diff options
Diffstat (limited to 'Master/texmf-dist')
| -rw-r--r-- | Master/texmf-dist/doc/generic/minifp/README | 36 | ||||
| -rw-r--r-- | Master/texmf-dist/doc/generic/minifp/minifp.pdf | bin | 419173 -> 421610 bytes | |||
| -rw-r--r-- | Master/texmf-dist/doc/generic/minifp/test1.tex | 11 | ||||
| -rw-r--r-- | Master/texmf-dist/doc/generic/minifp/test2.tex | 59 | ||||
| -rw-r--r-- | Master/texmf-dist/source/generic/minifp/minifp.dtx | 1388 | ||||
| -rw-r--r-- | Master/texmf-dist/tex/generic/minifp/mfpextra.tex | 416 | ||||
| -rw-r--r-- | Master/texmf-dist/tex/generic/minifp/minifp.sty | 268 |
7 files changed, 1114 insertions, 1064 deletions
diff --git a/Master/texmf-dist/doc/generic/minifp/README b/Master/texmf-dist/doc/generic/minifp/README index eec67f09a9e..065bac29198 100644 --- a/Master/texmf-dist/doc/generic/minifp/README +++ b/Master/texmf-dist/doc/generic/minifp/README @@ -11,10 +11,11 @@ Purpose: get additional operations (sine, cosine, angle, square root, log and exp), the file mfpextra.tex is provided. - Minifp should work in both latex and plaintex. + Minifp should work in both LaTeX and plainTeX. - This is version 0.92. It should work reasonably well, barring any bugs, - but I expect to spend some time fine-tuning it for version 1.0. + This is version 0.95. It should work reasonably well, barring any + bugs, but I expect to spend some time fine-tuning it before calling it + version 1.0. License: @@ -82,18 +83,23 @@ Manifest: <http://comp.uark.edu/~luecking/tex/tex.html>. History: - - Version 0.92 : Bugfixes: correct sign of floor and ceiling. Correct - occasional minus sign in front of 0 for sin or cos. - Version 0.9 : angle: near maximum accuracy, at some cost to speed. - Version 0.8 : exp: now more accurate for many cases. - Version 0.7 : sqrt: now exact when possible and much more accurate. - Version 0.6 : Added angle to mfpextra. Changed package name to minifp. - Version 0.5 : Added sqrt, deg, rad to mfpextra. - Version 0.4 : Added log, exp, pow to mfpextra. - Version 0.3 : Added mfpextra.tex, defines sin and cos. - Version 0.2 : Added macros for printing, formatting the results. - Version 0.1 : First working set of macros. Package named mfp.sty. + Version 0.95: More testing. Changed square root of negative from an + error to a warning. Documentation updated. Now mfpextra + checks for minifp.sty and inputs it if needed. + Version 0.94: Improved accuracy of log, sin and sqrt. + Version 0.93: Corrected mistyped data for logarithm. + Fixed bugs in degree/radian conversions. + Version 0.92: Corrected sign of floor and ceiling. Corrected + occasional minus sign in front of 0.0 for sin or cos. + Version 0.9 : angle: near maximum accuracy, at some cost to speed. + Version 0.8 : exp: now more accurate for many cases. + Version 0.7 : sqrt: now exact when possible and much more accurate. + Version 0.6 : Added angle to mfpextra. Changed package name to minifp. + Version 0.5 : Added sqrt, deg, rad to mfpextra. + Version 0.4 : Added log, exp, pow to mfpextra. + Version 0.3 : Added mfpextra.tex, defines sin and cos. + Version 0.2 : Added macros for printing, formatting the results. + Version 0.1 : First working set of macros. Package named mfp.sty. -- Dan Luecking <luecking (at) uark (dot) edu> diff --git a/Master/texmf-dist/doc/generic/minifp/minifp.pdf b/Master/texmf-dist/doc/generic/minifp/minifp.pdf Binary files differindex baca0c62f24..d08a688cf76 100644 --- a/Master/texmf-dist/doc/generic/minifp/minifp.pdf +++ b/Master/texmf-dist/doc/generic/minifp/minifp.pdf diff --git a/Master/texmf-dist/doc/generic/minifp/test1.tex b/Master/texmf-dist/doc/generic/minifp/test1.tex index c08127eb8b9..2956afa3fe6 100644 --- a/Master/texmf-dist/doc/generic/minifp/test1.tex +++ b/Master/texmf-dist/doc/generic/minifp/test1.tex @@ -1,7 +1,10 @@ \errorcontextlines999\relax %\def\MFPextra{} -X\input minifp.sty\relax X -X\MFPloadextra X +%X\input minifp.sty\relax X +%X\MFPloadextra X +X\input mfpextra\relax X +X\input mfpextra\relax X + \def\filbreak{\vskip 12pt plus 100pt\penalty 0 \vskip 0pt plus -100pt\relax} \def\meaningless#1>{} @@ -82,9 +85,7 @@ Square:\Rsq\y \Rpop\X\Rpush{21.34}% Inversion:\Rinv\y \Rpop\X\Rpush{21.34}% -\tracingmacros1 Floor:\Rfloor\y -\tracingmacros0 \Rpop\X\Rpush{21.34}% Ceiling:\Rceil\y \Rpop\X\Rpush{21.34}% @@ -381,7 +382,7 @@ large:^^J} \filbreak {\bf Extra tests of sqrt}\\ -\immediate\write16{^^J*** The following tests the error for a square root of a +\immediate\write16{^^J*** The following tests the warning for a square root of a negative:^^J} Square root of $-1$:\MFPsqrt{-1}\Z\w Square root of $0$:\MFPsqrt{0}\Z\w diff --git a/Master/texmf-dist/doc/generic/minifp/test2.tex b/Master/texmf-dist/doc/generic/minifp/test2.tex index 8cd7ab278c5..6d6398a8976 100644 --- a/Master/texmf-dist/doc/generic/minifp/test2.tex +++ b/Master/texmf-dist/doc/generic/minifp/test2.tex @@ -1,5 +1,9 @@ \errorcontextlines999\relax -X\input minifp.sty\relax X\MFPloadextra X +The various ``XX'' and ``xX'' pairs test whether there are errant spaces +in the macros or the macro files. + +X\input mfpextra\relax X +X\input mfpextra\relax X \def\empty{} \def\frac#1#2{{#1\over#2}} @@ -83,44 +87,45 @@ probably each equivalent to a dozen or more multiplications. Counting each such operation with the an estimated multiplicity, the tests probably perform 400 thousand or more basic operations. -Of the basic operations, multiplication is (by measurement) 4 times as -lengthy as addition, and division is about twice as lengthy as -multiplication. Actual times depend on the machine, but the ratios remain -pretty much the same. Here is a summary of timings on my fastest -machine; each operation is run 500 times in a loop: +Of the basic operations, multiplication is (by measurement) 2--4 times +as lengthy as addition, and division is 2--3 times as lengthy as +multiplication. Actual times depend on the machine, but the ratios +remain pretty much the same. Here is a summary of timings on my fastest +machine; each operation is run 500 times in a loop. (A loop in which +an input number is processed and a value returned, but no calculations are +performed ({\tt\string\MFPzero}), times at $0.0\,$sec.) Timing obtained +with {\tt\string\pdfelapsedtime}. \medskip \indent\vtop{\halign{\hfil$#$&\quad$#\,$sec\cr \noalign{\hrule\smallskip} -2.54321+22432.87654321 &0.015\cr -2.54321\times22432.87654321 &0.06\cr -22432.87654321/2.54321 &0.14\cr -\sqrt{23456789.54321} &0.20\cr -1.00001234^{8000} &0.73\cr -\exp(2.54321) &0.41\cr -\sin(2.54321) &0.45\cr -\log(2.54321) &0.53\cr -\mathop{\fam0 angle}(254.321,100) &1.17\cr +2.54321+22432.87654321 &0.016\cr +2.54321\times22432.87654321 &0.046\cr +22432.87654321/2.54321 &0.11\cr +\sqrt{23456789.54321} &0.172\cr +1.00001234^{8000} &0.72\cr +\exp(2.54321) &0.42\cr +\sin(2.54321) &0.41\cr +\log(2.54321) &0.73\cr +\mathop{\fam0 angle}(254.321,100) &1.14\cr \noalign{\smallskip\hrule} }} \medskip -Originally, all the tests combined took 21 seconds on a 4-year-old +Originally, all the tests below combined took 21 seconds on a 4-year-old Windows XP under TeX Live 2011. But since then I have changed angle and power computations so that they are considerably more accurate, but with -a possible reduction in speed. Of course, some of the speed loss may be -due to the operating system. - -On a Windows 7 machine, 64-bit, laptop, it takes 32 seconds to process -this file. On another Win7, machine, 32-bit, desktop, it takes 10 -seconds. - -The difference could be explained partially by the fact that the second -machine is newer and partially by the fact that TeX is a 32-bit program, -and therefore a better match to the operating system of the second -system. +a possible reduction in speed. I cannot test the speed reduction, since +I no longer have that machine. + +For my current machines: On a Windows 7 machine, 64-bit, laptop, it +takes 32 seconds to process this file. On another Windows 7 machine, +32-bit, desktop, it takes about 10 seconds. (This difference could be +explained partly by the fact that the last machine is newer and partly +by the fact that TeX is a 32-bit program and therefore a better match to +the 32-bit hardware.) \def\testi{% stack forms \startMFPprogram diff --git a/Master/texmf-dist/source/generic/minifp/minifp.dtx b/Master/texmf-dist/source/generic/minifp/minifp.dtx index 77f56099f70..c3621aef2ba 100644 --- a/Master/texmf-dist/source/generic/minifp/minifp.dtx +++ b/Master/texmf-dist/source/generic/minifp/minifp.dtx @@ -13,8 +13,8 @@ % minifp has maintenance status "author-maintained". The Current Maintainer % is Daniel H. Luecking. The Base Interpreter is TeX (plain TeX or LaTeX). %<*driver|sty> -\def\MFPfiledate{2013/02/01}% -\def\MFPfileversion{0.92}% +\def\MFPfiledate{2013/05/28}% +\def\MFPfileversion{0.95}% %</driver|sty> % %<*driver> @@ -49,8 +49,11 @@ \let\env\file \def\sgn{\mathop{\mathrm{sgn}}\nolimits} % \op is for abstract operations (e.g., \op{add}) as opposed to -% the macro that performs it (e.g., \cs{Radd}). +% the macro that performs it (e.g., \cs{Radd}). And \reg is for +% a "register" (e.g., the 3 macros \MFP@x@Sgn, \MFP@x@Int and \MFP@x@Frc) +% conceived of as a single entity. \let\op\textit +\def\reg#1{$#1$} % The occasional bare \tt braces \renewcommand\{{\char`\{} \renewcommand\}{\char`\}} @@ -60,7 +63,7 @@ \makeatletter \newcommand\bsl{{\mytt\@backslashchar}} -% Stupid lists! +% better lists \def\@listi{\leftmargin\leftmargini \parsep \z@ \@plus\p@ \@minus\z@ \topsep 4\p@ \@plus\p@ \@minus2\p@ @@ -70,11 +73,10 @@ \renewcommand\labelitemii{\textasteriskcentered} \renewcommand\labelitemiii{\textperiodcentered} \leftmargini\parindent -% Stupid index! +% better index \def\usage#1{\textrm{#1}} \def\index@prologue{\section*{Index}\markboth{Index}{Index}% - Numbers refer to the page(s) where the corresponding entry is described. -} + Numbers refer to the page(s) where the corresponding entry is described.} \def\IndexParms{% \parindent \z@ \columnsep 15pt \parskip 0pt plus 1pt @@ -102,7 +104,7 @@ \end{document} %</driver> %\fi -% \CheckSum{3339} +% \CheckSum{3402} % \CharacterTable % {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z % Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z @@ -138,13 +140,13 @@ % to make simple calculations with real numbers. What \TeX{} provides is % far too limited. In fact, its only native user-level support for real % numbers is as factors for dimensions. For example one can ``multiply'' -% $3.1\times 0.2$ by \verb$\dimen0=0.2pt \dimen0=3.1\dimen0 $. +% $3.1\times 0.2$ by the following: \verb$\dimen0=0.2pt \dimen0=3.1\dimen0$. % % Unfortunately \TeX{} stores dimensions as integer multiples of the -% ``scaled point'' (\dim{sp}) with \dim{sp}${}=2^{-16}$\dim{pt}, and +% ``scaled points'' (\dim{sp}) with \dim{sp}${}=2^{-16}$\dim{pt}, and % therefore \dim{.2pt} is approximated by $\frac{13107}{65536}$, which is % not exact. Then mutiplying by $3.1$ produces $\frac{40631}{65536}$. If -% we ask for five digit accuracy, this produces $0.61998$\dim{pt} and not the +% we ask \TeX{} to display this, it produces $0.61998$\dim{pt} and not the % exact value $0.62$. This is sufficiently accurate for positioning % elements on a page, but not for displaying automatically computed axis % labels if five digit accuracy is needed. @@ -205,30 +207,35 @@ % exceed the allowed eight digits) is always possible, but is much more % likely with multiplication and division. % -% Multiplication is carried out internally to an exact 16-digit answer, -% which is then rounded to an 8-digit result. Overflow (more than 8 -% digits in the integer part) is discarded. Division is internally -% carried to nine digits after the decimal, which is then also rounded to -% an 8-digit result. +% Multiplication is carried out internally to an exact answer, with 16 +% digits on each side of the decimal point. The underflow digits (places 9 +% through 16 after the decimal point) are used to round to an 8-digit +% result. Overflow digits (those to the left of the lowest 8 in the +% integer part) are discarded, usually without warning. Division is +% internally carried to nine digits after the decimal, which is then also +% rounded to an 8-digit result. Overflow digits are ignored for division +% also. % % We supply two kinds of operations in this package. There are stack-based -% operations, in which the operands are popped from a stack and the -% results pushed onto it, and argument-based, in which the operands (and a +% operations, in which the operands are \op{popped} from a stack and the +% results \op{pushed} onto it, and argument-based, in which the operands (and a % macro to hold the result^^A -% \footnote{Unlike most other packages for floating point +% \footnote{Unlike most other packages for decimal % arithmetic, \mfp{} puts the macro to hold the result % last. This allows the calculation to be performed before the -% macro is even read, and makes it somewhat easier for the +% macro is even read, and this makes it somewhat easier for the % stack- and argument-based versions to share code.}^^A % ) are arguments of a macro. Both types load the arguments into internal -% macros (think ``registers''), then call internal commands +% macros (think of them as ``registers''), then call internal commands % (think ``microcode'') which return the results in internal macros. -% These results are then pushed onto the stack (stack-based operations) or -% stored in a supplied macro argument (think ``variable''). +% These results are then \op{pushed} onto the stack (stack-based +% operations) or stored in a supplied macro argument (think ``variable''). +% The difference lies entirely in where the operands come from (arguments +% or stack) and where they go (macro or stack). % % The stack is implemented as an internal macro which is redefined with -% each command. The binary operations act on the last two pushed objects -% in the order they were pushed. For example, the sequence ``\op{push} 5, +% each command. The binary operations act on the last two \op{pushed} objects +% in the order they were \op{pushed}. For example, the sequence ``\op{push} 5, % \op{push} 3, \op{subtract}'' performs $5-3$ by popping $3$ and $5$ into % registers (thereby removing them from the stack), subtracting them % and then pushing the result ($2$) onto the stack. @@ -269,9 +276,14 @@ % \def\MFPextra{} \input minifp.sty \end{verbatim} % The extras can also be loaded by means of the command % \cs{MFPloadextra}, issued after \file{minifp.sty} is loaded. +% As of version 0.95 \file{mfpextra} can be directly \cs{input}. +% It will detect whether \file{minifp.sty} has been loaded and input it +% if not. This will work only in plain \TeX{}. % % If the extra operations are not needed, some memory and time might be -% saved by using \file{minifp.sty} alone. +% saved by using \file{minifp.sty} alone. I have not seriously tried to +% keep \file{mfpextra.tex} as small or fast as possible, but I do try +% to improve the accuracy when I can. % % As previously mentioned, each of these operations come in two versions: % a version that acts on operands and stores the result in a macro, and a @@ -283,7 +295,7 @@ % because it is possible that stacks of other types will be implemented in % the future. % -% For example, \verb$\MFPadd{1.3}{3.4}\X$ will add $1.20000000$ to +% For example, \verb$\MFPadd{1.2}{3.4}\X$ will add $1.20000000$ to % $3.40000000$ and then define \cs{X} to be the resulting % \texttt{4.60000000}. These operand forms do not alter or even address % the stack in any way. The stack-based version of the same operation @@ -316,10 +328,10 @@ % % In the following tables, an argument designated % \meta{num} can be any decimal real number with at most 8 -% digits on each side of the decimal point, or they can be macros that -% contain such a number. If the decimal dot is absent, the fractional part -% will be taken to be $0$, if the integer part or the fractional part is -% absent, it will be taken to be $0$. (One consequence of these rules is +% digits on each side of the decimal point, or it can be a macro that +% contains such a number. If the decimal dot is absent, the fractional part +% will be taken to be zero, if the integer part or the fractional part is +% absent, it will be taken to be zero. (One consequence of these rules is % that all the following arguments produce the same internal % representation of zero: \marg{0.0}, \marg{0.}, \marg{.0}, % \marg{0}, \marg{.}, and \marg{}\,.) Spaces may appear anywhere in the @@ -381,7 +393,7 @@ % Stores $|$\meta{num}$|$ in \cs{macro}.\\ % \SpecialUsageIndex{\MFPdbl}^^A % \cs{MFPdbl}\mmarg{num}\cs{macro}& -% Stores 2\meta{num} in \cs{macro}.\\ +% Stores $2\times{}$\meta{num} in \cs{macro}.\\ % \SpecialUsageIndex{\MFPhalve}^^A % \cs{MFPhalve}\mmarg{num}\cs{macro}& % Stores \meta{num}/2, rounded to 8 places after the decimal point, in @@ -389,12 +401,12 @@ % \SpecialUsageIndex{\MFPint}^^A % \cs{MFPint}\mmarg{num}\cs{macro}& % Replaces the part of \meta{num} after the decimal point with zeros -% (keeps the sign unless the result is $0$) and stores the result in +% (keeps the sign unless the result is zero) and stores the result in % \cs{macro}.\\ % \SpecialUsageIndex{\MFPfrac}^^A % \cs{MFPfrac}\mmarg{num}\cs{macro}& -% Replaces the part of \meta{num} before the decimal point with $0$ -% (keeps the sign unless the result is $0$) and stores the result in +% Replaces the part of \meta{num} before the decimal point with zero +% (keeps the sign unless the result is zero) and stores the result in % \cs{macro}.\\ % \SpecialUsageIndex{\MFPfloor}^^A % \cs{MFPfloor}\mmarg{num}\cs{macro}& @@ -432,7 +444,8 @@ % The command \cs{MFPzero} is useful for ``macro programs''. If you want % to do something to a number depending on the outcome of a test, you may % occasionally want to simply absorbed the number and output a default -% result. (There are more efficient ways to simply store $0$ in a macro.) +% result. This is more efficient than multiplying by zero (but less +% efficient than simply defining the \cs{macro} to be zero.) % % Note that one could easily double, halve, square, increment, % decrement or invert a \meta{num} using the binary versions of @@ -445,17 +458,16 @@ % use the two argument versions, \cs{MFPmul}\mmarg{num}\marg{.5} is faster than % \cs{MFPdiv}\mmarg{num}\marg{2}. % -% There is one command that takes no argument: +% There is one command that takes no argument and returns no value: % % \medskip % \centerline{% % \begin{tabular}{lp{3.4in}} -% \textit{Nullary Operations}&\\[3pt] +% \textit{Do Nothing}&\\[3pt] % \hline\hline % \textbf{Command}&\textbf{operation}\\ % \hline -% \SpecialUsageIndex{\MFPnoop}^^A -% \cs{MFPnoop}& Does nothing. +% \SpecialUsageIndex{\MFPnoop}\cs{MFPnoop}& Does nothing. % \end{tabular}} % % \bigskip @@ -502,7 +514,7 @@ % \medskip % Issuing \verb$\MFPchk{\X}$ will check the sign of the number stored in % the macro \cs{X}. Then \verb$\IFneg{A}{B}$ will produce `\verb$A$' if it -% is negative and `\verb$B$' if it is $0$ or positive. Similarly, +% is negative and `\verb$B$' if it is zero or positive. Similarly, % \verb$\MFPcmp{\X}{1}$ will compare the number stored in \cs{X} to $1$. % Afterward, \verb$\IFlt{A}{B}$ will produce `\verb$A$' if \cs{X} is less % than $1$ and `\verb$B$' if \cs{X} is equal to or greater than $1$. @@ -689,20 +701,20 @@ % Decreases by $1$. Slightly more efficient than the equivalent % subtraction.\\ % \SpecialUsageIndex{\Rzero}\cs{Rzero}& -% Replaces the number with $0$. Slightly more convenient than the +% Replaces the number with zero. Slightly more convenient than the % equivalent \cs{Rpop}\cs{X} followed by a \cs{Rpush}\marg{0}.\\ % \end{tabular}} % %\bigskip % % -% There is one nullary operation, which does not read the stack nor -% change it. +% There is one operation, which does not read the stack nor change it +% (nor do anything else). % % \medskip % \centerline{% % \begin{tabular}{lp{3.8in}} -% \multicolumn2{c}{\textit{Nullary Operations}}\\ +% \multicolumn2{c}{\textit{Do Nothing}}\\ % \hline\hline % \textbf{Command}&\textbf{operation}\\ % \hline @@ -808,12 +820,12 @@ % % \subsection{Errors} % -% If one tries to \op{pop}from an empty stack, an error message will be +% If one tries to \op{pop} from an empty stack, an error message will be % issued. Ignoring the error causes the macro to have the value stored % in the macro \SpecialUsageIndex{\EndofStack}\verb$\EndofStack$. % Its default is \texttt{0.00000000}. % -% If one tries to divide by $0$, an error message will be issued. +% If one tries to divide by zero, an error message will be issued. % Ignoring the error causes the result to be one of the following: % \begin{itemize} % \item Dividing $0$ by $0$ gives a result whose integer part is stored @@ -859,10 +871,10 @@ \ifx \csname MFP@finish\endcsname\relax \else \expandafter\endinput \fi \expandafter\edef\csname MFP@finish\endcsname{% - \catcode64=\the\catcode64 \space % @ - \catcode46=\the\catcode46 \space % . - \catcode60=\the\catcode60 \space % < - \catcode62=\the\catcode62 \space}% > + \catcode64=\the\catcode64 \space + \catcode46=\the\catcode46 \space + \catcode60=\the\catcode60 \space + \catcode62=\the\catcode62 \space}% \ifx\ProvidesPackage\UndEfInEd \newlinechar`\^^J% \message{% @@ -876,33 +888,27 @@ \DeclareOption{extra}{\def\MFPextra{}}% \ProcessOptions\relax \fi -\catcode64=11 % @=letter (already is in LaTeX) +\catcode64=11 \ifx\MFPextra\UndEfInEd \def\MFP@loadextra{}% \else \def\MFP@loadextra{\input mfpextra\relax}% \fi -\def\MFPloadextra{% - \edef\MFP@load@extra{% - \catcode46=12 \catcode60=12 \catcode62=12 \catcode64=11 - \noexpand\input mfpextra\relax - \catcode46=\the\catcode46\relax\catcode60=\the\catcode60\relax - \catcode62=\the\catcode62\relax\catcode64=\the\catcode64\relax}% - \MFP@load@extra}% -\catcode46=12 % . -\catcode60=12 % < -\catcode62=12 % > +\def\MFPloadextra{\input mfpextra\relax}% +\catcode46=12 +\catcode60=12 +\catcode62=12 % \end{macrocode} % -% We check for \LaTeX{} (ignoring \LaTeX209); \cs{MFP@ifnoLaTeX}\dots\cs{mfp@end} -% is skipped in LateX and executed otherwise. +% We check for \LaTeX{} (ignoring \LaTeX209); \cs{MFP@ifnoLaTeX}\dots\cs{MFP@end} +% is skipped in \LaTeX{} and executed otherwise. % \begin{macrocode} -\long\def\gobbleto@mfp@end#1\mfp@end{}% -\ifx\mfp@end\UndEfInEd\def\mfp@end{\@empty}\fi +\long\def\gobbleto@MFP@end#1\MFP@end{}% +\def\MFP@end{\@empty}% \ifx\documentclass\UndEfInEd \def\MFP@ifnoLaTeX{}% \else - \let\MFP@ifnoLaTeX\gobbleto@mfp@end + \let\MFP@ifnoLaTeX\gobbleto@MFP@end \fi % \end{macrocode} % @@ -945,7 +951,7 @@ \long\def\@firstofone #1{#1}% \long\def\@firstoftwo #1#2{#1}% \long\def\@secondoftwo#1#2{#2}% -\mfp@end +\MFP@end % \end{macrocode} % % We need to divide by both $10^4$ and $10^8$ several times. I could @@ -958,15 +964,15 @@ % % These are for manipulating digits. The \verb$\...ofmany$ commands % require a sequence of arguments (brace groups or tokens) followed by -% \verb$\mfp@end$. The minimum number of required parameters is surely +% \verb$\MFP@end$. The minimum number of required parameters is surely % obvious. For example, \cs{MFP@ninthofmany} must be used like\\ -% \indent\cs{MFP@ninthofmany}\meta{9 or more arguments}\cs{mfp@end} +% \indent\cs{MFP@ninthofmany}\meta{9 or more arguments}\cs{MFP@end}\\ % All these are fully expandable. % \begin{macrocode} -\def\MFP@firstofmany#1#2\mfp@end{#1}% -\def\MFP@fifthofmany#1#2#3#4#5#6\mfp@end{#5}% -\def\MFP@ninthofmany#1#2#3#4#5#6#7#8{\MFP@firstofmany}% -\def\MFP@firsteightofmany#1#2#3#4#5#6#7#8#9\mfp@end{#1#2#3#4#5#6#7#8}% +\def\MFP@oneofmany#1#2\MFP@end{#1}% +\def\MFP@fifthofmany#1#2#3#4#5#6\MFP@end{#5}% +\def\MFP@ninthofmany#1#2#3#4#5#6#7#8{\MFP@oneofmany}% +\def\MFP@eightofmany#1#2#3#4#5#6#7#8#9\MFP@end{#1#2#3#4#5#6#7#8}% % \end{macrocode} % % \subsection{Processing numbers and the stack} @@ -1011,7 +1017,8 @@ % \indent % \cs{endgroup}\cs{def}\cs{MFP@z@Val}\marg{\meta{expansion-of-\cs{MFP@z@Val}}}\\ % which defines \cs{MFP@z@Val} outside the current group to equal its expansion -% within the current group, provided it was started with \cs{begingroup}. +% within the current group (provided the group was started with +% \cs{begingroup}). % % We define a \cs{MFP@returned@values} to make all the conceivable produced % values survive the group. The \cs{MFPcurr@Sgn} part is to permit testing @@ -1020,30 +1027,27 @@ % I have been lax at making sure \cs{MFP@z@Ovr} is properly initiallized % and properly checked whenever it could be relevant, and properly % passed on. I think every internal command \cs{MFP@R}\textit{xxx} -% should ensure it starts being $0$ and ends with a numerical value. I -% notice that division might make it empty. +% should ensure it starts being zero and ends with a numerical value. At +% one time division could leave it undefined. % % \cs{MFP@subroutine} executes its argument (typically a single command) with % a wrapper that initializes all the macros that might need initializing, % and returns the necessary results. % \begin{macrocode} \def\MFP@endgroup@after#1{\edef\x{\endgroup#1}\x}% -\def\MFP@endgroup@return{\MFP@endgroup@after\MFP@returned@values}% -\def\MFP@def@after{\def\noexpand}% +\def\MFP@afterdef{\def\noexpand}% \def\MFP@returned@values{% - \MFP@def@after\MFP@z@Val{\MFP@z@Sign\MFP@z@Int.\MFP@z@Frc}% - \MFP@def@after\MFP@z@Ovr{\MFP@z@Ovr}% - \MFP@def@after\MFP@z@Und{\MFP@z@Und}% - \MFP@def@after\MFPcurr@Sgn{\MFP@z@Sgn}}% + \MFP@afterdef\MFP@z@Val{\MFP@z@Sign\MFP@z@Int.\MFP@z@Frc}% + \MFP@afterdef\MFP@z@Ovr{\MFP@z@Ovr}% + \MFP@afterdef\MFP@z@Und{\MFP@z@Und}% + \MFP@afterdef\MFPcurr@Sgn{\MFP@z@Sgn}}% \def\MFP@subroutine#1{% \begingroup - \MFP@basic@init@z + \MFP@Rzero + \def\MFP@z@Ovr{0}% + \def\MFP@z@Und{0}% #1% - \MFP@endgroup@return}% -\def\MFP@basic@init@z{% - \MFP@Rzero - \def\MFP@z@Ovr{0}% - \def\MFP@z@Und{0}}% + \MFP@endgroup@after\MFP@returned@values}% \def\MFP@Rzero{% \def\MFP@z@Sgn{0}% \def\MFP@z@Int{0}% @@ -1052,7 +1056,7 @@ % % \DescribeMacro{\EndofStack} % We define here the error messages: popping from an empty stack and -% dividing by $0$. In addition to the error messages, we provide some +% dividing by zero. In addition to the error messages, we provide some % default values that hopefully allow some operations to continue. % % We also have a warning or two. @@ -1087,7 +1091,7 @@ % \DescribeMacro{\MaxRealInt}These are the largest possible integer and % fractional parts of a real % \DescribeMacro{\MaxRealFrac}number. They are returned for division by -% $0$, for logarithm of $0$, and when overflow is detected in the +% zero, for logarithm of zero, and when overflow is detected in the % exponential function. % \begin{macrocode} \def\MaxRealInt {99999999}% @@ -1096,11 +1100,11 @@ % % \SpecialUsageIndex{\MaxRealInt} % \SpecialUsageIndex{\MaxRealFrac} -% These are the results returned when trying to divide by $0$. Two are +% These are the results returned when trying to divide by zero. Two are % \DescribeMacro{\xOverZeroInt} % \DescribeMacro{\xOverZeroFrac} -% used when dividing a nonzero number by $0$ and and two when trying to -% divide $0$ by $0$. +% used when dividing a nonzero number by zero and and two when trying to +% divide zero by zero. % \DescribeMacro{\ZeroOverZeroInt} % \DescribeMacro{\ZeroOverZeroFrac} % \begin{macrocode} @@ -1130,7 +1134,7 @@ \def\MFPparse@real#1#2#3#4{% \MFPnospace@def\MFPtemp@Val{#4}% \MFPprocess@into@parts\MFPtemp@Val#1#2#3% - \MFPpadto@eight#3}% + \MFP@padtoeight#3}% \def\MFPparse@x{\MFPparse@real\MFP@x@Sgn\MFP@x@Int\MFP@x@Frc}% \def\MFPparse@y{\MFPparse@real\MFP@y@Sgn\MFP@y@Int\MFP@y@Frc}% % \end{macrocode} @@ -1149,29 +1153,27 @@ % the sign, integer and fractional parts. % \begin{macrocode} \def\MFPprocess@into@parts#1#2#3#4{% - \@xp\MFPsplit@dot#1..\mfp@end #3#4% + \@xp\MFPsplit@dot#1..\MFP@end #3#4% % \end{macrocode} % -% This is the first place where having at most eight digits simplifies things. -% At this point \arg3 could contain any number of consecutive signs -% followed by any eight digits. It could be $0$, so to avoid losing the sign -% we append a \texttt{1} (for up to nine digits). We temporarily define the -% sign based on the result, but may need to drop it if both the integer -% and fractional parts are $0$. +% At this point \arg3 holds the part before the dot (or the whole thing +% if there was no dot) and \arg4 holds the part after the dot, (or +% nothing). Now is the first place where having at most eight digits +% simplifies things. Note that \arg3 could contain any number of +% consecutive signs followed by up to eight digits. It could be zero or +% empty, so to avoid losing the sign we append a \texttt{1} (for up to +% nine digits). We temporarily define the sign based on the result, but +% may need to drop it if both the integer and fractional parts are zero. % -% Prepending a 0 to the fractional part permits it to be empty. +% Prepending a zero to the fractional part pemits it to be empty. % In the final \cs{edef}, \arg3 is made positive. % \begin{macrocode} - \ifnum#31<0 - \def#2{-1}% - \else - \def#2{1}% + \ifnum#31<0 \def#2{-1}% + \else \def#2{1}% \fi \ifnum #30=0 \def#3{0}% - \ifnum 0#4=0 - \def#2{0}% - \fi + \ifnum 0#4=0 \def#2{0}\fi \fi \edef#3{\number \ifnum #2<0 -\fi#3}}% % \end{macrocode} @@ -1179,15 +1181,15 @@ % This only copies the parts before and after the dot, \arg1 and \arg2, % into macros \arg4 and \arg5. % \begin{macrocode} -\def\MFPsplit@dot#1.#2.#3\mfp@end#4#5{\edef#4{#1}\edef#5{#2}}% +\def\MFPsplit@dot#1.#2.#3\MFP@end#4#5{\edef#4{#1}\edef#5{#2}}% % \end{macrocode} % % This is used to pad the fractional part to eight places with zeros. If % a number with more than eight digits survives to this point, it gets % truncated. % \begin{macrocode} -\def\MFPpadto@eight#1{% - \edef#1{\@xp\MFP@firsteightofmany#100000000\mfp@end}}% +\def\MFP@padtoeight#1{% + \edef#1{\@xp\MFP@eightofmany#100000000\MFP@end}}% % \end{macrocode} % % These take operands off the stack. We know already that there are no @@ -1215,6 +1217,7 @@ % \end{macrocode} % % Sometimes only parts of the number needs changing (used in CHS, ABS). +% This copies the integer and fractional parts of $x$ into $z$. % \begin{macrocode} \def\copyMFP@x{\edef\MFP@z@Int{\MFP@x@Int}\edef\MFP@z@Frc{\MFP@x@Frc}}% % \end{macrocode} @@ -1238,7 +1241,7 @@ % \end{macrocode} % % The macro \cs{Rpop} calls \cs{MFP@popit} followed by the contents of the -% stack, the token \cs{mfp@end} and the macro to \op{pop} into. If the stack is +% stack, the token \cs{MFP@end} and the macro to \op{pop} into. If the stack is % not empty, \cs{doMFP@popit} will read the first group \arg1 into that macro % \arg3, and then redefine the stack to be the rest of the argument \arg2. % If the stack is empty, \cs{doMFP@EOS} will equate the macro to @@ -1246,8 +1249,8 @@ % message. % \begin{macrocode} \def\MFP@popit{\if@EndofStack\doMFP@EOS\doMFP@popit}% -\def\doMFP@EOS#1\mfp@end#2{\MFP@popempty@err\let#2\EndofStack}% -\def\doMFP@popit#1#2\mfp@end#3{\edef\MFP@Rstack{#2}\edef#3{#1}}% +\def\doMFP@EOS#1\MFP@end#2{\MFP@popempty@err\let#2\EndofStack}% +\def\doMFP@popit#1#2\MFP@end#3{\edef\MFP@Rstack{#2}\edef#3{#1}}% % \end{macrocode} % % \subsection{The user-level operations} @@ -1365,7 +1368,7 @@ % \item[cmp] compare $x$ and $y$ (stack version does not change stack). % \item[chk] examine the sign of $x$ (stack version does not change stack). % \item[dup] stack only, duplicate the top element of the stack. -% \item[push] stack only, put a value on top of the stack. +% \item[push] stack only, put a value onto the top of the stack. % \item[pop] stack only, remove the top element of the stack, % store it in a variable. % \item[exch] stack only, exchange top two elements of the stack. @@ -1449,8 +1452,8 @@ % \begin{macrocode} \let\Rnoop\relax \def\Rcmp{% - \MFPgetoperand@y\MFPgetoperand@x % get operands (last pushed is y) - \MFP@Rcat\MFP@x@Val\MFP@Rcat\MFP@y@Val % put back: LOFI + \MFPgetoperand@y\MFPgetoperand@x + \MFP@Rcat\MFP@x@Val\MFP@Rcat\MFP@y@Val \MFP@Rcmp}% \def\Rchk{% \MFPgetoperand@x @@ -1461,7 +1464,7 @@ \edef\MFP@z@Val{\MFP@x@Sign\MFP@x@Int.\MFP@x@Frc}% \edef\MFPcurr@Sgn{\MFP@x@Sgn}% \MFPpush@result}% - \def\Rpop{\@xp\MFP@popit\MFP@Rstack\mfp@end}% + \def\Rpop{\@xp\MFP@popit\MFP@Rstack\MFP@end}% \def\Rexch{% \Rpop\MFP@x@Val\Rpop\MFP@y@Val \MFP@Rcattwo\MFP@y@Val\MFP@x@Val}% @@ -1546,6 +1549,17 @@ \def\MFPmax{\MFP@op@Binary\MFP@Rmax}% % \end{macrocode} % +% A \emph{nullary} operation is one that produces a result with no +% operand. Thus, it could return a fixed constant, or it could perform +% calculations that obtain input from the system (e.g., current time). At +% the moment we don't define any. +% \begin{macrocode} +\def\MFP@stack@Nullary#1{% + \MFP@subroutine{#1}\MFPpush@result}% +\def\MFP@op@Nullary#1{% + \MFP@subroutine{#1}\MFPstore@result}% +% \end{macrocode} +% % These are the wrappers for unary operations. The operand versions have a % second argument, the macro that stores the result. But this will be the % argument of \cs{MFPstore@result}. @@ -1604,7 +1618,7 @@ \MFP@tempb\MFP@x@Frc\relax \ifodd\MFP@tempb \def\MFP@z@Und{5}% - \advance\MFP@tempb 1 % round up + \advance\MFP@tempb 1 \ifnum\MFP@ttteight=\MFP@tempb \MFP@tempb0 \advance\MFP@tempa1 \fi @@ -1623,13 +1637,15 @@ % \end{macrocode} % % The squaring operation just calls \cs{MFP@Rmul} after copying $x$ to -% $y$. +% $y$. Its gain in efficiency over a multiplication is that it can skip +% preprocessing of the second (identical) operand. % \begin{macrocode} \def\MFP@Rsq{\MFP@Rcopy xy\MFP@Rmul}% % \end{macrocode} % % The inversion operation just calls \cs{MFP@Rdiv} after copying $x$ to -% $y$ and $1$ to $x$. +% $y$ and $1$ to $x$. Its advantage over a divide is it skips the +% preprocessing of $1$ as an operand. % \begin{macrocode} \def\MFP@Rinv{\MFP@Rcopy xy\MFP@Rload x110\MFP@Rdiv}% % \end{macrocode} @@ -1640,7 +1656,7 @@ \MFP@Rloadz {\ifnum\MFP@x@Int=0 0\else\MFP@x@Sgn\fi}\MFP@x@Int 0}% % \end{macrocode} % -% Fractional part: replace integer part with a $0$. +% Fractional part: replace integer part with a zero. % \begin{macrocode} \def\MFP@Rfrac{% \MFP@Rloadz {\ifnum\MFP@x@Frc=0 0\else\MFP@x@Sgn\fi}0\MFP@x@Frc}% @@ -1649,8 +1665,8 @@ % To increment and decrement by $1$, except in border cases, we need only % address the integer part of a number. This doesn't seem so simple % written out but, even so, it is more efficient than full-blown addition. -% It would be very slightly more efficient to repeat the increment code in -% decrementing, but it would be annoying to do so, +% It would be very slightly more efficient if \cs{MFP@Rdecr} did not call +% \cs{MFP@Rincr}, but instead was similarly coded. % \begin{macrocode} \def\MFP@Rincr{% \ifnum\MFP@x@Sgn<0 @@ -1685,17 +1701,17 @@ % We use the same code to get floor or ceiling, the % appropriate inequality character being its argument. % \begin{macrocode} -\def\MFP@Rfloororceil#1{% +\def\MFP@Rfloorceil#1{% \MFP@tempa\MFP@x@Int\relax - \ifnum 0#1\MFP@x@Sgn + \ifnum \MFP@x@Sgn #10 \ifnum\MFP@x@Frc=0 \else \advance\MFP@tempa1 \fi \fi \MFP@Rloadz{\ifnum\MFP@x@Int=0 0\else\MFP@x@Sgn\fi}\MFP@tempa0}% -\def\MFP@Rfloor{\MFP@Rfloororceil>}% -\def\MFP@Rceil {\MFP@Rfloororceil<}% +\def\MFP@Rfloor{\MFP@Rfloorceil<}% +\def\MFP@Rceil {\MFP@Rfloorceil>}% % \end{macrocode} % % For multiplication, after the usual break into integer and fractional @@ -1713,14 +1729,13 @@ \multiply\MFP@tempb by\MFP@tttfour \advance\MFP@tempa-\MFP@tempb \MFP@endgroup@after{% - \MFP@def@after#2{#2}% - \MFP@def@after#3{\number\MFP@tempa}% + \MFP@afterdef#2{#2}% + \MFP@afterdef#3{\number\MFP@tempa}% }}% % -\def\MFP@x@split{% +\def\MFP@@split{% \MFP@split\MFP@x@Int\MFP@x@Int@ii\MFP@x@Int@i - \MFP@split\MFP@x@Frc\MFP@x@Frc@i\MFP@x@Frc@ii}% -\def\MFP@y@split{% + \MFP@split\MFP@x@Frc\MFP@x@Frc@i\MFP@x@Frc@ii \MFP@split\MFP@y@Int\MFP@y@Int@ii\MFP@y@Int@i \MFP@split\MFP@y@Frc\MFP@y@Frc@i\MFP@y@Frc@ii}% % \end{macrocode} @@ -1728,39 +1743,27 @@ % We will store the intermediate and final products in \cs{MFP@z@*}. Each one % is ultimately reduced to four digits, like the parts of $x$ and $y$. As each % base-$10000$ digit of $y$ is multiplied by a digit of $x$, we add the -% result to the appropriate digit of the partial result $z$. Thus, we need -% to zero out $z$ at the start (or treat the first iteration differently): +% result to the appropriate digit of the partial result $z$. % % The underflow ends up in \cs{MFP@z@Frc@iv} and \cs{MFP@z@Frc@iii}. % Overflow will be in \cs{MFP@z@Int@iii}. Unlike the rest, it can be up to % eight digits because we do not need to carry results out of it. -% \begin{macrocode} -\def\MFPmore@init@z{% - \def\MFP@z@Frc@iv {0}% - \def\MFP@z@Frc@iii{0}% - \def\MFP@z@Frc@ii {0}% - \def\MFP@z@Frc@i {0}% - \def\MFP@z@Int@i {0}% - \def\MFP@z@Int@ii {0}% - \def\MFP@z@Int@iii{0}}% -% \end{macrocode} % -% This command prepends zeros so a number fills four slots. In the -% ``make'' version, \arg1 is a macro holding the value and is redefined to -% contain the result. A macro that calls these should ensure that \arg1 is -% not empty and is less than 10,000. +% This command prepends zeros so a number fills four slots. Here \arg1 is +% a macro holding the value and it is redefined to contain the result. A +% macro that calls this should ensure that \arg1 is not empty and is less +% than 10,000. % \begin{macrocode} -\def\MFP@fourdigits#1{% - \@xp\MFP@fifthofmany\number#1{}{0}{00}{000}\mfp@end\number#1}% -\def\makeMFP@fourdigits#1{\edef#1{\MFP@fourdigits{#1}}}% +\def\makeMFP@fourdigits#1{% + \edef#1{\@xp\MFP@fifthofmany\number#1{}{0}{00}{000}\MFP@end\number#1}}% % \end{macrocode} % -% This is the same, but produce eight digits. +% This is the same, but produces eight digits. Similarly \arg1 should be +% nonempty and less than 100,000,000. % \begin{macrocode} -\def\MFP@eightdigits#1{% - \@xp\MFP@ninthofmany\number#1% - {}{0}{00}{000}{0000}{00000}{000000}{0000000}\mfp@end\number#1}% -\def\makeMFP@eightdigits#1{\edef#1{\MFP@eightdigits{#1}}}% +\def\makeMFP@eightdigits#1{% + \edef#1{\@xp\MFP@ninthofmany\number#1% + {}{0}{00}{000}{0000}{00000}{000000}{0000000}\MFP@end\number#1}}% % \end{macrocode} % % The following macros implement carrying. The macros \cs{MFP@carrya} and @@ -1779,22 +1782,19 @@ \begingroup \MFP@carryi{#1}#2#3% \MFP@endgroup@after{% - \MFP@def@after#3{\number\MFP@tempa}% - \MFP@def@after#2{\number\MFP@tempb}% + \MFP@afterdef#3{\number\MFP@tempa}% + \MFP@afterdef#2{\number\MFP@tempb}% }}% % \end{macrocode} % % This is the ``internal'' carry. \arg1, \arg2, and \arg3 are as in -% \cs{MFP@carry}. Its advantage is that it can be used used where \arg2 and -% \arg3 are not macros, leaving the result in \cs{MFP@tempa} and \cs{MFP@tempb} -% with \cs{MFP@tempb} in the correct range, $[0,\mbox{\arg1})$. Its -% disadvantage is it does not protect temporary registers. Warning: -% never use it in the form \cs{MFP@carryi}\meta{num}\cs{MFP@tempa}, -% because this would copy \meta{num} to \cs{MFP@tempa}, losing the value -% in the second argument before anything can be done. The other order is -% okay, and \cs{MFP@tempb} can be used in either slot. Do not use it -% without grouping if you want the values in the temp registers \texttt{a}, -% \texttt{b} or \texttt{c} preserved. +% \cs{MFP@carry}. Its advantage is that it can be used used where \arg2 +% and \arg3 are not macros, leaving the result in \cs{MFP@tempa} and +% \cs{MFP@tempb} with \cs{MFP@tempb} in the correct range, +% $[0,\mbox{\arg1})$. Its disadvantage is it does not protect temporary +% registers. Warning: do not use it with \arg2=\cs{MFP@tempa} and do not +% use it without grouping if you want to preserve the values in these +% temporary count registers. % \begin{macrocode} \def\MFP@carryi#1#2#3{% \MFP@tempa=#3\relax @@ -1814,7 +1814,7 @@ \MFP@tempa#1% \advance\MFP@tempa#2\relax \MFP@endgroup@after{% - \MFP@def@after#3{\number\MFP@tempa}% + \MFP@afterdef#3{\number\MFP@tempa}% }}% % \end{macrocode} % @@ -1940,43 +1940,44 @@ % \end{macrocode} % % \cs{MFP@Rmul} first computes the (theoretical) sign of the product: if -% $0$, return $0$, otherwise provisionally set the sign of the product and -% call \cs{MFP@@Rmul}. +% it is zero, return zero, otherwise provisionally set the sign of the product +% and call \cs{MFP@@Rmul}. % \begin{macrocode} \def\MFP@Rmul{% - \MFP@tempa\MFP@x@Sgn \multiply\MFP@tempa\MFP@y@Sgn\relax - \ifnum 0=\MFP@tempa - \MFP@Rzero - \else - \edef\MFP@z@Sgn{\number\MFP@tempa}% - \@xp\MFP@@Rmul - \fi}% + \ifnum\MFP@x@Sgn=0 \MFP@Rzero + \else\ifnum\MFP@y@Sgn=0 \MFP@Rzero + \else \edef\MFP@z@Sgn{\number\MFP@x@Sign\MFP@y@Sgn}% + \@XP\MFP@@Rmul + \fi\fi}% % \end{macrocode} % -% \cs{MFP@@Rmul} splits the four expected macros into eight macros -% considered to be four base-10000 digits for each of $x$ and $y$. +% \cs{MFP@@Rmul} first initializes the macros that will hold the +% base-10000 digits of $z$. Then it splits the four expected macros into +% eight macros that hold the base-10000 digits for each of $x$ and $y$. % Then each digit of $y$ is used to multiply the four digits of $x$ and the -% results are added to corresponding digits of $z$, which have been -% initialized to $0$ by \cs{MFPmore@init@z}. +% results are added to corresponding digits of $z$. % \begin{macrocode} \def\MFP@@Rmul{% - \MFPmore@init@z - \MFP@x@split\MFP@y@split + \def\MFP@z@Frc@iv {0}\def\MFP@z@Frc@iii{0}% + \def\MFP@z@Frc@ii {0}\def\MFP@z@Frc@i {0}% + \def\MFP@z@Int@i {0}\def\MFP@z@Int@ii {0}% + \def\MFP@z@Int@iii{0}% + \MFP@@split \MFP@multiplyfour \MFP@y@Frc@ii \MFP@z@Frc@i - \MFP@z@Frc@ii \MFP@z@Frc@iii\MFP@z@Frc@iv + \MFP@z@Frc@ii \MFP@z@Frc@iii \MFP@z@Frc@iv \MFP@multiplyfour \MFP@y@Frc@i \MFP@z@Int@i - \MFP@z@Frc@i \MFP@z@Frc@ii \MFP@z@Frc@iii + \MFP@z@Frc@i \MFP@z@Frc@ii \MFP@z@Frc@iii \MFP@multiplyfour \MFP@y@Int@i \MFP@z@Int@ii - \MFP@z@Int@i \MFP@z@Frc@i \MFP@z@Frc@ii + \MFP@z@Int@i \MFP@z@Frc@i \MFP@z@Frc@ii \MFP@multiplyfour \MFP@y@Int@ii \MFP@z@Int@iii - \MFP@z@Int@ii \MFP@z@Int@i \MFP@z@Frc@i + \MFP@z@Int@ii \MFP@z@Int@i \MFP@z@Frc@i % \end{macrocode} % Now apply the carry routines on the underflow digits\dots % \begin{macrocode} \MFP@carrym\MFP@z@Frc@iv\MFP@z@Frc@iii \MFP@carrym\MFP@z@Frc@iii\MFP@z@Frc@ii % \end{macrocode} -% \dots pause to round the lowest digit that will be kept\dots +% \dots and pause to round the lowest digit that will be kept\dots % \begin{macrocode} \ifnum\MFP@z@Frc@iii<5000 \else \MFP@tempb\MFP@z@Frc@ii @@ -1992,13 +1993,13 @@ \MFP@carrym\MFP@z@Int@ii\MFP@z@Int@iii % \end{macrocode} % To end, we arrange for all macros to hold four digits (except -% \cs{MFP@z@Int@ii} which doesn't need leading 0s, and \cs{MFP@z@Int@iii} -% which also doesn't) and load them into the appropriate 8-digit macros. -% The underflow digits are stored in \cs{MFP@z@Und} in case we ever need -% to examine them, and the overflow in \cs{MFP@z@Ovr} in case we ever need -% to implement an overflow error. Theoretically $z \ne 0$, but it is -% possible that $z=0$ after rounding to eight places. If so, we must reset -% \cs{MFP@z@Sgn}. +% \cs{MFP@z@Int@ii} and \cs{MFP@z@Int@iii} which don't need leading 0s) +% and load them into the appropriate 8-digit macros. The underflow digits +% are stored in \cs{MFP@z@Und} in case we ever need to examine them (we +% now do: in our unit conversion routine \cs{MFP@DPmul}), and the overflow +% in \cs{MFP@z@Ovr} in case we ever want to implement an overflow error. +% Theoretically $z \ne 0$, but it is possible that $z=0$ after reducing to +% eight places. If so, we must reset \cs{MFP@z@Sgn}. % \begin{macrocode} \makeMFP@fourdigits\MFP@z@Frc@iv \makeMFP@fourdigits\MFP@z@Frc@iii @@ -2011,8 +2012,7 @@ \edef\MFP@z@Und{\MFP@z@Frc@iii\MFP@z@Frc@iv}% \ifnum\MFP@z@Int>0 \else\ifnum\MFP@z@Frc>0 - \else - \def\MFP@z@Sgn{0}% + \else \def\MFP@z@Sgn{0}% \fi\fi}% % \end{macrocode} % @@ -2031,12 +2031,12 @@ % only $16$ significant digits should be retained in any case.) If $d$ is % $0$ and $n$ is $15$ we would need $-5$ digits. That means the first % nonzero digit is in the 15th or 16th place after the dot and the -% quotient is effectively $0$. +% quotient is effectively zero. % % Here I explain why we normalize the parts in this way. If a numerator % has the form $n_1.n_2$ and the denominator has the form $d_1.d_2$ then -% TeX can easily obtain the integer part of $n_1/d_1$, because these are -% within its range for integers. The resulting quotient (let's call it +% \TeX{} can easily obtain the integer part of $n_1/d_1$, because these +% are within its range for integers. The resulting quotient (let's call it % $q_1$) is the largest integer satisfying $q_1d_1 \le n_1$. What we seek, % however is the largest integer $q$ such that $q(d_1.d_2) \le n_1.n_2$. % It can easily be shown that $q \le q_1$. It is true, but not so easily @@ -2058,7 +2058,7 @@ % Since $d_2$ is no more than eight digits, $q_1 d_2$ is less than $9 % (10)^8$. Inequality (\ref{crucial}) is therefore satisfied if $n_1 - q_1 % d_1 \ge 9$. If that is not the case then the right side of -% (\ref{crucial}) is computable within TeX's integer ranges and we can +% (\ref{crucial}) is computable within \TeX's integer ranges and we can % easily test the inequality. If the inequality holds, then $q = q_1$, % otherwise $q = q_1 - 1$. % @@ -2069,19 +2069,18 @@ % % Now I need to get it organized. \cs{MFP@Rdiv} will have \cs{MFP@x@*} and % \cs{MFP@y@*} available. One step (could be first or last). Is to calculate -% the sign. Let's do it first (because we need to check for $0$ anyway). +% the sign. Let's do it first (because we need to check for zero anyway). % -% We invoke an error message upon division by $0$, but nevertheless return +% We invoke an error message upon division by zero, but nevertheless return % a value. By default it is $0$ for $0/0$ and the maximum possible real -% for $x/0$ when $x$ is not $0$. If the numerator is $0$ and the -% denominator not, we do nothing as $z$ was initialized to be $0$. +% for $x/0$ when $x$ is not zero. If the numerator is zero and the +% denominator not, we do nothing as $z$ was initialized to be zero. % -% If neither is $0$, we calculate the sign of the result and call +% If neither is zero, we calculate the sign of the result and call % \cs{MFP@@Rdiv} to divide the absolute values. % \begin{macrocode} \def\MFP@Rdiv{% - \ifnum\MFP@y@Sgn=0 - \MFP@dividebyzero@err + \ifnum\MFP@y@Sgn=0 \MFP@dividebyzero@err \ifnum\MFP@x@Sgn=0 \edef\MFP@z@Int{\ZeroOverZeroInt}% \edef\MFP@z@Frc{\ZeroOverZeroFrac}% @@ -2090,13 +2089,8 @@ \edef\MFP@z@Frc{\xOverZeroFrac}% \fi \edef\MFP@z@Sgn{\MFP@x@Sgn}% - \else\ifnum\MFP@x@Sgn=0 - \MFP@Rzero - \else - \MFP@tempa\MFP@x@Sgn - \multiply\MFP@tempa\MFP@y@Sgn - \edef\MFP@z@Sgn{\number\MFP@tempa}% - \MFP@@Rdiv + \else\ifnum\MFP@x@Sgn=0 \MFP@Rzero + \else \edef\MFP@z@Sgn{\number\MFP@x@Sign\MFP@y@Sgn}\MFP@@Rdiv \fi\fi}% % \end{macrocode} % @@ -2106,7 +2100,7 @@ % but knowing the shift will give us the correct quotient in the end. % % We first arrange that \cs{MFP@y@Int} is nonzero by making it \cs{MFP@y@Frc} if -% it is $0$ (a shift of eight digits). Then the macro +% it is zero (a shift of eight digits). Then the macro % \cs{MFP@numdigits@toshift} computes $8$ minus the number of digits in % \cs{MFP@y@Int}, which is how many positions left $y$ will be shifted. % We then call \cs{MFP@doshift@y} on the concatenation of the digits in @@ -2123,7 +2117,7 @@ \MFP@tempa=0 \fi \advance\MFP@tempa\MFP@numdigits@toshift\MFP@y@Int\relax - \@XP\MFP@doshift@y\@xp\MFP@y@Int\MFP@y@Frc0000000\mfp@end + \@XP\MFP@doshift@y\@xp\MFP@y@Int\MFP@y@Frc0000000\MFP@end % \end{macrocode} % % We repeat all that on the numerator $x$, except shifting its digits @@ -2142,7 +2136,7 @@ \advance\MFP@tempa -8 \fi \advance\MFP@tempa-\MFP@numdigits@toshift\MFP@x@Int\relax - \@XP\MFP@doshift@x\@xp\MFP@x@Int\MFP@x@Frc0000000\mfp@end + \@XP\MFP@doshift@x\@xp\MFP@x@Int\MFP@x@Frc0000000\MFP@end % \end{macrocode} % % Since our result will have at most one digit in the integer part, a @@ -2184,13 +2178,13 @@ % from the rest (\arg9) inside \cs{MFP@x@Frc}. The same with % \cs{MFP@doshift@y}. % \begin{macrocode} -\def\MFP@numdigits@toshift#1{\@xp\MFP@ninthofmany#101234567\mfp@end}% -\def\MFP@doshift@x#1#2#3#4#5#6#7#8#9\mfp@end{% +\def\MFP@numdigits@toshift#1{\@xp\MFP@ninthofmany#101234567\MFP@end}% +\def\MFP@doshift@x#1#2#3#4#5#6#7#8#9\MFP@end{% \def\MFP@x@Int{#1#2#3#4#5#6#7#8}% - \edef\MFP@x@Frc{\MFP@firsteightofmany#9\mfp@end}}% -\def\MFP@doshift@y#1#2#3#4#5#6#7#8#9\mfp@end{% + \edef\MFP@x@Frc{\MFP@eightofmany#9\MFP@end}}% +\def\MFP@doshift@y#1#2#3#4#5#6#7#8#9\MFP@end{% \def \MFP@y@Int{#1#2#3#4#5#6#7#8}% - \edef\MFP@y@Frc{\MFP@firsteightofmany#9\mfp@end}}% + \edef\MFP@y@Frc{\MFP@eightofmany#9\MFP@end}}% % \end{macrocode} % % The loop counter is \cs{MFP@tempf}, \cs{MFP@tempa} is reserved for the @@ -2252,7 +2246,7 @@ \fi \advance\MFP@tempf -1 \ifnum\MFP@tempf>0 - \edef\MFP@x@Int{\MFP@x@Int0}% easy multiplications by 10 + \edef\MFP@x@Int{\MFP@x@Int0}% \edef\MFP@x@Frc{\MFP@x@Frc0}% \MFP@carrya\MFP@x@Frc\MFP@x@Int \@xp\MFP@Rdivloop @@ -2289,7 +2283,7 @@ \advance \MFP@tempa -7 \ifnum\MFP@tempa>0 \def\MFP@z@Ovr{}% - \@xp\MFPget@Ovrdigits\MFP@z@digits\mfp@end + \@xp\MFPget@Ovrdigits\MFP@z@digits\MFP@end \else \ifnum\MFP@tempa<-7 \edef\MFP@z@digits{00000000\MFP@z@digits}% @@ -2307,7 +2301,7 @@ 0000\else 00000% \fi \MFP@z@digits}% - \@xp\MFPget@Intdigits\MFP@z@digits\mfp@end + \@xp\MFPget@Intdigits\MFP@z@digits\MFP@end \fi}% % \end{macrocode} % @@ -2338,7 +2332,7 @@ \MFPget@Frcdigits}% \def\MFPget@Frcdigits#1#2#3#4#5#6#7#8#9{% \def\MFP@z@Frc{#1#2#3#4#5#6#7#8}% - \def\MFP@z@Und{#9}\gobbleto@mfp@end}% + \def\MFP@z@Und{#9}\gobbleto@MFP@end}% % \end{macrocode} % % The max amd min operations simply run the compare operation and use @@ -2358,7 +2352,7 @@ % keeps the right number. For negative truncations we prepend zeros to the % integer part so it too is exactly eight digits. These become the % arguments of \cs{MFP@@iRtrunc}, which substitutes 0 for the last -% \texttt{-\cs{MFP@tempa}} of them. +% \texttt{-}\cs{MFP@tempa} of them. % % The macro to store the result in follows \arg2. It is read and % defined by either \cs{MFP@Rtrunc} or \cs{MFP@iRtrunc}. @@ -2373,15 +2367,15 @@ \@xp\MFP@Rtrunc \fi}% \def\MFP@Rtrunc#1{% - \edef\MFP@x@Frc{\@xp\MFP@@Rtrunc\MFP@x@Frc\mfp@end}% - \ifnum\MFP@x@Int=0 % possibly returns 0 + \edef\MFP@x@Frc{\@xp\MFP@@Rtrunc\MFP@x@Frc\MFP@end}% + \ifnum\MFP@x@Int=0 \ifnum\MFP@x@Frc=0 \def\MFP@x@Sgn{0}% \fi \fi \MFP@endgroup@after{% - \MFP@def@after#1{\MFP@x@Sign\MFP@x@Int.\MFP@x@Frc}}}% -\def\MFP@@Rtrunc#1#2#3#4#5#6#7#8#9\mfp@end{% + \MFP@afterdef#1{\MFP@x@Sign\MFP@x@Int.\MFP@x@Frc}}}% +\def\MFP@@Rtrunc#1#2#3#4#5#6#7#8#9\MFP@end{% \ifcase\MFP@tempa\or #1\or #1#2\or @@ -2393,9 +2387,9 @@ #1#2#3#4#5#6#7#8\fi}% \def\MFP@iRtrunc#1{% \makeMFP@eightdigits\MFP@x@Int - \edef\MFP@x@Val{\number\MFP@x@Sign\@xp\MFP@@iRtrunc\MFP@x@Int\mfp@end}% - \MFP@endgroup@after{\MFP@def@after#1{\MFP@x@Val}}}% -\def\MFP@@iRtrunc#1#2#3#4#5#6#7#8#9\mfp@end{% + \edef\MFP@x@Val{\number\MFP@x@Sign\@xp\MFP@@iRtrunc\MFP@x@Int\MFP@end}% + \MFP@endgroup@after{\MFP@afterdef#1{\MFP@x@Val}}}% +\def\MFP@@iRtrunc#1#2#3#4#5#6#7#8#9\MFP@end{% \ifcase-\MFP@tempa #1#2#3#4#5#6#7#8\or #1#2#3#4#5#6#70\or @@ -2419,41 +2413,41 @@ \ifnum 0>\MFP@tempa \edef\MFP@y@Tmp{% \ifcase-\MFP@tempa\or - 5\or % .5 x 10^1 - 50\or % .5 x 10^2 - 500\or % .5 x 10^3 - 5000\or % .5 x 10^4 - 50000\or % .5 x 10^5 - 500000\or % .5 x 10^6 - 5000000\else % .5 x 10^7 - 50000000\fi % .5 x 10^8 + 5\or + 50\or + 500\or + 5000\or + 50000\or + 500000\or + 5000000\else + 50000000\fi }% \else \edef\MFP@y@Tmp{% \ifcase\MFP@tempa - .5\or % .5 x 10^0 - .05\or % .5 x 10^{-1} - .005\or % .5 x 10^{-2} - .0005\or % .5 x 10^{-3} - .00005\or % .5 x 10^{-4} - .000005\or % .5 x 10^{-5} - .0000005\or % .5 x 10^{-6} - .00000005\else% .5 x 10^{-7} - 0\fi % + .5\or + .05\or + .005\or + .0005\or + .00005\or + .000005\or + .0000005\or + .00000005\else + 0\fi }% \fi \MFPchk{#2}\ifMFP@neg\edef\MFP@y@Tmp{-\MFP@y@Tmp}\fi \MFPadd{#2}\MFP@y@Tmp\MFP@z@Tmp - \MFP@endgroup@after{\MFP@def@after\MFP@z@Tmp{\MFP@z@Tmp}}% + \MFP@endgroup@after{\MFP@afterdef\MFP@z@Tmp{\MFP@z@Tmp}}% \MFPtruncate{#1}\MFP@z@Tmp}% % \end{macrocode} % % \DescribeMacro{\MFPstrip} % Stripping zeros from the right end of the fractional part. The star form -% differs only in the handling of a $0$ fractional part. So we check -% whether it is $0$ and when it is, we either append `\texttt{.0}' or +% differs only in the handling of a zero fractional part. So we check +% whether it is zero and when it is, we either append `\texttt{.0}' or % nothing. The rest of the code grabs a digit at a time and stops when the -% rest are $0$. +% rest are zero. % \begin{macrocode} \def\MFPstrip{% \@ifstar{\MFP@strip{}}{\MFP@strip{.0}}}% @@ -2462,15 +2456,15 @@ \ifnum \MFP@x@Frc=0 \edef#3{\MFP@x@Sign\MFP@x@Int#1}% \else - \edef#3{\MFP@x@Sign\MFP@x@Int.\@xp\MFP@@strip\MFP@x@Frc\mfp@end}% + \edef#3{\MFP@x@Sign\MFP@x@Int.\@xp\MFP@@strip\MFP@x@Frc\MFP@end}% \fi}% -\def\MFP@@strip#1#2\mfp@end{% +\def\MFP@@strip#1#2\MFP@end{% #1% \ifnum 0#2>0 \@xp\MFP@@strip \else - \@xp\gobbleto@mfp@end - \fi#2\mfp@end}% + \@xp\gobbleto@MFP@end + \fi#2\MFP@end}% % \end{macrocode} % % \subsection{Miscellaneous} @@ -2489,7 +2483,7 @@ \begingroup \toks@\@xp{\MFPprogram@returns}% \MFP@endgroup@after{% - \MFP@def@after\MFPprogram@returns{\the\toks@ \MFP@def@after#1{#1}}% + \MFP@afterdef\MFPprogram@returns{\the\toks@ \MFP@afterdef#1{#1}}% }}% \def\MFP@ExportStack{\MFP@Export\MFP@Rstack}% % \end{macrocode} @@ -2514,7 +2508,7 @@ % strictly needed, allowing the parts to be specified as anything \TeX{} % recognizes as a number and allowing any register name. This generality % might reduce efficiency but it simplifies code. Because register -% \texttt{z} is by far the most common one to load, we make more efficient +% \reg{z} is by far the most common one to load, we make more efficient % version of it. % \begin{macrocode} \def\MFP@Rload #1#2#3#4{% @@ -2638,40 +2632,10 @@ % % \bigskip % The user could easily convert between radians and degrees using -% multiplication and/or division. The commands \cs{Rdeg}, \cs{Rrad}, -% etc., aim to be a little more accurate. -% -% \subsection{Additional errors} -% -% These extra commands come with a few possible warnings and errors. -% -% \DescribeMacro{\LogOfZeroInt} -% \DescribeMacro{\LogOfZeroFrac} -% Trying to take the logarithm of $0$ will result in an error message. -% If one allows \TeX{} to continue, the returned value will be negative, -% with an integer part equal to the contents of \cs{LogOfZeroInt} and a -% fractional part equal to the contents of \cs{LogOfZeroFrac}. The -% defaults are both $99999999$. -% -% Trying to take the logarithm of a negative number will produce the -% warning -% \begin{verbatim} -% MFP warning: Log of a negative number is complex. -% Computing real part only. \end{verbatim} -% The log of the absolute value is returned. -% -% Trying to take the square root of a negative number results in an -% error. If you continue after the error message, the value $0$ is -% returned. -% -% \SpecialUsageIndex{\MaxRealInt} -% \SpecialUsageIndex{\MaxRealFrac} -% Trying to take the exponential of a number larger than about $18.42$ -% will cause an error and the number returned has integer part -% $99999999$ and fractional part $99999999$. -% -% Trying to take a negative power of $0$ returns the same as trying -% to divide $1$ by $0$. +% multiplication and/or division. One could similarly convert between +% natural logarithms and base ten logarithms. The commands \cs{Rdeg}, +% \cs{Rrad}, \cs{Rlog} and \cs{Rln} (and their \cs{MFP...} counterparts) +% aim for more accurate results. % % \subsection{Loading the extras} % @@ -2682,10 +2646,34 @@ % \DescribeMacro{\Rexp}\DescribeMacro{\Rsqrt} % \DescribeMacro{\Rpow} % We start \file{mfpextra} with the hook \cs{MFP@Rextra} that -% \cs{starMFPprogram} will call to make available the extra operations -% defined here. +% \cs{startMFPprogram} will call to make available the extra operations +% defined here. If \file{minifp.sty} has been loaded, this macro is +% \cs{@empty}, otherwise it should be undefined. If it is undefined we +% load \file{minifp.sty}. If it is then not \cs{@empty} we assume +% \file{mfpextra.tex} was previously loaded and end input here. % \begin{macrocode} %<*extra> +% check if mfpextra already loaded: +\expandafter\ifx\csname MFP@xfinish\endcsname\relax +\else \expandafter\endinput\fi +\expandafter\edef\csname MFP@xfinish\endcsname{% + \catcode64=\the\catcode64 \space + \catcode46=\the\catcode46 \space + \catcode60=\the\catcode60 \space + \catcode62=\the\catcode62 \space}% +\catcode64=11 % @ +\catcode46=12 % . (period) +\catcode60=12 % < +\catcode62=12 % > +\ifx\MFP@Rextra\UndEfInEd \input minifp.sty \fi +\ifx\MFP@Rextra\@empty +\else + \immediate\write16{mfpextra.tex: already loaded.^^J}% + \MFP@xfinish + \expandafter\endinput +\fi +\immediate\write16{% + mfpextra.tex: extra operations for the MiniFP package.^^J}% \def\MFP@Rextra{% \def\Rcos {\MFP@stack@Unary\MFP@Rcos }% \def\Rsin {\MFP@stack@Unary\MFP@Rsin }% @@ -2720,7 +2708,36 @@ % % \subsection{Error messages} % -% Messages for errors related to impossible powers, roots and logarithms. +% These extra commands come with a few possible new warnings and errors. +% +% \DescribeMacro{\LogOfZeroInt} +% \DescribeMacro{\LogOfZeroFrac} +% Trying to take the logarithm of zero will result in an error message. +% If one allows \TeX{} to continue, the returned value will be negative, +% with an integer part whose absolute value is equal to the contents of +% \cs{LogOfZeroInt} and a fractional part equal to the contents of +% \cs{LogOfZeroFrac}. The defaults are both $99999999$. +% +% Trying to take the logarithm of a negative number will produce the +% warning +% \begin{verbatim} +% MFP warning: Log of a negative number is complex. +% Only the real part will be computed. \end{verbatim} +% The log of the absolute value is returned. +% +% Trying to take the square root of a negative number has similar +% behavior. It produces a warning and returns $0$. +% +% \SpecialUsageIndex{\MaxRealInt} +% \SpecialUsageIndex{\MaxRealFrac} +% Trying to take the exponential of a number larger than about $18.42$ +% will cause an error and the number returned has integer part +% $99999999$ and fractional part $99999999$. +% +% Trying to take a negative power of $0$ produces an error and returns +% the same value as trying to divide $1$ by $0$. +% +% Messages for errors related to impossible powers and logarithms. % \begin{macrocode} \def\MFP@logofzero@err{% \MFP@errmsg{logarithm of zero}% @@ -2731,13 +2748,9 @@ \def\LogOfZeroFrac{\MaxRealFrac}% \def\MFP@expoverflow@err{% \MFP@errmsg{Power too large}% - {The power you tried to calcualate is too large for % + {The power you tried to calculate is too large for % 8 digits. If you continue, ^^Jthe value assigned will be % \MaxRealInt.\MaxRealFrac.}}% -\def\MFP@sqrtofneg@err{% - \MFP@errmsg{square root of a negative number}% - {You tried to take the square root of a negative value. What % - were you thinking? If you continue, zero will be returned.}}% \def\MFP@badpower@err{% \MFP@errmsg{negative power of zero}% {You tried to take a negative power of zero. What were you @@ -2745,10 +2758,26 @@ \xOverZeroInt.\xOverZeroFrac.}}% % \end{macrocode} % +% A debugging utility, \cs{MFPshowreg} displays the contents of a +% register. +% \begin{macrocode} +\def\MFPshowreg #1{% +\ifMFPdebug +\begingroup + \edef\theregister{% + #1 = \expandafter \MFP@Sign + \csname MFP@#1@Sgn\endcsname % + \csname MFP@#1@Int\endcsname.% + \csname MFP@#1@Frc\endcsname}% + \show\theregister +\endgroup +\fi}% +% \end{macrocode} +% % \subsection{Sine and Cosine} % -% For iterated code, the most common register to copy is \texttt{z} and -% the most common place to copy it is to \texttt{x} or \texttt{y} so we +% For iterated code, the most common register to copy is $z$ and +% the most common place to copy it is to $x$ or $y$ so we % make single commands to do those. % \begin{macrocode} \def\MFP@Rcopyz#1{\MFP@Rload {#1}\MFP@z@Sgn\MFP@z@Int\MFP@z@Frc}% @@ -2764,34 +2793,72 @@ % \begin{verbatim} % \MFPdeg\X\Y \MFPsin\Y\S \end{verbatim} % -% Our degree/radian conversions try to be more accurate than a simple -% multiplication by $57.2957 7951$ or $0.0174 5329$. These conversion -% factors are accurate to only eight digits, and the rounding error is -% magnified by multiplication. Thus we will use 16 digits for these -% constants. That is, we multiply first by $57.2957 7951$, then by the -% next eight digits ($.30823208\times 10^{-8}$), performing the ``${}\times -% 10^{-8}$'' by using the (rounded) integer part as the fractional part. -% -% The copying of \texttt{x} to \texttt{t} beforehand is so that we don't have to -% remember which operations (in this case only \cs{MFP@Rmul} and -% \cs{MFP@Radd}) leave register \texttt{x} unchanged. All operations -% defined in \file{mfpextra.tex} overwrite the \texttt{x} register, as do -% a few others. -% \begin{macrocode} -\def\MFP@Rdeg{% - \MFP@Rcopy xt\MFP@Rload y1{57}{29577951}\MFP@Rmul - \MFP@Rcopyz s\MFP@Rcopy tx\MFP@Rload y10{30823209}\MFP@Rmul - \MFP@tempa\MFP@z@Int\relax - \ifnum\MFP@z@Frc<50000000 \else \advance\MFP@tempa 1 \fi - \MFP@Rload x{\ifnum\MFP@tempa>0 \MFP@z@Sgn\else0\fi}0\MFP@tempa - \MFP@Rcopy sy\MFP@Radd}% -\def\MFP@Rrad{% - \MFP@Rcopy xt\MFP@Rload y10{0174 5329}\MFP@Rmul - \MFP@Rcopyz s\MFP@Rcopy tx\MFP@Rload y10{25199433}\MFP@Rmul - \MFP@tempa\MFP@z@Int\relax - \ifnum\MFP@z@Frc<50000000 \else \advance\MFP@tempa 1 \fi - \MFP@Rload x{\ifnum\MFP@tempa>0 \MFP@z@Sgn\else0\fi}0\MFP@tempa - \MFP@Rcopy sy\MFP@Radd}% +% For unit conversions such as radian to degree we try to be more accurate +% than a multiplication by an eight-digit conversion factor allows. +% If $x$ is large and the factor is off by $0.5\times 10^{-8}$, then the +% result can be significantly off. But if we are able to give the +% conversion factor 16 digits precision, then only the imprecision of $x$ +% will significantly affect the result. +% +% We express the conversion factor as an integer part and two eight-digit +% fractional parts. We multiply $x$ by the integer and first fractional +% part (\arg1 and \arg2) with a normal \cs{MFP@Rmul}, but we save the +% underflow digits and undo the rounding that occured at the 8th digit. +% Together these give us an essentially exact result. Then we multiply $x$ +% by the second fractional part (\arg3) and add the saved underflow to the +% result. Finally, we round and add the result to the first product. +% Argument \arg3, as well as the underflow digits, represent numbers less +% than $10^{-8}$, so we effectively scale them up by $10^8$, round the +% result to an integer and scale that back down. +% +% The registers $w$ and $v$ are used to save intermediate results. +% The ``\texttt{DP}'' in \cs{MFP@DPmul} refers to the fact that we are +% multiplying by a ``double precision'' real. +% \begin{macrocode} +\def\MFP@DPmul#1#2#3{% + \ifnum\MFP@x@Sgn=0 + \MFP@Rzero + \else + \MFP@Rcopy xv% + \MFP@Rload y1{#1}{#2}\MFP@Rmul + \edef\MFP@w@Und{\MFP@z@Und}% + \ifnum\MFP@z@Frc@iii>4999 + \MFP@tempa\MFP@z@Frc \advance\MFP@tempa-1 + \edef\MFP@z@Frc{\number\MFP@tempa}% + \makeMFP@eightdigits\MFP@z@Frc + \fi + \MFP@Rcopyz w% + \MFP@Rcopy vx\MFP@Rload y10{#3}\MFP@Rmul + \MFP@Rcopyzx\MFP@Rload y\MFP@v@Sgn 0{\MFP@w@Und}\MFP@Radd + \MFP@tempa\MFP@z@Int\relax + \ifnum\MFP@z@Frc<50000000 \else \advance\MFP@tempa 1 \fi + \ifnum\MFP@tempa<\MFP@ttteight\relax + \MFP@Rload x{\ifnum\MFP@tempa>0 \MFP@z@Sgn\else0\fi}0\MFP@tempa + \else + \MFP@Rload x\MFP@z@Sgn10% + \fi + \MFP@Rcopy wy\MFP@Radd + \fi}% +% \end{macrocode} +% +% Conversion factors: +% \begin{itemize} +% \item radians to degrees: $57.2957795130823209$ +% \item degrees to radians: $0.0174532925199433$ +% \item natural log to common log: $0.4342944819032518$ +% \item common log to natural log: $2.3025850929940457$ +% \end{itemize} +% +% Note that the comparatively large size of the first number means that +% the $\pm0.5\cdot10^{-8}$ imprecision that $x$ implicitly carries will +% be multiplied to approximately $\pm29.6\cdot 10^{-8}$ in the result. +% The only way around this would be to operate with higher precision +% internally. We do that in the code for computing angles. +% \begin{macrocode} +\def\MFP@Rdeg{\MFP@DPmul{57}{29577951}{30823209}}% +\def\MFP@Rrad{\MFP@DPmul{0}{01745329}{25199433}}% +\def\MFP@RbaseX{\MFP@DPmul{0}{43429448}{19032518}}% +\def\MFP@RbaseE{\MFP@DPmul{2}{30258509}{29940457}}% % \end{macrocode} % % There are very few angles that are expressible in eight digits whose sine @@ -2813,71 +2880,59 @@ \MFP@Rcopyzx\MFP@Rsin}% % \end{macrocode} % -% Return $0$ if $x$ is $0$. Then reduce $|x|$ by subtracting $360$ from -% the integer part until it is less than $360$. Of course, -% $\sin x = \sgn(x)\sin(|x|)$ so we only need to compute $\sin(|x|)$. For -% $|x| < 180$, the sign will be that of $x$. For $|x| >= 180$, calculate -% $\sin(|x| - 180)$ and the sign will be opposite that of $x$. Then again -% return $0$ if $x$ is $0$. Finally, call \cs{MFP@@Rsin} to finish. +% Reduce $|x|$ by subtracting $180$ from the integer part until it is less +% than $180$. Of course, $\sin x = \sgn(x)\sin(|x|)$ so we only need to +% compute $\sin(|x|)$. The sign will be that of $x$; each reduction by +% $180$ changes the sign, but the reduction code keeps track of that. If +% $x$ is 0 after the reduction, return zero. % \begin{macrocode} \def\MFP@Rsin{% - \ifnum \MFP@x@Sgn=0 % return 0 - \MFP@Rzero - \else - \MFP@tempa\MFP@x@Int\relax - \MFP@reduce@angle - \ifnum \MFP@tempa<180 - \edef\MFP@sin@Sgn{\MFP@x@Sgn}% - \else - \edef\MFP@sin@Sgn{\number-\MFP@x@Sgn}% - \advance \MFP@tempa-180 - \fi - \MFP@tempb\MFP@x@Frc\relax - \ifnum\MFP@tempa>0 - \MFP@@Rsin - \else\ifnum\MFP@tempb>0 - \MFP@@Rsin - \else - \MFP@Rzero - \fi\fi - \fi}% + \MFP@tempa\MFP@x@Int + \MFP@tempb\MFP@x@Frc + \MFP@tempc\MFP@x@Sgn\relax + \MFP@reduce@angle + \ifnum\MFP@tempa>0 \MFP@@Rsin + \else\ifnum\MFP@tempb>0 \MFP@@Rsin + \else \MFP@Rzero + \fi\fi}% % \end{macrocode} % -% This following reduces $|x|$ to the case $0 <= |x| < 360$. It assumes -% the integer part is in count register \cs{MFP@tempa}. +% This following reduces $|x|$ to the case $0 \le |x| < 180$. It assumes +% the integer part is in count register \cs{MFP@tempa}, the sign in +% \cs{MFP@tempc}. % \begin{macrocode} \def\MFP@reduce@angle{% - \ifnum\MFP@tempa<360 + \ifnum\MFP@tempa<180 \else - \advance\MFP@tempa-360 + \advance\MFP@tempa-180 + \MFP@tempc-\MFP@tempc \@xp\MFP@reduce@angle \fi}% % \end{macrocode} % % At this point, $|x|$ is represented by \cs{MFP@tempa} (integer part) and -% \cs{MFP@tempb} (fractional part). Also, we already know the sign, stored -% in \cs{MFP@sin@Sgn}. Moreover $0 < {}$\cs{MFP@tempa}${} < 180$. We now +% \cs{MFP@tempb} (fractional part). Also, we already know the sign stored +% in \cs{MFP@tempc}. Moreover $0 < {}$\cs{MFP@tempa}${} < 180$. We now % reduce to $0 < |x| \le 90$ using $\sin(x) = \sin(180-|x|)$, and return % $1$ if equal to $90$. % % The calculation of $180-x$ is optimized, taking advantage of the fact % that both $x$ and the result are known to be positive. If the fractional -% part is positive, we would normally borrow $1$ by reducing $180$, but -% instead we increase the integer part of $x$ by one. +% part is positive, we borrow $1$ by reducing $180$ to $179$. % \begin{macrocode} \def\MFP@@Rsin{% \ifnum\MFP@tempa<90 \else + \MFP@tempa -\MFP@tempa \ifnum\MFP@tempb>0 - \advance\MFP@tempa 1 \MFP@tempb -\MFP@tempb \advance\MFP@tempb \MFP@ttteight\relax + \advance\MFP@tempa 179 + \else \advance\MFP@tempa 180 \fi - \MFP@tempa -\MFP@tempa - \advance\MFP@tempa 180 \fi \ifnum\MFP@tempa=90 - \MFP@Rloadz \MFP@sin@Sgn10% + \MFP@Rloadz \MFP@tempc10% \else % \end{macrocode} % @@ -2900,11 +2955,12 @@ % computations amount to concatenating the top six digits of % \cs{MFP@tempb} to the digits of \cs{MFP@tempa}. This will produce the % integer form of the fractional part of $x/100$ (the integer part of -% $x/100$ is $0$). +% $x/100$ is zero). % % Division by $100$ can turn a number into $0$. This is one place we can -% lose accuracy in the last digit of the result. In compensation, the rest -% of the calculations become extremely accurate. +% lose accuracy (up to $\pm1$ in the last digit of the result). In +% compensation, the rest of the calculations become very much more +% accurate. % \begin{macrocode} \advance\MFP@tempb 50 \divide\MFP@tempb 100 \multiply\MFP@tempa 1000000 \advance\MFP@tempb\MFP@tempa @@ -2915,43 +2971,34 @@ % % We save some multiplications by working with $t=x^2$. As we don't need % the original $x$ anymore, we simply replace it with the newly reduced -% value. We also save this reduced $x$ in another register, \texttt{s}, as +% value. We also save this reduced $x$ in another register, $s$, as % we will need it again at the end, and our intermediate calculations do -% not preserve the \texttt{x} register. Then we square $x$ and, if that +% not preserve the $x$ register. Then we square $x$ and, if that % square is $0$ we can skip all the power series and simply return $x$ -% converted to radians (that's the last multiplication). If $x^2$ is not -% $0$, we save it in temporary register \texttt{t} and call our power -% series. When this program is finished, all that remains is a final -% multiplication by a conversion factor\dots +% converted to radians. If $x^2$ is not zero, we save it in temporary +% register $t$ and call our power series. When this program is +% finished, all that remains is the final multiplication by a conversion +% factor (\cs{MFP@DPmul}). % \begin{macrocode} - \MFP@Rload x10\MFP@tempb - \MFP@Rcopy xs% + \MFP@Rload s\MFP@tempc0\MFP@tempb + \MFP@Rcopy sx% \MFP@Rsq \ifnum \MFP@z@Frc>0 \MFP@Rcopyz t\MFP@Rsin@prog \else \MFP@Rcopy sx% \fi - \MFP@Rload y11{74532925}\MFP@Rmul -% \end{macrocode} -% \dots except this fiddle about the sign. Theoretically, all cases -% where $\sin x$ can be $0$ were previously weeded out. However, I am not -% 100 percent certain that rounding in \cs{MFP@Rsin@prog} will never -% lead to a value of $0$. -% \begin{macrocode} - \ifnum\MFP@z@Sgn=0 \else - \let\MFP@z@Sgn\MFP@sin@Sgn - \fi + \MFP@DPmul 1{74532925}{19943296}% \fi \fi}% % \end{macrocode} % % \cs{MFP@Rsin@prog} is the power series computation. The power series -% need only go to eight terms as the ninth would be less than $.5*10^{-8}$ and -% so our 8-place computations would return $0$. Our 8-term series is: +% need only go to the $x^{13}$ term as the next is less than $10^{-9}$ and +% in our 8-place computations is indistingushable from $0$. Our series is: % $$ % rx(1 - r^2t/3! + r^4t^2/5! - r^6t^3/7! + r^8t^4/9! - r^{10}t^5/11! + -% r^{12}t^6/13! - r^{14}t^7/15!) +% r^{12}t^6/13!) % $$ % where $r$ is the factor that converts $x$ to radian measure % (hectodegrees to radians). When $x$ is so small as to produce $t = 0$ we @@ -2959,32 +3006,37 @@ % % We minimize any multiplications of tiny numbers by computing this as % $$ -% r(1 - gt(1 - ft(1 - et(1 - dt(1 - ct(1 - bt(1 - at))))))). +% rx(1 - ft(1 - et(1 - dt(1 - ct(1 - bt(1 - at)))))). % $$ -% Now $r = 1.74532925$ and $a$, $b$, etc., have formulas: +% In this format, additional terms might actually make a difference, +% because $at$ is not particularly small. However, the more computations +% we have, the more errors accumulate. Therefore we take the fewest that +% produce acceptable accuracy. +% +% Now $r = 1.7453292519943296$ and $a$, $b$, etc., have formulas: % $$ % \vcenter{\centering -% $\displaystyle a = r^2/15/14,\ b = r^2/13/12,\ c = r^2/11/10,\ d = r^2/9/8$,\\ -% $\displaystyle e = r^2/7/6,\ f = r^2/5/4,\ g = r^2/3/2$.\par +% $\displaystyle a = r^2/13/12,\ b = r^2/11/10,\ c = r^2/9/8$,\\ +% $\displaystyle d = r^2/7/6,\ e = r^2/5/4,\ f = r^2/3/2$.\par % } % $$ % An alternative method would be to accumulate a sum, computing each term % from the previous one (e.g., if $u = t^3/7!$ is the fourth term, the next % one is $u*t*(1/(8*9))$). This is a bit more complicated to code and requires % moving values around more. It would have the advantage that we can stop -% whenever a term evaluates to $0$, making computation faster for small -% values of $x$. +% whenever a term evaluates to zero, making computation faster for small +% values of $x$. I have not determined whether it would compromise +% accuracy. % % We avoid divisions by precomputing the coefficients $a$, $b$, $c$, etc. % Note that without the reduction in $x$, the value of $a$ for example -% would be $0.00000145$, with only three significant figures of accuracy. +% would be $0.00000195$, with only three significant figures of accuracy. % Now we can have seven, and the accuracy is more-or-less determined by that % of the reduced x. % $$ % \vcenter{\centering -% $\displaystyle a = 0.01450559,\ b = 0.01952675,\ c = 0.02769249$,\\ -% $\displaystyle d = 0.04230797,\ e = 0.07252796,\ f = 0.15230871$,\\ -% $\displaystyle g = 0.50769570$.\par +% $\displaystyle a = 0.01952676,\ b = 0.02769249,\ c = 0.04230797,$,\\ +% $\displaystyle d = 0.07252796,\ e = 0.15230871,\ f = 0.50769570$.\par % } % $$ % It is important to note that the following operations step all over @@ -2995,28 +3047,20 @@ % previous operation. Instead of simply subtracting, we optimize based % on the fact that $z$ is known to be nonnegative and not larger than $1$. % -% The macro \cs{MFP@com@iter} `flips' the previous result then multiplies +% The macro \cs{MFP@com@iter} `flipz' the previous result then multiplies % by $t$ and the indicated coefficient. (The name of this macro stands for % ``common iterated'' code; it is reused for some other power series.) % % For extra efficiency, the power series uses a ``small'' version of % multiplication \cs{MFP@Rsmul}, used only when the factors are sure to -% lie in $[0,1]$. -% -% Despite what I said above, our chosen method of computation has a -% slightly improved accuracy (in numerical experiments) if we take it one -% step further, but adding only half the last term. This splits the -% difference between stopping at the 15th power or continuing to the 17th. -% It has provably better worst-case accuracy, but on average, who knows? -% We are right at the edge of our 8-digit accuracy anyway. The constant -% \texttt{00559959} corresponds to half of $r^2/16/17$. +% lie in $[0,1]$. This does not take into account the sign of $x$, +% whence the ending \cs{edef}. % \begin{macrocode} \def\MFP@Rsin@prog{% - \MFP@Rcopy tx% - \MFP@Rload y10{00559959}\MFP@Rsmul\MFP@com@iter{01450559}% - \MFP@com@iter{01952675}\MFP@com@iter{02769249}\MFP@com@iter{04230797}% - \MFP@com@iter{07252796}\MFP@com@iter{15230871}\MFP@com@iter{50769570}% - \MFP@flipz \MFP@Rcopyzx \MFP@Rcopy sy\MFP@Rsmul \MFP@Rcopyzx}% + \MFP@Rcopy tx\MFP@Rload y10{01952676}\MFP@Rsmul% + \MFP@com@iter{02769249}\MFP@com@iter{04230797}\MFP@com@iter{07252796}% + \MFP@com@iter{15230871}\MFP@com@iter{50769570}\MFP@flipz \MFP@Rcopyzx + \MFP@Rcopy sy\MFP@Rsmul\MFP@Rcopyzx\edef\MFP@x@Sgn{\MFP@s@Sgn}}% \def\MFP@flipz{% \ifnum\MFP@z@Sgn=0 \MFP@Rloadz 110% @@ -3053,18 +3097,16 @@ % multiplication could round to $0$ and then, after subtraction, a $1$ % could occur. We handle those easy cases separately, so that in % \cs{MFP@@Rsmul} we don't have to worry about the integer parts at all. +% +% Also, since these are completely internal, we don't even define the +% overflow and underflow macros. % \begin{macrocode} \def\MFP@Rsmul{% - \ifnum \MFP@x@Sgn=0 - \MFP@Rzero - \else\ifnum \MFP@y@Sgn=0 - \MFP@Rzero - \else\ifnum\MFP@x@Int>0 - \MFP@Rcopy yz% - \else\ifnum\MFP@y@Int>0 - \MFP@Rcopy xz% - \else - \MFP@@Rsmul + \ifnum \MFP@x@Sgn=0 \MFP@Rzero + \else\ifnum \MFP@y@Sgn=0 \MFP@Rzero + \else\ifnum\MFP@x@Int>0 \MFP@Rcopy yz% + \else\ifnum\MFP@y@Int>0 \MFP@Rcopy xz% + \else \MFP@@Rsmul \fi\fi\fi\fi}% \def\MFP@@Rsmul{% \MFP@split\MFP@x@Frc\MFP@x@Frc@i\MFP@x@Frc@ii @@ -3084,15 +3126,11 @@ \advance\MFP@tempb1 \edef\MFP@z@Frc@ii{\number\MFP@tempb}\fi \MFP@carrym\MFP@z@Frc@ii\MFP@z@Frc@i - \makeMFP@fourdigits\MFP@z@Frc@iv - \makeMFP@fourdigits\MFP@z@Frc@iii \makeMFP@fourdigits\MFP@z@Frc@ii \makeMFP@fourdigits\MFP@z@Frc@i \def\MFP@z@Int{0}% \edef\MFP@z@Frc{\MFP@z@Frc@i\MFP@z@Frc@ii}% - \edef\MFP@z@Sgn{\ifnum\MFP@z@Frc=0 0\else 1\fi}% - \edef\MFP@z@Und{\MFP@z@Frc@iii\MFP@z@Frc@iv}% - \edef\MFP@z@Ovr{0}}% + \edef\MFP@z@Sgn{\ifnum\MFP@z@Frc=0 0\else 1\fi}}% % \end{macrocode} % % \subsection{Polar angle} @@ -3101,7 +3139,8 @@ % more general angle function. This is a binary operation that accepts % the two coordinates of a point and computes its angle in polar % coordinates. One then has, for example, $\arctan x = -% \mathop{\rm angle}(1,x)$ and $\arccos x = \angle(x, \sqrt{1-x^2})$. +% \mathop{\mathrm{angle}}(1,x)$ and $\arccos x = \mathop{\mathrm{angle}} +% (x, \sqrt{1-x^2})$. % % We start, as usual, with a few reductions. When the $y$-part is $0$, we % immediately return $0$ or $180$. If the $y$-part is negative, we compute @@ -3109,11 +3148,12 @@ % compute the angle for $|x|$ and subtract it from $180$. Finally, % reduced to both coordinates positive, if $y>x$ we compute the angle of % $(y,x)$ and subtract that from $90$. Ultimately, we apply a power -% series formula for $\angle(1,y/x)$ and get convergence when the -% argument is less than $1$, but convergence is poor unless the argument -% is less than $1/2$. When that is not the case, conceptually, we rotate -% the picture clockwise by the arctangent of $1/2$, compute the angle of -% the new point and then add a precomputed value of $\arctan(1/2)$. +% series formula for $mathop{\mathrm{angle}}(1,y/x)$ and get convergence +% when the argument is less than $1$, but convergence is poor unless the +% argument is less than $1/2$. When that is not the case, conceptually, we +% rotate the picture clockwise by the arctangent of $1/2$, compute the +% angle of the new point and then add a precomputed value of +% $\arctan(1/2)$. % \begin{macrocode} \def\MFP@Rangle{% \ifcase\MFP@y@Sgn\relax @@ -3139,18 +3179,16 @@ \def\MFP@x@Sgn{1}\MFP@@Rangle \MFP@Rcopyzy\MFP@Rload x1{180}0\MFP@Rsub \fi - \let\MFP@z@Sgn\MFP@angle@Sgn - }}% + \let\MFP@z@Sgn\MFP@angle@Sgn}}% \def\MFP@@Rangle{% \MFP@Rcmp \ifMFP@neg - \MFP@Rcopy xs\MFP@Rcopy yx\MFP@Rcopy sy% + \MFP@Rcopy xw\MFP@Rcopy yx\MFP@Rcopy wy% \MFP@@@Rangle \MFP@Rload x1{90}0\MFP@Rcopyzy\MFP@Rsub \else \MFP@@@Rangle - \fi -}% + \fi}% % \end{macrocode} % % Precisely what we do when we are finally in the case $0<y<x$ is perform @@ -3167,62 +3205,56 @@ % If we still have $y/x > 1/4$, we perform $(x'',y'') = (4x + y, 4y - x)$, % which then satisfies $y''/x'' \le 1/4$. When either of these % transformations is performed, we add the corresponding angle to the -% ``angle-so-far'' in register \texttt{a}. +% ``angle-so-far'' in register $a$. % % We could continue this iteration 32 times to get (theoretically) the % angle in degrees to $\pm 10^{-8}$. That seems a bit long, plus the % accumulation of errors over $32$ iterations could (in the worst case) % produce less than $\pm10^{-7}$ accuracy. % -% To get the accuracy we need we work in ``scaled reals''. That is, we -% get 10 decimal places of accuracy by letting two digits of the integer -% part represent the first two digits after the decimal point, and the -% eight digits of the fractional part represent digits 3 through 10 after the -% point. The macro \cs{MFP@RmulC} (around line 19 of the definition of -% \cs{MFP@@@Rangle}) is a quick multiplication by 100, converting the -% argument of the arctangent command to a scaled real. -% -% Since we ultimately take the arctangent of $y/x$ we get best accuracy if -% $y$ is first scaled, which means it must necessarily be smaller than -% $10^6$ to start with. Our initial reductions can double its size, so we -% first make sure it is smaller than $500\,000$, dividing by $200$. This -% can actually reduce accuracy, but as it is only done when $y$ already -% has $14$ significant figures, and since the quotient $100y/x$ has at -% most $10$, the loss is not significant. +% To get the accuracy we need, we work in ``scaled reals''. That is, we +% get 10 effective decimal places of accuracy by letting an $x$ in the +% range $0< x < 100$ stand for $0< x/100 < 1$. +% +% Our initial reductions can increase $x$ by a factor of about 13. +% Moreover, we ultimately need to scale y by 100 when we convert to +% scaled computations. Thus, if we make sure $x$ is less than +% $1\,000\,000$, we will prevent overflow in both cases. % \begin{macrocode} \def\MFP@Rquad{\MFP@Rdbl\MFP@Rcopyzx\MFP@Rdbl}% \def\MFP@@@Rangle{% \MFP@Rcopy xs\MFP@Rcopy yt% - \ifnum\MFP@y@Int<500000 + \ifnum\MFP@x@Int<1000000 \else - \MFP@Rload y1{200}0\MFP@Rdiv - \MFP@Rcopyz s\MFP@Rcopy tx% - \MFP@Rload y1{200}0\MFP@Rdiv - \MFP@Rcopyz t% + \MFP@RdivC \MFP@Rcopyz s% + \MFP@Rcopy tx\MFP@RdivC \MFP@Rcopyz t% \fi - \MFP@Rcopy tx\MFP@Rdbl\MFP@Rcopyzx\MFP@Rcopy sy\MFP@Rcmp - \ifMFP@pos - \MFP@Rsub\MFP@Rcopyz u\MFP@Rcopy sx\MFP@Rdbl - \MFP@Rcopyzx\MFP@Rcopy ty\MFP@Radd - \MFP@Rcopyz s\MFP@Rcopy ut% - \MFP@Rload a1{2656}{50511771}% + \ifnum\MFP@t@Sgn=0 \MFP@Rzero \else - \MFP@Rload a000% - \fi - \MFP@Rcopy tx\MFP@Rquad\MFP@Rcopyzx\MFP@Rcopy sy\MFP@Rcmp - \ifMFP@pos - \MFP@Rsub\MFP@Rcopyz u\MFP@Rcopy sx\MFP@Rquad - \MFP@Rcopyzx\MFP@Rcopy ty\MFP@Radd - \MFP@Rcopyz s\MFP@Rcopy ut% - \MFP@Rcopy ax\MFP@Rload y1{1403}{62434679}% - \MFP@Radd\MFP@Rcopy za% - \fi - \MFP@Rcopy tx\MFP@RmulC - \MFP@Rcopyzx\MFP@Rcopy sy\MFP@Rdiv - \MFP@Rcopyzx\MFP@Ratanc - \MFP@Rcopyzx\MFP@Rdeg - \MFP@Rcopyzx\MFP@Rcopy ay\MFP@Radd - \MFP@Rcopyzx\MFP@RdivC}% + \MFP@Rcopy tx\MFP@Rdbl\MFP@Rcopyzx\MFP@Rcopy sy\MFP@Rcmp + \ifMFP@pos + \MFP@Rsub\MFP@Rcopyz u\MFP@Rcopy sx\MFP@Rdbl + \MFP@Rcopyzx\MFP@Rcopy ty\MFP@Radd + \MFP@Rcopyz s\MFP@Rcopy ut% + \MFP@Rload a1{2656}{50511771}% + \else + \MFP@Rload a000% + \fi + \MFP@Rcopy tx\MFP@Rquad\MFP@Rcopyzx\MFP@Rcopy sy\MFP@Rcmp + \ifMFP@pos + \MFP@Rsub\MFP@Rcopyz u\MFP@Rcopy sx\MFP@Rquad + \MFP@Rcopyzx\MFP@Rcopy ty\MFP@Radd + \MFP@Rcopyz s\MFP@Rcopy ut% + \MFP@Rcopy ax\MFP@Rload y1{1403}{62434679}% + \MFP@Radd\MFP@Rcopy za% + \fi + \MFP@Rcopy tx\MFP@RmulC + \MFP@Rcopyzx\MFP@Rcopy sy\MFP@Rdiv + \MFP@Rcopyzx\MFP@Ratanc + \MFP@Rcopyzx\MFP@Rdeg + \MFP@Rcopyzx\MFP@Rcopy ay\MFP@Radd + \MFP@Rcopyzx\MFP@RdivC + \fi}% % \end{macrocode} % % Here are fast multiplication and division by 100. We need these because @@ -3232,25 +3264,28 @@ % division by 100: $.5\times.5 = .25$ would be computed as $(50\times50) / % 100 = 25$. % \begin{macrocode} +\def\MFP@twoofmany#1#2#3\MFP@end{#1#2}% +\def\MFP@gobbletwo#1#2{}% \def\MFP@RmulC{% - \makeMFP@eightdigits\MFP@x@Frc - \edef\MFP@Tmp{\number\MFP@x@Int.\MFP@x@Frc}% - \@xp\MFP@@RmulC\MFP@Tmp\mfp@end}% -\def\MFP@@RmulC#1.#2#3#4\mfp@end{% - \MFP@Rloadz\MFP@x@Sgn{#1#2#3}{#400}}% + \edef\MFP@z@Int{\MFP@x@Int\@xp\MFP@twoofmany\MFP@x@Frc\MFP@end}% + \edef\MFP@z@Frc{\@xp\MFP@gobbletwo\MFP@x@Frc00}% + \edef\MFP@z@Sgn{\MFP@x@Sgn}}% \def\MFP@RdivC{% \makeMFP@eightdigits\MFP@x@Int \makeMFP@eightdigits\MFP@x@Frc - \@XP\MFP@@RdivC\@xp\MFP@x@Int\MFP@x@Frc\mfp@end}% + \@XP\MFP@@RdivC\@xp\MFP@x@Int\MFP@x@Frc\MFP@end}% \def\MFP@@RdivC#1#2#3#4#5#6{% \edef\MFP@z@Int{\number#1#2#3#4#5#6}% \MFP@@@RdivC}% -\def\MFP@@@RdivC#1#2#3#4#5#6#7#8#9\mfp@end{% +\def\MFP@@@RdivC#1#2#3#4#5#6#7#8#9\MFP@end{% \MFP@tempa#1#2#3#4#5#6#7#8\relax \ifnum#9>49 \advance\MFP@tempa1 \fi + \edef\MFP@z@Frc{\number\MFP@tempa}% + \makeMFP@eightdigits\MFP@z@Frc \edef\MFP@z@Sgn{\MFP@x@Sgn}% - \ifnum\MFP@tempa=0 \ifnum\MFP@x@Int=0 \def\MFP@z@Sgn{0}\fi\fi - \MFP@Rloadz\MFP@z@Sgn\MFP@z@Int\MFP@tempa}% + \ifnum\MFP@tempa=0 + \ifnum\MFP@z@Int=0 \def\MFP@z@Sgn{0}\fi + \fi}% % \end{macrocode} % % Finally, we compute the arctan of a scaled real producing a result @@ -3264,12 +3299,12 @@ % the sum % $$ % x\left(1 - \frac{u}{3} + \frac{u^2}{5} - \frac{u^3}{7} + \cdots -% - \frac{u^7}{2\cdot 15}\right), +% - \frac{u^7}{15}\right), % $$ % where $u = x^2$. % % We start with the common iterated code. It assumes a scaled value in x -% to be multiplied by the saved (scaled) value of $x^2$ (in register u) +% to be multiplied by the saved (scaled) value of $x^2$ (in register $u$) % and by a coefficient (supplied in separate integer and fractional % parts). It ends with the new value in x. % \begin{macrocode} @@ -3280,22 +3315,25 @@ \MFP@Rcopyzy\MFP@Rload x1{100}{00000000}% \MFP@Rsub\MFP@Rcopyzx}% \def\MFP@Ratanc{% - \MFP@Rcopy xs% - \MFP@Rcopy xy\MFP@scaledmul\MFP@Rcopyz u% - \MFP@Rcopyzx\MFP@Rload y1{86}{66666667}% - \MFP@scaledmul - \MFP@Rcopyzy\MFP@Rload x1{100}{00000000}% - \MFP@Rsub\MFP@Rcopyzx - \MFP@atan@iter{84}{61538462}\MFP@atan@iter{81}{81818182}% - \MFP@atan@iter{77}{77777778}\MFP@atan@iter{71}{42857143}% - \MFP@atan@iter{60}{00000000}\MFP@atan@iter{33}{33333333}% - \MFP@Rcopy sy\MFP@scaledmul}% + \MFP@Rcopy xs\MFP@Rcopy xy\MFP@scaledmul + \ifnum \MFP@z@Sgn=0 + \MFP@Rcopy sz% + \else + \MFP@Rcopyz u\MFP@Rcopyzx + \MFP@Rload y1{86}{66666667}\MFP@scaledmul + \MFP@Rcopyzy\MFP@Rload x1{100}{00000000}\MFP@Rsub\MFP@Rcopyzx + \MFP@atan@iter{84}{61538462}\MFP@atan@iter{81}{81818182}% + \MFP@atan@iter{77}{77777778}\MFP@atan@iter{71}{42857143}% + \MFP@atan@iter{60}{00000000}\MFP@atan@iter{33}{33333333}% + \MFP@Rcopy sy\MFP@scaledmul + \fi}% % \end{macrocode} % % \subsection{Logarithms} % -% Now for logarithms. We are going to compute a base 10 logarithm. This -% allows the first step of the calculation to be essentially trivial: to +% Now for logarithms. We are going to compute both common logarithms +% (base $10$) and natural logarithms (base $e$). The first step of the +% calculation is be essentially trivial and works with base 10: to % get the integer part of the log for numbers with positive integer part, % count the digits in the integer part and subtract $1$. For numbers less % than one, count the number of zeros at the beginning of the fractional @@ -3306,18 +3344,27 @@ % an acceptable number of of terms. Then we proceed as in the code for % sine. % -% We keep the value-so-far in register \texttt{s} and the modified -% $x$-value in register \texttt{t}. The log of \texttt{t} need only be -% added to register \texttt{s} for the final result. +% The power series produces a logarithm in base $e$ so we ultimately get +% the answer in two parts, with the parts calculated for different bases. +% The last step is to multiply the second part by a conversion factor and +% add the first to it. For natural log, convert the first and add the +% second. Which one is to be returned is passed as a boolean +% +% We keep the value-so-far in register $s$ and the modified +% $x$-value in register $t$. % \begin{macrocode} -\def\MFP@Rlog{% +\newif\ifMFP@natural +\def\MFP@Rlog{\MFP@naturalfalse\MFP@Rlog@}% +\def\MFP@Rln{\MFP@naturaltrue\MFP@Rlog@}% +\def\MFP@Rlog@{% \ifnum\MFP@x@Sgn=0 \MFP@logofzero@err \MFP@Rloadz{-1}\LogOfZeroInt\LogOfZeroFrac \else \ifnum \MFP@x@Sgn<0 - \MFP@warn{The logarithm of a negative number is complex.% + \MFP@warn{The logarithm of a negative number is complex. \MFP@msgbreak Only the real part will be computed}% + \def\MFP@x@Sgn{1}% \fi \MFP@Rload s000% % \end{macrocode} @@ -3326,115 +3373,113 @@ % number of places that will be shifted in \cs{MFP@tempa}. We use % \cs{number} to strip the leading zeros and (essentially) we count % the number of digits that remain. Then we shift left, putting the first -% digit into the integer part of \texttt{s} and the rest into the -% fractional part. Despite its name, \cs{MFP@numzeros} actually returns -% one more than the number of zeros, which is the number of places to -% shift, and the integer part of the logarithm. +% digit into the integer part of \reg{s} and the rest into the +% fractional part. % \begin{macrocode} \ifnum \MFP@x@Int=0 \edef\MFP@x@Tmp{\number\MFP@x@Frc}% - \MFP@tempa=\MFP@numzeros\MFP@x@Tmp\relax + \MFP@tempa=\MFP@numshiftL\MFP@x@Tmp\relax \def\MFP@s@Sgn{-1}% - \edef\MFP@t@Int{\@xp\MFP@firstofmany\MFP@x@Tmp\mfp@end}% + \edef\MFP@t@Int{\@xp\MFP@oneofmany\MFP@x@Tmp\MFP@end}% \edef\MFP@t@Frc{\@xp\@gobble\MFP@x@Tmp0}% - \MFPpadto@eight\MFP@t@Frc + \MFP@padtoeight\MFP@t@Frc \else % \end{macrocode} % When the integer part is not $0$, we get the number of digits to -% shift again in \cs{MFP@tempa}. We actually want one less than the -% number of digits, so that is what \cs{MFP@numdigits} actually produces. +% shift again in \cs{MFP@tempa}. It will be one less than the number of +% integer digits. % \begin{macrocode} - \MFP@tempa\MFP@numdigits\MFP@x@Int - \edef\MFP@x@Tmp{\MFP@x@Int\MFP@x@Frc}% gather all digits - \edef\MFP@s@Sgn{1}% - \edef\MFP@t@Int{\@xp\MFP@firstofmany\MFP@x@Tmp\mfp@end}% + \MFP@tempa\MFP@numshiftR\MFP@x@Int + \edef\MFP@x@Tmp{\MFP@x@Int\MFP@x@Frc}% + \ifnum\MFP@tempa>0 \def\MFP@s@Sgn{1}\fi + \edef\MFP@t@Int{\@xp\MFP@oneofmany\MFP@x@Tmp\MFP@end}% \edef\MFP@x@Tmp{\@xp\@gobble\MFP@x@Tmp}% - \edef\MFP@t@Frc{\@xp\MFP@firsteightofmany\MFP@x@Tmp\mfp@end}% + \edef\MFP@t@Frc{\@xp\MFP@eightofmany\MFP@x@Tmp\MFP@end}% \fi % \end{macrocode} % -% The integer part of $\log x$ is now known, so save it in value-so-far. -% Also, set the sign of the reduced argument to positive. Then call -% \cs{MFP@log@reduce}, which reduces $x$ to at most $10^{1/16} \approx -% 1.155\,$. Finally, if the reduced $x$ is $1$, return the value so far, -% otherwise call the power series program. +% Now the integer part of $\log_{10} x$ is known. We save it in $s$ +% Also, set the sign of the reduced argument (positive). Then call +% \cs{MFP@Rlog@reduce}, which reduces $x$ to less than $1.161\,$ while +% possibly increasing $s$. For the natural log, we convert the value in +% $s$. +% +% If the reduced $x$ is $1$, return the value in $s$, otherwise call the +% power series program (discarding the integer part of $t$, which should +% be a $1$). Finally, convert the returned result if necessary and add +% register $s$ to it. % \begin{macrocode} \edef\MFP@s@Int{\number\MFP@tempa}% \def\MFP@t@Sgn{1}% \MFP@Rlog@reduce + \ifMFP@natural \MFP@Rcopy sx\MFP@RbaseE \MFP@Rcopy zs\fi \ifnum\MFP@t@Frc=0 \MFP@Rcopy sz% \else - \MFP@Rlog@prog + \def\MFP@t@Int{0}\MFP@Rlog@prog + \ifMFP@natural\else \MFP@Rcopyzx \MFP@RbaseX \fi + \MFP@Rcopy sy\MFP@Rcopyzx\MFP@Radd \fi \fi}% -\def\showreg #1{% for debugging -\begingroup - \edef\reg{% - #1 = (\csname MFP@#1@Sgn\endcsname)% - \csname MFP@#1@Int\endcsname.% - \csname MFP@#1@Frc\endcsname}% - \show\reg -\endgroup}% % \end{macrocode} % -% We count the number of digits by lining them up, followed by the -% possible numbers, and picking out the ninth argument. We count leading -% zeros in the fractional part by removing leading zeros, lining up the -% remaining digits, followed by the possible numbers, and again picking -% the ninth. In \cs{MFP@Rlog@reduce} we divide by the square root of 10 if -% the number is larger than that (adding $.5$ to value-so-far). We repeat -% with the 4th, 8th and 16th roots. For speed, instead of dividing, we -% actually multiply by the precalculated reciprocal, passed as the third -% argument. +% We determine the size of a right shift by lining up the digits in +% the integer part, followed by the possible numbers, and picking out the +% ninth argument. Similarly, to get a left shift we line up the digits +% of the fractional part (minus the leading zeros) followed by the +% possible numbers, and again picking the ninth. +% \begin{macrocode} +\def\MFP@numshiftR#1{\@xp\MFP@ninthofmany#176543210\MFP@end}% +\def\MFP@numshiftL#1{\@xp\MFP@ninthofmany#112345678\MFP@end}% +% \end{macrocode} +% +% In \cs{MFP@Rlog@reduce} we divide by the square root of 10 if the number +% is significantly larger than that (adding $.5$ to value-so-far). We +% repeat with the 4th, 8th and 16th roots. It seems that this could be +% where errors can accumulate, so the divisions are done with double +% precision multiplication and $x$ is scaled by 100. Our check whether +% $x > \sqrt{10}$ is rather rough: comparing the first three digits only, +% but even in the worst case, the final $x$ is less than $1.1605$, so at +% most $0.161$ is fed to the power series. % \begin{macrocode} -\def\MFP@numdigits#1{\@xp\MFP@ninthofmany#176543210\mfp@end}% -\def\MFP@numzeros#1{% - \@xp\MFP@ninthofmany#112345678\mfp@end}% \def\MFP@Rlog@reduce{% - \MFP@reduceonce 3{16227766}{31622777}{50000000}% - \MFP@reduceonce 1{77827941}{56234133}{25000000}% - \MFP@reduceonce 1{33352143}{74989421}{12500000}% - \MFP@reduceonce 1{15478198}{86596433}{06250000}}% + \MFP@Rcopy tx\MFP@RmulC\MFP@Rcopyz t% + \MFP@reduceonce {316}{31622776}{60168379}{50000000}% + \MFP@reduceonce {177}{56234132}{51903491}{25000000}% + \MFP@reduceonce {133}{74989420}{93324558}{12500000}% + \MFP@reduceonce {115}{86596432}{33600654}{06250000}% + \MFP@Rcopy tx\MFP@RdivC\MFP@Rcopyz t}% \def\MFP@reduceonce#1#2#3#4{% - \MFP@Rcopy tx\MFP@Rload y1{#1}{#2}\MFP@Rcmp - \ifMFP@neg\else - \MFP@Rload y10{#3}\MFP@Rmul - \MFP@Rcopyz t\MFP@Rcopy sx\MFP@Rload y10{#4}\MFP@Radd + \ifnum\MFP@t@Int>#1\relax + \MFP@Rcopy tx% + \MFP@DPmul 0{#2}{#3}\MFP@Rcopyz t% + \MFP@Rcopy sx\MFP@Rload y10{#4}\MFP@Radd \MFP@Rcopyz s% \fi}% % \end{macrocode} % -% Now we have a value for $t$ of the form $1 + u$ with $0\le u < 0.155$ -% (approx.) We will use the formula +% Now we have a value for $t$ of the form $1 + u$ with $0\le u < 0.161$. +% We will use the formula % $$ -% \log (1 + u) = \frac{1}{\ln 10} \sum_{n=0}^\infty (-1)^n \frac{u^{n+1}}{n+1}. +% \ln (1 + u) = \sum_{n=1}^\infty (-1)^{n-1} \frac{u^n}{n}. % $$ -% We only need to carry it far enough to assure that the next term would -% be $0$ in our finite resolution arithmetic, that is $.155^{k}/k/\ln10 < .5\times -% 10^{-8}$. This is satisfied by $k=9$, so we only need eight terms. +% We only need to carry it far enough to assure that the remaining terms +% would be zero in our finite resolution arithmetic, that is +% $(.161)^k/k < .5\times 10^{-8}$. This is satisfied by $k=10$. +% So we carry the sum to 9 places. % % Again, we compute this by % $$ -% ru(1 - au(1 - bu(1-cu(1-du(1-eu(1-fu(1-gu(1-hu)))))))) +% u(1-au(1-bu(1-cu(1-du(1-eu(1-fu(1-gu(1-hu)))))))) % $$ -% where $a= 1/2$, $b = 2/3$,\dots,$h=7/8/2$, and $r = 1/ln 10 = \log e$. -% This allows us to reuse \cs{MFP@com@iter}. -% -% The for the natural log $\ln x$ we multiply the common logarithm by -% $\ln 10$. It seems we are redundantly canceling the earlier division -% by $\ln 10$, but remember we are also multiplying the integer part, -% which did not have such a division in its code. +% where $a= 1/2$, $b = 2/3$,\dots, $g=7/8$, and $h=8/9$ +% This arrangement allows us to reuse \cs{MFP@com@iter}. % \begin{macrocode} \def\MFP@Rlog@prog{% - \MFP@Rcopy tx\MFP@Rdecr - \MFP@Rcopyz t% - \MFP@Rcopyzx\MFP@Rload y10{43750000}\MFP@Rsmul - \MFP@com@iter{85714286}\MFP@com@iter{83333333}\MFP@com@iter{80000000}% - \MFP@com@iter{75000000}\MFP@com@iter{66666667}\MFP@com@iter{50000000}% - \MFP@com@iter{43429448}\MFP@Rcopyzx\MFP@Rcopy sy\MFP@Radd}% -\def\MFP@Rln{% - \MFP@Rlog\MFP@Rcopyzx\MFP@Rload y12{30258509}\MFP@Rmul}% + \MFP@Rcopy tx\MFP@Rload y10{88888889}\MFP@Rsmul + \MFP@com@iter{87500000}\MFP@com@iter{85714286}\MFP@com@iter{83333333}% + \MFP@com@iter{80000000}\MFP@com@iter{75000000}\MFP@com@iter{66666667}% + \MFP@com@iter{50000000}\MFP@flipz\MFP@Rcopyzx\MFP@Rcopy ty\MFP@Rsmul}% % \end{macrocode} % % \subsection{Powers} @@ -3472,10 +3517,10 @@ % \begin{macrocode} \def\MFP@Rexp@pos{% \MFP@Rload y1{18}{42068074}\MFP@Rcmp - \ifMFP@pos % overflow + \ifMFP@pos \MFP@expoverflow@err \MFP@Rloadz 1\MaxRealInt\MaxRealFrac - \else % handle integer part + \else \MFP@tempa\MFP@x@Int \edef\MFP@powerof@e{% 1\ifcase\MFP@tempa @@ -3503,7 +3548,7 @@ \ifnum\MFP@x@Frc=0 \else \MFP@Rcopyz s% - \MFP@tempa=\@xp\MFP@firstofmany\MFP@x@Frc\mfp@end + \MFP@tempa=\@xp\MFP@oneofmany\MFP@x@Frc\MFP@end \edef\MFP@powerof@e{% y1\ifcase\MFP@tempa 10\or @@ -3520,7 +3565,7 @@ \edef\MFP@t@Frc{0\@xp\@gobble\MFP@x@Frc}% \MFP@Rcopy sx\@xp\MFP@Rload\MFP@powerof@e\MFP@Rmul \ifnum\MFP@t@Frc=0 - \else % handle the rest + \else \MFP@Rcopyz s\MFP@Rload t10\MFP@t@Frc \MFP@Rexp@pos@prog \MFP@Rcopy sx\MFP@Rcopyzy\MFP@Rmul @@ -3537,16 +3582,17 @@ % $$ % 1 + (x + x/2(x + x/3(x + x/4(x + x/5(x + x/6))))) % $$ -% We start by loading $x$ (now in register \texttt{t}) into register -% \texttt{z}, then repeatedly run \cs{MFP@Rexp@iter} feeding it the +% We start by loading $x$ (now in register $t$) into register +% $z$, then repeatedly run \cs{MFP@Rexp@iter} feeding it the % successive values of $1/n$. This iterator first multiplies the most -% recent result (the \texttt{z} register) by $1/n$, then that by $x$ and +% recent result (the $z$ register) by $1/n$, then that by $x$ and % then adds $x$ to that. The final step is to add $1$. % \begin{macrocode} \def\MFP@Rexp@pos@prog{% - \MFP@Rcopy tz\MFP@Rexp@iter{16666667}\MFP@Rexp@iter{20000000}% - \MFP@Rexp@iter{25000000}\MFP@Rexp@iter{33333333}% - \MFP@Rexp@iter{50000000}\MFP@Rcopyzx\MFP@Rincr}% + \MFP@Rcopy tz\MFP@Rexp@iter{14285714}\MFP@Rexp@iter{16666667}% + \MFP@Rexp@iter{20000000}\MFP@Rexp@iter{25000000}% + \MFP@Rexp@iter{33333333}\MFP@Rexp@iter{50000000}\MFP@Rcopyzx + \MFP@Rincr}% \def\MFP@Rexp@iter#1{% \MFP@Rcopyzx\MFP@Rload y10{#1}\MFP@Rsmul \MFP@Rcopyzx\MFP@Rcopy ty\MFP@Rsmul @@ -3556,16 +3602,14 @@ % This is because an absolute error in $x$ converts to a relative error % in $e^x$, That is, knowing $x$ only to $10^{-8}$ means $e^x$ is off by % (about) $e^x\cdot 10^{-8}$. Roughly speaking, this means only about $8$ -% places of $e^x$ are accurate, so if the integer part of $e^x$ has six -% places then only two places after the decimal are significant. Even if -% $x$ is exact, we can only represent $e$ itself to eight decimals and the -% repeated multiplications accumulate errors in such a way that one still -% cannot get more than eight significant figures. +% places of $e^x$ are accurate, so if (for example) the integer part of +% $e^x$ has six places then only two places after the decimal are +% significant. % % \bigskip % The first issue with negative exponents is that it doesn't take much to % produce a value of $e^{-x}$ that rounds to $0$. Any $x > 19.11382792$. So -% we start by comparing to that value and simply return $0$ if $x$ is +% we start by comparing to that value and simply return zero if $x$ is % larger. % % We perform exactly the same reductions as for positive exponents, @@ -3577,7 +3621,7 @@ \MFP@Rcmp \ifMFP@pos \MFP@Rloadz 000% - \else % handle integer part + \else \MFP@tempa\MFP@x@Int \edef\MFP@powerof@e{% \ifcase\MFP@tempa @@ -3604,9 +3648,9 @@ 000\fi}% \@xp\MFP@Rloadz\MFP@powerof@e \ifnum\MFP@x@Frc=0 - \else % handle first decimal digit + \else \MFP@Rcopyz s% - \MFP@tempa=\@xp\MFP@firstofmany\MFP@x@Frc\mfp@end + \MFP@tempa=\@xp\MFP@oneofmany\MFP@x@Frc\MFP@end \edef\MFP@powerof@e{% y1\ifcase\MFP@tempa 10\or @@ -3623,10 +3667,10 @@ \edef\MFP@t@Frc{0\@xp\@gobble\MFP@x@Frc}% \MFP@Rcopy sx\@xp\MFP@Rload\MFP@powerof@e\MFP@Rmul \ifnum\MFP@t@Frc=0 - \else % handle the rest + \else \MFP@Rcopyz s\MFP@Rload t10\MFP@t@Frc \MFP@Rexp@neg@prog - \MFP@Rcopy sx\MFP@Rcopyzy\MFP@Rmul + \MFP@Rcopyzx\MFP@Rcopy sy\MFP@Rmul \fi \fi \fi}% @@ -3643,10 +3687,11 @@ % rather than call \cs{MFP@com@iter} with a useless multiplication by $1$. % \begin{macrocode} \def\MFP@Rexp@neg@prog{% - \MFP@Rcopy tx\MFP@Rload y10{16666667}\MFP@Rsmul - \MFP@com@iter{20000000}\MFP@com@iter{25000000}% - \MFP@com@iter{33333333}\MFP@com@iter{50000000}% - \MFP@flipz\MFP@Rcopyzx\MFP@Rcopy ty\MFP@Rsmul\MFP@flipz}% + \MFP@Rcopy tx\MFP@Rload y10{14285712}\MFP@Rsmul + \MFP@com@iter{16666667}\MFP@com@iter{20000000}% + \MFP@com@iter{25000000}\MFP@com@iter{33333333}% + \MFP@com@iter{50000000}\MFP@flipz\MFP@Rcopyzx + \MFP@Rcopy ty\MFP@Rsmul\MFP@flipz}% % \end{macrocode} % % The most efficient way to take an integer power of a number $x$ is to @@ -3675,11 +3720,14 @@ % The above scheme requires at most $\lfloor\log_2 n\rfloor$ squarings % and at most $\lceil \log_2 n \rceil$ multiplications for $x^n$, while % directly multiplying $x\cdot x \cdots x$ would require $n-1$ -% multiplications. +% multiplications. I have tested with an exponent equal to $8000$ and it +% takes only about $25$ times as long as a single multiplication (rather +% than $7999$ times). % -% For negative powers we can either find the positive power and take its -% reciprocal or take the reciprocal of $x$ and find its positive power. We -% do the first so that overflow can be detected in \cs{MFP@@Rpow}. +% For negative powers we can either find the positive power of $x$ and +% take its reciprocal or take the reciprocal of $x$ and find its positive +% power. We do the second so that overflow can be detected in +% \cs{MFP@@Rpow}. % \begin{macrocode} \def\MFP@Rpow{% \ifnum\MFP@y@Frc>0 @@ -3697,34 +3745,27 @@ \MFP@badpower@err \MFP@Rloadz 1\xOverZeroInt\xOverZeroFrac \fi - \else % integer power of nonzero number - % get the sign in case an overflow interrupts the calculations. + \else \ifnum\MFP@x@Sgn>0 \def\MFP@power@Sgn{1}% \else \edef\MFP@power@Sgn{\ifodd\MFP@loopctr -\fi 1}% -\ifMFPdebug - \show\MFP@power@Sgn -\fi \fi \ifnum\MFP@y@Sgn<0 \MFP@Rinv \MFP@Rcopyzx\fi \ifnum\MFP@loopctr=1 \MFP@Rloadz \MFP@power@Sgn\MFP@x@Int\MFP@x@Frc \else \MFP@@Rpow -\ifMFPdebug - \showreg z% -\fi \fi \fi - \fi}% % + \fi}% % \end{macrocode} % % This implements the algorithm discussed above. We save $x$ in register -% \texttt{q}, initialize the starting value of $1$ in \texttt{p} and then +% $q$, initialize the starting value of $1$ in \reg{p} and then % run the loop. If the binary digit just read is a 1 (i.e., \cs{ifodd} is % true), it multiplies $p$ and $q$. It also saves the last product (copies -% \texttt{z} to \texttt{p}). This need not be done on the last iteration, +% \reg{z} to \reg{p}). This need not be done on the last iteration, % but must not be moved out of the \cs{ifodd} conditional because % intervening computations modify $z$. If there are more iterations to do % (i.e., the \cs{ifnum} is true), this squares $q$ and reduces the @@ -3735,8 +3776,8 @@ % break the loop and return $\pm\infty$. % \begin{macrocode} \def\MFP@@Rpow{% - \MFP@Rcopy xq% initialize register to be squared - \MFP@Rload p110% initialize register to hold partial products + \MFP@Rcopy xq% + \MFP@Rload p110% \MFP@Rpow@loop}% \def\MFP@Rpow@loop{% \ifodd\MFP@loopctr @@ -3758,13 +3799,12 @@ \def\MFP@handle@expoverflow{% \MFP@expoverflow@err \MFP@loopctr=0 - \MFP@Rloadz\MFP@power@Sgn\MaxRealInt\MaxRealFrac -}% + \MFP@Rloadz\MFP@power@Sgn\MaxRealInt\MaxRealFrac}% % \end{macrocode} % % \subsection{The square root} % -% One can combine logarithms and exponentials to can get any power: to get +% One can combine logarithms and exponentials to get any power: to get % $x^y$, compute $e^{y\ln x}$. This has the disadvantage that it doesn't % work if $x$ is negative. Most powers of negative numbers are not % defined, but certainly integer powers are. Thus we have defined @@ -3778,11 +3818,15 @@ % $\sqrt{9} = 3$. In fact, if a square root can be expressed exactly % within our 8-digit precision, our code will find it. % -% For the square root we return $0$ if $x$ is not positive. If the integer +% For the square root we return zero if $x$ is not positive. If the integer % part of $x$ is $0$, we copy the fractional part to the integer part % (that is, we multiply by $10^{8}$, remembering to multiply by $10^{-4}$ -% later). This makes the square root of such numbers slightly more -% accurate. We then compute the square root using an algorithm that will +% later). This makes the square root of such numbers rather more +% accurate. (To get around some other rare but annoying inaccuracies, we +% go through a similar process when the integer part of $x$ is at most $4$ +% digits, multiplying by $10^4$ before and by $10^{-2}$ after.) +% +% We then compute the square root using an algorithm that will % be exact whenever possible. We perform one additional processing step. % To explain it, note that our algorithm actually produces the largest % number $s$ with four digits right of the decimal place that satisfies $s^2 @@ -3801,46 +3845,46 @@ % With this value, $s + \bar\epsilon$ misses the exact square root by at % most $\epsilon^2/(2s) < .5\cdot 10^{-8}$, because $s \ge 1$. % The final result $s + \bar\epsilon$ is equivalent to computing the -% average $s$ and $x/s$. This, possibly divided by $10^4$ is the +% average $s$ and $x/s$. This, possibly divided by $10^4$ or $10^2$ is the % returned value. % -% I originally tried power series methods, but they failed to produce -% exact answers when they existed (unless they were inconveniently carried -% to nine decimals and then rounded to eight). Then I tried the ``exact when -% possible'' algorithm to get $s$, but correcting it as follows: find -% $\sqrt{x/s^2}$ by power series and multiply by $s$. But this turned out -% to be remarkably inaccurate, being paradoxically worst when $s$ is -% already close, but not equal, to $\sqrt{x}$. Finally, I hit on the above -% simple and efficient correction, which turns out to be also the most -% accurate. By tests it produces a result correct in all but the last -% (eighth) decimal place, and within $1$ of the correct value in that -% place. Of course, it still produces exact results when that is possible -% within our digit limits. +% By tests, with rare exceptions, our computations produces a result +% correct in all eight decimal places. In the rare exception, the last +% place is within $1$ of the correct value. % \begin{macrocode} \def\MFP@Rsqrt{% \ifcase\MFP@x@Sgn\relax \MFP@Rzero \or \ifnum\MFP@x@Int=0 - \let\MFP@sqrt@reduce=Y% + \def\MFP@sqrt@reduce{2}% \edef\MFP@x@Int{\number\MFP@x@Frc}% \edef\MFP@x@Frc{00000000}% + \else\ifnum\MFP@x@Int<10000 + \def\MFP@sqrt@reduce{1}% + \edef\MFP@x@Int{\MFP@x@Int\@xp\MFP@fourofmany\MFP@x@Frc\MFP@end}% + \edef\MFP@x@Frc{\@xp\MFP@gobblefour\MFP@x@Frc0000}% \else - \let\MFP@sqrt@reduce=N% - \fi + \def\MFP@sqrt@reduce{0}% + \fi\fi \MFP@Rcopy xt% \MFP@Isqrt \MFP@Rcopyz s\MFP@Rcopyzy \MFP@Rcopy tx\MFP@Rdiv \MFP@Rcopy sx\MFP@Rcopyzy\MFP@Radd \MFP@Rcopyzx\MFP@Rhalve - \ifx Y\MFP@sqrt@reduce - \MFP@Rcopyzx\MFP@Rload y10{10000}\MFP@Rmul + \ifcase \MFP@sqrt@reduce\relax + \or + \MFP@Rcopyzx\MFP@Rload y10{01000000}\MFP@Rmul + \or + \MFP@Rcopyzx\MFP@Rload y10{00010000}\MFP@Rmul \fi \else - \MFP@sqrtofneg@err + \MFP@warn{Square root of a negative number. Zero will be returned.}% \MFP@Rzero \fi}% +\def\MFP@fourofmany#1#2#3#4#5\MFP@end{#1#2#3#4}% +\def\MFP@gobblefour#1#2#3#4{}% % \end{macrocode} % % There is a rather straightforward pencil and paper algorithm that @@ -3907,7 +3951,7 @@ \def\MFP@Isqrt{% \MFP@ItoQ\MFP@x@Int\MFP@x@Frc \MFP@tempa=0 \MFP@tempb=0 \MFP@tempc=0 - \expandafter\MFP@Isqrt@loop\MFP@ItoQ@Tmp\mfp@end + \expandafter\MFP@Isqrt@loop\MFP@ItoQ@Tmp\MFP@end \MFP@tempa=\MFP@tempc \divide\MFP@tempc\MFP@tttfour \edef\MFP@z@Int{\number\MFP@tempc}% @@ -3925,7 +3969,7 @@ % it written out, I am surprise at how concise and elegant it is! % \begin{macrocode} \def\MFP@Isqrt@loop#1{% - \ifx\mfp@end #1% + \ifx\MFP@end #1% \else \multiply\MFP@tempa 2 \multiply\MFP@tempb 4 \multiply\MFP@tempc 2 \advance \MFP@tempb#1\relax @@ -3935,18 +3979,18 @@ \advance\MFP@tempa 1 \fi \expandafter\MFP@Isqrt@loop - \fi -}% + \fi}% +\MFP@xfinish %</extra> % \end{macrocode} -% For my own benefit: this finds the next binary digit and updates the -% square root (in \cs{MFP@tempc}) by appending that digit. The new digit -% is also appended to the end of \cs{MFP@tempa}. This is subtracted from -% \cs{MFP@tempb}, but only if the last digit is a 1. Then the next -% quadrenary digit is appended to \cs{MFP@tempb}. Finally, the last binary -% digit found is added (not appended) to \cs{MFP@tempa}. The ``appending'' -% of a digit means a multiplication by $2$ (or $4$) and the addition of the -% digit. We perform such additions only if the digit is a 1, and we -% determine if the digit is 1 or 0 by the \cs{ifnum} test. +% For my own benefit: the above code finds the next binary digit and +% updates the square root (in \cs{MFP@tempc}) by appending that digit. The +% new digit is also appended to the end of \cs{MFP@tempa}. This is +% subtracted from \cs{MFP@tempb}, but only if the last digit is a 1. Then +% the next quadrenary digit is appended to \cs{MFP@tempb}. Finally, the +% last binary digit found is added (not appended) to \cs{MFP@tempa}. The +% ``appending'' of a digit means a multiplication by $2$ (or $4$) and the +% addition of the digit. We perform such additions only if the digit is a +% 1, and we determine if the digit is 1 or 0 by the \cs{ifnum} test. %\Finale % diff --git a/Master/texmf-dist/tex/generic/minifp/mfpextra.tex b/Master/texmf-dist/tex/generic/minifp/mfpextra.tex index 2e9a6d88bf0..9d65d0ff1aa 100644 --- a/Master/texmf-dist/tex/generic/minifp/mfpextra.tex +++ b/Master/texmf-dist/tex/generic/minifp/mfpextra.tex @@ -21,6 +21,26 @@ %% is Daniel H. Luecking. The Base Interpreters associated %% with minifp are plain TeX and LaTeX. %% +\expandafter\ifx\csname MFP@xfinish\endcsname\relax +\else \expandafter\endinput\fi +\expandafter\edef\csname MFP@xfinish\endcsname{% + \catcode64=\the\catcode64 \space + \catcode46=\the\catcode46 \space + \catcode60=\the\catcode60 \space + \catcode62=\the\catcode62 \space}% +\catcode64=11 % @ +\catcode46=12 % . (period) +\catcode60=12 % < +\catcode62=12 % > +\ifx\MFP@Rextra\UndEfInEd \input minifp.sty \fi +\ifx\MFP@Rextra\@empty +\else + \immediate\write16{mfpextra.tex: already loaded.^^J}% + \MFP@xfinish + \expandafter\endinput +\fi +\immediate\write16{% + mfpextra.tex: extra operations for the MiniFP package.^^J}% \def\MFP@Rextra{% \def\Rcos {\MFP@stack@Unary\MFP@Rcos }% \def\Rsin {\MFP@stack@Unary\MFP@Rsin }% @@ -51,104 +71,110 @@ \def\LogOfZeroFrac{\MaxRealFrac}% \def\MFP@expoverflow@err{% \MFP@errmsg{Power too large}% - {The power you tried to calcualate is too large for % + {The power you tried to calculate is too large for % 8 digits. If you continue, ^^Jthe value assigned will be % \MaxRealInt.\MaxRealFrac.}}% -\def\MFP@sqrtofneg@err{% - \MFP@errmsg{square root of a negative number}% - {You tried to take the square root of a negative value. What % - were you thinking? If you continue, zero will be returned.}}% \def\MFP@badpower@err{% \MFP@errmsg{negative power of zero}% {You tried to take a negative power of zero. What were you thinking? If you ^^Jcontinue, the value assigned will be % \xOverZeroInt.\xOverZeroFrac.}}% +\def\MFPshowreg #1{% +\ifMFPdebug +\begingroup + \edef\theregister{% + #1 = \expandafter \MFP@Sign + \csname MFP@#1@Sgn\endcsname % + \csname MFP@#1@Int\endcsname.% + \csname MFP@#1@Frc\endcsname}% + \show\theregister +\endgroup +\fi}% \def\MFP@Rcopyz#1{\MFP@Rload {#1}\MFP@z@Sgn\MFP@z@Int\MFP@z@Frc}% \def\MFP@Rcopyzx{\MFP@Rcopyz x}% \def\MFP@Rcopyzy{\MFP@Rcopyz y}% -\def\MFP@Rdeg{% - \MFP@Rcopy xt\MFP@Rload y1{57}{29577951}\MFP@Rmul - \MFP@Rcopyz s\MFP@Rcopy tx\MFP@Rload y10{30823209}\MFP@Rmul - \MFP@tempa\MFP@z@Int\relax - \ifnum\MFP@z@Frc<50000000 \else \advance\MFP@tempa 1 \fi - \MFP@Rload x{\ifnum\MFP@tempa>0 \MFP@z@Sgn\else0\fi}0\MFP@tempa - \MFP@Rcopy sy\MFP@Radd}% -\def\MFP@Rrad{% - \MFP@Rcopy xt\MFP@Rload y10{0174 5329}\MFP@Rmul - \MFP@Rcopyz s\MFP@Rcopy tx\MFP@Rload y10{25199433}\MFP@Rmul - \MFP@tempa\MFP@z@Int\relax - \ifnum\MFP@z@Frc<50000000 \else \advance\MFP@tempa 1 \fi - \MFP@Rload x{\ifnum\MFP@tempa>0 \MFP@z@Sgn\else0\fi}0\MFP@tempa - \MFP@Rcopy sy\MFP@Radd}% -\def\MFP@Rcos{% - \MFP@Rcopy xy\MFP@Rload x1{90}0\MFP@Rsub - \MFP@Rcopyzx\MFP@Rsin}% -\def\MFP@Rsin{% - \ifnum \MFP@x@Sgn=0 % return 0 +\def\MFP@DPmul#1#2#3{% + \ifnum\MFP@x@Sgn=0 \MFP@Rzero \else - \MFP@tempa\MFP@x@Int\relax - \MFP@reduce@angle - \ifnum \MFP@tempa<180 - \edef\MFP@sin@Sgn{\MFP@x@Sgn}% - \else - \edef\MFP@sin@Sgn{\number-\MFP@x@Sgn}% - \advance \MFP@tempa-180 + \MFP@Rcopy xv% + \MFP@Rload y1{#1}{#2}\MFP@Rmul + \edef\MFP@w@Und{\MFP@z@Und}% + \ifnum\MFP@z@Frc@iii>4999 + \MFP@tempa\MFP@z@Frc \advance\MFP@tempa-1 + \edef\MFP@z@Frc{\number\MFP@tempa}% + \makeMFP@eightdigits\MFP@z@Frc \fi - \MFP@tempb\MFP@x@Frc\relax - \ifnum\MFP@tempa>0 - \MFP@@Rsin - \else\ifnum\MFP@tempb>0 - \MFP@@Rsin + \MFP@Rcopyz w% + \MFP@Rcopy vx\MFP@Rload y10{#3}\MFP@Rmul + \MFP@Rcopyzx\MFP@Rload y\MFP@v@Sgn 0{\MFP@w@Und}\MFP@Radd + \MFP@tempa\MFP@z@Int\relax + \ifnum\MFP@z@Frc<50000000 \else \advance\MFP@tempa 1 \fi + \ifnum\MFP@tempa<\MFP@ttteight\relax + \MFP@Rload x{\ifnum\MFP@tempa>0 \MFP@z@Sgn\else0\fi}0\MFP@tempa \else - \MFP@Rzero - \fi\fi + \MFP@Rload x\MFP@z@Sgn10% + \fi + \MFP@Rcopy wy\MFP@Radd \fi}% +\def\MFP@Rdeg{\MFP@DPmul{57}{29577951}{30823209}}% +\def\MFP@Rrad{\MFP@DPmul{0}{01745329}{25199433}}% +\def\MFP@RbaseX{\MFP@DPmul{0}{43429448}{19032518}}% +\def\MFP@RbaseE{\MFP@DPmul{2}{30258509}{29940457}}% +\def\MFP@Rcos{% + \MFP@Rcopy xy\MFP@Rload x1{90}0\MFP@Rsub + \MFP@Rcopyzx\MFP@Rsin}% +\def\MFP@Rsin{% + \MFP@tempa\MFP@x@Int + \MFP@tempb\MFP@x@Frc + \MFP@tempc\MFP@x@Sgn\relax + \MFP@reduce@angle + \ifnum\MFP@tempa>0 \MFP@@Rsin + \else\ifnum\MFP@tempb>0 \MFP@@Rsin + \else \MFP@Rzero + \fi\fi}% \def\MFP@reduce@angle{% - \ifnum\MFP@tempa<360 + \ifnum\MFP@tempa<180 \else - \advance\MFP@tempa-360 + \advance\MFP@tempa-180 + \MFP@tempc-\MFP@tempc \@xp\MFP@reduce@angle \fi}% \def\MFP@@Rsin{% \ifnum\MFP@tempa<90 \else + \MFP@tempa -\MFP@tempa \ifnum\MFP@tempb>0 - \advance\MFP@tempa 1 \MFP@tempb -\MFP@tempb \advance\MFP@tempb \MFP@ttteight\relax + \advance\MFP@tempa 179 + \else \advance\MFP@tempa 180 \fi - \MFP@tempa -\MFP@tempa - \advance\MFP@tempa 180 \fi \ifnum\MFP@tempa=90 - \MFP@Rloadz \MFP@sin@Sgn10% + \MFP@Rloadz \MFP@tempc10% \else \advance\MFP@tempb 50 \divide\MFP@tempb 100 \multiply\MFP@tempa 1000000 \advance\MFP@tempb\MFP@tempa \ifnum\MFP@tempb=0 \MFP@Rzero \else - \MFP@Rload x10\MFP@tempb - \MFP@Rcopy xs% + \MFP@Rload s\MFP@tempc0\MFP@tempb + \MFP@Rcopy sx% \MFP@Rsq \ifnum \MFP@z@Frc>0 \MFP@Rcopyz t\MFP@Rsin@prog \else \MFP@Rcopy sx% \fi - \MFP@Rload y11{74532925}\MFP@Rmul - \ifnum\MFP@z@Sgn=0 \else - \let\MFP@z@Sgn\MFP@sin@Sgn - \fi + \MFP@DPmul 1{74532925}{19943296}% \fi \fi}% \def\MFP@Rsin@prog{% - \MFP@Rcopy tx% - \MFP@Rload y10{00559959}\MFP@Rsmul\MFP@com@iter{01450559}% - \MFP@com@iter{01952675}\MFP@com@iter{02769249}\MFP@com@iter{04230797}% - \MFP@com@iter{07252796}\MFP@com@iter{15230871}\MFP@com@iter{50769570}% - \MFP@flipz \MFP@Rcopyzx \MFP@Rcopy sy\MFP@Rsmul \MFP@Rcopyzx}% + \MFP@Rcopy tx\MFP@Rload y10{01952676}\MFP@Rsmul% + \MFP@com@iter{02769249}\MFP@com@iter{04230797}\MFP@com@iter{07252796}% + \MFP@com@iter{15230871}\MFP@com@iter{50769570}\MFP@flipz \MFP@Rcopyzx + \MFP@Rcopy sy\MFP@Rsmul\MFP@Rcopyzx\edef\MFP@x@Sgn{\MFP@s@Sgn}}% \def\MFP@flipz{% \ifnum\MFP@z@Sgn=0 \MFP@Rloadz 110% @@ -161,16 +187,11 @@ \MFP@Rcopyzx\MFP@Rcopy ty\MFP@Rsmul \MFP@Rcopyzx\MFP@Rload y10{#1}\MFP@Rsmul}% \def\MFP@Rsmul{% - \ifnum \MFP@x@Sgn=0 - \MFP@Rzero - \else\ifnum \MFP@y@Sgn=0 - \MFP@Rzero - \else\ifnum\MFP@x@Int>0 - \MFP@Rcopy yz% - \else\ifnum\MFP@y@Int>0 - \MFP@Rcopy xz% - \else - \MFP@@Rsmul + \ifnum \MFP@x@Sgn=0 \MFP@Rzero + \else\ifnum \MFP@y@Sgn=0 \MFP@Rzero + \else\ifnum\MFP@x@Int>0 \MFP@Rcopy yz% + \else\ifnum\MFP@y@Int>0 \MFP@Rcopy xz% + \else \MFP@@Rsmul \fi\fi\fi\fi}% \def\MFP@@Rsmul{% \MFP@split\MFP@x@Frc\MFP@x@Frc@i\MFP@x@Frc@ii @@ -190,15 +211,11 @@ \advance\MFP@tempb1 \edef\MFP@z@Frc@ii{\number\MFP@tempb}\fi \MFP@carrym\MFP@z@Frc@ii\MFP@z@Frc@i - \makeMFP@fourdigits\MFP@z@Frc@iv - \makeMFP@fourdigits\MFP@z@Frc@iii \makeMFP@fourdigits\MFP@z@Frc@ii \makeMFP@fourdigits\MFP@z@Frc@i \def\MFP@z@Int{0}% \edef\MFP@z@Frc{\MFP@z@Frc@i\MFP@z@Frc@ii}% - \edef\MFP@z@Sgn{\ifnum\MFP@z@Frc=0 0\else 1\fi}% - \edef\MFP@z@Und{\MFP@z@Frc@iii\MFP@z@Frc@iv}% - \edef\MFP@z@Ovr{0}}% + \edef\MFP@z@Sgn{\ifnum\MFP@z@Frc=0 0\else 1\fi}}% \def\MFP@Rangle{% \ifcase\MFP@y@Sgn\relax \ifcase\MFP@x@Sgn\relax @@ -223,70 +240,72 @@ \def\MFP@x@Sgn{1}\MFP@@Rangle \MFP@Rcopyzy\MFP@Rload x1{180}0\MFP@Rsub \fi - \let\MFP@z@Sgn\MFP@angle@Sgn - }}% + \let\MFP@z@Sgn\MFP@angle@Sgn}}% \def\MFP@@Rangle{% \MFP@Rcmp \ifMFP@neg - \MFP@Rcopy xs\MFP@Rcopy yx\MFP@Rcopy sy% + \MFP@Rcopy xw\MFP@Rcopy yx\MFP@Rcopy wy% \MFP@@@Rangle \MFP@Rload x1{90}0\MFP@Rcopyzy\MFP@Rsub \else \MFP@@@Rangle - \fi -}% + \fi}% \def\MFP@Rquad{\MFP@Rdbl\MFP@Rcopyzx\MFP@Rdbl}% \def\MFP@@@Rangle{% \MFP@Rcopy xs\MFP@Rcopy yt% - \ifnum\MFP@y@Int<500000 + \ifnum\MFP@x@Int<1000000 \else - \MFP@Rload y1{200}0\MFP@Rdiv - \MFP@Rcopyz s\MFP@Rcopy tx% - \MFP@Rload y1{200}0\MFP@Rdiv - \MFP@Rcopyz t% + \MFP@RdivC \MFP@Rcopyz s% + \MFP@Rcopy tx\MFP@RdivC \MFP@Rcopyz t% \fi - \MFP@Rcopy tx\MFP@Rdbl\MFP@Rcopyzx\MFP@Rcopy sy\MFP@Rcmp - \ifMFP@pos - \MFP@Rsub\MFP@Rcopyz u\MFP@Rcopy sx\MFP@Rdbl - \MFP@Rcopyzx\MFP@Rcopy ty\MFP@Radd - \MFP@Rcopyz s\MFP@Rcopy ut% - \MFP@Rload a1{2656}{50511771}% + \ifnum\MFP@t@Sgn=0 \MFP@Rzero \else - \MFP@Rload a000% - \fi - \MFP@Rcopy tx\MFP@Rquad\MFP@Rcopyzx\MFP@Rcopy sy\MFP@Rcmp - \ifMFP@pos - \MFP@Rsub\MFP@Rcopyz u\MFP@Rcopy sx\MFP@Rquad - \MFP@Rcopyzx\MFP@Rcopy ty\MFP@Radd - \MFP@Rcopyz s\MFP@Rcopy ut% - \MFP@Rcopy ax\MFP@Rload y1{1403}{62434679}% - \MFP@Radd\MFP@Rcopy za% - \fi - \MFP@Rcopy tx\MFP@RmulC - \MFP@Rcopyzx\MFP@Rcopy sy\MFP@Rdiv - \MFP@Rcopyzx\MFP@Ratanc - \MFP@Rcopyzx\MFP@Rdeg - \MFP@Rcopyzx\MFP@Rcopy ay\MFP@Radd - \MFP@Rcopyzx\MFP@RdivC}% + \MFP@Rcopy tx\MFP@Rdbl\MFP@Rcopyzx\MFP@Rcopy sy\MFP@Rcmp + \ifMFP@pos + \MFP@Rsub\MFP@Rcopyz u\MFP@Rcopy sx\MFP@Rdbl + \MFP@Rcopyzx\MFP@Rcopy ty\MFP@Radd + \MFP@Rcopyz s\MFP@Rcopy ut% + \MFP@Rload a1{2656}{50511771}% + \else + \MFP@Rload a000% + \fi + \MFP@Rcopy tx\MFP@Rquad\MFP@Rcopyzx\MFP@Rcopy sy\MFP@Rcmp + \ifMFP@pos + \MFP@Rsub\MFP@Rcopyz u\MFP@Rcopy sx\MFP@Rquad + \MFP@Rcopyzx\MFP@Rcopy ty\MFP@Radd + \MFP@Rcopyz s\MFP@Rcopy ut% + \MFP@Rcopy ax\MFP@Rload y1{1403}{62434679}% + \MFP@Radd\MFP@Rcopy za% + \fi + \MFP@Rcopy tx\MFP@RmulC + \MFP@Rcopyzx\MFP@Rcopy sy\MFP@Rdiv + \MFP@Rcopyzx\MFP@Ratanc + \MFP@Rcopyzx\MFP@Rdeg + \MFP@Rcopyzx\MFP@Rcopy ay\MFP@Radd + \MFP@Rcopyzx\MFP@RdivC + \fi}% +\def\MFP@twoofmany#1#2#3\MFP@end{#1#2}% +\def\MFP@gobbletwo#1#2{}% \def\MFP@RmulC{% - \makeMFP@eightdigits\MFP@x@Frc - \edef\MFP@Tmp{\number\MFP@x@Int.\MFP@x@Frc}% - \@xp\MFP@@RmulC\MFP@Tmp\mfp@end}% -\def\MFP@@RmulC#1.#2#3#4\mfp@end{% - \MFP@Rloadz\MFP@x@Sgn{#1#2#3}{#400}}% + \edef\MFP@z@Int{\MFP@x@Int\@xp\MFP@twoofmany\MFP@x@Frc\MFP@end}% + \edef\MFP@z@Frc{\@xp\MFP@gobbletwo\MFP@x@Frc00}% + \edef\MFP@z@Sgn{\MFP@x@Sgn}}% \def\MFP@RdivC{% \makeMFP@eightdigits\MFP@x@Int \makeMFP@eightdigits\MFP@x@Frc - \@XP\MFP@@RdivC\@xp\MFP@x@Int\MFP@x@Frc\mfp@end}% + \@XP\MFP@@RdivC\@xp\MFP@x@Int\MFP@x@Frc\MFP@end}% \def\MFP@@RdivC#1#2#3#4#5#6{% \edef\MFP@z@Int{\number#1#2#3#4#5#6}% \MFP@@@RdivC}% -\def\MFP@@@RdivC#1#2#3#4#5#6#7#8#9\mfp@end{% +\def\MFP@@@RdivC#1#2#3#4#5#6#7#8#9\MFP@end{% \MFP@tempa#1#2#3#4#5#6#7#8\relax \ifnum#9>49 \advance\MFP@tempa1 \fi + \edef\MFP@z@Frc{\number\MFP@tempa}% + \makeMFP@eightdigits\MFP@z@Frc \edef\MFP@z@Sgn{\MFP@x@Sgn}% - \ifnum\MFP@tempa=0 \ifnum\MFP@x@Int=0 \def\MFP@z@Sgn{0}\fi\fi - \MFP@Rloadz\MFP@z@Sgn\MFP@z@Int\MFP@tempa}% + \ifnum\MFP@tempa=0 + \ifnum\MFP@z@Int=0 \def\MFP@z@Sgn{0}\fi + \fi}% \def\MFP@scaledmul{\MFP@Rmul\MFP@Rcopyzx\MFP@RdivC}% \def\MFP@atan@iter#1#2{% \MFP@Rcopy uy\MFP@scaledmul @@ -294,82 +313,80 @@ \MFP@Rcopyzy\MFP@Rload x1{100}{00000000}% \MFP@Rsub\MFP@Rcopyzx}% \def\MFP@Ratanc{% - \MFP@Rcopy xs% - \MFP@Rcopy xy\MFP@scaledmul\MFP@Rcopyz u% - \MFP@Rcopyzx\MFP@Rload y1{86}{66666667}% - \MFP@scaledmul - \MFP@Rcopyzy\MFP@Rload x1{100}{00000000}% - \MFP@Rsub\MFP@Rcopyzx - \MFP@atan@iter{84}{61538462}\MFP@atan@iter{81}{81818182}% - \MFP@atan@iter{77}{77777778}\MFP@atan@iter{71}{42857143}% - \MFP@atan@iter{60}{00000000}\MFP@atan@iter{33}{33333333}% - \MFP@Rcopy sy\MFP@scaledmul}% -\def\MFP@Rlog{% + \MFP@Rcopy xs\MFP@Rcopy xy\MFP@scaledmul + \ifnum \MFP@z@Sgn=0 + \MFP@Rcopy sz% + \else + \MFP@Rcopyz u\MFP@Rcopyzx + \MFP@Rload y1{86}{66666667}\MFP@scaledmul + \MFP@Rcopyzy\MFP@Rload x1{100}{00000000}\MFP@Rsub\MFP@Rcopyzx + \MFP@atan@iter{84}{61538462}\MFP@atan@iter{81}{81818182}% + \MFP@atan@iter{77}{77777778}\MFP@atan@iter{71}{42857143}% + \MFP@atan@iter{60}{00000000}\MFP@atan@iter{33}{33333333}% + \MFP@Rcopy sy\MFP@scaledmul + \fi}% +\newif\ifMFP@natural +\def\MFP@Rlog{\MFP@naturalfalse\MFP@Rlog@}% +\def\MFP@Rln{\MFP@naturaltrue\MFP@Rlog@}% +\def\MFP@Rlog@{% \ifnum\MFP@x@Sgn=0 \MFP@logofzero@err \MFP@Rloadz{-1}\LogOfZeroInt\LogOfZeroFrac \else \ifnum \MFP@x@Sgn<0 - \MFP@warn{The logarithm of a negative number is complex.% + \MFP@warn{The logarithm of a negative number is complex. \MFP@msgbreak Only the real part will be computed}% + \def\MFP@x@Sgn{1}% \fi \MFP@Rload s000% \ifnum \MFP@x@Int=0 \edef\MFP@x@Tmp{\number\MFP@x@Frc}% - \MFP@tempa=\MFP@numzeros\MFP@x@Tmp\relax + \MFP@tempa=\MFP@numshiftL\MFP@x@Tmp\relax \def\MFP@s@Sgn{-1}% - \edef\MFP@t@Int{\@xp\MFP@firstofmany\MFP@x@Tmp\mfp@end}% + \edef\MFP@t@Int{\@xp\MFP@oneofmany\MFP@x@Tmp\MFP@end}% \edef\MFP@t@Frc{\@xp\@gobble\MFP@x@Tmp0}% - \MFPpadto@eight\MFP@t@Frc + \MFP@padtoeight\MFP@t@Frc \else - \MFP@tempa\MFP@numdigits\MFP@x@Int - \edef\MFP@x@Tmp{\MFP@x@Int\MFP@x@Frc}% gather all digits - \edef\MFP@s@Sgn{1}% - \edef\MFP@t@Int{\@xp\MFP@firstofmany\MFP@x@Tmp\mfp@end}% + \MFP@tempa\MFP@numshiftR\MFP@x@Int + \edef\MFP@x@Tmp{\MFP@x@Int\MFP@x@Frc}% + \ifnum\MFP@tempa>0 \def\MFP@s@Sgn{1}\fi + \edef\MFP@t@Int{\@xp\MFP@oneofmany\MFP@x@Tmp\MFP@end}% \edef\MFP@x@Tmp{\@xp\@gobble\MFP@x@Tmp}% - \edef\MFP@t@Frc{\@xp\MFP@firsteightofmany\MFP@x@Tmp\mfp@end}% + \edef\MFP@t@Frc{\@xp\MFP@eightofmany\MFP@x@Tmp\MFP@end}% \fi \edef\MFP@s@Int{\number\MFP@tempa}% \def\MFP@t@Sgn{1}% \MFP@Rlog@reduce + \ifMFP@natural \MFP@Rcopy sx\MFP@RbaseE \MFP@Rcopy zs\fi \ifnum\MFP@t@Frc=0 \MFP@Rcopy sz% \else - \MFP@Rlog@prog + \def\MFP@t@Int{0}\MFP@Rlog@prog + \ifMFP@natural\else \MFP@Rcopyzx \MFP@RbaseX \fi + \MFP@Rcopy sy\MFP@Rcopyzx\MFP@Radd \fi \fi}% -\def\showreg #1{% for debugging -\begingroup - \edef\reg{% - #1 = (\csname MFP@#1@Sgn\endcsname)% - \csname MFP@#1@Int\endcsname.% - \csname MFP@#1@Frc\endcsname}% - \show\reg -\endgroup}% -\def\MFP@numdigits#1{\@xp\MFP@ninthofmany#176543210\mfp@end}% -\def\MFP@numzeros#1{% - \@xp\MFP@ninthofmany#112345678\mfp@end}% +\def\MFP@numshiftR#1{\@xp\MFP@ninthofmany#176543210\MFP@end}% +\def\MFP@numshiftL#1{\@xp\MFP@ninthofmany#112345678\MFP@end}% \def\MFP@Rlog@reduce{% - \MFP@reduceonce 3{16227766}{31622777}{50000000}% - \MFP@reduceonce 1{77827941}{56234133}{25000000}% - \MFP@reduceonce 1{33352143}{74989421}{12500000}% - \MFP@reduceonce 1{15478198}{86596433}{06250000}}% + \MFP@Rcopy tx\MFP@RmulC\MFP@Rcopyz t% + \MFP@reduceonce {316}{31622776}{60168379}{50000000}% + \MFP@reduceonce {177}{56234132}{51903491}{25000000}% + \MFP@reduceonce {133}{74989420}{93324558}{12500000}% + \MFP@reduceonce {115}{86596432}{33600654}{06250000}% + \MFP@Rcopy tx\MFP@RdivC\MFP@Rcopyz t}% \def\MFP@reduceonce#1#2#3#4{% - \MFP@Rcopy tx\MFP@Rload y1{#1}{#2}\MFP@Rcmp - \ifMFP@neg\else - \MFP@Rload y10{#3}\MFP@Rmul - \MFP@Rcopyz t\MFP@Rcopy sx\MFP@Rload y10{#4}\MFP@Radd + \ifnum\MFP@t@Int>#1\relax + \MFP@Rcopy tx% + \MFP@DPmul 0{#2}{#3}\MFP@Rcopyz t% + \MFP@Rcopy sx\MFP@Rload y10{#4}\MFP@Radd \MFP@Rcopyz s% \fi}% \def\MFP@Rlog@prog{% - \MFP@Rcopy tx\MFP@Rdecr - \MFP@Rcopyz t% - \MFP@Rcopyzx\MFP@Rload y10{43750000}\MFP@Rsmul - \MFP@com@iter{85714286}\MFP@com@iter{83333333}\MFP@com@iter{80000000}% - \MFP@com@iter{75000000}\MFP@com@iter{66666667}\MFP@com@iter{50000000}% - \MFP@com@iter{43429448}\MFP@Rcopyzx\MFP@Rcopy sy\MFP@Radd}% -\def\MFP@Rln{% - \MFP@Rlog\MFP@Rcopyzx\MFP@Rload y12{30258509}\MFP@Rmul}% + \MFP@Rcopy tx\MFP@Rload y10{88888889}\MFP@Rsmul + \MFP@com@iter{87500000}\MFP@com@iter{85714286}\MFP@com@iter{83333333}% + \MFP@com@iter{80000000}\MFP@com@iter{75000000}\MFP@com@iter{66666667}% + \MFP@com@iter{50000000}\MFP@flipz\MFP@Rcopyzx\MFP@Rcopy ty\MFP@Rsmul}% \def\MFP@Rexp{% \ifcase\MFP@x@Sgn\relax \MFP@Rloadz 110% @@ -381,10 +398,10 @@ \fi}% \def\MFP@Rexp@pos{% \MFP@Rload y1{18}{42068074}\MFP@Rcmp - \ifMFP@pos % overflow + \ifMFP@pos \MFP@expoverflow@err \MFP@Rloadz 1\MaxRealInt\MaxRealFrac - \else % handle integer part + \else \MFP@tempa\MFP@x@Int \edef\MFP@powerof@e{% 1\ifcase\MFP@tempa @@ -412,7 +429,7 @@ \ifnum\MFP@x@Frc=0 \else \MFP@Rcopyz s% - \MFP@tempa=\@xp\MFP@firstofmany\MFP@x@Frc\mfp@end + \MFP@tempa=\@xp\MFP@oneofmany\MFP@x@Frc\MFP@end \edef\MFP@powerof@e{% y1\ifcase\MFP@tempa 10\or @@ -429,7 +446,7 @@ \edef\MFP@t@Frc{0\@xp\@gobble\MFP@x@Frc}% \MFP@Rcopy sx\@xp\MFP@Rload\MFP@powerof@e\MFP@Rmul \ifnum\MFP@t@Frc=0 - \else % handle the rest + \else \MFP@Rcopyz s\MFP@Rload t10\MFP@t@Frc \MFP@Rexp@pos@prog \MFP@Rcopy sx\MFP@Rcopyzy\MFP@Rmul @@ -437,9 +454,10 @@ \fi \fi}% \def\MFP@Rexp@pos@prog{% - \MFP@Rcopy tz\MFP@Rexp@iter{16666667}\MFP@Rexp@iter{20000000}% - \MFP@Rexp@iter{25000000}\MFP@Rexp@iter{33333333}% - \MFP@Rexp@iter{50000000}\MFP@Rcopyzx\MFP@Rincr}% + \MFP@Rcopy tz\MFP@Rexp@iter{14285714}\MFP@Rexp@iter{16666667}% + \MFP@Rexp@iter{20000000}\MFP@Rexp@iter{25000000}% + \MFP@Rexp@iter{33333333}\MFP@Rexp@iter{50000000}\MFP@Rcopyzx + \MFP@Rincr}% \def\MFP@Rexp@iter#1{% \MFP@Rcopyzx\MFP@Rload y10{#1}\MFP@Rsmul \MFP@Rcopyzx\MFP@Rcopy ty\MFP@Rsmul @@ -449,7 +467,7 @@ \MFP@Rcmp \ifMFP@pos \MFP@Rloadz 000% - \else % handle integer part + \else \MFP@tempa\MFP@x@Int \edef\MFP@powerof@e{% \ifcase\MFP@tempa @@ -476,9 +494,9 @@ 000\fi}% \@xp\MFP@Rloadz\MFP@powerof@e \ifnum\MFP@x@Frc=0 - \else % handle first decimal digit + \else \MFP@Rcopyz s% - \MFP@tempa=\@xp\MFP@firstofmany\MFP@x@Frc\mfp@end + \MFP@tempa=\@xp\MFP@oneofmany\MFP@x@Frc\MFP@end \edef\MFP@powerof@e{% y1\ifcase\MFP@tempa 10\or @@ -495,18 +513,19 @@ \edef\MFP@t@Frc{0\@xp\@gobble\MFP@x@Frc}% \MFP@Rcopy sx\@xp\MFP@Rload\MFP@powerof@e\MFP@Rmul \ifnum\MFP@t@Frc=0 - \else % handle the rest + \else \MFP@Rcopyz s\MFP@Rload t10\MFP@t@Frc \MFP@Rexp@neg@prog - \MFP@Rcopy sx\MFP@Rcopyzy\MFP@Rmul + \MFP@Rcopyzx\MFP@Rcopy sy\MFP@Rmul \fi \fi \fi}% \def\MFP@Rexp@neg@prog{% - \MFP@Rcopy tx\MFP@Rload y10{16666667}\MFP@Rsmul - \MFP@com@iter{20000000}\MFP@com@iter{25000000}% - \MFP@com@iter{33333333}\MFP@com@iter{50000000}% - \MFP@flipz\MFP@Rcopyzx\MFP@Rcopy ty\MFP@Rsmul\MFP@flipz}% + \MFP@Rcopy tx\MFP@Rload y10{14285712}\MFP@Rsmul + \MFP@com@iter{16666667}\MFP@com@iter{20000000}% + \MFP@com@iter{25000000}\MFP@com@iter{33333333}% + \MFP@com@iter{50000000}\MFP@flipz\MFP@Rcopyzx + \MFP@Rcopy ty\MFP@Rsmul\MFP@flipz}% \def\MFP@Rpow{% \ifnum\MFP@y@Frc>0 \MFP@warn{The "pow" function requires an integer power. @@ -523,30 +542,23 @@ \MFP@badpower@err \MFP@Rloadz 1\xOverZeroInt\xOverZeroFrac \fi - \else % integer power of nonzero number - % get the sign in case an overflow interrupts the calculations. + \else \ifnum\MFP@x@Sgn>0 \def\MFP@power@Sgn{1}% \else \edef\MFP@power@Sgn{\ifodd\MFP@loopctr -\fi 1}% -\ifMFPdebug - \show\MFP@power@Sgn -\fi \fi \ifnum\MFP@y@Sgn<0 \MFP@Rinv \MFP@Rcopyzx\fi \ifnum\MFP@loopctr=1 \MFP@Rloadz \MFP@power@Sgn\MFP@x@Int\MFP@x@Frc \else \MFP@@Rpow -\ifMFPdebug - \showreg z% -\fi \fi \fi - \fi}% % + \fi}% \def\MFP@@Rpow{% - \MFP@Rcopy xq% initialize register to be squared - \MFP@Rload p110% initialize register to hold partial products + \MFP@Rcopy xq% + \MFP@Rload p110% \MFP@Rpow@loop}% \def\MFP@Rpow@loop{% \ifodd\MFP@loopctr @@ -568,32 +580,40 @@ \def\MFP@handle@expoverflow{% \MFP@expoverflow@err \MFP@loopctr=0 - \MFP@Rloadz\MFP@power@Sgn\MaxRealInt\MaxRealFrac -}% + \MFP@Rloadz\MFP@power@Sgn\MaxRealInt\MaxRealFrac}% \def\MFP@Rsqrt{% \ifcase\MFP@x@Sgn\relax \MFP@Rzero \or \ifnum\MFP@x@Int=0 - \let\MFP@sqrt@reduce=Y% + \def\MFP@sqrt@reduce{2}% \edef\MFP@x@Int{\number\MFP@x@Frc}% \edef\MFP@x@Frc{00000000}% + \else\ifnum\MFP@x@Int<10000 + \def\MFP@sqrt@reduce{1}% + \edef\MFP@x@Int{\MFP@x@Int\@xp\MFP@fourofmany\MFP@x@Frc\MFP@end}% + \edef\MFP@x@Frc{\@xp\MFP@gobblefour\MFP@x@Frc0000}% \else - \let\MFP@sqrt@reduce=N% - \fi + \def\MFP@sqrt@reduce{0}% + \fi\fi \MFP@Rcopy xt% \MFP@Isqrt \MFP@Rcopyz s\MFP@Rcopyzy \MFP@Rcopy tx\MFP@Rdiv \MFP@Rcopy sx\MFP@Rcopyzy\MFP@Radd \MFP@Rcopyzx\MFP@Rhalve - \ifx Y\MFP@sqrt@reduce - \MFP@Rcopyzx\MFP@Rload y10{10000}\MFP@Rmul + \ifcase \MFP@sqrt@reduce\relax + \or + \MFP@Rcopyzx\MFP@Rload y10{01000000}\MFP@Rmul + \or + \MFP@Rcopyzx\MFP@Rload y10{00010000}\MFP@Rmul \fi \else - \MFP@sqrtofneg@err + \MFP@warn{Square root of a negative number. Zero will be returned.}% \MFP@Rzero \fi}% +\def\MFP@fourofmany#1#2#3#4#5\MFP@end{#1#2#3#4}% +\def\MFP@gobblefour#1#2#3#4{}% \def\MFP@ItoQ#1#2{% \MFP@tempa#1\relax\MFP@tempb#2\relax \def\MFP@ItoQ@Tmp{}\MFP@ItoQ@loop}% @@ -617,7 +637,7 @@ \def\MFP@Isqrt{% \MFP@ItoQ\MFP@x@Int\MFP@x@Frc \MFP@tempa=0 \MFP@tempb=0 \MFP@tempc=0 - \expandafter\MFP@Isqrt@loop\MFP@ItoQ@Tmp\mfp@end + \expandafter\MFP@Isqrt@loop\MFP@ItoQ@Tmp\MFP@end \MFP@tempa=\MFP@tempc \divide\MFP@tempc\MFP@tttfour \edef\MFP@z@Int{\number\MFP@tempc}% @@ -628,7 +648,7 @@ \edef\MFP@z@Frc{\MFP@z@Frc0000}% \def\MFP@z@Sgn{1}}% \def\MFP@Isqrt@loop#1{% - \ifx\mfp@end #1% + \ifx\MFP@end #1% \else \multiply\MFP@tempa 2 \multiply\MFP@tempb 4 \multiply\MFP@tempc 2 \advance \MFP@tempb#1\relax @@ -638,8 +658,8 @@ \advance\MFP@tempa 1 \fi \expandafter\MFP@Isqrt@loop - \fi -}% + \fi}% +\MFP@xfinish \endinput %% %% End of file `mfpextra.tex'. diff --git a/Master/texmf-dist/tex/generic/minifp/minifp.sty b/Master/texmf-dist/tex/generic/minifp/minifp.sty index 321ec4197f1..0566ae33e1e 100644 --- a/Master/texmf-dist/tex/generic/minifp/minifp.sty +++ b/Master/texmf-dist/tex/generic/minifp/minifp.sty @@ -21,16 +21,16 @@ %% is Daniel H. Luecking. The Base Interpreters associated %% with minifp are plain TeX and LaTeX. %% -\def\MFPfiledate{2013/02/01}% -\def\MFPfileversion{0.92}% +\def\MFPfiledate{2013/05/28}% +\def\MFPfileversion{0.95}% \expandafter \ifx \csname MFP@finish\endcsname\relax \else \expandafter\endinput \fi \expandafter\edef\csname MFP@finish\endcsname{% - \catcode64=\the\catcode64 \space % @ - \catcode46=\the\catcode46 \space % . - \catcode60=\the\catcode60 \space % < - \catcode62=\the\catcode62 \space}% > + \catcode64=\the\catcode64 \space + \catcode46=\the\catcode46 \space + \catcode60=\the\catcode60 \space + \catcode62=\the\catcode62 \space}% \ifx\ProvidesPackage\UndEfInEd \newlinechar`\^^J% \message{% @@ -44,28 +44,22 @@ \DeclareOption{extra}{\def\MFPextra{}}% \ProcessOptions\relax \fi -\catcode64=11 % @=letter (already is in LaTeX) +\catcode64=11 \ifx\MFPextra\UndEfInEd \def\MFP@loadextra{}% \else \def\MFP@loadextra{\input mfpextra\relax}% \fi -\def\MFPloadextra{% - \edef\MFP@load@extra{% - \catcode46=12 \catcode60=12 \catcode62=12 \catcode64=11 - \noexpand\input mfpextra\relax - \catcode46=\the\catcode46\relax\catcode60=\the\catcode60\relax - \catcode62=\the\catcode62\relax\catcode64=\the\catcode64\relax}% - \MFP@load@extra}% -\catcode46=12 % . -\catcode60=12 % < -\catcode62=12 % > -\long\def\gobbleto@mfp@end#1\mfp@end{}% -\ifx\mfp@end\UndEfInEd\def\mfp@end{\@empty}\fi +\def\MFPloadextra{\input mfpextra\relax}% +\catcode46=12 +\catcode60=12 +\catcode62=12 +\long\def\gobbleto@MFP@end#1\MFP@end{}% +\def\MFP@end{\@empty}% \ifx\documentclass\UndEfInEd \def\MFP@ifnoLaTeX{}% \else - \let\MFP@ifnoLaTeX\gobbleto@mfp@end + \let\MFP@ifnoLaTeX\gobbleto@MFP@end \fi \let\@xp\expandafter \def\@XP{\@xp\@xp\@xp}% @@ -97,13 +91,13 @@ \long\def\@firstofone #1{#1}% \long\def\@firstoftwo #1#2{#1}% \long\def\@secondoftwo#1#2{#2}% -\mfp@end +\MFP@end \def\MFP@tttfour {10000}% ttt = Ten To The \def\MFP@ttteight{100000000}% -\def\MFP@firstofmany#1#2\mfp@end{#1}% -\def\MFP@fifthofmany#1#2#3#4#5#6\mfp@end{#5}% -\def\MFP@ninthofmany#1#2#3#4#5#6#7#8{\MFP@firstofmany}% -\def\MFP@firsteightofmany#1#2#3#4#5#6#7#8#9\mfp@end{#1#2#3#4#5#6#7#8}% +\def\MFP@oneofmany#1#2\MFP@end{#1}% +\def\MFP@fifthofmany#1#2#3#4#5#6\MFP@end{#5}% +\def\MFP@ninthofmany#1#2#3#4#5#6#7#8{\MFP@oneofmany}% +\def\MFP@eightofmany#1#2#3#4#5#6#7#8#9\MFP@end{#1#2#3#4#5#6#7#8}% \let\MFP@eos\relax \def\MFP@EOS{\MFP@eos}% \def\MFP@initRstack{\def\MFP@Rstack{\MFP@eos}}% @@ -116,22 +110,19 @@ \countdef \MFP@tempf 10 \newcount \MFP@loopctr \def\MFP@endgroup@after#1{\edef\x{\endgroup#1}\x}% -\def\MFP@endgroup@return{\MFP@endgroup@after\MFP@returned@values}% -\def\MFP@def@after{\def\noexpand}% +\def\MFP@afterdef{\def\noexpand}% \def\MFP@returned@values{% - \MFP@def@after\MFP@z@Val{\MFP@z@Sign\MFP@z@Int.\MFP@z@Frc}% - \MFP@def@after\MFP@z@Ovr{\MFP@z@Ovr}% - \MFP@def@after\MFP@z@Und{\MFP@z@Und}% - \MFP@def@after\MFPcurr@Sgn{\MFP@z@Sgn}}% + \MFP@afterdef\MFP@z@Val{\MFP@z@Sign\MFP@z@Int.\MFP@z@Frc}% + \MFP@afterdef\MFP@z@Ovr{\MFP@z@Ovr}% + \MFP@afterdef\MFP@z@Und{\MFP@z@Und}% + \MFP@afterdef\MFPcurr@Sgn{\MFP@z@Sgn}}% \def\MFP@subroutine#1{% \begingroup - \MFP@basic@init@z + \MFP@Rzero + \def\MFP@z@Ovr{0}% + \def\MFP@z@Und{0}% #1% - \MFP@endgroup@return}% -\def\MFP@basic@init@z{% - \MFP@Rzero - \def\MFP@z@Ovr{0}% - \def\MFP@z@Und{0}}% + \MFP@endgroup@after\MFP@returned@values}% \def\MFP@Rzero{% \def\MFP@z@Sgn{0}% \def\MFP@z@Int{0}% @@ -170,28 +161,24 @@ \def\MFPparse@real#1#2#3#4{% \MFPnospace@def\MFPtemp@Val{#4}% \MFPprocess@into@parts\MFPtemp@Val#1#2#3% - \MFPpadto@eight#3}% + \MFP@padtoeight#3}% \def\MFPparse@x{\MFPparse@real\MFP@x@Sgn\MFP@x@Int\MFP@x@Frc}% \def\MFPparse@y{\MFPparse@real\MFP@y@Sgn\MFP@y@Int\MFP@y@Frc}% \def\MFPnospace@def#1#2{% \edef#1{#2\space}\edef#1{\@xp\zap@space#1\@empty}}% \def\MFPprocess@into@parts#1#2#3#4{% - \@xp\MFPsplit@dot#1..\mfp@end #3#4% - \ifnum#31<0 - \def#2{-1}% - \else - \def#2{1}% + \@xp\MFPsplit@dot#1..\MFP@end #3#4% + \ifnum#31<0 \def#2{-1}% + \else \def#2{1}% \fi \ifnum #30=0 \def#3{0}% - \ifnum 0#4=0 - \def#2{0}% - \fi + \ifnum 0#4=0 \def#2{0}\fi \fi \edef#3{\number \ifnum #2<0 -\fi#3}}% -\def\MFPsplit@dot#1.#2.#3\mfp@end#4#5{\edef#4{#1}\edef#5{#2}}% -\def\MFPpadto@eight#1{% - \edef#1{\@xp\MFP@firsteightofmany#100000000\mfp@end}}% +\def\MFPsplit@dot#1.#2.#3\MFP@end#4#5{\edef#4{#1}\edef#5{#2}}% +\def\MFP@padtoeight#1{% + \edef#1{\@xp\MFP@eightofmany#100000000\MFP@end}}% \def\MFPgetoperand@x{\Rpop\MFP@x@Val \MFPprocess@into@parts\MFP@x@Val\MFP@x@Sgn\MFP@x@Int\MFP@x@Frc}% \def\MFPgetoperand@y{\Rpop\MFP@y@Val @@ -211,8 +198,8 @@ \@xp\@secondoftwo \fi}% \def\MFP@popit{\if@EndofStack\doMFP@EOS\doMFP@popit}% -\def\doMFP@EOS#1\mfp@end#2{\MFP@popempty@err\let#2\EndofStack}% -\def\doMFP@popit#1#2\mfp@end#3{\edef\MFP@Rstack{#2}\edef#3{#1}}% +\def\doMFP@EOS#1\MFP@end#2{\MFP@popempty@err\let#2\EndofStack}% +\def\doMFP@popit#1#2\MFP@end#3{\edef\MFP@Rstack{#2}\edef#3{#1}}% \def\MFPchk#1{% \MFPparse@x{#1}% \MFP@Rchk\MFP@x@Sgn}% @@ -267,8 +254,8 @@ \def\Rmax {\MFP@stack@Binary\MFP@Rmax}% \let\Rnoop\relax \def\Rcmp{% - \MFPgetoperand@y\MFPgetoperand@x % get operands (last pushed is y) - \MFP@Rcat\MFP@x@Val\MFP@Rcat\MFP@y@Val % put back: LOFI + \MFPgetoperand@y\MFPgetoperand@x + \MFP@Rcat\MFP@x@Val\MFP@Rcat\MFP@y@Val \MFP@Rcmp}% \def\Rchk{% \MFPgetoperand@x @@ -279,7 +266,7 @@ \edef\MFP@z@Val{\MFP@x@Sign\MFP@x@Int.\MFP@x@Frc}% \edef\MFPcurr@Sgn{\MFP@x@Sgn}% \MFPpush@result}% - \def\Rpop{\@xp\MFP@popit\MFP@Rstack\mfp@end}% + \def\Rpop{\@xp\MFP@popit\MFP@Rstack\MFP@end}% \def\Rexch{% \Rpop\MFP@x@Val\Rpop\MFP@y@Val \MFP@Rcattwo\MFP@y@Val\MFP@x@Val}% @@ -316,6 +303,10 @@ \def\MFPdiv{\MFP@op@Binary\MFP@Rdiv}% \def\MFPmin{\MFP@op@Binary\MFP@Rmin}% \def\MFPmax{\MFP@op@Binary\MFP@Rmax}% +\def\MFP@stack@Nullary#1{% + \MFP@subroutine{#1}\MFPpush@result}% +\def\MFP@op@Nullary#1{% + \MFP@subroutine{#1}\MFPstore@result}% \def\MFP@stack@Unary#1{% \MFPgetoperand@x \MFP@subroutine{#1}\MFPpush@result}% @@ -339,7 +330,7 @@ \MFP@tempb\MFP@x@Frc\relax \ifodd\MFP@tempb \def\MFP@z@Und{5}% - \advance\MFP@tempb 1 % round up + \advance\MFP@tempb 1 \ifnum\MFP@ttteight=\MFP@tempb \MFP@tempb0 \advance\MFP@tempa1 \fi @@ -379,17 +370,17 @@ \edef\MFP@x@Sgn{\number -\MFP@x@Sgn}\MFP@Rincr \edef\MFP@z@Sgn{\number -\MFP@z@Sgn}}% \def\MFP@Rstore{\MFP@Rcopy xz}% -\def\MFP@Rfloororceil#1{% +\def\MFP@Rfloorceil#1{% \MFP@tempa\MFP@x@Int\relax - \ifnum 0#1\MFP@x@Sgn + \ifnum \MFP@x@Sgn #10 \ifnum\MFP@x@Frc=0 \else \advance\MFP@tempa1 \fi \fi \MFP@Rloadz{\ifnum\MFP@x@Int=0 0\else\MFP@x@Sgn\fi}\MFP@tempa0}% -\def\MFP@Rfloor{\MFP@Rfloororceil>}% -\def\MFP@Rceil {\MFP@Rfloororceil<}% +\def\MFP@Rfloor{\MFP@Rfloorceil<}% +\def\MFP@Rceil {\MFP@Rfloorceil>}% \def\MFP@split#1#2#3{% \begingroup \MFP@tempa#1\relax @@ -399,38 +390,27 @@ \multiply\MFP@tempb by\MFP@tttfour \advance\MFP@tempa-\MFP@tempb \MFP@endgroup@after{% - \MFP@def@after#2{#2}% - \MFP@def@after#3{\number\MFP@tempa}% + \MFP@afterdef#2{#2}% + \MFP@afterdef#3{\number\MFP@tempa}% }}% -\def\MFP@x@split{% +\def\MFP@@split{% \MFP@split\MFP@x@Int\MFP@x@Int@ii\MFP@x@Int@i - \MFP@split\MFP@x@Frc\MFP@x@Frc@i\MFP@x@Frc@ii}% -\def\MFP@y@split{% + \MFP@split\MFP@x@Frc\MFP@x@Frc@i\MFP@x@Frc@ii \MFP@split\MFP@y@Int\MFP@y@Int@ii\MFP@y@Int@i \MFP@split\MFP@y@Frc\MFP@y@Frc@i\MFP@y@Frc@ii}% -\def\MFPmore@init@z{% - \def\MFP@z@Frc@iv {0}% - \def\MFP@z@Frc@iii{0}% - \def\MFP@z@Frc@ii {0}% - \def\MFP@z@Frc@i {0}% - \def\MFP@z@Int@i {0}% - \def\MFP@z@Int@ii {0}% - \def\MFP@z@Int@iii{0}}% -\def\MFP@fourdigits#1{% - \@xp\MFP@fifthofmany\number#1{}{0}{00}{000}\mfp@end\number#1}% -\def\makeMFP@fourdigits#1{\edef#1{\MFP@fourdigits{#1}}}% -\def\MFP@eightdigits#1{% - \@xp\MFP@ninthofmany\number#1% - {}{0}{00}{000}{0000}{00000}{000000}{0000000}\mfp@end\number#1}% -\def\makeMFP@eightdigits#1{\edef#1{\MFP@eightdigits{#1}}}% +\def\makeMFP@fourdigits#1{% + \edef#1{\@xp\MFP@fifthofmany\number#1{}{0}{00}{000}\MFP@end\number#1}}% +\def\makeMFP@eightdigits#1{% + \edef#1{\@xp\MFP@ninthofmany\number#1% + {}{0}{00}{000}{0000}{00000}{000000}{0000000}\MFP@end\number#1}}% \def\MFP@carrya{\MFP@carry\MFP@ttteight}% \def\MFP@carrym{\MFP@carry\MFP@tttfour}% \def\MFP@carry#1#2#3{% \begingroup \MFP@carryi{#1}#2#3% \MFP@endgroup@after{% - \MFP@def@after#3{\number\MFP@tempa}% - \MFP@def@after#2{\number\MFP@tempb}% + \MFP@afterdef#3{\number\MFP@tempa}% + \MFP@afterdef#2{\number\MFP@tempb}% }}% \def\MFP@carryi#1#2#3{% \MFP@tempa=#3\relax @@ -445,7 +425,7 @@ \MFP@tempa#1% \advance\MFP@tempa#2\relax \MFP@endgroup@after{% - \MFP@def@after#3{\number\MFP@tempa}% + \MFP@afterdef#3{\number\MFP@tempa}% }}% \def\MFP@multiplyone#1#2{% \MFP@tempa#1% @@ -500,24 +480,25 @@ \edef\MFP@z@Frc{\number\MFP@tempb}% \makeMFP@eightdigits\MFP@z@Frc}% \def\MFP@Rmul{% - \MFP@tempa\MFP@x@Sgn \multiply\MFP@tempa\MFP@y@Sgn\relax - \ifnum 0=\MFP@tempa - \MFP@Rzero - \else - \edef\MFP@z@Sgn{\number\MFP@tempa}% - \@xp\MFP@@Rmul - \fi}% + \ifnum\MFP@x@Sgn=0 \MFP@Rzero + \else\ifnum\MFP@y@Sgn=0 \MFP@Rzero + \else \edef\MFP@z@Sgn{\number\MFP@x@Sign\MFP@y@Sgn}% + \@XP\MFP@@Rmul + \fi\fi}% \def\MFP@@Rmul{% - \MFPmore@init@z - \MFP@x@split\MFP@y@split + \def\MFP@z@Frc@iv {0}\def\MFP@z@Frc@iii{0}% + \def\MFP@z@Frc@ii {0}\def\MFP@z@Frc@i {0}% + \def\MFP@z@Int@i {0}\def\MFP@z@Int@ii {0}% + \def\MFP@z@Int@iii{0}% + \MFP@@split \MFP@multiplyfour \MFP@y@Frc@ii \MFP@z@Frc@i - \MFP@z@Frc@ii \MFP@z@Frc@iii\MFP@z@Frc@iv + \MFP@z@Frc@ii \MFP@z@Frc@iii \MFP@z@Frc@iv \MFP@multiplyfour \MFP@y@Frc@i \MFP@z@Int@i - \MFP@z@Frc@i \MFP@z@Frc@ii \MFP@z@Frc@iii + \MFP@z@Frc@i \MFP@z@Frc@ii \MFP@z@Frc@iii \MFP@multiplyfour \MFP@y@Int@i \MFP@z@Int@ii - \MFP@z@Int@i \MFP@z@Frc@i \MFP@z@Frc@ii + \MFP@z@Int@i \MFP@z@Frc@i \MFP@z@Frc@ii \MFP@multiplyfour \MFP@y@Int@ii \MFP@z@Int@iii - \MFP@z@Int@ii \MFP@z@Int@i \MFP@z@Frc@i + \MFP@z@Int@ii \MFP@z@Int@i \MFP@z@Frc@i \MFP@carrym\MFP@z@Frc@iv\MFP@z@Frc@iii \MFP@carrym\MFP@z@Frc@iii\MFP@z@Frc@ii \ifnum\MFP@z@Frc@iii<5000 \else @@ -540,12 +521,10 @@ \edef\MFP@z@Und{\MFP@z@Frc@iii\MFP@z@Frc@iv}% \ifnum\MFP@z@Int>0 \else\ifnum\MFP@z@Frc>0 - \else - \def\MFP@z@Sgn{0}% + \else \def\MFP@z@Sgn{0}% \fi\fi}% \def\MFP@Rdiv{% - \ifnum\MFP@y@Sgn=0 - \MFP@dividebyzero@err + \ifnum\MFP@y@Sgn=0 \MFP@dividebyzero@err \ifnum\MFP@x@Sgn=0 \edef\MFP@z@Int{\ZeroOverZeroInt}% \edef\MFP@z@Frc{\ZeroOverZeroFrac}% @@ -554,13 +533,8 @@ \edef\MFP@z@Frc{\xOverZeroFrac}% \fi \edef\MFP@z@Sgn{\MFP@x@Sgn}% - \else\ifnum\MFP@x@Sgn=0 - \MFP@Rzero - \else - \MFP@tempa\MFP@x@Sgn - \multiply\MFP@tempa\MFP@y@Sgn - \edef\MFP@z@Sgn{\number\MFP@tempa}% - \MFP@@Rdiv + \else\ifnum\MFP@x@Sgn=0 \MFP@Rzero + \else \edef\MFP@z@Sgn{\number\MFP@x@Sign\MFP@y@Sgn}\MFP@@Rdiv \fi\fi}% \def\MFP@@Rdiv{% \ifnum\MFP@y@Int=0 @@ -571,14 +545,14 @@ \MFP@tempa=0 \fi \advance\MFP@tempa\MFP@numdigits@toshift\MFP@y@Int\relax - \@XP\MFP@doshift@y\@xp\MFP@y@Int\MFP@y@Frc0000000\mfp@end + \@XP\MFP@doshift@y\@xp\MFP@y@Int\MFP@y@Frc0000000\MFP@end \ifnum\MFP@x@Int=0 \edef\MFP@x@Int{\number\MFP@x@Frc}% \def\MFP@x@Frc{00000000}% \advance\MFP@tempa -8 \fi \advance\MFP@tempa-\MFP@numdigits@toshift\MFP@x@Int\relax - \@XP\MFP@doshift@x\@xp\MFP@x@Int\MFP@x@Frc0000000\mfp@end + \@XP\MFP@doshift@x\@xp\MFP@x@Int\MFP@x@Frc0000000\MFP@end \ifnum\MFP@tempa<-9 \else \MFP@tempf\MFP@tempa @@ -593,13 +567,13 @@ \makeMFP@eightdigits\MFP@z@Frc \fi \fi}% -\def\MFP@numdigits@toshift#1{\@xp\MFP@ninthofmany#101234567\mfp@end}% -\def\MFP@doshift@x#1#2#3#4#5#6#7#8#9\mfp@end{% +\def\MFP@numdigits@toshift#1{\@xp\MFP@ninthofmany#101234567\MFP@end}% +\def\MFP@doshift@x#1#2#3#4#5#6#7#8#9\MFP@end{% \def\MFP@x@Int{#1#2#3#4#5#6#7#8}% - \edef\MFP@x@Frc{\MFP@firsteightofmany#9\mfp@end}}% -\def\MFP@doshift@y#1#2#3#4#5#6#7#8#9\mfp@end{% + \edef\MFP@x@Frc{\MFP@eightofmany#9\MFP@end}}% +\def\MFP@doshift@y#1#2#3#4#5#6#7#8#9\MFP@end{% \def \MFP@y@Int{#1#2#3#4#5#6#7#8}% - \edef\MFP@y@Frc{\MFP@firsteightofmany#9\mfp@end}}% + \edef\MFP@y@Frc{\MFP@eightofmany#9\MFP@end}}% \def\MFP@Rdivloop{% \MFP@tempb\MFP@x@Int % \MFP@tempb = n_1 \MFP@tempc\MFP@y@Int % \MFP@tempc = d_1 @@ -634,7 +608,7 @@ \fi \advance\MFP@tempf -1 \ifnum\MFP@tempf>0 - \edef\MFP@x@Int{\MFP@x@Int0}% easy multiplications by 10 + \edef\MFP@x@Int{\MFP@x@Int0}% \edef\MFP@x@Frc{\MFP@x@Frc0}% \MFP@carrya\MFP@x@Frc\MFP@x@Int \@xp\MFP@Rdivloop @@ -643,7 +617,7 @@ \advance \MFP@tempa -7 \ifnum\MFP@tempa>0 \def\MFP@z@Ovr{}% - \@xp\MFPget@Ovrdigits\MFP@z@digits\mfp@end + \@xp\MFPget@Ovrdigits\MFP@z@digits\MFP@end \else \ifnum\MFP@tempa<-7 \edef\MFP@z@digits{00000000\MFP@z@digits}% @@ -661,7 +635,7 @@ 0000\else 00000% \fi \MFP@z@digits}% - \@xp\MFPget@Intdigits\MFP@z@digits\mfp@end + \@xp\MFPget@Intdigits\MFP@z@digits\MFP@end \fi}% \def\MFPget@Ovrdigits#1{% \edef\MFP@z@Ovr{\MFP@z@Ovr#1}% @@ -676,7 +650,7 @@ \MFPget@Frcdigits}% \def\MFPget@Frcdigits#1#2#3#4#5#6#7#8#9{% \def\MFP@z@Frc{#1#2#3#4#5#6#7#8}% - \def\MFP@z@Und{#9}\gobbleto@mfp@end}% + \def\MFP@z@Und{#9}\gobbleto@MFP@end}% \def\MFP@Rmax{% \MFP@Rcmp \ifMFP@neg \MFP@Rcopy yz\else\MFP@Rcopy xz\fi}% \def\MFP@Rmin{% @@ -691,15 +665,15 @@ \@xp\MFP@Rtrunc \fi}% \def\MFP@Rtrunc#1{% - \edef\MFP@x@Frc{\@xp\MFP@@Rtrunc\MFP@x@Frc\mfp@end}% - \ifnum\MFP@x@Int=0 % possibly returns 0 + \edef\MFP@x@Frc{\@xp\MFP@@Rtrunc\MFP@x@Frc\MFP@end}% + \ifnum\MFP@x@Int=0 \ifnum\MFP@x@Frc=0 \def\MFP@x@Sgn{0}% \fi \fi \MFP@endgroup@after{% - \MFP@def@after#1{\MFP@x@Sign\MFP@x@Int.\MFP@x@Frc}}}% -\def\MFP@@Rtrunc#1#2#3#4#5#6#7#8#9\mfp@end{% + \MFP@afterdef#1{\MFP@x@Sign\MFP@x@Int.\MFP@x@Frc}}}% +\def\MFP@@Rtrunc#1#2#3#4#5#6#7#8#9\MFP@end{% \ifcase\MFP@tempa\or #1\or #1#2\or @@ -711,9 +685,9 @@ #1#2#3#4#5#6#7#8\fi}% \def\MFP@iRtrunc#1{% \makeMFP@eightdigits\MFP@x@Int - \edef\MFP@x@Val{\number\MFP@x@Sign\@xp\MFP@@iRtrunc\MFP@x@Int\mfp@end}% - \MFP@endgroup@after{\MFP@def@after#1{\MFP@x@Val}}}% -\def\MFP@@iRtrunc#1#2#3#4#5#6#7#8#9\mfp@end{% + \edef\MFP@x@Val{\number\MFP@x@Sign\@xp\MFP@@iRtrunc\MFP@x@Int\MFP@end}% + \MFP@endgroup@after{\MFP@afterdef#1{\MFP@x@Val}}}% +\def\MFP@@iRtrunc#1#2#3#4#5#6#7#8#9\MFP@end{% \ifcase-\MFP@tempa #1#2#3#4#5#6#7#8\or #1#2#3#4#5#6#70\or @@ -730,32 +704,32 @@ \ifnum 0>\MFP@tempa \edef\MFP@y@Tmp{% \ifcase-\MFP@tempa\or - 5\or % .5 x 10^1 - 50\or % .5 x 10^2 - 500\or % .5 x 10^3 - 5000\or % .5 x 10^4 - 50000\or % .5 x 10^5 - 500000\or % .5 x 10^6 - 5000000\else % .5 x 10^7 - 50000000\fi % .5 x 10^8 + 5\or + 50\or + 500\or + 5000\or + 50000\or + 500000\or + 5000000\else + 50000000\fi }% \else \edef\MFP@y@Tmp{% \ifcase\MFP@tempa - .5\or % .5 x 10^0 - .05\or % .5 x 10^{-1} - .005\or % .5 x 10^{-2} - .0005\or % .5 x 10^{-3} - .00005\or % .5 x 10^{-4} - .000005\or % .5 x 10^{-5} - .0000005\or % .5 x 10^{-6} - .00000005\else% .5 x 10^{-7} - 0\fi % + .5\or + .05\or + .005\or + .0005\or + .00005\or + .000005\or + .0000005\or + .00000005\else + 0\fi }% \fi \MFPchk{#2}\ifMFP@neg\edef\MFP@y@Tmp{-\MFP@y@Tmp}\fi \MFPadd{#2}\MFP@y@Tmp\MFP@z@Tmp - \MFP@endgroup@after{\MFP@def@after\MFP@z@Tmp{\MFP@z@Tmp}}% + \MFP@endgroup@after{\MFP@afterdef\MFP@z@Tmp{\MFP@z@Tmp}}% \MFPtruncate{#1}\MFP@z@Tmp}% \def\MFPstrip{% \@ifstar{\MFP@strip{}}{\MFP@strip{.0}}}% @@ -764,22 +738,22 @@ \ifnum \MFP@x@Frc=0 \edef#3{\MFP@x@Sign\MFP@x@Int#1}% \else - \edef#3{\MFP@x@Sign\MFP@x@Int.\@xp\MFP@@strip\MFP@x@Frc\mfp@end}% + \edef#3{\MFP@x@Sign\MFP@x@Int.\@xp\MFP@@strip\MFP@x@Frc\MFP@end}% \fi}% -\def\MFP@@strip#1#2\mfp@end{% +\def\MFP@@strip#1#2\MFP@end{% #1% \ifnum 0#2>0 \@xp\MFP@@strip \else - \@xp\gobbleto@mfp@end - \fi#2\mfp@end}% + \@xp\gobbleto@MFP@end + \fi#2\MFP@end}% \def\MFP@Global#1{\toks@\@xp{#1}\xdef#1{\the\toks@}}% \def\MFP@GlobalStack{\MFP@Global\MFP@Rstack}% \def\MFP@Export#1{% \begingroup \toks@\@xp{\MFPprogram@returns}% \MFP@endgroup@after{% - \MFP@def@after\MFPprogram@returns{\the\toks@ \MFP@def@after#1{#1}}% + \MFP@afterdef\MFPprogram@returns{\the\toks@ \MFP@afterdef#1{#1}}% }}% \def\MFP@ExportStack{\MFP@Export\MFP@Rstack}% \def\MFP@Rload #1#2#3#4{% |