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It sometimes happens that an identity
includes two terms whose coefficients have the same value (with
the same, or opposite, signs) - this often happens when one of
the irreducibles in the "eliminated set" features in
only two of the reductions participating in the elimination process.
Included in the original Machination package were programs for
deriving alternative pairs of arccotangents equivalent to such
matching pairs of terms (or indeed individual terms); e.g. [x]
± [y] = [p] ± [q] or [x] = [m] ± [n], and
by exploiting these it was sometimes possible to "tweak"
an identity in order to improve its measure. An example is worked
out below:
Using the eliminated set [2], [5], [6], [49], [107], [109], it
is possible to obtain the following identity with measure 1.55905
(MRW, 6 Dec 94):
[1] = 581[1252]+764[5593]+1030[5832]+366[58898]+195[110443]
+537[4841182]+266[6826318]
Knowing that [5593]+[5832] = [2855], it is possible to rewrite this identity in the following way (its measure becoming 1.58160):
[1] = 581[1252]+764[2855]+266[5832]+366[58898]+195[110443]
+537[4841182]+266[6826318]
By observing that the coefficients of [5832] and [6826318] are now the same, and knowing that [5832] + [6826318] = [5827] - [1561886607], the identity can be "tweaked" to produce a new one with measure 1.54408 (HCL, 19 Jul 95):
[1] = 581[1252]+764[2855]+266[5827]+366[58898]+195[110443]
+537[4841182]-266[1561886607]
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Any combination of arccotangent terms present in an identity can be represented as the inverse cotangent of a single fractional value; e.g. the combination [110443] - [4841182] is equivalent to [34245479 / 303]. This is obtainable, using complex arithmetic, from the product (110443 + i)(4841182 - i). It is customary to represent such fractional inverse cotangents in the form: [Q + R/D], Q and R being respectively the integer quotient and remainder resulting from dividing the numerator by the denominator D (after first removing any factors common to the numerator and denominator). In this case, [34245479 / 303] = [113021 + (116/303)]. In the rare cases when the denominator of the fractional value obtained in this way is 3 or 4, it is easy to prove that equivalent pairs of terms can always be found in which one (or, when the denominator is 4, both) of the terms is a half-integer; and usually this provides the best measure improvement. Relevant formulae here are:
[(3x + 1) / 3] = [(2x + 1) / 2] + [6x² + 5x + 7]
[(3x - 1) / 3] = [(2x - 1) / 2] - [6x² - 5x + 7]
[(4x + 1) / 4] = [(2x + 1) / 2] + [(8x² + 6x + 9) / 2]
[(4x - 1) / 4] = [(2x - 1) / 2] - [(8x² - 6x + 9) / 2]
For example, the pair of terms [91093] + [103862] is equivalent to [145589 / 3], and, as Amrik Singh Nimbran has shown, an alternative representation for this is: [97059 / 2] - [14130722757]. Another example, where the denominator of the equivalent fractional representation is 4, is [1252] - [5832] = [6377 / 4] = [3189 / 2] + [20336261 / 2].
Our original "best" identity was that discovered by Hwang Chien-lih, with measure 1.51244 (HCL, 20 Sep 94), which includes the pair [110443] - [4841182]. Much effort went into trying to improve this by "tweaking". An improvement was finally effected by Hwang, who, after being unable to find a better pair of arccotangents to replace that pair of terms, and noticing that they are equivalent to [113021 + (116/303)] (see above), investigated alternative 3-term expressions ("triplets") in which the leading term was [113021] (i.e. [Q]) of the form:
[110443]-[4841182] = [113021] - ([A] ± [B])
Hwang's success in finding two [A] - [B] pairs which improved the measure 1.51244 (by 0.00404 in one case, 0.00047 in the other) anticipated, and led to, the enhancement of the "pairs" software to search for such triplets. In another case (improving the 3-digit identity with measure 1.31102 to the version with measure 1.26579) the task was to find alternative terms to replace the pair [105218] + [7167807]; here the enhanced software was able to find the following equivalences:
[105218]+[7167807] = [1762829/17] = [103695 + 14/17] = [103697]+[18280007883/2]
The processes for generating "pairs" and "triplets" are described in more detail on a separate page.
The triplet-finding software, which uses 96-bit arithmetic internally, originally accepted as parameters only pairs with matching coefficients (with the same, or opposite, signs) and only looked for triplets whose first term involved an integer cotangent (equal to, or in the neighbourhood of, [113021] in Hwang's example). It has now been further enhanced so that not only does it accept parameters specifying the numerator and denominator of a fractional value - e.g. in the above case it would accept the three parameters 105218, +, 7167807, or alternatively 1762829, /, 17 - but it is now also capable of seeking triplets whose leading terms (specifically, the 200 integer and half-integer terms "nearest" to the quotient of the fractional representation - i.e. [Q], [(2Q + 1) / 2], [(2Q - 1) / 2], etc.) may be half-integers (the 2nd and 3rd terms may also be half-integers).
Amrik Singh Nimbran has recently (August 2009) communicated an important discovery; he has established that when the fractional cotangent representation in lowest terms, expressed as [Q + R/D], is such that |(2R - D)| = 1 or 2 (i.e. R/D is one of the two fractions with denominator D nearest, but not equal, to 1/2), the inverse fractional cotangent can always be expressed as the sum or difference of a half-integer term [(2Q + 1) / 2] and another integer or half-integer term (the algebra needed to verify this is not difficult). Moreover, since [(2Q + 1) / 2] is now always included in the possible leading terms examined by the triplet-finding software, the latter should inevitably identify all such cases.
An early success of this new version has been finding three even better "triplet" equivalents for [110443] - [4841182]:
[110443]-[4841182] = [226043/2] + [109027476193] - [3375905320682366575989/2] [110443]-[4841182] = [113022] + [41395642009/2] + [160221523081015501672943] [110443]-[4841182] = [226041/2] - [28937877829/2] + [8960188273719119420807]
The measure improvements over the original [110443] - [48411182] are 0.01227 in the first case, 0.00995 in the second, and 0.00601 in the third - the original triplets with leading term [113021] have therefore been superseded.
In this connection, the following valid equation also been observed by Amrik Singh Nimbran:
[110443]-[4841182] = [105218] - [2513489/2] + [7167807] (as may be verified either from the reductions of these terms, or by comparing the two 3-digit identities with measures 1.31390 and 1.31102).
As demonstrated above, the combination of the first and last terms on the right-hand side may be "improved", transforming the equation to become:
[110443]-[4841182] = [103697] - [2513489/2] + [18280007883/2]
The combined measures of the three terms on the RHS of the above equation exceed the measure of the two LHS terms. However, if the LHS combination are present in an identity in which [2513489/2] is already present (which may well be the case if [15] is in the "eliminated set"), and the three RHS terms are now substituted for the two on the LHS, the measure of the resultant identity is actually decreased - by 0.04812, assuming [2513489/2] is still present, albeit with a different coefficient; by 0.31207, if it is no longer present. The 3-digit identities with measures 1.31390 and 1.26579, respectively, provide "Before" and "After" (retrospective) examples of this process.
Unless having a lower measure than the original one, an identity which is generated by this sort of "tweaking" will not usually be included in our lists; however some identities thus "superseded" (e.g. 1.58160) may remain listed, albeit omitted from the sets of "best 20" measures, and the improved identity will refer back to the original by carrying an asterisked copy of the latter's measure. It will be seen that most of these identities feature at least one pair of terms with matching coefficients, these being the terms replaced in the "tweaked" version. "Tweaking" often leads to the replacement of one or more of the irreducibles in the original eliminated set with others - for example, in the identity with measure 1.54408 generated above, the irreducible [5827], besides actually appearing in the identity, has displaced [109] from the eliminated set. When a pair of terms is replaced by a "triplet", the new eliminated set is usually not listed at all, and a set of three terms with matching coefficients will be present. As a matter of principle, if an identity is shown without an eliminated set, it will usually include an asterisked back-reference to an identity (possibly artificially created) with an eliminated set from which it could have been derived.
"Tweaking" is sometimes feasible when a pair of inverse integral cotangents occur in an identity with non-matching coefficients. For example, knowing that the expression ([53] - 2[107]) = [303479 / 53] = {[5726] - [1737720807]}, if a particular identity involves both [53] and [107], then the [53] term can obviously be eliminated at the cost of changing the coefficient of [107] and introducing [5726] and [1737720807] terms into the identity - the last two together will contribute less to the measure than the [53] term did. This can be generalised: for any integer n, ([n] - 2[2n+1]) = [C + (1/n)], where C = 2(n² + n + 1). This fractional inverse cotangent [(nC + 1) / n] can always be expressed as the difference of a pair of inverse integral cotangents, viz. [C] - [n(C²+1) + C].