitrary Python programs.

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Memoization arithmetic and stack costs 2026-10-13

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Under constant-cost lookup and integer addition, Memoized Fibonacci evaluation has time complexity

O (n)

in Big O notation, retaining

O (n)

values. A hash table needs its own lookup-cost assumptions; an array indexed by

0, \dots, n

avoids hash collisions in this example. Memoization removes repeated bodies, but also pays for keys, lookups and retained answers. A workload without reuse can pay those costs without saving evaluations.

The call stack remains. Left-first recursion descends through arguments

n, n - 1, \dots, 1

before returning, giving

n

simultaneously active frames for

n \geq 2

. Therefore the auxiliary space complexity includes

O (n)

frames as well as

O (n)

table entries in the constant-size-value model. An iterative bottom-up dynamic programming calculation removes these recursive frames; retaining only the two previous Fibonacci numbers suffices when only the final value is wanted, but discards the table useful for later requests.

Exact Fibonacci numbers do not remain constant-size values. For

k \geq 2

, the recurrence relation gives

2^{(k - 2) /2} \leq F_{k} \leq 2^{k - 1}

, so the representation of

F_{k}

as a binary number has order

k

bits. With ordinary addition costing order

k

bit operations on such operands, the

n - 1

additions cost order

n^{2}

bit operations in total. Retaining all values also requires order

n^{2}

bits of value storage, before keys and frame overhead. The linear bound counts additions under its stated cost model; it is not a linear bound on exact integer arithmetic in bits.^[ref]

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Python (programming language) 2026-10-13

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A programming language whose standard-library ast module provides access to its abstract syntax trees. The examples here use its version 3.14 schema and literal integer arithmetic.

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