I am trying to understand the memory expansion cost.
We can see in the Ethereum Yellow Paper that there's this formula to count the memory cost:
Now, it's also said that this formula doesn't include the expansion cost.
Can anyone in simple terms explain what expansion cost means?
The formula in your question describes a gas cost for the total amount of memory allocated in a contract call (i.e. the biggest memory location that contains a nonzero value. Zeroing memory after using it does not decrease the total amount of allocated memory). Note this is in addition to the base 3 gas of an mstore opcode.
In the above formula, a is the maximum memory location written to in a contract call. Note that a is denominated in 32 byte words.
For example, if your contract uses 1,024 bytes of memory, a = 32.
From the Ethereum yellow paper (https://ethereum.github.io/yellowpaper/paper.pdf), the G_memory = 3.
Putting it all together, the extra gas required by your contract's memory consumption is:
3 * (max_memory / 32) + floor(max_memory^2 / 524,288)
If you use <=724 bytes of memory the second part of this equation is 0. The first term is the linear part of the memory expansion cost (3 times the number of 32-byte words used).
You have to use a very large amount of memory (dozens of kilobytes) for the memory expansion cost to significantly deviate from being linear. Here's a table with some examples:
| Memory used (in kb) | Memory expansion cost |
|---|---|
| 1 | 98 |
| 2 | 200 |
| 4 | 416 |
| 8 | 896 |
| 16 | 2048 |
| 32 | 5120 |
| 64 | 14336 |
| 128 | 45056 |
| 256 | 155648 |
Most contract calls use a few kb at most, making the memory expansion cost small vs the cost of modifying a storage variable.
N.B. It's difficult to see just how much memory a Solidity contract will consume due to memory management being handled by the Solidity compiler. As a general rule of thumb using structs and arrays will increase memory consumption but basic variables like bytes32, uint256 won't. Last time I checked the Solidity compiler does not re-use allocated memory; every time a new memory variable is created, additional memory will be allocated.
Can anyone in simple terms explain what expansion cost means?
When your contract writes to memory, you pay for the costs associated with the number of bytes written.
However, if you are writing to an area of memory that hasn't been written to before, there is an associated additional cost with using it for the first time. Think of it as an additional tax levied for using a pristine piece of memory.
From the "Storage, Memory and the Stack" section of the Solidity docs:
Memory is expanded by a word (256-bit), when accessing (either reading or writing) a previously untouched memory word (i.e. any offset within a word). At the time of expansion, the cost in gas must be paid. Memory is more costly the larger it grows (it scales quadratically).
(To add to this: it scales linearly for the first 724 bytes [Appendix H, Yellow Paper] and quadratically after that.)
