Memory expansion gas cost calculation.

EVM memory is a byte-addressable array that starts empty and expands on demand. Expanding memory is not free — the cost grows quadratically to discourage excessive memory usage.

Formula (Yellow Paper, Appendix H)

cost(words) = 3 × words + words² ÷ 512

where words = ceil(byte_size / 32).

The expansion cost is the difference between the cost at the new size and the cost at the current size. If the access fits within already-allocated memory, the cost is 0.

Example

# Expanding from 0 to 32 bytes (1 word): 3×1 + 1²/512 = 3 gas
# Expanding from 0 to 1024 bytes (32 words): 3×32 + 32²/512 = 98 gas
# Expanding from 0 to 32768 bytes (1024 words): 3×1024 + 1024²/512 = 5120 gas

The quadratic term makes very large memory allocations prohibitively expensive, providing a natural bound on memory usage.