0395MediumMathBit Manipulation

Bitwise ALU Total

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Trains the technique from

This is an original problem, written from a brief that listed the technique, the difficulty, the topics, the function shape and the input bounds — none of that problem's wording, examples, hints or editorials. The link is there so you can map your practice onto the standard set.

Same function shape, different story and different numbers.

You are writing microcode for a teaching processor whose arithmetic unit was left off the die. The only opcodes available to you are bitwise AND, OR, XOR, NOT, left shift and right shift, plus comparisons and loops.

Two signed sensor readings a and b arrive in registers. Report their combined value.

Your routine must not apply the + or - operators anywhere, and it must not reach for a library routine that performs arithmetic on your behalf. Readings may be negative; the machine stores a register as a 32-bit two's complement word, and every legal total fits in such a word.

Examples

Example 1

Input: a = 14, b = 9
Output: 23

The two readings come to 23 together.

Example 2

Input: a = -7, b = 4
Output: -3

Readings may be negative; sitting 4 above -7 lands on -3.

Example 3

Input: a = -18, b = -25
Output: -43

Both registers hold negative readings, and together they come to -43.

Example 4

Input: a = 640, b = -640
Output: 0

The two readings cancel, so the routine reports 0.

Constraints

-1000 <= a, b <= 1000

The signature

The editor is preloaded with this. It matches the parent problem's shape, so a solution that works here transfers to a judge unchanged.

Python

def get_sum(a: int, b: int) -> int:

Java

public int getSum(int a, int b)

The intended approach

Time O(1) - at most 32 carry roundsSpace O(1)

Hold both registers as 32-bit patterns. Each round, XOR gives the bits of the total that need no carry, while AND shifted left one place gives the bits that still have to be carried; repeat on those two words until the carry word is empty. Because the carry word shifts left each round it empties within 32 rounds, and a set top bit at the end is turned back into a negative Python integer with NOT and XOR rather than arithmetic.

Pythonentry point: get_sum

def get_sum(a, b):
    # Time O(1): the carry word shifts left every round, so at most 32 rounds
    # run for 32-bit operands. Space O(1) - four scalars, nothing allocated.
    mask = 0xFFFFFFFF
    x = a & mask
    y = b & mask
    while y:
        carry = ((x & y) << 1) & mask
        x = (x ^ y) & mask
        y = carry
    if x & 0x80000000:
        return ~(x ^ mask)
    return x

The obvious version

Walk the 32 bit positions from the bottom up, applying a one-bit full-adder truth table at each position and threading the carry along by hand.

This is not filler. It was written independently of the fast solution and the two were run against thousands of random inputs to confirm they agree — which is how the answer above is known to be right. Read it when the intended approach feels like a rabbit pulled out of a hat.

Python

def bf_get_sum(a, b):
    mask = 0xFFFFFFFF
    x = a & mask
    y = b & mask
    result = 0
    carry = 0
    for i in range(32):
        p = (x >> i) & 1
        q = (y >> i) & 1
        if p ^ q ^ carry:
            result |= 1 << i
        if (p & q) | (p & carry) | (q & carry):
            carry = 1
        else:
            carry = 0
    if result & 0x80000000:
        return ~(result ^ mask)
    return result

How this problem was checked

Reference solution executed against 10 test cases — 10 passed
4 worked examples confirmed by running the code
0 eight-word sequences shared with the parent problem's text
Agrees with an independent brute-force implementation on random inputs
Test set proven to catch deliberately broken solutions
Reviewed by a separate quality pass
Function arity matches the parent problem's

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