Sum Two Numbers Without + or sum | Dataford Interview Questions

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Problem

Reply data-processing code may need a low-level integer operation that does not rely on Python's arithmetic addition. Implement a function that adds two signed integers without using the + operator, sum(), .sum(), operator.add, or equivalent arithmetic helpers.

Formal Specification

Implement add_integers(a, b):

Input: Two Python integers a and b in the signed 32-bit range.
Output: Their mathematical sum as a Python integer.
Use bitwise operations such as XOR, AND, left shift, and masking to simulate binary addition.
The implementation must correctly handle positive values, negative values, zero, and sums outside the signed 32-bit range.
Do not convert the integers to strings or lists of bits.

Python integers use unlimited precision and conceptually extend negative values with infinite leading ones. Explain how your implementation uses a fixed-width two's-complement representation to ensure termination and then converts the result back to a Python integer.

Constraints

-2^31 <= a, b <= 2^31 - 1
The output must be the exact mathematical sum.
The + operator, sum(), .sum(), and arithmetic addition helpers are prohibited.
String conversion and conversion to a list of bits are prohibited.

Function Signature

def add_integers(a, b):