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What is a Flop

From [What Is a Flop? – Nick Higham]:

Moler and Van Loan give a different definition of “flop” in their classic paper “Nineteen Dubious Ways to Compute the Exponential of a Matrix” (1978), and their definition was adopted in the flop command in the original Fortran version of MATLAB, which counts the number of flops executed in the most recent command or since MATLAB started. In the 1981 MATLAB Users’ Guide, Cleve Moler explains that

One flop is one execution of a Fortran statement like S = S + X(I)*Y(I) or Y(I) = Y(I) + T*X(I). In other words, it consists of one floating point multiplication, together with one floating point addition and the associated indexing and storage reference operations.

This original definition attempted to account for indexing and storage costs in the computer implementation of an algorithm as well as the cost of the floating-point arithmetic. It was motivated by the fact that in linear algebra computations most arithmetic appears in statements of the form s = s + x*y, which appear in evaluating inner products and in adding one multiple of a vector to another.

In the LINPACK Users’ Guide (1979, App. B) flops were used to normalize timing data, but the word “flop” was not used.

The widespread adoption of the flop was ensured by its use in Golub and Van Loan’s book Matrix Computations (1981). In the 1989 second edition of the book a flop was redefined as in our first paragraph and this definition quickly became the standard usage. The Fortran statements given by Moler now involve two flops.

The MATLAB flop function survived until around 2000, when MATLAB started using LAPACK in place of LINPACK for its core linear algebra computations and it became no longer feasible to keep track of the number of flops executed.

It is interesting to note that an operation of the form S + X(I)*Y(I) that was used in the original definition of flop is now supported in the hardware of certain processors. The operation is called a fused multiply-add and is done in the same time as one multiplication or addition and with one rounding error.