What all block ciphers can achieve encryption speed similar/near to AES on general purpose computers with AES-NI enabled and disabled.
Camellia seems to be near to AES, and Twofish seems to be much slower than AES.
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Although one would like to think that AES-NI can speed up Camellia because the s-box structure is shared, there are differences because of the fact that Camellia is a Feistel-Network structure. AES-NI does not expose the intermediate layers of the circuits, so you'd have to use Camellia-NI instructions type architectures. I know my NTT phone has an IC that has Camellia instructions in it (or so says the manual)
One thing that you need to be mindful of when comparing ciphers is that people generally compare things to be favorable for their results. My Simon implementations were asynchronous and could complete a round in a "single cycle" due to the nature of the hardware. Regarding the speedup due to hardware, all ciphers can be optimized for a hardware implementation to make them "fast". You need a proper constraint set to compare them well, but in a synchronous system for the same clock speed, you need to do better than 14 instructions to beat AES-256.