sequential approximation to exponentials, trigonometrics, multiplicative-like algorithms ... by progressive cumulation |

Multiplication on early digital computers was typically by sequential approximation and repetition of addition, progressive cumulation like manual long-multiplication and long-division implemented in hardware (equipment) for integers, and in software emulation aided by floating-point addition, -or fully hardware depending on cost, space, availability, factors-... In the present era it has become replaced by parallel micro-circuitry.

But by freeing this procedure from fixed multiplier and multiplicand, and speed-ups with simple fast bit-scaler ("barrel-shifter") and sequential-access memory ("SAM") or other, more functions can be efficiently calculated:-- Exponentiation; circular-function sinusoid trigonometric; hypertrigonometric; arithmetic-like data-compression, ... by algorithm. (Even square-root is such an algorithm.)

Exponentiation could be accomplished by cumulative successive partial multiplications,
Π(1 + d x_{i}) where x_{i} are chosen to simplify the multiplication
to just addition of d=2^{-i}, a bit-scaling (shift) and add or subtract ...
A table of x_{i} is typically n-bits long ... It might be even practical
to build the calculation directly in hardware, even as multiplication was so built--
and useful, as sinusoids are useful in video construction.

The fastest bit-scaler is built on base-8 (near base e² ~7.39), -requiring the least hardware reduplication and substrate 'realestate'.

* [Author's note: My original article was more inclusive than the ca-1950's utility, CORDIC CoOrdinate Rotation Digital Computer [algorithm] used for circular and hyperbolic x-y pair rotations,- but as it was conceptually related as progressive cumulation of additions, I kept the other's -title- as a subject-reference: believing it had meant the -subject- of successive narrowing of measure along a cord aka chord, a progressive binary-search-like calculus-analytic method for measuring nonlinear arcs by computation, and especially for reducing its error coefficient. (cf Then-new college subject, my UCSD Department of Applied Physics and Information Sciences APIS, also-noted Egyption bull, only years after I graduated, became EECS Electronic Engineering and Computer Sciences; but we'd built computers and interfaces from small-scale-integration in that era; and SAR Successive Approximation Register was ordinary technology, decades before calling that, pyramidal.) In my algorithm discovered-on-demand for myself in college I'd used the process for exponentials, and later heard and read of the CORDIC describing the same method with paired complex rotation angle increments chosen similarly to simplify the whole to multiplication by successively selected single-additions]

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