The answer is 00000000000000001e8...

[flailing Junk]

Then again, do remember the BTC network has GPU power equaling 146 petaFLOPS (146,407 gigaFLOPS) average as of July 2011. That's over a hundred times faster than the Tianhe-1 supercluster that held the #1 position on the top500.org list of the world's most powerful computers.

Well not exactly. Bitcoin does not actually do floating point calculations so the FLOPS value must be some kind of estimate. If your source is bitcoin watch then I cant find any confirmation of the number or even how it is calculated. So it is remarkable if true but needs to be taken with a grain of salt I think.

I am very interested in any type of confirmation or methodology for calculating the power of the bitcoin network in floating point, integer calculations or whatever.

[Jack of Diamonds]

For example a Radeon 6990 has 5.2 gigaFLOPS of computing power[1] and yields roughly 800 megahash/s in bitcoin mining.

[1]Sources:
http://www.asus.com/Graphics_Cards/AMD_Series/EAH69903DI4S4GD5/
http://www.pcmag.com/article2/0,2817,2381557,00.asp

[J.P. Larocque]

I wrestled through the non-standard use of little-endian hashes and input values in bitcoin, and dug out the block input which results in this tiny tiny hash output.  I documented it all as an example in the wiki:

  https://en.bitcoin.it/wiki/Block_hashing_algorithm

Kudos, this answered a few nagging questions I had with the mining process.

[J.P. Larocque]

For example a Radeon 6990 has 5.2 gigaFLOPS of computing power[1] and yields roughly 800 megahash/s in bitcoin mining.

Apples to oranges.  If all Bitcoin miners everywhere used Radeon 6990s, then you could say that the mining collective could compute at (big number)FLOP/s if they directed their computational capacity at floating-point problems, but right now we're computing at 0 FLOP/s, simply because it's all integer based.  You can't exchange floating-point and integer operations like that.

[Jack of Diamonds]

For example a Radeon 6990 has 5.2 gigaFLOPS of computing power[1] and yields roughly 800 megahash/s in bitcoin mining.

Apples to oranges.  If all Bitcoin miners everywhere used Radeon 6990s, then you could say that the mining collective could compute at (big number)FLOP/s if they directed their computational capacity at floating-point problems, but right now we're computing at 0 FLOP/s, simply because it's all integer based.  You can't exchange floating-point and integer operations like that.

And the Tianhe-1 is computing at 0 khash/s because it only does floating point operations. That doesn't mean it can't be utilized to mine at 1,433,600 mhash/s. (7168 Nvidia Tesla S1070 GPUs at 200 mhash/s each)
Of course it's theoretical, why would anyone assume otherwise

[nealmcb]

00000000000000001e8d6829a8a21adc5d38d0a473b144b6765798e61f98bd1d 125552

So is this the equivalent of brute forcing a 64 bit password?

edit:

I mean a password that is 64 bits long, encrypted with SHA-256.

This is actually quite a different thing - it is finding a password which would hash to the same value, if the output of the hash was only 64 bits (really 67, FWIW).

If the password hashing algorithm was really awful, without salt or iterations (i.e. worse than what we had in 1978), these might require similar amounts of effort.  But a modern hashing algorithm like PBKDF2 or bcrypt or scrypt would make things much much harder.

See more on how password hashing should really be done at How to securely hash passwords? - IT Security - Stack Exchange