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The compiler for solidity supports the flag --optimize which performs some set of optimizations to the compiled solidity code, typically resulting in a reduction of gas costs.

How does the optimizer work and what are some examples of the optimizations will it make?

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The optimizer seems like magic even to me. chriseth (He is the main DEV developer of solidity) would probably be in a better position to answer your question than myself. Here is an interesting high level overview though, which I hope answers your question.

Everything below is from the solidity docs.

Internals - the Optimizer

The Solidity optimizer operates on assembly, so it can be and also is used by other languages. It splits the sequence of instructions into basic blocks at JUMPs and JUMPDESTs. Inside these blocks, the instructions are analysed and every modification to the stack, to memory or storage is recorded as an expression which consists of an instruction and a list of arguments which are essentially pointers to other expressions. The main idea is now to find expressions that are always equal (on every input) and combine them into an expression class. The optimizer first tries to find each new expression in a list of already known expressions. If this does not work, the expression is simplified according to rules like constant + constant = sum_of_constants or X * 1 = X. Since this is done recursively, we can also apply the latter rule if the second factor is a more complex expression where we know that it will always evaluate to one. Modifications to storage and memory locations have to erase knowledge about storage and memory locations which are not known to be different: If we first write to location x and then to location y and both are input variables, the second could overwrite the first, so we actually do not know what is stored at x after we wrote to y. On the other hand, if a simplification of the expression x - y evaluates to a non-zero constant, we know that we can keep our knowledge about what is stored at x.

At the end of this process, we know which expressions have to be on the stack in the end and have a list of modifications to memory and storage. This information is stored together with the basic blocks and is used to link them. Furthermore, knowledge about the stack, storage and memory configuration is forwarded to the next block(s). If we know the targets of all JUMP and JUMPI instructions, we can build a complete control flow graph of the program. If there is only one target we do not know (this can happen as in principle, jump targets can be computed from inputs), we have to erase all knowledge about the input state of a block as it can be the target of the unknown JUMP. If a JUMPI is found whose condition evaluates to a constant, it is transformed to an unconditional jump.

As the last step, the code in each block is completely re-generated. A dependency graph is created from the expressions on the stack at the end of the block and every operation that is not part of this graph is essentially dropped. Now code is generated that applies the modifications to memory and storage in the order they were made in the original code (dropping modifications which were found not to be needed) and finally, generates all values that are required to be on the stack in the correct place.

These steps are applied to each basic block and the newly generated code is used as replacement if it is smaller. If a basic block is split at a JUMPI and during the analysis, the condition evaluates to a constant, the JUMPI is replaced depending on the value of the constant, and thus code like

var x = 7;
data[7] = 9;
if (data[x] != x + 2)
  return 2;
else
  return 1;

is simplified to code which can also be compiled from

data[7] = 9;
return 1;

even though the instructions contained a jump in the beginning.

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