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12.1.1.1.1 Examples of Associativity and Commutativity in Numeric Operations

Consider the following expression, in which we assume that `1.0` and `1.0e-15` both denote *single floats*:

(+ 1/3 2/3 1.0d0 1.0 1.0e-15)

One *conforming implementation* might process the *arguments* from left to right, first adding `1/3` and `2/3` to get `1`, then converting that to a *double float* for combination with `1.0d0`, then successively converting and adding `1.0` and `1.0e-15`.

Another *conforming implementation* might process the *arguments* from right to left, first performing a *single float* addition of `1.0` and `1.0e-15` (perhaps losing accuracy in the process), then converting the sum to a *double float* and adding `1.0d0`, then converting `2/3` to a *double float* and adding it, and then converting `1/3` and adding that.

A third *conforming implementation* might first scan all the *arguments*, process all the *rationals* first to keep that part of the computation exact, then find an *argument* of the largest floating-point format among all the *arguments* and add that, and then add in all other *arguments*, converting each in turn (all in a perhaps misguided attempt to make the computation as accurate as possible).

In any case, all three strategies are legitimate.

A *conforming program* could control the order by writing, for example,

(+ (+ 1/3 2/3) (+ 1.0d0 1.0e-15) 1.0)

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