diff options
Diffstat (limited to 'Master/texmf-dist/tex/generic')
| -rw-r--r-- | Master/texmf-dist/tex/generic/texdimens/texdimens.sty | 2 | ||||
| -rw-r--r-- | Master/texmf-dist/tex/generic/texdimens/texdimens.tex | 937 |
2 files changed, 674 insertions, 265 deletions
diff --git a/Master/texmf-dist/tex/generic/texdimens/texdimens.sty b/Master/texmf-dist/tex/generic/texdimens/texdimens.sty index e906d11fa3d..db0220f1db4 100644 --- a/Master/texmf-dist/tex/generic/texdimens/texdimens.sty +++ b/Master/texmf-dist/tex/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}[2021/11/04 v0.99d conversion of TeX dimensions to decimals (JFB)] +\ProvidesPackage{texdimens}[2021/11/10 v1.0 conversion of TeX dimensions to decimals (JFB)] \@@input texdimens.tex\relax \endinput
\ No newline at end of file diff --git a/Master/texmf-dist/tex/generic/texdimens/texdimens.tex b/Master/texmf-dist/tex/generic/texdimens/texdimens.tex index 61086d138dd..de6f20e58fe 100644 --- a/Master/texmf-dist/tex/generic/texdimens/texdimens.tex +++ b/Master/texmf-dist/tex/generic/texdimens/texdimens.tex @@ -1,170 +1,207 @@ % This is file texdimens.tex, part of texdimens package, which % is distributed under the LPPL 1.3c. Copyright (c) 2021 Jean-François Burnol -% 2021/11/04 v0.99d +% 2021/11/10 v1.0 \edef\texdimensendinput{\endlinechar\the\endlinechar% \catcode`\noexpand _=\the\catcode`\_% \catcode`\noexpand @=\the\catcode`\@\relax\noexpand\endinput}% \endlinechar13\relax% -\catcode`\_=11 \catcode`\@=11 % only for using \p@ of Plain. Check exists? +\catcode`\_=11 \catcode`\@=11 % only for using \p@ (also \z@ now) of Plain. Check exists? % -% Mathematics ("down" and "up" macros) +% Mathematics (the challenge of the "down" and "up" macros) % =========== % % In the entire discussion here, "uu" stands for some core unit, -% or some unit corresponding to a dimension > 1pt. For the case -% of a unit corresponding to a dimension < 1pt, i.e. to -% \texdimenwithunit macro added at 0.99, refer to the -% comments of issue #2 on the tracker site. +% or some unit corresponding to an internal dimension > 1pt. % % Is T sp attainable from unit "uu"?. % If not, what is largest dimension < Tsp which is attainable? % Here we suppose T>0. % -% phi>1, psi=1/phi, psi<1. +% D uu --> N = round(D * 65536) --> T = trunc (N * phi) % -% U(N,phi)=trunc(N phi) is the strictly increasing sequence, -% indexed by non-negative integers, of attainable dimensions. -% (in sp unit) +% phi>1 is the conversion factor associated to "uu" +% psi=1/phi, psi<1. Define U(N, phi) = trunc (N * phi) +% +% U(N,phi) is thus the strictly increasing sequence, +% indexed by non-negative integers, of non-negative +% attainable dimensions. (in sp unit) +% +% T>0, then: % % 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 % -% Stumbling block -% --------------- +% In other words: +% +% - the largest attainable dimension not exceeding T sp +% is obtained via the integer "Zd = ceil((T+1)psi) - 1 = N", +% (i.e. find D with Zd=round(65536 D) then "D uu" is "down" approximation) +% +% - the smallest attainable dimension at least equal to T sp +% is obtained from the integer "Zu = ceil(T psi) = M + 1" +% +% - the two "Z"'s are either equal (i.e. T is attained) or Zu=Zd+1. +% +% The round((T+0.5)*psi) based approach (basis of the "\texdimenUU" macros) +% ------------------------------------- +% +% Recall in all of this T > 0. And phi>1, psi=1/phi<1. +% +% Let's return to our analysis of the +% +% U(N)<= T < U(N+1) and U(M)< T <= U(M+1) +% +% equations. We will also use the N=Zd, and M+1=Zu notations. +% +% case1: M = N, i.e. Zd<Zu, i.e. T is not attainable: +% M=N=Zd < T psi < (T+1) psi <= N+1=Zu +% +% Then clearly R = round((T+0.5)psi) is either Zd or Zu. +% We will not know which one before computing trunc(R phi) +% and check if it is < T or > T. +% +% As will be explained later trunc(R phi) can be computed very +% easily by hijacking TeX's handling of dimensions, we don't +% have to launch into \numexpr evaluations for that. +% +% case2: M = N - 1, i.e. T = Zd = Zu is attained: +% T psi <= N < (T+1) psi, T = trunc(N phi) +% +% Let v=(T+0.5)psi. As v = T psi + 0.5 psi it is < N+0.5 +% And as v = (T+1)psi - 0.5psi it is > N - 0.5. +% So R = round(v) = N. +% +% We thus have the initial observation which was at the core of this +% package initial release: +% +% - compute R = round((T+0.5) psi) +% +% - if T is attained, then T = trunc(R * phi) +% +% - if T is not attained then either { Zd = R and Zu = R+1 } or +% {Zd = R-1 and Zu = R}. +% +% How do we check if R = Zd or Zu? We need to evaluate trunc(R phi) and +% compare it with T. This trunc(R phi) can be computed the following way: % -% The stumbling block is that computing "ceil((T+1)psi) - 1" without -% overflow is not obvious: yes \numexpr/\dimexpr allow so-called -% "scaling operations" but only in the "rounding up" variant. +% - obtain D pt from \the\dimexpr R sp. Knuth's algorithm guarantees +% that R = round(D * 65536) % -% If we attempt computing the ceil(x) function via round(x+0.5), -% for example with psi=100/7227 which corresponds to the unit "in", -% this necessitates evaluating: +% - then D uu where uu is the unit with conversion factor phi is +% converted by TeX into "trunc(R phi) sp", i.e. trunc(R phi) = +% \number\dimexpr Duu\relax, where D pt = \the\dimexpr Rsp\relax. % -% round((((T+1)*200)+7227)/14454) +% Conclusion: % -% But as far as I can tell currently, for this we need to be able -% to evaluate without overflow (T+1)*200+7227 and this limits to -% T's which are (roughly) such that 100 T is less than \maxdimen. +% 1. the macro \texdimenuu does the one-liner R=round((T+0.5) psi) +% then \the\dimexpr Rsp\relax and strips the "pt" unit % -% A work-around -% ------------- +% 2. macros \texdimenuuup and \texdimenuudown go further and check +% which one of Zd or Zu is R, obtaining thus Zd or Zu. +% [update: this describes the macros prior to 1.0 release] % -% The rest of the discussion is about an algorithm providing an -% alternative route to N, using \numexpr/\dimexpr/TeX facilities, -% and working with (almost, as we will see) the full range of allowed -% T's, 0 < T <= \maxdimen. (that the algorithm works for T=0 is to be -% checked manually after the main discussion). +% For units with conversion factor phi>2, a simplification is possible. +% In that case let X = round(T psi) (it has the advantage compared to +% R that we can apply the formula without checking the sign of T). % -% Let's return to the U(N)<= T < U(N+1) and U(M)< T <= U(M+1) equations. +% Going back to our earlier analyis, now with psi < 0.5 (1uu>2pt) % -% Either (recall in all of this T > 0): +% case1: T is not attainable +% M=N=Zd < T psi < (T+1) psi <= N+1=Zu +% As Zd < T psi < Zu, we have round(T psi) = Zd or Zu % -% 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 = trunc(N phi) +% case2: T is attained, i.e. T psi <= N < (T+1) psi. +% As psi<0.5, and T psi + psi > N, we have T psi > N - 0.5. +% And T psi <= N so N = round(T psi). % -% Let X = round(T psi). And let Y = trunc(X phi). We will explain later -% how X and Y can be computed using \numexpr/\dimexpr/TeX. +% So, for psi < 0.5, the X=round(T psi) can play the same role as +% R=round((T+0.5)psi). If T is attained, we get the decimal D from this +% X and if T is not attained we know that X is either Zd or Zu. % -% 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. +% The computations of X and Y=trunc(X phi) 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 out the sign of the +% input. % -% This is not convenient: if Y < T it could be that we are in case 2 -% but to decide we must check if trunc((X+1) phi) = T or not, so -% this means a second computation. +% Going back to the 1<phi<2 case, psi>0.5, then it would be 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, i.e. we don't know if Zd =X or X+1, +% and we can not tell yet if T is attained or not. % -% If psi < 0.5 -% ------------ +% In contrast if we find out that trunc(R phi)<T, we then know for sure +% that Zd=R, Zu=R+1 and that T is not attained. % -% The situation then simplifies: +% Problems with \maxdimen in the obtention of Zu and Zd +% ----------------------------------------------------- % -% case1: X can be N or N+1. It will be N+1 iff Y = trunc(X phi) > T. -% case2: X is necessarily N. +% Obtaining R = round((T+0.5)psi) has no risk of overflow. +% But checking as described above which one of Zd or Zu (or both) +% is R goes via a test computation which will cause overflow +% if by bad luck R = Zu and Zu will give rise to a decimal D +% such that D uu > \maxdimen. % -% 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, and we leave X as it is. -% We have thus found N. +% For T=\maxdimen (or very close) this is what happens for the units +% "dd", "nc", and "in". % -% 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. +% Besides, it turns out that this test which is done to decide whether +% R=Zu or R=Zd, and on which the initial implementation of the macros +% "up" and "down" was done at 0.9 gamma release is a bit costly. % -% This way we find the maximal dimension at most T sp exactly -% representable in "uu" unit. +% At 1.0 release, all the "up" and "down" macros were re-implemented +% via a more stubborn usage of the ceil() based formulae for Zd and Zu. +% This made all usable even with \maxdimen input and besided, proved +% on average slightly faster. % -% 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 out the sign of the input. +% Overcoming the ceil() stumbling block (i.e. the 1.0 manner for "up/down") +% ------------------------------------- % -% 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 attainable we don't know if this X will give -% a D such that D uu < T sp or D uu > T sp. +% I will in what follows refer to trunc(), floor() or ceil() only for +% positive arguments, obtained as ratios x/y or sometimes as a numexpr +% "scaling" operation" x*y/z which uses temporarily use doubled +% precision. % -% If psi > 0.5 -% ------------ +% As \numexpr's x/y is round(x/y), with rounding away from zero, we have +% access to floor(t) for t>=0 as round(t+0.5)-1 and for t>0 also as +% round(t-0.5). The former may cause overflow as it involves +% (2x+y)/(2y) but the latter (2x-y)/(2y) will not overflow if x comes +% from a dimension as 2x<2**31 then. % -% For example unit "bp" has phi=803/800. +% ceil(t) is more complex as it is floor(t)+1 only for t not an integer. +% Let's explain how to overcome the challenge for Zd and the "in" unit, +% i.e. a conversion factor of 7227/100. % -% It is then not true that if T sp is attainable, the X = round(T psi) -% will always work. +% We want Zd = ceil((T+1)*100/7227) - 1, with T assumed positive. % -% But it is true that R = round((T + 0.5) psi) will always work. -% Here we must use -0.5 if T < 0, though. +% Let T = k*7227 + r with 0<= r < 7227, 0<=k, and r>0 if k=0. % -% This R=round((T+0.5) psi) can always be computed via \numexpr because 2T+1 -% will not trigger arithmetic overflow. +% (T+1)*100/7227 becomes 100*k + (r+1)*100/7227 and thus % -% So this gives an approach to find a D such that "D uu" is exactly -% T sp when this is possible. +% Zd = 100 * k + ceil(x) - 1 % -% If Tsp (positive) is not attainable, this R however can produce -% either N or N+1. +% with x = n*100/7227, and n = 1+r, so 0<n<=7227 % -% But we can decide what happened by computing Z = trunc(R phi). -% If and only if Z > T this means R was N+1. +% Here we have a nice situation 0 < x <= 100. Then: +% +% ceil(x) = 100 - floor(100 - x) +% = 100 - (round(100 - x + 0.5) - 1) +% = 101 - round(100 * (1 - n/7227) + 0.5) +% = 101 - round((200 * (7227 - n) + 7227)/14454) % -% 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. So we proceed via R, not X, -% to not have to make a second computation if a dimension comparison -% test goes awry. +% We can thus achieve the computation of Zd = ceil((T+1)*100/7227) - 1 +% for T>0 without overflow in \numexpr this way: % -% To recapitulate: we have our algorithm for all units to find out -% maximal dimension exactly attainable in "uu" unit and at most equal -% to (positive) T sp. +% k = floor(T/7227) = round(T/7227 - 0.5) +% = round((2*T - 7227) / 14454) (T>0 used here) % -% Unfortunately the check that Y (in case psi < 0.5) or Z (in case psi > -% 0.5) verifies or not Y > T may trigger a Dimension too large error if -% T sp was near non-attainable \maxdimen. It turns out this sad -% situation happens only for the units `dd`, `nc`, and `in`, and T sp -% very close to \maxdimen (like for all units apart from `pt`, `bp`, -% `nd`, the \maxdimen is not attainable, and by bad luck for `dd`, `nc`, -% and `in`, the X will correspond to a decimal D such that Duu>\maxdimen -% is the nearest virtually attaible dimensions from above not from -% below; see the README.md for the tabulation of the maximal usable inputs). -% -% Regarding the \texdimen<uu> macros, and units with phi > 2, I -% hesitated using either the round((T+0.5)psi) or round(T psi), but for -% Tsp = \maxdimen, both formulas turned out to give the same result for -% all such units, so I chose for these \texdimen<uu> macros and the -% units with phi>2 to use the simpler round(T psi) which does not need -% to check the sign of T. -% -% For the "up" and "down" macros, we again use the round(T psi), but do -% have to check the sign anyhow. We could also have used the -% round((T+0.5)psi) which requires a sign check too, but it costs a bit -% more. It would have allowed though to share the same codebase for all -% units, here we have to prepare some slightly different shared macros -% for the first batch bp, nd, dd and the second batch mm, pc, nc, cc, -% cm, in. +% r = T - 7227 * k = T modulo 7227 +% +% Zd = 100 * k + 100 - round( (201*7227 - 200*(r+1))/14454 ) +% +% Everything here is computable within \numexpr and has absolutely no +% potential overflow problem at all. The same analysis can be done for +% Zu = ceil(T*100/7227) and for all core TeX units. See the comments +% below for all obtained formulae and some additional details. % % Implementation % ============== @@ -173,45 +210,6 @@ {\catcode`p 12\catcode`t 12 \csname expandafter\endcsname\gdef\csname texdimenstrippt\endcsname#1pt{#1}}% % -% down macros: -% for units with phi < 2: -\def\texdimendown_A#1{\if-#1\texdimendown_neg\fi\texdimendown_B#1}% -\def\texdimendown_B#1;#2;{\expandafter\texdimendown_c\the\numexpr(2*#1+1)#2;#1;}% -% for units with phi > 2: -\def\texdimendown_a#1{\if-#1\texdimendown_neg\fi\texdimendown_b#1}% -\def\texdimendown_b#1;#2;{\expandafter\texdimendown_c\the\numexpr#1#2;#1;}% -% shared macros: -\def\texdimendown_c#1;{\expandafter\texdimendown_d\the\dimexpr#1sp;#1;}% -{\catcode`P 12\catcode`T 12\lowercase{\gdef\texdimendown_d#1PT};#2;#3;#4;% - {\ifdim#1#4>#3sp \texdimendown_e{#2}\fi\texdimenfirstofone{#1}}% -}% -% this #2 will be \fi -\def\texdimendown_e#1#2#3#4{#2\expandafter\texdimenstrippt\the\dimexpr\numexpr#1-1sp\relax}% -% negative branch: -% The problem here is that if input very small, output can be 0.0, and we -% do not want -0.0 as output. -% So let's do this somewhat brutally and non-efficiently. -% Anyhow, negative inputs are not our priority. -% #1 is \fi here and #2 is \texdimendown_b or _B: -\def\texdimendown_neg#1#2-#3;#4;#5;{#1\expandafter\texdimenstrippt\the\dimexpr-#2#3;#4;#5;pt\relax}% -% -% up macros: -\def\texdimenup_A#1{\if-#1\texdimenup_neg\fi\texdimenup_B#1}% -\def\texdimenup_B#1;#2;{\expandafter\texdimenup_c\the\numexpr(2*#1+1)#2;#1;}% -\def\texdimenup_a#1{\if-#1\texdimenup_neg\fi\texdimenup_b#1}% -\def\texdimenup_b#1;#2;{\expandafter\texdimenup_c\the\numexpr#1#2;#1;}% -\def\texdimenup_c#1;{\expandafter\texdimenup_d\the\dimexpr#1sp;#1;}% -{\catcode`P 12\catcode`T 12\lowercase{\gdef\texdimenup_d#1PT};#2;#3;#4;% - {\ifdim#1#4<#3sp \texdimenup_e{#2}\fi\texdimenfirstofone{#1}}% -}% -% this #2 will be \fi -\def\texdimenup_e#1#2#3#4{#2\expandafter\texdimenstrippt\the\dimexpr\numexpr#1+1sp\relax}% -% negative branch: -% Here we can me more expeditive than for the "down" macros. -% But this breaks f-expandability. -% #1 will be \fi and #2 is \texdimenup_b or _B: -\def\texdimenup_neg#1#2-{#1-#2}% -% % pt % \def\texdimenpt#1{\expandafter\texdimenstrippt\the\dimexpr#1\relax}% @@ -222,10 +220,6 @@ \def\texdimenbp_#1#2;{% \expandafter\texdimenstrippt\the\dimexpr\numexpr(2*#1#2+\if-#1-\fi1)*400/803sp\relax }% -% \texdimenbpdown: maximal dim exactly expressible in bp and at most equal to input -\def\texdimenbpdown#1{\expandafter\texdimendown_A\the\numexpr\dimexpr#1;*400/803;bp;}% -% \texdimenbpup: minimal dim exactly expressible in bp and at least equal to input -\def\texdimenbpup#1{\expandafter\texdimenup_A\the\numexpr\dimexpr#1;*400/803;bp;}% % % nd 685/642 % @@ -233,10 +227,6 @@ \def\texdimennd_#1#2;{% \expandafter\texdimenstrippt\the\dimexpr\numexpr(2*#1#2+\if-#1-\fi1)*321/685sp\relax }% -% \texdimennddown: maximal dim exactly expressible in nd and at most equal to input -\def\texdimennddown#1{\expandafter\texdimendown_A\the\numexpr\dimexpr#1;*321/685;nd;}% -% \texdimenndup: minimal dim exactly expressible in nd and at least equal to input -\def\texdimenndup#1{\expandafter\texdimenup_A\the\numexpr\dimexpr#1;*321/685;nd;}% % % dd 1238/1157 % @@ -244,60 +234,296 @@ \def\texdimendd_#1#2;{% \expandafter\texdimenstrippt\the\dimexpr\numexpr(2*#1#2+\if-#1-\fi1)*1157/2476sp\relax }% -% \texdimendddown: maximal dim exactly expressible in dd and at most equal to input -\def\texdimendddown#1{\expandafter\texdimendown_A\the\numexpr\dimexpr#1;*1157/2476;dd;}% -% \texdimenddup: minimal dim exactly expressible in dd and at least equal to input -\def\texdimenddup#1{\expandafter\texdimenup_A\the\numexpr\dimexpr#1;*1157/2476;dd;}% % -% mm 7227/2540 phi now >2, use from here on the simpler approach +% mm 7227/2540 phi now >2, use from here on the X = round(T psi) approach % \def\texdimenmm#1{\expandafter\texdimenstrippt\the\dimexpr(#1)*2540/7227\relax}% -% \texdimenmmdown: maximal dim exactly expressible in mm and at most equal to input -\def\texdimenmmdown#1{\expandafter\texdimendown_a\the\numexpr\dimexpr#1;*2540/7227;mm;}% -% \texdimenmmup: minimal dim exactly expressible in mm and at least equal to input -\def\texdimenmmup#1{\expandafter\texdimenup_a\the\numexpr\dimexpr#1;*2540/7227;mm;}% % % pc 12/1 % \def\texdimenpc#1{\expandafter\texdimenstrippt\the\dimexpr(#1)/12\relax}% -% \texdimenpcdown: maximal dim exactly expressible in pc and at most equal to input -\def\texdimenpcdown#1{\expandafter\texdimendown_a\the\numexpr\dimexpr#1;/12;pc;}% -% \texdimenpcup: minimal dim exactly expressible in pc and at least equal to input -\def\texdimenpcup#1{\expandafter\texdimenup_a\the\numexpr\dimexpr#1;/12;pc;}% % % nc 1370/107 % \def\texdimennc#1{\expandafter\texdimenstrippt\the\dimexpr(#1)*107/1370\relax}% -% \texdimenncdown: maximal dim exactly expressible in nc and at most equal to input -\def\texdimenncdown#1{\expandafter\texdimendown_a\the\numexpr\dimexpr#1;*107/1370;nc;}% -% \texdimenncup: minimal dim exactly expressible in nc and at least equal to input -\def\texdimenncup#1{\expandafter\texdimenup_a\the\numexpr\dimexpr#1;*107/1370;nc;}% % % cc 14856/1157 % \def\texdimencc#1{\expandafter\texdimenstrippt\the\dimexpr(#1)*1157/14856\relax}% -% \texdimenccdown: maximal dim exactly expressible in cc and at most equal to input -\def\texdimenccdown#1{\expandafter\texdimendown_a\the\numexpr\dimexpr#1;*1157/14856;cc;}% -% \texdimenccup: minimal dim exactly expressible in cc and at least equal to input -\def\texdimenccup#1{\expandafter\texdimenup_a\the\numexpr\dimexpr#1;*1157/14856;cc;}% % % cm 7227/254 % \def\texdimencm#1{\expandafter\texdimenstrippt\the\dimexpr(#1)*254/7227\relax}% -% \texdimencmdown: maximal dim exactly expressible in cm and at most equal to input -\def\texdimencmdown#1{\expandafter\texdimendown_a\the\numexpr\dimexpr#1;*254/7227;cm;}% -% \texdimencmup: minimal dim exactly expressible in cm and at least equal to input -\def\texdimencmup#1{\expandafter\texdimenup_a\the\numexpr\dimexpr#1;*254/7227;cm;}% % % in 7227/100 % \def\texdimenin#1{\expandafter\texdimenstrippt\the\dimexpr(#1)*100/7227\relax}% -% \texdimenindown: maximal dim exactly expressible in in and at most equal to input -\def\texdimenindown#1{\expandafter\texdimendown_a\the\numexpr\dimexpr#1;*100/7227;in;}% -% \texdimeninup: minimal dim exactly expressible in in and at least equal to input -\def\texdimeninup#1{\expandafter\texdimenup_a\the\numexpr\dimexpr#1;*100/7227;in;}% -% both in and cm -% Mathematics ("both" macros) +% +% "no test" ceil()-based approach to up and down macros (release 1.0) +% ===================================================== +% +% The notation <u/v> means u/v in numexpr, which does rounding +% away from zero. It is essential that the argument be >-0.5 else <x+1> +% not same as <x>+1. All formulae are overflow free. +% +% The comments are for T > 0. +% +% Roughly such an approach works for phi = a/b > 1, such that: +% +% a*(2b+1)<2**31 if a is odd, <2**32 if a is even +% +% This is true for all core units with quite some margin, the one with +% largest a*b being phi=7227/2540 for "mm". +% +% Note: for a unit such as "ex" or "em" where morally b=65536=2**16, +% this limits to a<=16383 if a is odd and to a<=32766 if a is even. +% Thus the general \texdimenwithunit{dim1}{dim2} (which for dim2<1pt +% computes basically an "up" value) can *not imitate fully* this scheme. +% +% The macros and formulas in the comments were obtained from a template +% (see file generateupdownmacros.py at the project repository), +% and we could actually combine them into a generic macro handling +% general a/b (assuming above bounds are verified). +% But for the the sake of efficiency, this is "rolled-out" here unit per unit. +% +\def\texdimenuudownup_zero#1;{\z@\relax}% +\def\texdimenuudownup_neg#1-{-#1}% +% bp 803/800 +% T = 803 k + r +% Zd = 800 k + 800 - <(1284003 - 1600 r)/1606> +\def\texdimenbpdown#1{\expandafter\texdimenstrippt\the\dimexpr + \expandafter\texdimenbpdown_a\the\numexpr\dimexpr#1;% +}% +\def\texdimenbpdown_a#1{\texdimenzerominusfork + #1-\texdimenuudownup_zero + 0#1\texdimenuudownup_neg + 0-{}% + \krof \texdimenbpdown_b#1}% +\def\texdimenbpdown_b#1;{\expandafter\texdimenbpdown_c\the\numexpr(2*#1-803)/1606;#1;}% +\def\texdimenbpdown_c#1;#2;{\expandafter\texdimenbpdown_d\the\numexpr#2-803*#1;#1;}% +\def\texdimenbpdown_d#1;#2;{\numexpr800*#2+800-(1284003-1600*#1)/1606sp\relax}% +% Zu = 800 k + 800 + 1 - <(1285603 - 1600 r)/1606> +\def\texdimenbpup#1{\expandafter\texdimenstrippt\the\dimexpr + \expandafter\texdimenbpup_a\the\numexpr\dimexpr#1;% +}% +\def\texdimenbpup_a#1{\texdimenzerominusfork + #1-\texdimenuudownup_zero + 0#1\texdimenuudownup_neg + 0-{}% + \krof \texdimenbpup_b#1}% +\def\texdimenbpup_b#1;{\expandafter\texdimenbpup_c\the\numexpr(2*#1-803)/1606;#1;}% +\def\texdimenbpup_c#1;#2;{\expandafter\texdimenbpup_d\the\numexpr#2-803*#1;#1;}% +\def\texdimenbpup_d#1;#2;{\numexpr800*#2+801-(1285603-1600*#1)/1606sp\relax}% +% nd 685/642 +% T = 685 k + r +% Zd = 642 k + 642 - <(878941 - 1284 r)/1370> +\def\texdimennddown#1{\expandafter\texdimenstrippt\the\dimexpr + \expandafter\texdimennddown_a\the\numexpr\dimexpr#1;% +}% +\def\texdimennddown_a#1{\texdimenzerominusfork + #1-\texdimenuudownup_zero + 0#1\texdimenuudownup_neg + 0-{}% + \krof \texdimennddown_b#1}% +\def\texdimennddown_b#1;{\expandafter\texdimennddown_c\the\numexpr(2*#1-685)/1370;#1;}% +\def\texdimennddown_c#1;#2;{\expandafter\texdimennddown_d\the\numexpr#2-685*#1;#1;}% +\def\texdimennddown_d#1;#2;{\numexpr642*#2+642-(878941-1284*#1)/1370sp\relax}% +% Zu = 642 k + 642 + 1 - <(880225 - 1284 r)/1370> +\def\texdimenndup#1{\expandafter\texdimenstrippt\the\dimexpr + \expandafter\texdimenndup_a\the\numexpr\dimexpr#1;% +}% +\def\texdimenndup_a#1{\texdimenzerominusfork + #1-\texdimenuudownup_zero + 0#1\texdimenuudownup_neg + 0-{}% + \krof \texdimenndup_b#1}% +\def\texdimenndup_b#1;{\expandafter\texdimenndup_c\the\numexpr(2*#1-685)/1370;#1;}% +\def\texdimenndup_c#1;#2;{\expandafter\texdimenndup_d\the\numexpr#2-685*#1;#1;}% +\def\texdimenndup_d#1;#2;{\numexpr642*#2+643-(880225-1284*#1)/1370sp\relax}% +% dd 1238/1157 +% T = 1238 k + r +% Zd = 1157 k + 1157 - <(1431828 - 1157 r)/1238> +\def\texdimendddown#1{\expandafter\texdimenstrippt\the\dimexpr + \expandafter\texdimendddown_a\the\numexpr\dimexpr#1;% +}% +\def\texdimendddown_a#1{\texdimenzerominusfork + #1-\texdimenuudownup_zero + 0#1\texdimenuudownup_neg + 0-{}% + \krof \texdimendddown_b#1}% +\def\texdimendddown_b#1;{\expandafter\texdimendddown_c\the\numexpr(#1-619)/1238;#1;}% +\def\texdimendddown_c#1;#2;{\expandafter\texdimendddown_d\the\numexpr#2-1238*#1;#1;}% +\def\texdimendddown_d#1;#2;{\numexpr1157*#2+1157-(1431828-1157*#1)/1238sp\relax}% +% Zu = 1157 k + 1157 + 1 - <(1432985 - 1157 r)/1238> +\def\texdimenddup#1{\expandafter\texdimenstrippt\the\dimexpr + \expandafter\texdimenddup_a\the\numexpr\dimexpr#1;% +}% +\def\texdimenddup_a#1{\texdimenzerominusfork + #1-\texdimenuudownup_zero + 0#1\texdimenuudownup_neg + 0-{}% + \krof \texdimenddup_b#1}% +\def\texdimenddup_b#1;{\expandafter\texdimenddup_c\the\numexpr(#1-619)/1238;#1;}% +\def\texdimenddup_c#1;#2;{\expandafter\texdimenddup_d\the\numexpr#2-1238*#1;#1;}% +\def\texdimenddup_d#1;#2;{\numexpr1157*#2+1158-(1432985-1157*#1)/1238sp\relax}% +% mm 7227/2540 +% T = 7227 k + r +% Zd = 2540 k + 2540 - <(36715307 - 5080 r)/14454> +\def\texdimenmmdown#1{\expandafter\texdimenstrippt\the\dimexpr + \expandafter\texdimenmmdown_a\the\numexpr\dimexpr#1;% +}% +\def\texdimenmmdown_a#1{\texdimenzerominusfork + #1-\texdimenuudownup_zero + 0#1\texdimenuudownup_neg + 0-{}% + \krof \texdimenmmdown_b#1}% +\def\texdimenmmdown_b#1;{\expandafter\texdimenmmdown_c\the\numexpr(2*#1-7227)/14454;#1;}% +\def\texdimenmmdown_c#1;#2;{\expandafter\texdimenmmdown_d\the\numexpr#2-7227*#1;#1;}% +\def\texdimenmmdown_d#1;#2;{\numexpr2540*#2+2540-(36715307-5080*#1)/14454sp\relax}% +% Zu = 2540 k + 2540 + 1 - <(36720387 - 5080 r)/14454> +\def\texdimenmmup#1{\expandafter\texdimenstrippt\the\dimexpr + \expandafter\texdimenmmup_a\the\numexpr\dimexpr#1;% +}% +\def\texdimenmmup_a#1{\texdimenzerominusfork + #1-\texdimenuudownup_zero + 0#1\texdimenuudownup_neg + 0-{}% + \krof \texdimenmmup_b#1}% +\def\texdimenmmup_b#1;{\expandafter\texdimenmmup_c\the\numexpr(2*#1-7227)/14454;#1;}% +\def\texdimenmmup_c#1;#2;{\expandafter\texdimenmmup_d\the\numexpr#2-7227*#1;#1;}% +\def\texdimenmmup_d#1;#2;{\numexpr2540*#2+2541-(36720387-5080*#1)/14454sp\relax}% +% pc 12/1 +% T = 12 k + r +% Zd = 1 k + 1 - <(17 - 1 r)/12> +\def\texdimenpcdown#1{\expandafter\texdimenstrippt\the\dimexpr + \expandafter\texdimenpcdown_a\the\numexpr\dimexpr#1;% +}% +\def\texdimenpcdown_a#1{\texdimenzerominusfork + #1-\texdimenuudownup_zero + 0#1\texdimenuudownup_neg + 0-{}% + \krof \texdimenpcdown_b#1}% +\def\texdimenpcdown_b#1;{\expandafter\texdimenpcdown_c\the\numexpr(#1-6)/12;#1;}% +\def\texdimenpcdown_c#1;#2;{\expandafter\texdimenpcdown_d\the\numexpr#2-12*#1;#1;}% +\def\texdimenpcdown_d#1;#2;{\numexpr#2+1-(17-#1)/12sp\relax}% +% Zu = 1 k + 1 + 1 - <(18 - 1 r)/12> +\def\texdimenpcup#1{\expandafter\texdimenstrippt\the\dimexpr + \expandafter\texdimenpcup_a\the\numexpr\dimexpr#1;% +}% +\def\texdimenpcup_a#1{\texdimenzerominusfork + #1-\texdimenuudownup_zero + 0#1\texdimenuudownup_neg + 0-{}% + \krof \texdimenpcup_b#1}% +\def\texdimenpcup_b#1;{\expandafter\texdimenpcup_c\the\numexpr(#1-6)/12;#1;}% +\def\texdimenpcup_c#1;#2;{\expandafter\texdimenpcup_d\the\numexpr#2-12*#1;#1;}% +\def\texdimenpcup_d#1;#2;{\numexpr#2+2-(18-#1)/12sp\relax}% +% nc 1370/107 +% T = 1370 k + r +% Zd = 107 k + 107 - <(147168 - 107 r)/1370> +\def\texdimenncdown#1{\expandafter\texdimenstrippt\the\dimexpr + \expandafter\texdimenncdown_a\the\numexpr\dimexpr#1;% +}% +\def\texdimenncdown_a#1{\texdimenzerominusfork + #1-\texdimenuudownup_zero + 0#1\texdimenuudownup_neg + 0-{}% + \krof \texdimenncdown_b#1}% +\def\texdimenncdown_b#1;{\expandafter\texdimenncdown_c\the\numexpr(#1-685)/1370;#1;}% +\def\texdimenncdown_c#1;#2;{\expandafter\texdimenncdown_d\the\numexpr#2-1370*#1;#1;}% +\def\texdimenncdown_d#1;#2;{\numexpr107*#2+107-(147168-107*#1)/1370sp\relax}% +% Zu = 107 k + 107 + 1 - <(147275 - 107 r)/1370> +\def\texdimenncup#1{\expandafter\texdimenstrippt\the\dimexpr + \expandafter\texdimenncup_a\the\numexpr\dimexpr#1;% +}% +\def\texdimenncup_a#1{\texdimenzerominusfork + #1-\texdimenuudownup_zero + 0#1\texdimenuudownup_neg + 0-{}% + \krof \texdimenncup_b#1}% +\def\texdimenncup_b#1;{\expandafter\texdimenncup_c\the\numexpr(#1-685)/1370;#1;}% +\def\texdimenncup_c#1;#2;{\expandafter\texdimenncup_d\the\numexpr#2-1370*#1;#1;}% +\def\texdimenncup_d#1;#2;{\numexpr107*#2+108-(147275-107*#1)/1370sp\relax}% +% cc 14856/1157 +% T = 14856 k + r +% Zd = 1157 k + 1157 - <(17194663 - 1157 r)/14856> +\def\texdimenccdown#1{\expandafter\texdimenstrippt\the\dimexpr + \expandafter\texdimenccdown_a\the\numexpr\dimexpr#1;% +}% +\def\texdimenccdown_a#1{\texdimenzerominusfork + #1-\texdimenuudownup_zero + 0#1\texdimenuudownup_neg + 0-{}% + \krof \texdimenccdown_b#1}% +\def\texdimenccdown_b#1;{\expandafter\texdimenccdown_c\the\numexpr(#1-7428)/14856;#1;}% +\def\texdimenccdown_c#1;#2;{\expandafter\texdimenccdown_d\the\numexpr#2-14856*#1;#1;}% +\def\texdimenccdown_d#1;#2;{\numexpr1157*#2+1157-(17194663-1157*#1)/14856sp\relax}% +% Zu = 1157 k + 1157 + 1 - <(17195820 - 1157 r)/14856> +\def\texdimenccup#1{\expandafter\texdimenstrippt\the\dimexpr + \expandafter\texdimenccup_a\the\numexpr\dimexpr#1;% +}% +\def\texdimenccup_a#1{\texdimenzerominusfork + #1-\texdimenuudownup_zero + 0#1\texdimenuudownup_neg + 0-{}% + \krof \texdimenccup_b#1}% +\def\texdimenccup_b#1;{\expandafter\texdimenccup_c\the\numexpr(#1-7428)/14856;#1;}% +\def\texdimenccup_c#1;#2;{\expandafter\texdimenccup_d\the\numexpr#2-14856*#1;#1;}% +\def\texdimenccup_d#1;#2;{\numexpr1157*#2+1158-(17195820-1157*#1)/14856sp\relax}% +% cm 7227/254 +% T = 7227 k + r +% Zd = 254 k + 254 - <(3678035 - 508 r)/14454> +\def\texdimencmdown#1{\expandafter\texdimenstrippt\the\dimexpr + \expandafter\texdimencmdown_a\the\numexpr\dimexpr#1;% +}% +\def\texdimencmdown_a#1{\texdimenzerominusfork + #1-\texdimenuudownup_zero + 0#1\texdimenuudownup_neg + 0-{}% + \krof \texdimencmdown_b#1}% +\def\texdimencmdown_b#1;{\expandafter\texdimencmdown_c\the\numexpr(2*#1-7227)/14454;#1;}% +\def\texdimencmdown_c#1;#2;{\expandafter\texdimencmdown_d\the\numexpr#2-7227*#1;#1;}% +\def\texdimencmdown_d#1;#2;{\numexpr254*#2+254-(3678035-508*#1)/14454sp\relax}% +% Zu = 254 k + 254 + 1 - <(3678543 - 508 r)/14454> +\def\texdimencmup#1{\expandafter\texdimenstrippt\the\dimexpr + \expandafter\texdimencmup_a\the\numexpr\dimexpr#1;% +}% +\def\texdimencmup_a#1{\texdimenzerominusfork + #1-\texdimenuudownup_zero + 0#1\texdimenuudownup_neg + 0-{}% + \krof \texdimencmup_b#1}% +\def\texdimencmup_b#1;{\expandafter\texdimencmup_c\the\numexpr(2*#1-7227)/14454;#1;}% +\def\texdimencmup_c#1;#2;{\expandafter\texdimencmup_d\the\numexpr#2-7227*#1;#1;}% +\def\texdimencmup_d#1;#2;{\numexpr254*#2+255-(3678543-508*#1)/14454sp\relax}% +% in 7227/100 +% T = 7227 k + r +% Zd = 100 k + 100 - <(1452427 - 200 r)/14454> +\def\texdimenindown#1{\expandafter\texdimenstrippt\the\dimexpr + \expandafter\texdimenindown_a\the\numexpr\dimexpr#1;% +}% +\def\texdimenindown_a#1{\texdimenzerominusfork + #1-\texdimenuudownup_zero + 0#1\texdimenuudownup_neg + 0-{}% + \krof \texdimenindown_b#1}% +\def\texdimenindown_b#1;{\expandafter\texdimenindown_c\the\numexpr(2*#1-7227)/14454;#1;}% +\def\texdimenindown_c#1;#2;{\expandafter\texdimenindown_d\the\numexpr#2-7227*#1;#1;}% +\def\texdimenindown_d#1;#2;{\numexpr#200+100-(1452427-2*#100)/14454sp\relax}% +% Zu = 100 k + 100 + 1 - <(1452627 - 200 r)/14454> +\def\texdimeninup#1{\expandafter\texdimenstrippt\the\dimexpr + \expandafter\texdimeninup_a\the\numexpr\dimexpr#1;% +}% +\def\texdimeninup_a#1{\texdimenzerominusfork + #1-\texdimenuudownup_zero + 0#1\texdimenuudownup_neg + 0-{}% + \krof \texdimeninup_b#1}% +\def\texdimeninup_b#1;{\expandafter\texdimeninup_c\the\numexpr(2*#1-7227)/14454;#1;}% +\def\texdimeninup_c#1;#2;{\expandafter\texdimeninup_d\the\numexpr#2-7227*#1;#1;}% +\def\texdimeninup_d#1;#2;{\numexpr#200+101-(1452627-2*#100)/14454sp\relax}% +% +% Mathematics ("both in and cm", added at release 0.9) % =========== % % Let a and b be two non-negative integers such that U = floor(a 7227/100) = @@ -374,8 +600,110 @@ \def\texdimenbothsp_neg#1#2-#3;{#1-\numexpr(((2*#3-3612)/7227)*7227+1)/2-1\relax\relax}% % \let\texdimenbothcminsp\texdimenbothincmsp -% (\texdimenwithunit, added at 0.99) -% Mathematics +% +% Mathematics and Algorithm ("both mm and bp", added at release 1.0) +% ========================= +% We start from a dimension expressed in sp unit, "T sp". Assume T positive. +% We know how to get largest "X sp <= T sp" which is exactly expressible +% in mm unit +% i.e. can be written X=trunc(a 7227/2540) for some non-negative integer a. +% We want to achieve X=trunc(b 803/800) for some b. +% +% Only the congruence of X modulo 803 matters for this. +% It turns out that the mod 803 impossible values are 267, 535, 802. +% As pointed out by Ruixi Zhang on the package repo issue #10, +% when a<--a+2540, X increases by 7227=9*803 hence the value +% modulo 803 does not change. Thus only "a modulo 2540" matters +% to check if X(a) is attainable with bp unit. Ruixi Zhang found by +% brute force that there are modulo 2540 nine excluded a-values +% +% Rather than checking if "a mod. 2540" avoids the 9 Ruixi Zhang values +% or if "X mod. 803" avoids 267, 535, 802, we will simply basically +% check if X sp = \texdimenbp{X sp}bp, as this approach is probably +% about the same cost or even less than computing "X mod. 803" and +% correspondingly branching. +% +% The key is that if "a" is bad, then "a-1" is automatically good as +% pointed out by R.Z. on #10, which can be seen without knowing the 9 +% bad congruences, simply by noticing that a<--a-1 modifies X either to +% X-2 or X-3, so if X was bad certainly the new one is not. +% +% Once "a" has gotten its final value, we apply "\the\dimexpr a sp +% = D pt" trick to recover the D such that "D mm" gives rise to the found +% dimension. We go via this "Dmm" intermediary also to express the final +% result as "X sp", because anyhow the "X" we worked with and had in +% our token stream has to be recomputed if a<--a-1, so lets always +% recompute it from final "a", and this goes via "D mm" (but see +% the paragraph MEMO for alternative for this trunc(a 7227/2540) step). +% +% I will copy here the style I used for bothincm expansion triggering +% via an already positioned \dimexpr waiting to output final result. +\def\texdimenbothbpmm#1{\expandafter\texdimenstrippt\the\dimexpr + \expandafter\texdimenbothbpmm_fork\the\numexpr\dimexpr#1;}% +% the \texdimenzerominusfork is defined in the \texdimenwithunit section +\def\texdimenbothbpmm_fork#1{\texdimenzerominusfork + #1-\texdimenbothbpmm_zero + 0#1\texdimenbothbpmm_neg + 0-\texdimenbothbpmm_a + \krof#1}% +% because this is *inside* a pre-positioned \dimexpr, we don't have +% to worry about zero output ending up as -0.0 +\def\texdimenbothbpmm_neg-{-\texdimenbothbpmm_a}% +\def\texdimenbothbpmm_zero#1;{\z@\relax}% +% now, find X sp <= T sp maximal and expressible in mm unit +% it will be X=trunc(a 7227/2540), we first get a candidate for "a" +\def\texdimenbothbpmm_a#1;% + {\expandafter\texdimenbothbpmm_b\the\numexpr#1*2540/7227;#1;}% +% we get in a single line the X from this candidate, hijacking TeX's +% built-in *7227/2540... the "MEMO" above explains one could do this +% purely within \numexpr, working around its division rounds, and +% avoiding overflow, but I suspect this would be more costly. +\def\texdimenbothbpmm_b#1;{\expandafter\texdimenbothbpmm_c + \the\numexpr\dimexpr\expandafter\texdimenstrippt\the\dimexpr#1spmm;#1;}% +% now we have X;a;T; +\def\texdimenbothbpmm_c#1;#2;#3;{% +% If X>T, our candidate "a=#2" must be decreased by 1 and we go to _ca +% The original #3 is not needed anymore + \ifnum#1>#3 \expandafter\texdimenbothbpmm_ca\fi +% Else we decide whether it is "a" or "a-1" we must use. I preferred +% to induce a re-grabbing cost here, rather than have \texdimenbothbpmm_ca +% re-grab its arguments from \texdimenbothbpmm_d replacement text. + \texdimenbothbpmm_d#1;#2;% +}% +% Here, dynamically at the time of the concluding \dimexpr, we +% check if X sp is expressible in bp unit and then use "a" or "a-1" +% accordingly +\def\texdimenbothbpmm_d#1;#2;{#2sp% + \ifnum\dimexpr + \expandafter\texdimenstrippt\the\dimexpr\numexpr(2*#1+1)*400/803spbp=#1 + \else-1sp\fi +% and a \relax to stop the concluding \dimexpr + \relax +}% +% Here we must decrease "a=#2" by 1, recompute X=#1, then loop +% back to \texdimenbothbpmm_d. Hesitation between forcing a +% re-grab or doing it in one step with the subtraction of 1 done twice +\def\texdimenbothbpmm_ca\texdimenbothbpmm_d#1;#2;% + {\expandafter\texdimenbothbpmm_cb\the\numexpr#2-1;}% +\def\texdimenbothbpmm_cb#1;{% + \expandafter\texdimenbothbpmm_d + \the\numexpr\dimexpr\expandafter\texdimenstrippt\the\dimexpr#1spmm;#1;% +}% +% done... +% now the lazy way for \texdimenbothmmbp +\def\texdimenbothmmbp#1{\expandafter\texdimenstrippt\the\dimexpr + \expandafter\texdimenbothmmbp_a\the\numexpr\dimexpr\texdimenbothbpmm{#1}mm;}% +% or remove the + and do \if-#1-\else+\fi1 ? +% If zero at this stage, we will correctly get 0.0 in the end +\def\texdimenbothmmbp_a#1#2;{\numexpr(2*#1#2+\if-#1-\fi1)*400/803sp\relax}% +% \texdimenbothbpmmpt and its alias \texdimenbothmmbppt +\def\texdimenbothbpmmpt#1{\texdimenpt{\texdimenbothbpmm{#1}mm}}% +\let\texdimenbothmmbppt\texdimenbothbpmmpt +% \texdimenbothbpmmsp and its alias \texdimenbothmmbpsp +\def\texdimenbothbpmmsp#1{\the\numexpr\dimexpr\texdimenbothbpmm{#1}mm\relax\relax}% +\let\texdimenbothmmbpsp\texdimenbothbpmmsp +% +% Mathematics (\texdimenwithunit, added at release 0.99, modified at 1.0) % =========== % % As explained in the README.md, the ex and em units are @@ -404,96 +732,177 @@ % If f/65536<1, all dimensions Tsp are attainable as D uu. Indeed % D uu is parsed by TeX via N=round(D*65536), then T=trunc(N*phi), % with phi=f/65536. Starting from T we need to find an N such that -% T/phi <= N< (T+1)/phi. We first consider v=(T+0.5)/phi. As its -% distance to the extremities is 0.5/phi>0.5,(*) its rounding M +% T/phi <= N< (T+1)/phi. +% +% This is equivalent to ceil(T/phi)<= N < ceil((T+1)/phi) +% +% Now obsolete remark: let v=(T+0.5)/phi. As its +% distance to the extremities is 0.5/phi>0.5, (phi>1) its rounding M % to an integer verifies automatically T/phi < M < (T+1)/phi, so -% is a candidate. The TeX core conversion of Msp to a Dpt with -% D a decimal of at most 5 fractional digits will provide a D -% such that indeed M=round(D*65536). +% is a valid candidate. This was used at 0.99 release. +% (it is funny that N=round((2T+1)*32768/f) works for all f>0 +% *except* f=65536). % -% (*) attention that this fails if phi=1, we definitely can not -% use the N=round((2T+1)*32768/f) formula for f=65536, it gives -% N=T+1 in place of N=T... It is funny the formula works for all -% f apart from f=65536... +% The 1.0 release chooses to implement the ceil(T/phi) formula rather as +% it is closer to naive expectation "dim1/dim2" of a division. % -% Attention that if T=0, the M=round(0.5/phi) will be at least 1, -% so the produced decimal D will not be 0.0, although it will -% be true that D \dimexpr f sp\relax produces the zero dimension, -% the above analysis being correct even for T=0. +% It is not obvious to compute this ceil(T/phi) without overflow. % % Implementation % ============== % -% \texdimenwithunit{dim1}{dim2}. dim2>0 assumed. +% \texdimenwithunit{dim1}{dim2} +% +% First done at 0.99, then refactored at 1.0: +% - to add support for dim2<0pt +% - to handle differently the dim2<1pt case and make the output +% closer to mathematical dim1/dim2 +% +% To handle dim2<0pt, we simply simultaneously do +% dim1<-- (-dim1) and dim2<-- (-dim2). +% +% dim2=0pt is not intercepted and will cause division by zero low-level +% error. Code comments below were not adjusted and handle only +% dim2>0pt. +% % We first get f from dim2 and branch according to whether f>65536, -% f=65536 or f<65536. +% or f<=65536. % We will also need to check the sign of T (dim1=T sp). -% We then compute in both branches round((2T+1)*32768/f), but -% in an indirect way in the f<65536 branch to avoid overflow. -% We will need also to intercept T=0 to avoid producing a non -% pleasing positive D in that case (still correct for f<65536; -% this is issue #3). -% -% In the f<65536 branch we first do the Euclidean division -% 2T+1 = k*2*f + R with 0<=R<2f. The k is obtained as round((2T+1-f)/(2f)) -% which can be computed in a numexpr (and never gives -1 even for T=0) -% Then (2T+1)*32768/f=65536*k + R*32768/f -% Then R*32768/f<=65536-32768/f<65536-32768/65536=65536-0.5 -% Hence the numexpr evaluation gives an integer B<65536. -% -% N.B.: if f>=65536, we still have R*32768/f<65536 as R<2f -% so the only difference is that the B could be here 65536 -% -% From \the\dimexpr Bsp, we get a decimal E < 1, so E=0.ddddd -% (or less digits) and the looked for D will be the contatenation -% k.ddddd with k as obtained earlier. This procedure has no possible -% arithmetic overflow. -% -% #2 is assumed positive. The case #2=1pt=65536sp is special, and -% must be filtered out (issues #4, #6). -% pre-multiplication of #1 by 2 here -\def\texdimenwithunit#1#2{\expandafter\texdimenwithunit_ - \the\numexpr\dimexpr#2\expandafter;\the\numexpr2*\dimexpr#1;}% +% f>65536: we compute round((2T+1)*32768/f) +% f=65536: merged with f<65536 branch +% f<65536: we compute ceil(T*65536/f) +% +% rationale: round((2T+1)*32768/f) which was used at 0.99 +% would be ok [if f<65536 not f=65536 ! cf #3, #4] +% also for this branch +% +% BUT it diverges noticeably from naive expectation +% dim1/dim2 "=" T*65536/f the more so when f is small. +% See issue #16 and also the discussion at #13. +% +% As was explained in comments to issue #2 which asked for a +% \texdimenwithunit the ceil(T*65536/f) is the smallest +% allowable choice +% +% To avoid arithmetic overflow issues we first do the euclidean +% division T = k f + r, 0<= r < f +% +% The final result in sp units will be k*65536 + ceil(r * 65536/f) +% We don't do this k*65536 explicitly as it may overflow +% but output the decimal k + E where E is the conversion +% to a decimal 0.ddddd or 1.0 of "ceil(r * 65536/f) sp" +% (which is at most 65536sp=1pt so E is at most 1.0). +% +% To compute the Euclidean quotient k in \numexpr we use there +% (2T-f)/(2f) i.e. round((2T-f)/2f) = trunc(T/f) +% as we are careful to never have T=0 inthere... +% +% Computing C = ceil(r * 65536/f) in \numexpr is the delicate +% part, as r can be as large as f-1 hence 65535 and r*65536 would +% overflow. We could compute R=round(r*65536/f) ("scaling operation") +% then C=R+1 if R*f-65536*r<0, else C=R. +% +% The problem is then: how to get the sign of R*f-65536*r without +% overflow? I considered various ways. +% +% But in the end, opted for simply this: +% - get R=round(r*65536/f) as \the\numexpr r*65536/f ("scaling" no overflow) +% - hence get E pt=\the\dimexpr R sp +% - let TeX compute E<f sp>: if it turns out < r sp, +% then C=R+1, +% else C=R. Done. +% +\def\texdimenwithunit#1#2{\expandafter\texdimenwithunit_i +% no premultiplication of dim1 by 2 as was done for technical +% reasons when dim2<1pt branch used round((2T+1)*32768/f) + \the\numexpr\dimexpr#2\expandafter;\the\numexpr\dimexpr#1;}% +% +% so tempted to do \input xintkernel.sty to have some utilities... +% not even a \@gobble in Plain... \let\texdimenorthat\texdimenfirstofone \def\texdimendothis#1#2\texdimenorthat#3{\fi#1}% -\def\texdimenwithunit_#1;#2{% - \ifnum#1=\p@\texdimendothis\texdimenwithunit_p@\fi +\def\texdimengobtominus#1-{}% +\def\texdimenwithunit_i#1{% + \texdimengobtominus#1\texdimenwithunit_switchsigns-% + \texdimenwithunit_j#1% +}% +\def\texdimenwithunit_switchsigns-\texdimenwithunit_j-#1;#2% +{% +% due to \texdimenwithunit_Bneg we can not simply prefix dim1 +% with -, as -0 is bad there. So let's check also if #2 is 0 + \texdimenzerominusfork + #2-\texdimenwithunit_Bzero % abusive double usage + 0#2\texdimenwithunit_j % abusive shortcut + 0-{\texdimenwithunit_ic#2}% + \krof + #1;% +}% +\def\texdimenwithunit_ic#1#2;{\texdimenwithunit_j#2;-#1}% +\def\texdimenwithunit_j#1;#2{% + % \ifnum#1=\p@\texdimendothis\texdimenwithunit_p@\fi \ifnum#1>\p@\texdimendothis\texdimenwithunit_A\fi \texdimenorthat\texdimenwithunit_B#2#1;% }% -% we premultiplied the first argument by 2... must be undone now! -% and we must avoid overflow. Very underoptimal, but user is not -% supposed to do something as silly as actually using this unit=1pt case! -\def\texdimenwithunit_p@#1#2;#3;{\expandafter\texdimenstrippt\the\dimexpr\numexpr#1#3/2sp\relax}% -% unit>1pt, handle this as for bp. Attention it would be wrong for unit=1pt! -\def\texdimenwithunit_A#1#2;#3;{\expandafter\texdimenstrippt - \the\dimexpr\numexpr(#1#3+\if-#1-\fi1)*32768/#2sp\relax +% not needed, as the "ceil" branch is fine to use for f = 65536 +% \def\texdimenwithunit_p@#1#2;#3;{% +% \expandafter\texdimenstrippt\the\dimexpr#1#3sp/2\relax +% }% +% unit>1pt, handle this as for bp. Attention it would be wrong for +% unit=1pt! +\def\texdimenwithunit_A#1#2;#3;{% + \expandafter\texdimenstrippt + \the\dimexpr\numexpr(2*#1#3+\if-#1-\fi1)*32768/#2sp\relax + % - fine if dim1>0, <0, or =0 + % - with *\p@ better but an early doubled dim2 would complicate 1pt + % test and not sure if doing \p@/(2*#2) here advantageous }% -% unit<1pt. Attention it would be wrong for unit=1pt! +% unit<=1pt. % if dim1<0, simply negate result for dim1>0 as it can not possibly be 0.0 -% Indeed (2T+1)*32768/f will be at least 3*32768/f so its rounding at least 2. -% Add a check for dim1=0 to fix issue #3 +% Indeed T*65536/f will be at least 1 so its ceil also (in fact ceil +% will even be at least 2 if f<65536). +% The dim1=0 case must get filtered out due to way of calculating the +% "ceil" in \numexpr \def\texdimenzerominusfork #10-#2#3\krof {#2}% \def\texdimenwithunit_B#1{\texdimenzerominusfork #1-\texdimenwithunit_Bzero 0#1\texdimenwithunit_Bneg 0-\texdimenwithunit_Ba \krof#1}% -\def\texdimenwithunit_Ba#1#2;#3;{\expandafter\texdimenwithunit_Bb\the\numexpr#1#3+1;#2;}% -\def\texdimenwithunit_Bb#1;#2;{\expandafter\texdimenwithunit_Bc\the\numexpr(#1-#2)/(2*#2);#1;#2;}% -% I was hesitating between leaving k in the stream (breaking f-expandability) -% and then remove the "0" and trailing "pt" from 0.ddddd pt, but opted -% finally for doing \the\numexpr..+0.ddddd which is f-expandable and allows -% recycling strippt here with no need of another utility. -% -% This means (see the nota bene above) that we could apply this procedure -% also for f>65536, because at worst we will get a \the\numexprk+1.0, which -% gives the correct result. I tested and found about 39% longer execution time -% if dim2>1pt does same calculations as for dim2<1pt, and at the same time -% dim2<1pt was improved about 11% from skipping the conditional -\def\texdimenwithunit_Bc#1;#2;#3;{\the\numexpr#1+\expandafter\texdimenstrippt - \the\dimexpr\numexpr(#2-#1*2*#3)*32768/#3sp\relax}% -% Here, definitely not caring about f-expandability. Or efficiency. +\def\texdimenwithunit_Ba#1#2;#3;{% + % no overflow possible from 2*#1#3=2*dim1 + \expandafter\texdimenwithunit_Bb\the\numexpr(2*#1#3-#2)/(2*#2);#1#3;#2;% +}% +% now k;T;f;. Get the remainder r=T-k*f +\def\texdimenwithunit_Bb#1;#2;#3;{% + \expandafter\texdimenwithunit_Bc\the\numexpr#2-#1*#3;#1;#3;% +}% +% now r;k;f;. We can start \the\numexpr k+ .... +% and there we will need to get R=round(r*65536/f), Ept=Rsp, +% check if E"f sp"<"r sp", if yes replace R by R+1 else keep E etc. +\def\texdimenwithunit_Bc#1;#2;#3;{% +% \the\numexpr k+0.ddddd is handy because it can well be actually +% \the\numexpr k+1.0, now that we use ceil approach in this branch + \the\numexpr#2+\expandafter\texdimenwithunit_Bd + \the\numexpr #1*\p@/#3;#1;#3;% +}% +% R;r;f; +\def\texdimenwithunit_Bd#1;{% + \expandafter\texdimenwithunit_Be\the\dimexpr#1sp;#1;% +}% +% Ept;R;r;f; +% #1=0.ddd... or 1.0 but has no end marker hence the +% \texdimenfirstofone{#1} as in \texdimendown_d and \texdimenup_d +{\catcode`P 12\catcode`T 12 +\lowercase{\gdef\texdimenwithunit_Be#1PT};#2;#3;#4;{% + \ifdim#1\dimexpr#4sp<#3sp \texdimenwithunit_Bf{#2}\fi + \texdimenfirstofone{#1}% + }% +}% +% #2 is \fi. Add a dimen storage \onesp for 1sp? +\def\texdimenwithunit_Bf#1#2\texdimenfirstofone#3{#2% + \expandafter\texdimenstrippt\the\dimexpr#1sp+1sp\relax +}% +% Here definitely not caring about f-expandability. Or efficiency. \def\texdimenwithunit_Bneg-{-\texdimenwithunit_Ba{}}% \def\texdimenwithunit_Bzero#1;#2;{0.0}% \texdimensendinput |