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### 16.5 Scalar evolutions

Scalar evolutions (SCEV) are used to represent results of induction variable analysis on GIMPLE. They enable us to represent variables with complicated behavior in a simple and consistent way (we only use it to express values of polynomial induction variables, but it is possible to extend it). The interfaces to SCEV analysis are declared in `tree-scalar-evolution.h`. To use scalar evolutions analysis, `scev_initialize` must be used. To stop using SCEV, `scev_finalize` should be used. SCEV analysis caches results in order to save time and memory. This cache however is made invalid by most of the loop transformations, including removal of code. If such a transformation is performed, `scev_reset` must be called to clean the caches.

Given an SSA name, its behavior in loops can be analyzed using the `analyze_scalar_evolution` function. The returned SCEV however does not have to be fully analyzed and it may contain references to other SSA names defined in the loop. To resolve these (potentially recursive) references, `instantiate_parameters` or `resolve_mixers` functions must be used. `instantiate_parameters` is useful when you use the results of SCEV only for some analysis, and when you work with whole nest of loops at once. It will try replacing all SSA names by their SCEV in all loops, including the super-loops of the current loop, thus providing a complete information about the behavior of the variable in the loop nest. `resolve_mixers` is useful if you work with only one loop at a time, and if you possibly need to create code based on the value of the induction variable. It will only resolve the SSA names defined in the current loop, leaving the SSA names defined outside unchanged, even if their evolution in the outer loops is known.

The SCEV is a normal tree expression, except for the fact that it may contain several special tree nodes. One of them is `SCEV_NOT_KNOWN`, used for SSA names whose value cannot be expressed. The other one is `POLYNOMIAL_CHREC`. Polynomial chrec has three arguments – base, step and loop (both base and step may contain further polynomial chrecs). Type of the expression and of base and step must be the same. A variable has evolution `POLYNOMIAL_CHREC(base, step, loop)` if it is (in the specified loop) equivalent to `x_1` in the following example

    while (…)
      {
        x_1 = phi (base, x_2);
        x_2 = x_1 + step;
      }

Note that this includes the language restrictions on the operations. For example, if we compile C code and `x` has signed type, then the overflow in addition would cause undefined behavior, and we may assume that this does not happen. Hence, the value with this SCEV cannot overflow (which restricts the number of iterations of such a loop).

In many cases, one wants to restrict the attention just to affine induction variables. In this case, the extra expressive power of SCEV is not useful, and may complicate the optimizations. In this case, `simple_iv` function may be used to analyze a value – the result is a loop-invariant base and step.

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