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I looked at https://github.com/ethereum/web3.js/blob/develop/dist/web3.js but can not find the method which it uses to estimate the amount of gas a transaction is going to consume.

Does anybody know how this works or can point to a reference?

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The web3.js library makes a call eth_estimategas to the ethereum client.

Geth will execute the transaction and will return the gas consumed, but it will discard any modification.

A note from the documentation

Note that the estimate may be significantly more than the amount of gas actually used by the transaction, for a variety of reasons including EVM mechanics and node performance.

Metamask's VM subprovider also implements estimate gas by executing the transaction and measuring the gas consumed.

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