c/c++ left shift unsigned vs signed

Question

I have this code.

#include <iostream>

int main()
{
    unsigned long int i = 1U << 31;
    std::cout << i << std::endl;
    unsigned long int uwantsum = 1 << 31;
    std::cout << uwantsum << std::endl;
    return 0;
}

It prints out.

2147483648
18446744071562067968

on Arch Linux 64 bit, gcc, ivy bridge architecture.

The first result makes sense, but I don't understand where the second number came from. 1 represented as a 4byte int signed or unsigned is

00000000000000000000000000000001

When you shift it 31 times to the left, you end up with

10000000000000000000000000000000

no? I know shifting left for positive numbers is essentially 2^k where k is how many times you shift it, assuming it still fits within bounds. Why is it I get such a bizarre number?

Answer 1

Presumably you're interested in why this: unsigned long int uwantsum = 1 << 31; produces a "strange" value.

The problem is pretty simple: 1 is a plain int, so the shift is done on a plain int, and only after it's complete is the result converted to unsigned long.

In this case, however, 1<<31 overflows the range of a 32-bit signed int, so the result is undefined¹. After conversion to unsigned, the result remains undefined.

That said, in most typical cases, what's likely to happen is that 1<<31 will give a bit pattern of 10000000000000000000000000000000. When viewed as a signed 2's complement² number, this is -2147483648. Since that's negative, when it's converted to a 64-bit type, it'll be sign extended, so the top 32 bits will be filled with copies of what's in bit 31. That gives: 1111111111111111111111111111111110000000000000000000000000000000 (33 1-bits followed by 31 0-bits).

If we then treat that as an unsigned 64-bit number, we get 18446744071562067968.

---

1. §5.8/2:

   The value of E1 << E2 is E1 left-shifted E2 bit positions; vacated bits are zero-filled. If E1 has an unsigned type, the value of the result is E1 × 2E2, reduced modulo one more than the maximum value representable in the result type. Otherwise, if E1 has a signed type and non-negative value, and E1×2E2 is representable in the corresponding unsigned type of the result type, then that value, converted to the result type, is the resulting value; otherwise, the behavior is undefined.

2. In theory, the computer could use 1's complement or signed magnitude for signed numbers--but 2's complement is currently much more common than either of those. If it did use one of those, we'd expect a different final result.