Analytically finding magic constant in reversible hash function

Question

The below integer "hash" functions form a pair of reversible hash functions:

uint32_t hash ( uint32_t x ) {

    x = ( ( x >> 16 ) ^ x ) * 0x45d9f3b;
    x = ( ( x >> 16 ) ^ x ) * 0x45d9f3b;
    x = ( ( x >> 16 ) ^ x );

    return x;
}

uint32_t unhash ( uint32_t x ) {

    x = ( ( x >> 16 ) ^ x ) * 0x119de1f3;
    x = ( ( x >> 16 ) ^ x ) * 0x119de1f3;
    x = ( ( x >> 16 ) ^ x );

    return x;
}

The reversibility obviously depends on the "magic values". In the case of uint32_t's, given 0x45d9f3b, the value 0x119de1f3 can be easily found by exhaustive search.

I would like to design a 64-bit equivalent. Exhaustive search is in the 64-bit case no longer an option.

The question is: "Can this 'reversing magical constant' be found analytically?"

PS: I do realise that this is not a hash function in a cryptograhical sense.

Hint: $\mathrm{0x119de1f3} = \mathrm{0x45d9f3b}^{-1} \pmod {2^{32}}$ — gammatester, Nov 24 '16 at 07:55
After understanding the above hint, you might want to head there. — fgrieu, Nov 24 '16 at 09:10
@gammatester Thanks for that answer. Cannot accept a comment as "the accepted answer" though... — degski, Nov 24 '16 at 10:30
@fgrieu Thanks for that implementation, easy to adapt to the 32-bit case, just for checking... — degski, Nov 24 '16 at 10:30
https://www.wolframalpha.com/input/?i=PowerMod%5B0x45d9f3b,+-1,+2%5E32%5D — CodesInChaos, Nov 24 '16 at 12:05
@degski You can copy / expand the comments of gammatester + fgrieu if they are unwilling to answer (contribution to them would be nice of course). As long as the answer is indeed an answer, it should be accepted by the mods. — Maarten Bodewes, Nov 24 '16 at 19:25

Analytically finding magic constant in reversible hash function

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