% This is file texdimens.tex, part of texdimens package, which % is distributed under the LPPL 1.3c. Copyright (c) 2021 Jean-François Burnol % 2021/07/15 v0.9delta \edef\texdimensendinput{\endlinechar\the\endlinechar\catcode`\noexpand _=\the\catcode`\_\relax\noexpand\endinput}% \endlinechar13\relax% \catcode`\_=11 % % Mathematics % =========== % % Is T sp attainable from unit "uu"?. Here we suppose T>0. % phi>1, psi=1/phi, psi<1 % 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: % 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) % % Let X = round(T psi). And let Y = trunc(X phi). % % 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 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. % % 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. % % 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. % % This way we find the maximal dimension at most T sp exactly % representable in "uu" unit. % % 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. % % 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. % % If psi > 0.5 % ------------ % % 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. % % 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. % % 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, % % there was some hesitation about whether or not using the simpler % round(T psi) approach for units > 2pt and the \texdimin 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. % % Implementation % ============== % \def\texdimenfirstofone#1{#1}% {\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\texdiminpt#1{\expandafter\texdimenstrippt\the\dimexpr#1\relax}% % % bp 7227/7200 = 803/800 % \def\texdiminbp#1{\expandafter\texdiminbp_\the\numexpr\dimexpr#1;}% \def\texdiminbp_#1#2;{% \expandafter\texdimenstrippt\the\dimexpr\numexpr(2*#1#2+\if-#1-\fi1)*400/803sp\relax }% % \texdiminbpdown: maximal dim exactly expressible in bp and at most equal to input \def\texdiminbpdown#1{\expandafter\texdimendown_A\the\numexpr\dimexpr#1;*400/803;bp;}% % \texdiminbpup: minimal dim exactly expressible in bp and at least equal to input \def\texdiminbpup#1{\expandafter\texdimenup_A\the\numexpr\dimexpr#1;*400/803;bp;}% % % nd 685/642 % \def\texdiminnd#1{\expandafter\texdiminnd_\the\numexpr\dimexpr#1;}% \def\texdiminnd_#1#2;{% \expandafter\texdimenstrippt\the\dimexpr\numexpr(2*#1#2+\if-#1-\fi1)*321/685sp\relax }% % \texdiminnddown: maximal dim exactly expressible in nd and at most equal to input \def\texdiminnddown#1{\expandafter\texdimendown_A\the\numexpr\dimexpr#1;*321/685;nd;}% % \texdiminndup: minimal dim exactly expressible in nd and at least equal to input \def\texdiminndup#1{\expandafter\texdimenup_A\the\numexpr\dimexpr#1;*321/685;nd;}% % % dd 1238/1157 % \def\texdimindd#1{\expandafter\texdimindd_\the\numexpr\dimexpr#1;}% \def\texdimindd_#1#2;{% \expandafter\texdimenstrippt\the\dimexpr\numexpr(2*#1#2+\if-#1-\fi1)*1157/2476sp\relax }% % \texdimindddown: maximal dim exactly expressible in dd and at most equal to input \def\texdimindddown#1{\expandafter\texdimendown_A\the\numexpr\dimexpr#1;*1157/2476;dd;}% % \texdiminddup: minimal dim exactly expressible in dd and at least equal to input \def\texdiminddup#1{\expandafter\texdimenup_A\the\numexpr\dimexpr#1;*1157/2476;dd;}% % % mm 7227/2540 phi now >2, use from here on the simpler approach % \def\texdiminmm#1{\expandafter\texdimenstrippt\the\dimexpr(#1)*2540/7227\relax}% % \texdiminmmdown: maximal dim exactly expressible in mm and at most equal to input \def\texdiminmmdown#1{\expandafter\texdimendown_a\the\numexpr\dimexpr#1;*2540/7227;mm;}% % \texdiminmmup: minimal dim exactly expressible in mm and at least equal to input \def\texdiminmmup#1{\expandafter\texdimenup_a\the\numexpr\dimexpr#1;*2540/7227;mm;}% % % pc 12/1 % \def\texdiminpc#1{\expandafter\texdimenstrippt\the\dimexpr(#1)/12\relax}% % \texdiminpcdown: maximal dim exactly expressible in pc and at most equal to input \def\texdiminpcdown#1{\expandafter\texdimendown_a\the\numexpr\dimexpr#1;/12;pc;}% % \texdiminpcup: minimal dim exactly expressible in pc and at least equal to input \def\texdiminpcup#1{\expandafter\texdimenup_a\the\numexpr\dimexpr#1;/12;pc;}% % % nc 1370/107 % \def\texdiminnc#1{\expandafter\texdimenstrippt\the\dimexpr(#1)*107/1370\relax}% % \texdiminncdown: maximal dim exactly expressible in nc and at most equal to input \def\texdiminncdown#1{\expandafter\texdimendown_a\the\numexpr\dimexpr#1;*107/1370;nc;}% % \texdiminncup: minimal dim exactly expressible in nc and at least equal to input \def\texdiminncup#1{\expandafter\texdimenup_a\the\numexpr\dimexpr#1;*107/1370;nc;}% % % cc 14856/1157 % \def\texdimincc#1{\expandafter\texdimenstrippt\the\dimexpr(#1)*1157/14856\relax}% % \texdiminccdown: maximal dim exactly expressible in cc and at most equal to input \def\texdiminccdown#1{\expandafter\texdimendown_a\the\numexpr\dimexpr#1;*1157/14856;cc;}% % \texdiminccup: minimal dim exactly expressible in cc and at least equal to input \def\texdiminccup#1{\expandafter\texdimenup_a\the\numexpr\dimexpr#1;*1157/14856;cc;}% % % cm 7227/254 % \def\texdimincm#1{\expandafter\texdimenstrippt\the\dimexpr(#1)*254/7227\relax}% % \texdimincmdown: maximal dim exactly expressible in cm and at most equal to input \def\texdimincmdown#1{\expandafter\texdimendown_a\the\numexpr\dimexpr#1;*254/7227;cm;}% % \texdimincmup: minimal dim exactly expressible in cm and at least equal to input \def\texdimincmup#1{\expandafter\texdimenup_a\the\numexpr\dimexpr#1;*254/7227;cm;}% % % in 7227/100 % \def\texdiminin#1{\expandafter\texdimenstrippt\the\dimexpr(#1)*100/7227\relax}% % \texdiminindown: maximal dim exactly expressible in in and at most equal to input \def\texdiminindown#1{\expandafter\texdimendown_a\the\numexpr\dimexpr#1;*100/7227;in;}% % \texdimininup: minimal dim exactly expressible in in and at least equal to input \def\texdimininup#1{\expandafter\texdimenup_a\the\numexpr\dimexpr#1;*100/7227;in;}% % \texdimensendinput