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authorNorbert Preining <norbert@preining.info>2021-07-15 03:01:01 +0000
committerNorbert Preining <norbert@preining.info>2021-07-15 03:01:01 +0000
commit55ae461252a599d6bc951065a37c16a8971fc560 (patch)
tree1b73269f29ae053c3fa9f00a365b009569df721a /macros/generic
parent3303a89b11161d66ec78811056b85306f696cc09 (diff)
CTAN sync 202107150300
Diffstat (limited to 'macros/generic')
-rw-r--r--macros/generic/texdimens/README.md192
-rw-r--r--macros/generic/texdimens/texdimens.sty2
-rw-r--r--macros/generic/texdimens/texdimens.tex340
3 files changed, 402 insertions, 132 deletions
diff --git a/macros/generic/texdimens/README.md b/macros/generic/texdimens/README.md
index 54f5a530dc..401dea828f 100644
--- a/macros/generic/texdimens/README.md
+++ b/macros/generic/texdimens/README.md
@@ -13,7 +13,7 @@ Development: https://github.com/jfbu/texdimens
## Aim of this package
Utilities and documentation related to TeX dimensional units, usable
-both with Plain (\input texdimens) and with LaTeX (\usepackage{texdimens}).
+both with Plain (`\input texdimens`) and with LaTeX (`\usepackage{texdimens}`).
The aim of this package is to address the issue of expressing dimensions
(or dimension expressions evaluated by `\dimexpr`) in the various TeX
@@ -25,9 +25,15 @@ in a faithful manner the exact outcome of those internal procedures.
The reader is supposed to be familiar with TeX basics.
This project requires the e-TeX extensions `\dimexpr` and `\numexpr`.
+The notation `<dim. expr.>` in the macro descriptions refers to a
+*dimensional expression* as accepted by `\dimexpr`. The syntax has some
+peculiarities: among them the fact that `-(...)` (for example `-(3pt)`)
+is illegal, one must use alternatives such as `0pt-(...)` or a
+sub-expression `-\dimexpr...\relax` for example.
Notice that this is WIP and inaccuracies may exist even relative to
-descriptions of TeX handlings due to limited time available for the project.
+descriptions of TeX handlings due to limited time available for the
+project.
## TeX points and scaled points
@@ -57,17 +63,16 @@ When `\foo` has been assigned as `D pt`, `\the\foo` will produce some
`E pt` where `E` is not necessarily the same as `D`. But it is guaranteed
that `E pt` defines the same dimension as `D pt̀`.
-
## Further units known to TeX on input
TeX understands on input further units: `bp`, `cm`, `mm`, `in`, `pc`,
-`cc`, `dd` and `nd`. It also understands font-dependent units `ex` and
+`cc`, `nc`, `dd` and `nd`. It also understands font-dependent units `ex` and
`em`, and PDFTeX adds the `px` dimension unit. Japanese engines also
add specific units.
`ex`, `em`, `px` and other engine-dependent units, are all currently
excluded from consideration in this project and the rest of this
-documentation is only for the `bp`, `cm`, `mm`, `in`, `pc`, `cc`, `dd`
+documentation is only for the `bp`, `cm`, `mm`, `in`, `pc`, `cc`, `nc`, `dd`
and `nd` units. When we say "unit" we mean one of those or the `pt` (the
`sp` is a special case and will be included or not tacitly in the "unit"
denomination depending on the case).
@@ -76,7 +81,7 @@ TeX associates to each such unit `uu` a fraction `phi` which is a conversion
factor. It is always `>1`:
uu phi reduced real approximation 1uu in sp= x=[65536phi]/65536 \the<1uu>
- [65536phi] (real approximation)
+ (Python output) [65536phi] (real approximation)
-- ---------- ------- ------------------ --------- -------------------- ----------
bp 7227/7200 803/800 1.00375 65781 1.0037384033203125 1.00374pt
nd 685/642 same 1.0669781931464175 69925 1.0669708251953125 1.06697pt
@@ -116,11 +121,18 @@ to an integer ratio `12`: only dimensions which in scaled points are
multiple of `12` are exactly representable in the `pc` unit.
This also means that some dimensions expressible in one unit may not be
-available with another unit. For example, it turns out that `0.6in` can
-not be expressed as `D cm`, whatever the `D`. More surprisingly perhaps
-is that `1in==2.54cm` is **false** in TeX! But it is true that
-`100in==254cm`! It is also false that `10in==25.4cm` but it is true that
-`10in==254mm`! It is false though that `1in==25.4mm`!
+available with another unit. For example, and perhaps surprisingly,
+there is no decimal `D` which would achieve `1in==Dcm`: the "step"
+between attainable dimensions is `72--73sp` for the `in` and `28--29sp`
+for the `cm`, and as `1in` differs internally from `2.54cm` by only
+`12sp` (see below the `xintsession` verbatim) it is impossible to adjust
+either the `in` side or the `cm` side to obtain equality. See in the
+[TODO] section the closest dimension attainable both via `in` and via
+`cm`.
+
+In particular `1in==2.54cm` is **false** in TeX, but it is true that
+`100in==254cm`... It is also false that `10in==25.4cm` but it is true that
+`10in==254mm`... It is false though that `1in==25.4mm`!
>>> (\dimexpr1in, \dimexpr2.54cm);
@_1 4736286, 4736274
@@ -135,17 +147,22 @@ is that `1in==2.54cm` is **false** in TeX! But it is true that
@_5 47362867, 47362867
`\maxdimen` can be expressed only with `pt`, `bp`, and `nd`. For the
-other units the maximal attainable dimensions are given in this table:
-
- 16322.78954 bp (\maxdimen = 1073741823 sp)
- 15355.51532 nd (\maxdimen = 1073741823 sp)
- 15312.02583 dd (1073741822 sp)
- 5758.31741 mm (1073741822 sp)
- 1365.33333 pc (1073741820 sp)
- 1279.62627 nc (1073741814 sp)
- 1276.00215 cc (1073741821 sp)
- 575.83174 cm (1073741822 sp)
- 226.7054 in (1073741768 sp)
+other units the maximal attainable dimensions in `sp` unit are given in
+the middle column of the next table.
+
+ maximal allowed the corresponding minimal TeX dimen denotation
+ (with 5 places) maximal attainable dim. causing "Dimension too large"
+ --------------- -------------------------- --------------------------
+ 16383.99999 pt 1073741823 sp (=\maxdimen) 16383.99999237060546875 pt
+ 16322.78954 bp 1073741823 sp (=\maxdimen) 16322.78954315185546875 bp
+ 15355.51532 nd 1073741823 sp (=\maxdimen) 15355.51532745361328125 nd
+ 15312.02584 dd 1073741822 sp 15312.02584075927734375 dd
+ 5758.31742 mm 1073741822 sp 5758.31742095947265625 mm
+ 1365.33333 pc 1073741820 sp 1365.33333587646484375 pc
+ 1279.62627 nc 1073741814 sp 1279.62627410888671875 nc
+ 1276.00215 cc 1073741821 sp 1276.00215911865234375 cc
+ 575.83174 cm 1073741822 sp 575.83174896240234375 cm
+ 226.70540 in 1073741768 sp 226.70540618896484375 in
Perhaps for these various peculiarities with dimensional units, TeX does
not provide an output facility for them similar to what `\the` achieves for
@@ -158,29 +175,28 @@ f-expandable, but (perhaps) in future final versions will expand fully
in two steps. This refinement is anyhow not really important as TeX
engines now support the `\expanded` primitive.
-All macros handle negative dimensions via their absolute value then
-taking the opposite.
-
-1. At time of writing only the `\texdimin<uu>` macros are implemented,
-The envisioned "down" and "up" variants are not done yet.
-
-2. For `dd`, `nc` and `in`, input equal to (or sufficiently close to)
-`\maxdimen` will produce also with `\texdimin<uu>` an output `D`
-representing the next "attainable" dimension above `\maxdimen` hence
-using `Duu` will trigger "Dimension too large error".
-
-3. (not yet) For input equal to (or sufficiently close to) `\maxdimen` and those
-units `uu` for which `\maxdimen` is not exactly representable, i.e. all
-units except `pt`, `bp` and `nd`, the output `D` of the "up" variants
-`\texdimin<uu>u` if used as `Duu` in a dimension assignment or
-expression will (naturally) trigger "Dimension too large" error.
-
-4. (not yet) For some units the "down" and "up" macros may trigger "Dimension too
-large" during their execution if used with an input too close to
-`\maxdimen`. "Safe" variants which are guaranteed never to trigger this
-error but have some extra overhead to filter out inputs too close to
-`\maxdimen` will *perhaps* be provided. But see 2. and 3. regarding the
-usability of the output anyhow.
+Negative dimensions behave as if replaced by their absolute value, then
+at last step the sign (if result is not zero) is applied.
+
+1. For input `X` equal to (or sufficiently close to)
+`\maxdimen` and those units `uu` for which `\maxdimen` is not exactly
+representable (i.e. all units except `pt`, `bp` and `nd`), the output `D`
+of the "up" macros `\texdimin<uu>up{X}`, if used as `Duu` in a dimension
+assignment or expression, will (naturally) trigger a "Dimension too large"
+error.
+
+2. For `dd`, `nc` and `in`, and input `X` equal to (or sufficiently
+close to) `\maxdimen` it turns out that `\texdimin<uu>{X}` produces an
+output `D` such that `Duu` is the first "virtually attainable" TeX
+dimension *beyond* `\maxdimen`. Hence `Duu` will trigger on use
+"Dimension too large error".
+
+3. For some units the "down" and "up" macros may trigger
+"Dimension too large" during their execution if used with an input too
+close to `\maxdimen`. "Safe" variants which are guaranteed never to
+trigger this error but have some extra overhead to filter out inputs too
+close to `\maxdimen` will *perhaps* be provided. But
+see 1. and 2. regarding the usability of the output anyhow.
`\texdiminpt{<dim. expr.>}`
@@ -195,19 +211,19 @@ usability of the output anyhow.
> `\maxdimen` on input produces `16322.78954` and indeed is realized as `16322.78954bp`.
-`\texdiminbpd{<dim. expr.>}` NOT YET
+`\texdiminbpdown{<dim. expr.>}`
> Produces a decimal (with up to five decimal places) `D` such that `D
> bp` represents the dimension exactly if possible. If not possible it
> will be smaller by `1sp` from the original dimension.
-`\texdiminbpu{<dim. expr.>}` NOT YET
+`\texdiminbpup{<dim. expr.>}`
> Produces a decimal (with up to five decimal places) `D` such that `D
> bp` represents the dimension exactly if possible. If not possible it
> will be larger by `1sp` from the original dimension.
-`\texdiminnd{<dim. expr.>}`
+`\texdiminndown{<dim. expr.>}`
> Produces a decimal (with up to five decimal places) `D` such that `D
> nd` represents the dimension exactly if possible. If not possible it
@@ -216,19 +232,19 @@ usability of the output anyhow.
> `\maxdimen` on input produces `15355.51532` and indeed is realized as `15355.51532nd`.
-`\texdiminndd{<dim. expr.>}` NOT YET
+`\texdiminnddown{<dim. expr.>}`
> Produces a decimal (with up to five decimal places) `D` such that `D
> nd` represents the dimension exactly if possible. If not possible it
> will be smaller by `1sp` from the original dimension.
-`\texdiminndu{<dim. expr.>}` NOT YET
+`\texdiminndup{<dim. expr.>}`
> Produces a decimal (with up to five decimal places) `D` such that `D
> nd` represents the dimension exactly if possible. If not possible it
> will be larger by `1sp` from the original dimension.
-`\texdimindd{<dim. expr.>}`
+`\texdiminddown{<dim. expr.>}`
> Produces a decimal (with up to five decimal places) `D` such that `D
> dd` represents the dimension exactly if possible. If not possible it
@@ -237,15 +253,15 @@ usability of the output anyhow.
> Warning: the output for `\maxdimen` is `15312.02585` but `15312.02585dd`
> will trigger "Dimension too large" error.
-> `\maxdimen-1sp` is atteignable via `15312.02583dd`.
+> `\maxdimen-1sp` is attainable via `15312.02584dd`.
-`\texdiminddd{<dim. expr.>}` NOT YET
+`\texdimindddown{<dim. expr.>}`
> Produces a decimal (with up to five decimal places) `D` such that `D
> dd` represents the dimension exactly if possible. If not possible it
> will be smaller by `1sp` from the original dimension.
-`\texdiminddu{<dim. expr.>}` NOT YET
+`\texdiminddup{<dim. expr.>}`
> Produces a decimal (with up to five decimal places) `D` such that `D
> dd` represents the dimension exactly if possible. If not possible it
@@ -262,13 +278,13 @@ usability of the output anyhow.
> `\maxdimen` as input produces on output `5758.31741` and indeed the
> maximal attainable dimension is `5758.31741mm` (`1073741822sp`).
-`\texdiminmmd{<dim. expr.>}` NOT YET
+`\texdiminmmdown{<dim. expr.>}`
> Produces a decimal (with up to five decimal places) `D` such that `D
> mm` represents the dimension exactly if possible. If not possible it
> will be largest representable dimension smaller than the original one.
-`\texdiminmmu{<dim. expr.>}` NOT YET
+`\texdiminmmup{<dim. expr.>}`
> Produces a decimal (with up to five decimal places) `D` such that `D
> mm` represents the dimension exactly if possible. If not possible it
@@ -284,13 +300,13 @@ usability of the output anyhow.
> `\maxdimen` as input produces on output `1365.33333` and indeed the
> maximal attainable dimension is `1365.33333pc` (`1073741820sp`).
-`\texdiminpcd{<dim. expr.>}` NOT YET
+`\texdiminpcdown{<dim. expr.>}`
> Produces a decimal (with up to five decimal places) `D` such that `D
> pc` represents the dimension exactly if possible. If not possible it
> will be largest representable dimension smaller than the original one.
-`\texdiminpcu{<dim. expr.>}` NOT YET
+`\texdiminpcup{<dim. expr.>}`
> Produces a decimal (with up to five decimal places) `D` such that `D
> pc` represents the dimension exactly if possible. If not possible it
@@ -308,13 +324,13 @@ usability of the output anyhow.
> will trigger "Dimension too large" error.
> `\maxdimen-9sp` is attainable via `1279.62627nc`.
-`\texdiminncd{<dim. expr.>}` NOT YET
+`\texdiminncdown{<dim. expr.>}`
> Produces a decimal (with up to five decimal places) `D` such that `D
> nc` represents the dimension exactly if possible. If not possible it
> will be largest representable dimension smaller than the original one.
-`\texdiminncu{<dim. expr.>}` NOT YET
+`\texdiminncup{<dim. expr.>}`
> Produces a decimal (with up to five decimal places) `D` such that `D
> nc` represents the dimension exactly if possible. If not possible it
@@ -325,19 +341,19 @@ usability of the output anyhow.
> Produces a decimal (with up to five decimal places) `D` such that `D
> cc` represents the dimension exactly if possible. If not possible it
> will either be the closest from below or from above, but it is not
-> known in advacce which one (and it is not known if the other choice
+> known in advance which one (and it is not known if the other choice
> would have been closer).
> `\maxdimen` as input produces on output `1276.00215` and indeed the
> maximal attainable dimension is `1276.00215cc` (`1073741821sp`).
-`\texdiminccd{<dim. expr.>}` NOT YET
+`\texdiminccdown{<dim. expr.>}`
> Produces a decimal (with up to five decimal places) `D` such that `D
> cc` represents the dimension exactly if possible. If not possible it
> will be largest representable dimension smaller than the original one.
-`\texdiminccu{<dim. expr.>}` NOT YET
+`\texdiminccup{<dim. expr.>}`
> Produces a decimal (with up to five decimal places) `D` such that `D
> cc` represents the dimension exactly if possible. If not possible it
@@ -354,13 +370,13 @@ usability of the output anyhow.
> `\maxdimen` as input produces on output `575.83174` and indeed the
> maximal attainable dimension is `575.83174cm` (`1073741822sp`).
-`\texdimincmd{<dim. expr.>}` NOT YET
+`\texdimincmdown{<dim. expr.>}`
> Produces a decimal (with up to five decimal places) `D` such that `D
> cm` represents the dimension exactly if possible. If not possible it
> will be largest representable dimension smaller than the original one.
-`\texdimincmu{<dim. expr.>}` NOT YET
+`\texdimincmup{<dim. expr.>}`
> Produces a decimal (with up to five decimal places) `D` such that `D
> cm` represents the dimension exactly if possible. If not possible it
@@ -378,31 +394,57 @@ usability of the output anyhow.
> will trigger "Dimension too large" error.
> `\maxdimen-55sp` is maximal attainable dimension (via `226.7054in`).
-`\texdiminind{<dim. expr.>}` NOT YET
+`\texdiminindown{<dim. expr.>}`
> Produces a decimal (with up to five decimal places) `D` such that `D
> in` represents the dimension exactly if possible. If not possible it
> will be largest representable dimension smaller than the original one.
-`\texdimininu{<dim. expr.>}` NOT YET
+`\texdimininup{<dim. expr.>}`
> Produces a decimal (with up to five decimal places) `D` such that `D
> in` represents the dimension exactly if possible. If not possible it
> will be smallest representable dimension larger than the original one.
-
## TODO
-Implement the "up" and "down" variants.
+Currently (most) macros require exhaustive expansion.
+
+Provide a macro `\texdimnearest{in,cm}{<dim.expr.>}` which would output
+the nearest dimension simultaneously representable both in `in` and in
+`cm`?
-Provide a macro `\texdimnearest{in,cm}{<dim.expr.>}` which provides the
-nearest dimension simultaneously representable both in `in` and in `cm`?
According to a reference on the web by an anonymous contributor the
-available positive dimensions in scaled points have the shape
-`floor(3613.5*k) sp` for some integer `k`. So we basically may have a
+dimensions representable with both `in` and `cm` units have the shape
+`trunc(3613.5*k) sp` for some integer `k`. So we basically may have a
delta up to about `1800sp` which is about `0.0275pt` and is still small
-(less than one hundredth of a millimeter), so perhaps such a utility for
-"safe dimensions" may be useful. Simpler however and more efficient
+(less than one hundredth of a millimeter, i.e. less than ten micron),
+so perhaps such a utility for
+"safe dimensions" may be useful. Here are for example the dimensions
+nearest to `1in` and realizable both in `in` and `cm` units:
+
+ >>> \input texdimens.tex\relax
+ (executing \input texdimens.tex\relax in background)
+ (./texdimens.tex)
+ >>> &exact
+ exact mode (floating point evaluations use 16 digits)
+ >>> (\texdiminin{4737298sp});
+ @_10 1.00021
+ >>> (\texdimincm{4737298sp});
+ @_11 2.54054
+ >>> (\dimexpr1.00021in, \dimexpr2.54054cm);
+ @_12 4737298, 4737298
+ >>> (\texdiminin{4733685sp});
+ @_13 0.99945
+ >>> (\texdimincm{4733685sp});
+ @_14 2.5386
+ >>> (\dimexpr0.99945in, \dimexpr2.5386cm);
+ @_15 4733685, 4733685
+
+As promised, one of them, the upper approximation, is at less than
+one hundredth of millimeter from the two nearby targets.
+
+Simpler however and more efficient
would be for people to finally adopt the French revolution Système
Métrique (rather than setting up giant financial paradises).
diff --git a/macros/generic/texdimens/texdimens.sty b/macros/generic/texdimens/texdimens.sty
index c29f1458e0..9aa422cb55 100644
--- a/macros/generic/texdimens/texdimens.sty
+++ b/macros/generic/texdimens/texdimens.sty
@@ -1,5 +1,5 @@
% This is file texdimens.tex, part of texdimens package, which
% is distributed under the LPPL 1.3c. Copyright (c) 2021 Jean-François Burnol
-\ProvidesPackage{texdimens}[2021/06/30 v0.9beta conversion of TeX dimensions to decimals (JFB)]
+\ProvidesPackage{texdimens}[2021/07/14 v0.9gamma conversion of TeX dimensions to decimals (JFB)]
\@@input texdimens.tex\relax
\endinput \ No newline at end of file
diff --git a/macros/generic/texdimens/texdimens.tex b/macros/generic/texdimens/texdimens.tex
index 471a07f807..7068d57db9 100644
--- a/macros/generic/texdimens/texdimens.tex
+++ b/macros/generic/texdimens/texdimens.tex
@@ -1,101 +1,329 @@
% This is file texdimens.tex, part of texdimens package, which
% is distributed under the LPPL 1.3c. Copyright (c) 2021 Jean-François Burnol
-% 2021/06/30 v0.9beta
+% 2021/07/14 v0.9gamma
\edef\texdimensendinput{\endlinechar\the\endlinechar\catcode`\noexpand _=\the\catcode`\_\relax\noexpand\endinput}%
\endlinechar13\relax%
\catcode`\_=11
% Is T sp attainable from unit "uu"?. Here we suppose T>0.
% phi>1, psi=1/phi, psi<1
-% U(N,phi)=floor(N phi) is strictly increasing
+% U(N,phi)=trunc(N phi) is strictly increasing
% U(N)<= T < U(N+1) iff N = ceil((T+1)psi) - 1
% U(M)< T <= U(M+1) iff M = ceil(T psi) - 1
% Either:
-% - M = N, i.e. T is not attainable, M=N < T psi < (T+1) psi <= N+1
-% - M = N - 1, i.e. T is attained, T psi <= N < (T+1) psi, T = floor(N phi)
+% case1: M = N, i.e. T is not attainable, M=N < T psi < (T+1) psi <= N+1
+% case2: M = N - 1, i.e. T is attained, T psi <= N < (T+1) psi, T = floor(N phi)
%
-% In the latter case:
-% - as psi<1, |N - (T+0.5) psi| < 0.5, hence N = R := round((T+0.5) psi).
-% Also works for T=0 but T<0 would need -0.5.
+% Let X = round(T psi). And let Y = trunc(X phi).
%
-% - if psi<1/2, then N = round(T psi) is simpler formula which works
-% also for attainable T<0.
+% case1: X can be N or N+1. It will be N+1 iff Y > T.
+% case2: X can be N or N-1. It will be N iff trunc((X+1)phi)>T.
%
-% This R=round((T+0.5) psi) can always be computed via \numexpr because 2T+1
-% will not trigger arithmetic overflow.
+% This is not convenient: if Y <= T it might still be that we are in case 2
+% and we must check then if trunc((X+1) phi) > T or not.
+%
+% If psi < 0.5
+% ------------
+%
+% The situation then simplifies:
+%
+% case1: X can be N or N+1. It will be N+1 iff Y = trunc(X phi) > T.
+% case2: X is necessarily N.
%
-% If Tsp>0 is not attainable, this R can produce either N or N+1 (=M+1).
+% Thus:
+% a) compute X = round(T psi)
+% b) compute Y = trunc(X phi) and test if Y > T. If true, we
+% were in case 1, replace X by X - 1, else we were either
+% in case 1 or case 2, but we can leave X as is.
+% We have thus found N.
%
-% If we try computing ceil(x) via round(x+0.5) (\numexpr rounds up)
-% this means for N, we need round((T+1)psi + 0.5), for example with
-% psi = 100/7227 for "in", this gives round((((T+1)200)+7227)/14454)
-% feasible via \numexpr only for (circa) 100 T less than \maxdimen.
+% The operation Y = trunc(X phi) can be achieved this way:
+% i) use \the\dimexpr to convert X sp into D pt,
+% ii) use \the\numexpr\dimexpr to convert "D uu" into sp.
+% These steps give Y.
%
-% We could rather compute round(T psi) but we don't know if it gives
-% N or N+1. We know it is N if round(T psi)< round((T+1) psi)
-% but if the two are the same we don't know if they are both N or
-% both N+1.
+% This way we find the maximal dimension at most T sp exactly
+% representable in "uu" unit.
%
-% It is slightly less costly to compute X = round(T psi) than R,
-% but if we then realize that X uu < T sp we do not yet know
-% if (X+1) uu = T sp or is > Tsp, except if psi<1/2, because
-% if T sp is attainable then X = round(T psi) is then necessarily N
-% so if X uu < T sp we now that T sp was not attainable.
+% The computations of X and Y can be done independently of sign of T.
+% But the final test has to be changed to Y < T if T < 0 and then
+% one must replace X by X+1. So we must filter sign.
%
-% We decide with some hesitation to not split whether psi<1/2 or
-% psi>1/2. Reverted! Currently following done only for psi>1/2,
-% i.e. bp, nd, dd.
+% If the goal is only to find a decimal D such that "D uu" is
+% exactly T sp in the case this is possible, then things are simpler
+% because from X = round(T psi) we get D such as X sp is same as D pt
+% and "D uu" will work.
+% We don't have to take sign into account for this computation.
+% But if T sp was not atteignable we don't know if this X will give
+% a D such that D uu < T sp or D uu > T sp.
%
-% 1. compute R = round((T+0.5) psi) in \numexpr. This forces
-% to check for negative T because then we would want here (T-0.5)psi
+% If psi > 0.5
+% ------------
%
-% 2. check for the "up" and "down" variants whether R uu is <, =, or > T sp.
-% But we have to choose here what "up" and "down" mean for T<0.
-% Also, computation of R uu
-% may trigger Dimension too large if T sp is not attainable,
-% close to \maxdimen, and \maxdimen itself is not attainable.
+% For example unit "bp" has phi=803/800.
+%
+% It is then not true that if T sp is atteignable, the X = round(T psi)
+% will always work.
+%
+% But it is true that R = round((T + 0.5) psi) will always work.
+% Here we must use -0.5 if T < 0, though.
+%
+% This R=round((T+0.5) psi) can always be computed via \numexpr because 2T+1
+% will not trigger arithmetic overflow.
%
-% For the envisioned "safe versions" we would tabulate first per unit
-% what is Rmax such that Rmax uu <= \maxdimen. Then the "safe" versions
-% would have an extra check of R. But for \texdimeninuuu it will
-% then not be compliant to its definition for inputs close to
-% non-attainable \maxdimen.
+% So this gives an approach to find a D such that "D uu" is exactly
+% T sp when this is possible.
+%
+% If Tsp (positive) is not attainable, this R however can produce
+% either N or N+1.
+%
+% But we can decide what happened by computing Z = trunc(R phi).
+% If and only if Z > T this means R was N+1.
+%
+% It is slightly less costly to compute X = round(T psi) than
+% R = round((T + 0.5) psi),
+% but if we then realize that trunc(X phi) < T we do not yet know
+% if trunc((X+1) phi) = T or is > T.
+%
+% To recapitulate: we have our algorithm for all units to find out
+% maximal dimension exactly atteignable in "uu" unit and at most equal
+% to (positive) T sp.
+%
+% Unfortunately the check that Y (in case psi < 0.5) or Z (in case psi >
+% 0.5) may trigger a Dimension too large error if T sp was near
+% non-atteignable \maxdimen.
+%
+% For additional envisioned "safe versions" we would tabulate first per unit
+% what is the integer Rmax such that trunc(Rmax phi) <= \maxdimen. Then
+% the "safe" versions would have an extra check of X or R before
+% proceeding further. But the "up macros" supposed to give the next
+% dimension above Tsp and exactly atteignable in "uu" unit, if compliant
+% to their description can not avoid "Dimension too large" for inputs
+% close to non-attainable \maxdimen.
%
% After having written the macros we will tabulate what is for each unit
% the maximal attainable dimension.
%
-% Hesitation about whether using simpler round(T psi) approach
-% for units > 2pt and the \texdimin<uu> macros.
+% About the macros such as \texdiminbp whose constraints are:
+% - give a decimal D such that "Duu" = "T sp" for TeX if possible
+% - else give nearest from below or above without knowing
+% which one,
%
-% Testing shows that this would not change output for \maxdimen
-% with "nc" and "in": still N+1 is returned... but it has great
-% advantage to not have to check the sign.
+% there was some hesitation about whether or not using the simpler
+% round(T psi) approach for units > 2pt and the \texdimin<uu> macros.
+% Testing showed that this did not change the output for \maxdimen
+% with the units "nc" and "in": still N+1 is returned...
+%
+% As it has great
+% advantage to not have to check the sign of the input, the
+% "simpler" approach was chosen for those units to which it
+% applies, i.e. the units uu > 2pt (phi>2, psi<1/2), i.e.
+% all units except bp, nd and dd.
+%
+\def\texdimfirstofone#1{#1}%
+% this #2 will be \fi
+\def\texdiminuudown_e#1#2#3#4{#2\expandafter\texdiminpt_\the\dimexpr\numexpr(#1-1)sp\relax}%
+\def\texdiminuuup_e#1#2#3#4{#2\expandafter\texdiminpt_\the\dimexpr\numexpr(#1+1)sp\relax}%
+% this #1 will be \fi
+\def\texdiminuuup_neg#1#2-{#1-#2}%
+%
+% pt
%
-% OK let's do this,
-% especially as I don't know if I will ever implement "up" and "down".
\def\texdiminpt#1{\expandafter\texdiminpt_\the\dimexpr#1\relax}%
{\catcode`p 12\catcode`t 12\csname expandafter\endcsname\gdef\csname texdiminpt_\endcsname#1pt{#1}}%
+%
% bp 7227/7200 = 803/800
-% complications and annoying overhead caused by sign
-% and we don't want to evaluate #1 twice in a \dimexpr; if #1 was
-% restricted to be a dimen register, we would avoid "\the and re-grab" step.
+%
\def\texdiminbp#1{\expandafter\texdiminbp_\the\numexpr\dimexpr#1;}%
\def\texdiminbp_#1#2;{\texdiminpt{\numexpr(2*#1#2+\if-#1-\fi1)*400/803sp}}%
+% \texdiminbpdown: maximal dim exactly expressible in bp and at most equal to input
+\def\texdiminbpdown#1{\expandafter\texdiminbpdown_a\the\numexpr\dimexpr#1;}%
+\def\texdiminbpdown_a#1{\if-#1\texdiminbpdown_neg\fi\texdiminbpdown_b#1}%
+\def\texdiminbpdown_b#1;{\expandafter\texdiminbpdown_c\the\numexpr(2*#1+1)*400/803;#1;}%
+\def\texdiminbpdown_c#1;{\expandafter\texdiminbpdown_d\the\dimexpr#1sp;#1;}%
+{\catcode`P 12\catcode`T 12\lowercase{\gdef\texdiminbpdown_d#1PT};#2;#3;%
+ {\ifdim#1bp>#3sp \texdiminuudown_e{#2}\fi\texdimfirstofone{#1}}%
+}%
+% The problem here is that if close to 0sp, output can be 0.0, and we do not want
+% -0.0 as output. So let's do this somewhat brutally. Anyhow, negative inputs are
+% not our priority. #1 is \fi here:
+\def\texdiminbpdown_neg#1#2-#3;{#1\expandafter\texdiminpt_\the\dimexpr-\texdiminbpdown_b#3;pt\relax}%
+% \texdiminbpup: minimal dim exactly expressible in bp and at least equal to input
+\def\texdiminbpup#1{\expandafter\texdiminbpup_a\the\numexpr\dimexpr#1;}%
+\def\texdiminbpup_a#1{\if-#1\texdiminuuup_neg\fi\texdiminbpup_b#1}%
+\def\texdiminbpup_b#1;{\expandafter\texdiminbpup_c\the\numexpr(2*#1+1)*400/803;#1;}%
+\def\texdiminbpup_c#1;{\expandafter\texdiminbpup_d\the\dimexpr#1sp;#1;}%
+{\catcode`P 12\catcode`T 12\lowercase{\gdef\texdiminbpup_d#1PT};#2;#3;%
+ {\ifdim#1bp<#3sp \texdiminuuup_e{#2}\fi\texdimfirstofone{#1}}%
+}%
+%
% nd 685/642
+%
\def\texdiminnd#1{\expandafter\texdiminnd_\the\numexpr\dimexpr#1;}%
\def\texdiminnd_#1#2;{\texdiminpt{\numexpr(2*#1#2+\if-#1-\fi1)*321/685sp}}%
+% \texdiminnddown: maximal dim exactly expressible in nd and at most equal to input
+\def\texdiminnddown#1{\expandafter\texdiminnddown_a\the\numexpr\dimexpr#1;}%
+\def\texdiminnddown_a#1{\if-#1\texdiminnddown_neg\fi\texdiminnddown_b#1}%
+\def\texdiminnddown_b#1;{\expandafter\texdiminnddown_c\the\numexpr(2*#1+1)*321/685;#1;}%
+\def\texdiminnddown_c#1;{\expandafter\texdiminnddown_d\the\dimexpr#1sp;#1;}%
+{\catcode`P 12\catcode`T 12\lowercase{\gdef\texdiminnddown_d#1PT};#2;#3;%
+ {\ifdim#1nd>#3sp \texdiminuudown_e{#2}\fi\texdimfirstofone{#1}}%
+}%
+\def\texdiminnddown_neg#1#2-#3;{#1\expandafter\texdiminpt_\the\dimexpr-\texdiminnddown_b#3;pt\relax}%
+% \texdiminndup: minimal dim exactly expressible in nd and at least equal to input
+\def\texdiminndup#1{\expandafter\texdiminndup_a\the\numexpr\dimexpr#1;}%
+\def\texdiminndup_a#1{\if-#1\texdiminuuup_neg\fi\texdiminndup_b#1}%
+\def\texdiminndup_b#1;{\expandafter\texdiminndup_c\the\numexpr(2*#1+1)*321/685;#1;}%
+\def\texdiminndup_c#1;{\expandafter\texdiminndup_d\the\dimexpr#1sp;#1;}%
+{\catcode`P 12\catcode`T 12\lowercase{\gdef\texdiminndup_d#1PT};#2;#3;%
+ {\ifdim#1nd<#3sp \texdiminuuup_e{#2}\fi\texdimfirstofone{#1}}%
+}%
+%
% dd 1238/1157
+%
\def\texdimindd#1{\expandafter\texdimindd_\the\numexpr\dimexpr#1;}%
\def\texdimindd_#1#2;{\texdiminpt{\numexpr(2*#1#2+\if-#1-\fi1)*1157/2476sp}}%
+% \texdimindddown: maximal dim exactly expressible in dd and at most equal to input
+\def\texdimindddown#1{\expandafter\texdimindddown_a\the\numexpr\dimexpr#1;}%
+\def\texdimindddown_a#1{\if-#1\texdimindddown_neg\fi\texdimindddown_b#1}%
+\def\texdimindddown_b#1;{\expandafter\texdimindddown_c\the\numexpr(2*#1+1)*1157/2476;#1;}%
+\def\texdimindddown_c#1;{\expandafter\texdimindddown_d\the\dimexpr#1sp;#1;}%
+{\catcode`P 12\catcode`T 12\lowercase{\gdef\texdimindddown_d#1PT};#2;#3;%
+ {\ifdim#1dd>#3sp \texdiminuudown_e{#2}\fi\texdimfirstofone{#1}}%
+}%
+\def\texdimindddown_neg#1#2-#3;{#1\expandafter\texdiminpt_\the\dimexpr-\texdimindddown_b#3;pt\relax}%
+% \texdiminddup: minimal dim exactly expressible in dd and at least equal to input
+\def\texdiminddup#1{\expandafter\texdiminddup_a\the\numexpr\dimexpr#1;}%
+\def\texdiminddup_a#1{\if-#1\texdiminuuup_neg\fi\texdiminddup_b#1}%
+\def\texdiminddup_b#1;{\expandafter\texdiminddup_c\the\numexpr(2*#1+1)*1157/2476;#1;}%
+\def\texdiminddup_c#1;{\expandafter\texdiminddup_d\the\dimexpr#1sp;#1;}%
+{\catcode`P 12\catcode`T 12\lowercase{\gdef\texdiminddup_d#1PT};#2;#3;%
+ {\ifdim#1dd<#3sp \texdiminuuup_e{#2}\fi\texdimfirstofone{#1}}%
+}%
+%
% mm 7227/2540 phi now >2, use from here on the simpler approach
-\def\texdiminmm#1{\texdiminpt{(#1)*2540/7227}}%
+%
+\def\texdiminmm#1{\expandafter\texdiminpt_\the\dimexpr(#1)*2540/7227\relax}%
+% \texdiminmmdown: maximal dim exactly expressible in mm and at most equal to input
+\def\texdiminmmdown#1{\expandafter\texdiminmmdown_a\the\numexpr\dimexpr#1;}%
+\def\texdiminmmdown_a#1{\if-#1\texdiminmmdown_neg\fi\texdiminmmdown_b#1}%
+\def\texdiminmmdown_b#1;{\expandafter\texdiminmmdown_c\the\numexpr#1*2540/7227;#1;}%
+\def\texdiminmmdown_c#1;{\expandafter\texdiminmmdown_d\the\dimexpr#1sp;#1;}%
+{\catcode`P 12\catcode`T 12\lowercase{\gdef\texdiminmmdown_d#1PT};#2;#3;%
+ {\ifdim#1mm>#3sp \texdiminuudown_e{#2}\fi\texdimfirstofone{#1}}%
+}%
+\def\texdiminmmdown_neg#1#2-#3;{#1\expandafter\texdiminpt_\the\dimexpr-\texdiminmmdown_b#3;pt\relax}%
+% \texdiminmmup: minimal dim exactly expressible in mm and at least equal to input
+\def\texdiminmmup#1{\expandafter\texdiminmmup_a\the\numexpr\dimexpr#1;}%
+\def\texdiminmmup_a#1{\if-#1\texdiminuuup_neg\fi\texdiminmmup_b#1}%
+\def\texdiminmmup_b#1;{\expandafter\texdiminmmup_c\the\numexpr#1*2540/7227;#1;}%
+\def\texdiminmmup_c#1;{\expandafter\texdiminmmup_d\the\dimexpr#1sp;#1;}%
+{\catcode`P 12\catcode`T 12\lowercase{\gdef\texdiminmmup_d#1PT};#2;#3;%
+ {\ifdim#1mm<#3sp \texdiminuuup_e{#2}\fi\texdimfirstofone{#1}}%
+}%
+%
% pc 12/1
-\def\texdiminpc#1{\texdiminpt{(#1)/12}}%
+%
+\def\texdiminpc#1{\expandafter\texdiminpt_\the\dimexpr(#1)/12\relax}%
+% \texdiminpcdown: maximal dim exactly expressible in pc and at most equal to input
+\def\texdiminpcdown#1{\expandafter\texdiminpcdown_a\the\numexpr\dimexpr#1;}%
+\def\texdiminpcdown_a#1{\if-#1\texdiminpcdown_neg\fi\texdiminpcdown_b#1}%
+\def\texdiminpcdown_b#1;{\expandafter\texdiminpcdown_c\the\numexpr#1/12;#1;}%
+\def\texdiminpcdown_c#1;{\expandafter\texdiminpcdown_d\the\dimexpr#1sp;#1;}%
+{\catcode`P 12\catcode`T 12\lowercase{\gdef\texdiminpcdown_d#1PT};#2;#3;%
+ {\ifdim#1pc>#3sp \texdiminuudown_e{#2}\fi\texdimfirstofone{#1}}%
+}%
+\def\texdiminpcdown_neg#1#2-#3;{#1\expandafter\texdiminpt_\the\dimexpr-\texdiminpcdown_b#3;pt\relax}%
+% \texdiminpcup: minimal dim exactly expressible in pc and at least equal to input
+\def\texdiminpcup#1{\expandafter\texdiminpcup_a\the\numexpr\dimexpr#1;}%
+\def\texdiminpcup_a#1{\if-#1\texdiminuuup_neg\fi\texdiminpcup_b#1}%
+\def\texdiminpcup_b#1;{\expandafter\texdiminpcup_c\the\numexpr#1/12;#1;}%
+\def\texdiminpcup_c#1;{\expandafter\texdiminpcup_d\the\dimexpr#1sp;#1;}%
+{\catcode`P 12\catcode`T 12\lowercase{\gdef\texdiminpcup_d#1PT};#2;#3;%
+ {\ifdim#1pc<#3sp \texdiminuuup_e{#2}\fi\texdimfirstofone{#1}}%
+}%
+%
% nc 1370/107
-\def\texdiminnc#1{\texdiminpt{(#1)*107/1370}}%
+%
+\def\texdiminnc#1{\expandafter\texdiminpt_\the\dimexpr(#1)*107/1370\relax}%
+% \texdiminncdown: maximal dim exactly expressible in nc and at most equal to input
+\def\texdiminncdown#1{\expandafter\texdiminncdown_a\the\numexpr\dimexpr#1;}%
+\def\texdiminncdown_a#1{\if-#1\texdiminncdown_neg\fi\texdiminncdown_b#1}%
+\def\texdiminncdown_b#1;{\expandafter\texdiminncdown_c\the\numexpr#1*107/1370;#1;}%
+\def\texdiminncdown_c#1;{\expandafter\texdiminncdown_d\the\dimexpr#1sp;#1;}%
+{\catcode`P 12\catcode`T 12\lowercase{\gdef\texdiminncdown_d#1PT};#2;#3;%
+ {\ifdim#1nc>#3sp \texdiminuudown_e{#2}\fi\texdimfirstofone{#1}}%
+}%
+\def\texdiminncdown_neg#1#2-#3;{#1\expandafter\texdiminpt_\the\dimexpr-\texdiminncdown_b#3;pt\relax}%
+% \texdiminncup: minimal dim exactly expressible in nc and at least equal to input
+\def\texdiminncup#1{\expandafter\texdiminncup_a\the\numexpr\dimexpr#1;}%
+\def\texdiminncup_a#1{\if-#1\texdiminuuup_neg\fi\texdiminncup_b#1}%
+\def\texdiminncup_b#1;{\expandafter\texdiminncup_c\the\numexpr#1*107/1370;#1;}%
+\def\texdiminncup_c#1;{\expandafter\texdiminncup_d\the\dimexpr#1sp;#1;}%
+{\catcode`P 12\catcode`T 12\lowercase{\gdef\texdiminncup_d#1PT};#2;#3;%
+ {\ifdim#1nc<#3sp \texdiminuuup_e{#2}\fi\texdimfirstofone{#1}}%
+}%
+%
% cc 14856/1157
-\def\texdimincc#1{\texdiminpt{(#1)*1157/14856}}%
+%
+\def\texdimincc#1{\expandafter\texdiminpt_\the\dimexpr(#1)*1157/14856\relax}%
+% \texdiminccdown: maximal dim exactly expressible in cc and at most equal to input
+\def\texdiminccdown#1{\expandafter\texdiminccdown_a\the\numexpr\dimexpr#1;}%
+\def\texdiminccdown_a#1{\if-#1\texdiminccdown_neg\fi\texdiminccdown_b#1}%
+\def\texdiminccdown_b#1;{\expandafter\texdiminccdown_c\the\numexpr#1*1157/14856;#1;}%
+\def\texdiminccdown_c#1;{\expandafter\texdiminccdown_d\the\dimexpr#1sp;#1;}%
+{\catcode`P 12\catcode`T 12\lowercase{\gdef\texdiminccdown_d#1PT};#2;#3;%
+ {\ifdim#1cc>#3sp \texdiminuudown_e{#2}\fi\texdimfirstofone{#1}}%
+}%
+\def\texdiminccdown_neg#1#2-#3;{#1\expandafter\texdiminpt_\the\dimexpr-\texdiminccdown_b#3;pt\relax}%
+% \texdiminccup: minimal dim exactly expressible in cc and at least equal to input
+\def\texdiminccup#1{\expandafter\texdiminccup_a\the\numexpr\dimexpr#1;}%
+\def\texdiminccup_a#1{\if-#1\texdiminuuup_neg\fi\texdiminccup_b#1}%
+\def\texdiminccup_b#1;{\expandafter\texdiminccup_c\the\numexpr#1*1157/14856;#1;}%
+\def\texdiminccup_c#1;{\expandafter\texdiminccup_d\the\dimexpr#1sp;#1;}%
+{\catcode`P 12\catcode`T 12\lowercase{\gdef\texdiminccup_d#1PT};#2;#3;%
+ {\ifdim#1cc<#3sp \texdiminuuup_e{#2}\fi\texdimfirstofone{#1}}%
+}%
+%
% cm 7227/254
-\def\texdimincm#1{\texdiminpt{(#1)*254/7227}}%
+%
+\def\texdimincm#1{\expandafter\texdiminpt_\the\dimexpr(#1)*254/7227\relax}%
+% \texdimincmdown: maximal dim exactly expressible in cm and at most equal to input
+\def\texdimincmdown#1{\expandafter\texdimincmdown_a\the\numexpr\dimexpr#1;}%
+\def\texdimincmdown_a#1{\if-#1\texdimincmdown_neg\fi\texdimincmdown_b#1}%
+\def\texdimincmdown_b#1;{\expandafter\texdimincmdown_c\the\numexpr#1*254/7227;#1;}%
+\def\texdimincmdown_c#1;{\expandafter\texdimincmdown_d\the\dimexpr#1sp;#1;}%
+{\catcode`P 12\catcode`T 12\lowercase{\gdef\texdimincmdown_d#1PT};#2;#3;%
+ {\ifdim#1cm>#3sp \texdiminuudown_e{#2}\fi\texdimfirstofone{#1}}%
+}%
+\def\texdimincmdown_neg#1#2-#3;{#1\expandafter\texdiminpt_\the\dimexpr-\texdimincmdown_b#3;pt\relax}%
+% \texdimincmup: minimal dim exactly expressible in cm and at least equal to input
+\def\texdimincmup#1{\expandafter\texdimincmup_a\the\numexpr\dimexpr#1;}%
+\def\texdimincmup_a#1{\if-#1\texdiminuuup_neg\fi\texdimincmup_b#1}%
+\def\texdimincmup_b#1;{\expandafter\texdimincmup_c\the\numexpr#1*254/7227;#1;}%
+\def\texdimincmup_c#1;{\expandafter\texdimincmup_d\the\dimexpr#1sp;#1;}%
+{\catcode`P 12\catcode`T 12\lowercase{\gdef\texdimincmup_d#1PT};#2;#3;%
+ {\ifdim#1cm<#3sp \texdiminuuup_e{#2}\fi\texdimfirstofone{#1}}%
+}%
+%
% in 7227/100
-\def\texdiminin#1{\texdiminpt{(#1)*100/7227}}%
+%
+\def\texdiminin#1{\expandafter\texdiminpt_\the\dimexpr(#1)*100/7227\relax}%
+% \texdiminindown: maximal dim exactly expressible in in and at most equal to input
+\def\texdiminindown#1{\expandafter\texdiminindown_a\the\numexpr\dimexpr#1;}%
+\def\texdiminindown_a#1{\if-#1\texdiminindown_neg\fi\texdiminindown_b#1}%
+\def\texdiminindown_b#1;{\expandafter\texdiminindown_c\the\numexpr#1*100/7227;#1;}%
+\def\texdiminindown_c#1;{\expandafter\texdiminindown_d\the\dimexpr#1sp;#1;}%
+{\catcode`P 12\catcode`T 12\lowercase{\gdef\texdiminindown_d#1PT};#2;#3;%
+ {\ifdim#1in>#3sp \texdiminuudown_e{#2}\fi\texdimfirstofone{#1}}%
+}%
+\def\texdiminindown_neg#1#2-#3;{#1\expandafter\texdiminpt_\the\dimexpr-\texdiminindown_b#3;pt\relax}%
+% \texdimininup: minimal dim exactly expressible in in and at least equal to input
+\def\texdimininup#1{\expandafter\texdimininup_a\the\numexpr\dimexpr#1;}%
+\def\texdimininup_a#1{\if-#1\texdiminuuup_neg\fi\texdimininup_b#1}%
+\def\texdimininup_b#1;{\expandafter\texdimininup_c\the\numexpr#1*100/7227;#1;}%
+\def\texdimininup_c#1;{\expandafter\texdimininup_d\the\dimexpr#1sp;#1;}%
+{\catcode`P 12\catcode`T 12\lowercase{\gdef\texdimininup_d#1PT};#2;#3;%
+ {\ifdim#1in<#3sp \texdiminuuup_e{#2}\fi\texdimfirstofone{#1}}%
+}%
+%
\texdimensendinput