The Karatsuba formula is the simplest case of a general approach to splitting inputs that leads to both Toom and FFT algorithms. A description of Toom can be found in Knuth section 4.3.3, with an example 3-way calculation after Theorem A. The 3-way form used in GMP is described here.
The operands are each considered split into 3 pieces of equal length (or the most significant part 1 or 2 limbs shorter than the other two).
high low +----------+----------+----------+ | x2 | x1 | x0 | +----------+----------+----------+ +----------+----------+----------+ | y2 | y1 | y0 | +----------+----------+----------+
These parts are treated as the coefficients of two polynomials
X(t) = x2*t^2 + x1*t + x0 Y(t) = y2*t^2 + y1*t + y0
Let b equal the power of 2 which is the size of the x0, x1, y0 and y1 pieces, i.e. if they’re k limbs each then b=2^(k*mp_bits_per_limb). With this x=X(b) and y=Y(b).
Let a polynomial W(t)=X(t)*Y(t) and suppose its coefficients are
W(t) = w4*t^4 + w3*t^3 + w2*t^2 + w1*t + w0
The w[i] are going to be determined, and when they are they’ll give the final result using w=W(b), since x*y=X(b)*Y(b)=W(b). The coefficients will be roughly b^2 each, and the final W(b) will be an addition like,
high low +-------+-------+ | w4 | +-------+-------+ +--------+-------+ | w3 | +--------+-------+ +--------+-------+ | w2 | +--------+-------+ +--------+-------+ | w1 | +--------+-------+ +-------+-------+ | w0 | +-------+-------+
The w[i] coefficients could be formed by a simple set of cross products, like w4=x2*y2, w3=x2*y1+x1*y2, w2=x2*y0+x1*y1+x0*y2 etc, but this would need all nine x[i]*y[j] for i,j=0,1,2, and would be equivalent merely to a basecase multiply. Instead the following approach is used.
X(t) and Y(t) are evaluated and multiplied at 5 points, giving values of W(t) at those points. In GMP the following points are used,
Point Value t=0 x0 * y0, which gives w0 immediately t=1 (x2+x1+x0) * (y2+y1+y0) t=-1 (x2-x1+x0) * (y2-y1+y0) t=2 (4*x2+2*x1+x0) * (4*y2+2*y1+y0) t=inf x2 * y2, which gives w4 immediately
At t=-1 the values can be negative and that’s handled using the absolute values and tracking the sign separately. At t=inf the value is actually X(t)*Y(t)/t^4 in the limit as t approaches infinity, but it’s much easier to think of as simply x2*y2 giving w4 immediately (much like x0*y0 at t=0 gives w0 immediately).
Each of the points substituted into W(t)=w4*t^4+…+w0 gives a linear combination of the w[i] coefficients, and the value of those combinations has just been calculated.
W(0) = w0 W(1) = w4 + w3 + w2 + w1 + w0 W(-1) = w4 - w3 + w2 - w1 + w0 W(2) = 16*w4 + 8*w3 + 4*w2 + 2*w1 + w0 W(inf) = w4
This is a set of five equations in five unknowns, and some elementary linear
algebra quickly isolates each w[i]. This involves adding or
subtracting one W(t) value from another, and a couple of divisions by
powers of 2 and one division by 3, the latter using the special
mpn_divexact_by3 (see Exact Division).
The conversion of W(t) values to the coefficients is interpolation. A polynomial of degree 4 like W(t) is uniquely determined by values known at 5 different points. The points are arbitrary and can be chosen to make the linear equations come out with a convenient set of steps for quickly isolating the w[i].
Squaring follows the same procedure as multiplication, but there’s only one
X(t) and it’s evaluated at the 5 points, and those values squared to
give values of W(t). The interpolation is then identical, and in fact
toom_interpolate_5pts subroutine is used for both squaring and
Toom-3 is asymptotically O(N^1.465), the exponent being log(5)/log(3), representing 5 recursive multiplies of 1/3 the original size each. This is an improvement over Karatsuba at O(N^1.585), though Toom does more work in the evaluation and interpolation and so it only realizes its advantage above a certain size.
Near the crossover between Toom-3 and Karatsuba there’s generally a range of
sizes where the difference between the two is small.
MUL_TOOM33_THRESHOLD is a somewhat arbitrary point in that range and
successive runs of the tune program can give different values due to small
variations in measuring. A graph of time versus size for the two shows the
effect, see tune/README.
At the fairly small sizes where the Toom-3 thresholds occur it’s worth remembering that the asymptotic behaviour for Karatsuba and Toom-3 can’t be expected to make accurate predictions, due of course to the big influence of all sorts of overheads, and the fact that only a few recursions of each are being performed. Even at large sizes there’s a good chance machine dependent effects like cache architecture will mean actual performance deviates from what might be predicted.
The formula given for the Karatsuba algorithm (see Karatsuba Multiplication) has an equivalent for Toom-3 involving only five multiplies, but this would be complicated and unenlightening.
An alternate view of Toom-3 can be found in Zuras (see References), using a vector to represent the x and y splits and a matrix multiplication for the evaluation and interpolation stages. The matrix inverses are not meant to be actually used, and they have elements with values much greater than in fact arise in the interpolation steps. The diagram shown for the 3-way is attractive, but again doesn’t have to be implemented that way and for example with a bit of rearrangement just one division by 6 can be done.