Almost Integer Rank

Definition

The quality function

For an almost integer $x$ define

$$\text{surplus}(x) = n(x) - \frac{\text{complexity}(x)}{10},\qquad n(x) = -\log_{10}\left|x - \operatorname{nint}(x)\right|$$

$n$ uses the absolute error and is the continuous version of the run of 0s or 9s after the decimal point: $e^{\pi\sqrt{163}}$ misses the nearest integer by $7.5\times10^{-13}$, so $n = 12.12$. Relative precision is not used, because the number of digits before the decimal point has nothing to do with the coincidence: once the magnitude is fixed, the fractional part of a random real is uniformly distributed, and the probability of landing within $\varepsilon$ of an integer is simply $2\varepsilon$.

Complexity: RIES symbol weights

Complexity is borrowed from the complexity() function in Robert Munafo's RIES. An expression is written in postfix form and its complexity is the sum of the weights of its symbols, where weight = base 10 + the symbol's individual weight. The base keeps every symbol weight positive, so that a search ordered by increasing complexity terminates; it also makes complexity/10 roughly the number of decimal digits needed to describe the whole expression.

RIES only has single digits; 10 must be written 2 5 *. A comment in ries.c says constant weights are close to $10\log_{10}x$, and this site extrapolates from it: an integer $N \ge 10$ weighs $10 + \operatorname{round}(10\log_{10}N)$, so 163 costs 32. Symbols RIES lacks are weighted by analogy: $\cosh,\sinh,\tanh$ cost 13 like the trigonometric functions; $\Gamma$, digamma, erfi and $\zeta$ cost 15 like Lambert $W$; Euler's $\gamma$, Catalan's constant, Khinchin's constant and the supergolden ratio $\psi$ cost 18 like $\phi$. Each entry is scored on its cheapest form: $e^{\pi}$ is pi exp (27) rather than e pi ^ (36), and $\pi^4$ is pi sq sq (32).

Calibration

Why n − complexity/10

The ries.c log records 2 236 462 expressions of complexity $\le 64$; $\log_{10}$ of that is 6.35, about $64/10$. So complexity $c$ corresponds to roughly $10^{c/10}$ equally simple expressions. A random real lands within $10^{-n}$ of an integer with probability about $10^{-n}$, so among expressions of complexity at most $c$ one expects $10^{c/10-n}$ almost integers as good as a given entry. The surplus is the negative log of that expected count: above 0 means rarer than chance, below 0 means such precision is bound to appear at that complexity anyway. This is the same test Borwein used when discussing "high precision fraud".

By comparison, $n/c$ and $n/c^2$ carry no probabilistic meaning: $n/c$ rates $\sin 11$ (5 digits for 33) and the nested cosine (34 digits for 93) as differing by a factor of 2.4, when their rarity differs by $10^{23}$; the ranking under $n/c^2$ reshuffles with the arbitrary choice of the RIES base weight.

Stripping

Amplifiers

A symbol-counting complexity cannot see three mechanical constructions that buy arbitrary precision at almost no cost, so they are handled separately:

• Trigonometric functions at a critical point. $\cos y \approx \pm1$ is the same as $y \approx k\pi$, and $\cos(k\pi+\varepsilon) = \pm(1 - \varepsilon^2/2)$ squares the error of the underlying approximation $y\approx k\pi$. The 34 digits of $\cos(\pi\cos(\pi\cos(\ln(\pi+20))))$ are all manufactured from the 3 digits of $e^{\pi} - \pi \approx 20$. Such entries are scored on their core almost integer.
• Powers of Pisot numbers. If an algebraic integer $x$ has all its other conjugates inside the unit circle, then $x^k$ approaches integers geometrically while complexity grows only like $\log k$, so the surplus rises without bound. The base's minimal polynomial is found with findpoly and matching entries are dropped from the ranking, e.g. the supergolden ratio $\psi^{11}$.
• Algebraic values of modular functions at CM points. Complex multiplication guarantees that a modular function ($j$, the Weber functions, …) takes an algebraic value $y$ at a CM point with $y = e^{\pi\sqrt r}\,(1 + O(e^{-\pi\sqrt r}))$, $r$ rational; for example $j = e^{\pi\sqrt D} - 744 + O(e^{-\pi\sqrt D})$. This is a theorem, not a coincidence: a class invariant assembled from products of units $a + \sqrt{a^2-1}$ gains precision like $\sqrt D$ while its complexity grows slowly. Wrapping it in $\ln y/\pi$ turns the relative precision of $y$ into absolute precision, which brings the rejected relative-precision metric back through the side door. Detection: for every algebraic subexpression (integers, arithmetic, roots, rational powers, $\phi$, $\psi$) and every rational combination of logarithms of algebraic numbers (whose exponential is again algebraic, so splitting $\ln y$ into $\sum \ln u_i$ does not help), if $|y| \ge 10^6$ and $(\ln y/\pi)^2$ lies within $10^{-10}$ of a rational $r$ with denominator $\le 576$, or within $|\ln y - \pi\sqrt r| < 1000/|y|$ for denominator $\le 4$ (allowing constant terms such as 744 or 24), the entry is a CM identity. When $y$ is an integer the entry is scored on its core $e^{\pi\sqrt r}$, so $(\ln(640320^3+744)/\pi)^2$ is merged into $e^{\pi\sqrt{163}}$; otherwise there is no almost-integer core and the entry is dropped, e.g. $(\sqrt2\,\psi)^{24} - e^{\pi\sqrt{31}}$. $e^{\pi\sqrt{163}}$ itself is unaffected, since $\pi\sqrt{163}$ is not the logarithm of an algebraic number.

Admission

Threshold and stability test

The surplus only answers "how much better than chance"; it does not decide whether a number is an almost integer at all. Two further rules do that, and the calculator adds a precision guard:

• Threshold n ≥ 2. The distance to the nearest integer must be below 0.01. $\pi$ has $n = 0.85$ and surplus $-0.55$, better than most entries in the table, yet it is not an almost integer. Eight MathWorld entries fall below the threshold; they are listed below and not ranked.
• Structural-stability test. Each constant in the expression is multiplied by 1.01 and by 0.99 in turn and the value recomputed. A genuine coincidence has at least one critical constant that shatters it: moving the 163 in $e^{\pi\sqrt{163}}$ by 1% leaves 0.3 digits. Limiting structures such as $1/\pi^{100}$, $\cos(\pi/8^8)$ or $2^{1/\pi^{25}}$ stay near an integer whichever constant is moved. If every perturbation still leaves $\ge 3$ digits, the entry is structural and not ranked. Integers added or subtracted at the top level are exempt, otherwise $1 + 1/\pi^{100}$ would escape by letting the 1 become 1.01 and shift the target. The calibration assumption "fractional part uniform given the magnitude" fails for such structures: $1/\pi^{100}$ is near 0 because its magnitude was built to be $10^{-50}$, not because of its fractional part. The test produces no false positives on the existing entries and also catches nested-cosine amplifiers; it does not catch Pisot powers (perturbing $\phi$ destroys $\phi^{64}$), so the Pisot rule stays.
• Precision guard in the calculator. The value is first computed at 90 significant digits while recording the largest number of integer digits $I$ among all intermediate values; if $I + 70$ digits are needed the value is recomputed at that precision, and above 1000 digits (the limit of decimal.js's built-in $\pi$) the input is refused. An exact integer is only declared when a recomputation with 40 more digits shrinks the distance further, so an out-of-precision number such as $\pi^{1000}$ is no longer reported as an integer.

Scope

Included and excluded

Entries come from MathWorld's Almost Integer page, including the 13 values of the $e^{\pi\sqrt n}$ table and the Hickerson numbers $h_0$ to $h_{17}$. Excluded are entries requiring infinite sums or products (the Borwein series, the theta sum, the Maze–Minder $u_k$, the q-Pochhammer symbol), non-closed forms (an elliptic integral of the third kind, a Student $t$ quantile) and anything involving physical constants or units. All values were computed with mpmath at 120 digits and agree with the figures MathWorld prints; three formula images on that page have ambiguous or wrong alt text, corrected here by numerical check.

Appendix

RIES built-in symbols

RIES writes every symbol as a single character and an expression as a postfix string of them: pE is $e^{\pi}$, 2q is $\sqrt2$, 5 2 v is $\sqrt[5]{2}$. Symbols fall into three "stack effect types" (seft): constants push a value, unary functions replace the top of the stack, binary operators pop two values and push one. The tables list every built-in symbol with its weight including the base of 10.

Constants (seft a)

Unary functions (seft b)

Binary operators (seft c)

A few internal symbols take no part in the search: a space is a no-op for --eval-expression; !/, !-, !^ and >< are "reversed operation" placeholders used when converting postfix to infix, with weight 0; parentheses, comma and equals sign are output formatting only; and when the user disables every unary function, RIES temporarily adds an identity symbol I of weight 20 to keep expression shapes intact. Users can add constants with a chosen weight via --define, override any weight with --symbol-weights, and print the value and complexity of a postfix string with --eval-expression.