path: root/Build/source/texk/web2c/luatexdir/tex/arithmetic.c

/*

arithmetic.w

Copyright 2009-2010 Taco Hoekwater <taco@@luatex.org>

This file is part of LuaTeX.

LuaTeX is free software; you can redistribute it and/or modify it under
the terms of the GNU General Public License as published by the Free
Software Foundation; either version 2 of the License, or (at your
option) any later version.

LuaTeX is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
FITNESS FOR A PARTICULAR PURPOSE.  See the GNU Lesser General Public
License for more details.

You should have received a copy of the GNU General Public License along
with LuaTeX; if not, see <http://www.gnu.org/licenses/>.

*/

#include "ptexlib.h"

/*tex

The principal computations performed by \TeX\ are done entirely in terms of
integers less than $2^{31}$ in magnitude; and divisions are done only when both
dividend and divisor are nonnegative. Thus, the arithmetic specified in this
program can be carried out in exactly the same way on a wide variety of
computers, including some small ones. Why? Because the arithmetic calculations
need to be spelled out precisely in order to guarantee that \TeX\ will produce
identical output on different machines. If some quantities were rounded
differently in different implementations, we would find that line breaks and even
page breaks might occur in different places. Hence the arithmetic of \TeX\ has
been designed with care, and systems that claim to be implementations of \TeX82
should follow precisely the @:TeX82}{\TeX82@> calculations as they appear in the
present program.

Actually there are three places where \TeX\ uses |div| with a possibly negative
numerator. These are harmless; see |div| in the index. Also if the user sets the
\.{\\time} or the \.{\\year} to a negative value, some diagnostic information
will involve negative-numerator division. The same remarks apply for |mod| as
well as for |div|.

Here is a routine that calculates half of an integer, using an unambiguous
convention with respect to signed odd numbers.

*/

int half(int x)
{
    if (odd(x))
        return ((x + 1) / 2);
    else
        return (x / 2);
}

/*tex

The following function is used to create a scaled integer from a given decimal
fraction $(.d_0d_1\ldots d_{k-1})$, where |0<=k<=17|. The digit $d_i$ is
given in |dig[i]|, and the calculation produces a correctly rounded result.

*/

scaled round_decimals(int k)
{
    int a = 0;
    while (k-- > 0) {
        a = (a + dig[k] * two) / 10;
    }
    return ((a + 1) / 2);
}

/*tex

Conversely, here is a procedure analogous to |print_int|. If the output of this
procedure is subsequently read by \TeX\ and converted by the |round_decimals|
routine above, it turns out that the original value will be reproduced exactly;
the ``simplest'' such decimal number is output, but there is always at least one
digit following the decimal point.

The invariant relation in the \&{repeat} loop is that a sequence of decimal
digits yet to be printed will yield the original number if and only if they form
a fraction~$f$ in the range $s-\delta\L10\cdot2^{16}f<s$. We can stop if and only
if $f=0$ satisfies this condition; the loop will terminate before $s$ can
possibly become zero.

The next one prints a scaled real, rounded to five digits.

*/

void print_scaled(scaled s)
{
    /*tex The amount of allowable inaccuracy: */
    scaled delta;
    char buffer[20];
    int i = 0;
    if (s < 0) {
        /*tex Print the sign, if negative. */
        print_char('-');
        negate(s);
    }
    /*tex Print the integer part. */
    print_int(s / unity);
    buffer[i++] = '.';
    s = 10 * (s % unity) + 5;
    delta = 10;
    do {
        if (delta > unity) {
            /*tex Round the last digit. */
            s = s + 0100000 - 50000;
        }
        buffer[i++] = '0' + (s / unity);
        s = 10 * (s % unity);
        delta = delta * 10;
    } while (s > delta);
    buffer[i++] = '\0';
    tprint(buffer);
}

/*tex

Physical sizes that a \TeX\ user specifies for portions of documents are
represented internally as scaled points. Thus, if we define an `sp' (scaled
@^sp@> point) as a unit equal to $2^{-16}$ printer's points, every dimension
inside of \TeX\ is an integer number of sp. There are exactly 4,736,286.72 sp per
inch. Users are not allowed to specify dimensions larger than $2^{30}-1$ sp,
which is a distance of about 18.892 feet (5.7583 meters); two such quantities can
be added without overflow on a 32-bit computer.

The present implementation of \TeX\ does not check for overflow when @^overflow
in arithmetic@> dimensions are added or subtracted. This could be done by
inserting a few dozen tests of the form `\ignorespaces|if x>=010000000000 then
@t\\{report\_overflow}@>|', but the chance of overflow is so remote that such
tests do not seem worthwhile.

\TeX\ needs to do only a few arithmetic operations on scaled quantities, other
than addition and subtraction, and the following subroutines do most of the work.
A single computation might use several subroutine calls, and it is desirable to
avoid producing multiple error messages in case of arithmetic overflow; so the
routines set the global variable |arith_error| to |true| instead of reporting
errors directly to the user. Another global variable, |tex_remainder|, holds the
remainder after a division.

*/

/*tex Has arithmetic overflow occurred recently? */

boolean arith_error;

/*tex The amount subtracted to get an exact division. */

scaled tex_remainder;

/*tex

 The first arithmetical subroutine we need computes $nx+y$, where |x|
and~|y| are |scaled| and |n| is an integer. We will also use it to
multiply integers.

*/

scaled mult_and_add(int n, scaled x, scaled y, scaled max_answer)
{
    if (n == 0)
        return y;
    if (n < 0) {
        negate(x);
        negate(n);
    }
    if (((x <= (max_answer - y) / n) && (-x <= (max_answer + y) / n))) {
        return (n * x + y);
    } else {
        arith_error = true;
        return 0;
    }
}

/*tex

We also need to divide scaled dimensions by integers.

*/

scaled x_over_n(scaled x, int n)
{
    /*tex Should |tex_remainder| be negated? */
    boolean negative = false;
    if (n == 0) {
        arith_error = true;
        tex_remainder = x;
        return 0;
    } else {
        if (n < 0) {
            negate(x);
            negate(n);
            negative = true;
        }
        if (x >= 0) {
            tex_remainder = x % n;
            if (negative)
                negate(tex_remainder);
            return (x / n);
        } else {
            tex_remainder = -((-x) % n);
            if (negative)
                negate(tex_remainder);
            return (-((-x) / n));
        }
    }
}

/*tex

Then comes the multiplication of a scaled number by a fraction |n/d|, where |n|
and |d| are nonnegative integers |<=@t$2^{16}$@>| and |d| is positive. It would
be too dangerous to multiply by~|n| and then divide by~|d|, in separate
operations, since overflow might well occur; and it would be too inaccurate to
divide by |d| and then multiply by |n|. Hence this subroutine simulates
1.5-precision arithmetic.

*/

scaled xn_over_d(scaled x, int n, int d)
{
    nonnegative_integer t, u, v, xx, dd;
    boolean positive = true;
    if (x < 0) {
        negate(x);
        positive = false;
    }
    xx = (nonnegative_integer) x;
    dd = (nonnegative_integer) d;
    t = ((xx % 0100000) * (nonnegative_integer) n);
    u = ((xx / 0100000) * (nonnegative_integer) n + (t / 0100000));
    v = (u % dd) * 0100000 + (t % 0100000);
    if (u / dd >= 0100000)
        arith_error = true;
    else
        u = 0100000 * (u / dd) + (v / dd);
    if (positive) {
        tex_remainder = (int) (v % dd);
        return (scaled) u;
    } else {
        /*tex The casts are for ms cl. */
        tex_remainder = -(int) (v % dd);
        return -(scaled) (u);
    }
}

/*tex

The next subroutine is used to compute the ``badness'' of glue, when a total~|t|
is supposed to be made from amounts that sum to~|s|. According to {\sl The \TeX
book}, the badness of this situation is $100(t/s)^3$; however, badness is simply
a heuristic, so we need not squeeze out the last drop of accuracy when computing
it. All we really want is an approximation that has similar properties.
@:TeXbook}{\sl The \TeX book@>

The actual method used to compute the badness is easier to read from the program
than to describe in words. It produces an integer value that is a reasonably
close approximation to $100(t/s)^3$, and all implementations of \TeX\ should use
precisely this method. Any badness of $2^{13}$ or more is treated as infinitely
bad, and represented by 10000.

It is not difficult to prove that $$\hbox{|badness(t+1,s)>=badness(t,s)
>= badness(t,s+1)|}.$$ The badness function defined here is capable of computing
at most 1095 distinct values, but that is plenty.

*/

halfword badness(scaled t, scaled s)
{
    /*tex Approximation to $\alpha t/s$, where $\alpha^3\approx 100\cdot2^{18}$ */
    int r;
    if (t == 0) {
        return 0;
    } else if (s <= 0) {
        return inf_bad;
    } else {
        /*tex $297^3=99.94\times2^{18}$ */
        if (t <= 7230584) {
            r = (t * 297) / s;
        } else if (s >= 1663497) {
            r = t / (s / 297);
        } else {
            r = t;
        }
        if (r > 1290) {
            /*tex $1290^3<2^{31}<1291^3$ */
            return inf_bad;
        } else {
            /*tex This is $r^3/2^{18}$, rounded to the nearest integer. */
            return ((r * r * r + 0400000) / 01000000);
        }
    }
}

/*tex

When \TeX\ ``packages'' a list into a box, it needs to calculate the
proportionality ratio by which the glue inside the box should stretch or shrink.
This calculation does not affect \TeX's decision making, so the precise details
of rounding, etc., in the glue calculation are not of critical importance for the
consistency of results on different computers.

We shall use the type |glue_ratio| for such proportionality ratios. A glue ratio
should take the same amount of memory as an |integer| (usually 32 bits) if it is
to blend smoothly with \TeX's other data structures. Thus |glue_ratio| should be
equivalent to |short_real| in some implementations of PASCAL. Alternatively, it
is possible to deal with glue ratios using nothing but fixed-point arithmetic;
see {\sl TUGboat \bf3},1 (March 1982), 10--27. (But the routines cited there must
be modified to allow negative glue ratios.) @^system dependencies@>

*/

/*

This section is (almost) straight from MetaPost. I (Taco) had to change the types
(use |integer| instead of |fraction|), but that should not have any influence on
the actual calculations (the original comments refer to quantities like
|fraction_four| ($2^{30}$), and that is the same as the numeric representation of
|max_dimen|).

I've copied the low-level variables and routines that are needed, but only those
(e.g. |m_log|), not the accompanying ones like |m_exp|. Most of the following
low-level numeric routines are only needed within the calculation of |norm_rand|.
I've been forced to rename |make_fraction| to |make_frac| because TeX already has
a routine by that name with a wholly different function (it creates a
|fraction_noad| for math typesetting)

And now let's complete our collection of numeric utility routines by considering
random number generation. \MP{} generates pseudo-random numbers with the additive
scheme recommended in Section 3.6 of {\sl The Art of Computer Programming};
however, the results are random fractions between 0 and |fraction_one-1|,
inclusive.

There's an auxiliary array |randoms| that contains 55 pseudo-random fractions.
Using the recurrence $x_n=(x_{n-55}-x_{n-31})\bmod 2^{28}$, we generate batches
of 55 new $x_n$'s at a time by calling |new_randoms|. The global variable
|j_random| tells which element has most recently been consumed.

*/

/*tex The last 55 random values generated: */

static int randoms[55];

/*tex The number of unused |randoms|: */

static int j_random;

/*tex The default random seed: */

scaled random_seed;

/*tex A small bit of \METAPOST\ is needed. */

#define fraction_half  01000000000  /* $2^{27}  $, represents 0.50000000 */
#define fraction_one   02000000000  /* $2^{28}  $, represents 1.00000000 */
#define fraction_four 010000000000  /* $2^{30}  $, represents 4.00000000 */
#define el_gordo      017777777777  /* $2^{31}-1$, the largest value that \MP\ likes */

/*tex

The |make_frac| routine produces the |fraction| equivalent of |p/q|, given
integers |p| and~|q|; it computes the integer
$f=\lfloor2^{28}p/q+{1\over2}\rfloor$, when $p$ and $q$ are positive. If |p| and
|q| are both of the same scaled type |t|, the ``type relation''
|make_frac(t,t)=fraction| is valid; and it's also possible to use the subroutine
``backwards,'' using the relation |make_frac(t,fraction)=t| between scaled types.

If the result would have magnitude $2^{31}$ or more, |make_frac| sets
|arith_error:=true|. Most of \MP's internal computations have been designed to
avoid this sort of error.

If this subroutine were programmed in assembly language on a typical machine, we
could simply compute |(@t$2^{28}$@>*p)div q|, since a double-precision product
can often be input to a fixed-point division instruction. But when we are
restricted to PASCAL arithmetic it is necessary either to resort to
multiple-precision maneuvering or to use a simple but slow iteration. The
multiple-precision technique would be about three times faster than the code
adopted here, but it would be comparatively long and tricky, involving about
sixteen additional multiplications and divisions.

This operation is part of \MP's ``inner loop''; indeed, it will consume nearly
10\%! of the running time (exclusive of input and output) if the code below is
left unchanged. A machine-dependent recoding will therefore make \MP\ run faster.
The present implementation is highly portable, but slow; it avoids multiplication
and division except in the initial stage. System wizards should be careful to
replace it with a routine that is guaranteed to produce identical results in all
cases. @^system dependencies@>

As noted below, a few more routines should also be replaced by machine-dependent
code, for efficiency. But when a procedure is not part of the ``inner loop,''
such changes aren't advisable; simplicity and robustness are preferable to
trickery, unless the cost is too high.

*/

static int make_frac(int p, int q)
{
    /*tex The fraction bits, with a leading 1 bit: */
    int f;
    /*tex The integer part of $\vert p/q\vert$: */
    int n;
    /*tex Disables certain compiler optimizations: */
    register int be_careful;
    /*tex Should the result be negated? */
    boolean negative = false;
    if (p < 0) {
        negate(p);
        negative = true;
    }
    if (q <= 0) {
        negate(q);
        negative = !negative;
    }
    n = p / q;
    p = p % q;
    if (n >= 8) {
        arith_error = true;
        if (negative)
            return (-el_gordo);
        else
            return el_gordo;
    } else {
        n = (n - 1) * fraction_one;
        /*tex_remainder

            Compute $f=\lfloor 2^{28}(1+p/q)+{1\over2}\rfloor$. The |repeat| loop
            here preserves the following invariant relations between |f|, |p|,
            and~|q|: (i)~|0<=p<q|; (ii)~$fq+p=2^k(q+p_0)$, where $k$ is an
            integer and $p_0$ is the original value of~$p$.

            Notice that the computation specifies |(p-q)+p| instead of |(p+p)-q|,
            because the latter could overflow. Let us hope that optimizing
            compilers do not miss this point; a special variable |be_careful| is
            used to emphasize the necessary order of computation. Optimizing
            compilers should keep |be_careful| in a register, not store it in
            memory.

        */
        f = 1;
        do {
            be_careful = p - q;
            p = be_careful + p;
            if (p >= 0)
                f = f + f + 1;
            else {
                f += f;
                p = p + q;
            }
        } while (f < fraction_one);
        be_careful = p - q;
        if (be_careful + p >= 0)
            incr(f);

        if (negative)
            return (-(f + n));
        else
            return (f + n);
    }
}

static int take_frac(int q, int f)
{
    /*tex The fraction so far: */
    int p;
    /*tex Additional multiple of $q$: */
    int n;
    /*tex Disables certain compiler optimizations. */
    register int be_careful;
    /*tex Should the result be negated? */
    boolean negative = false;
    /*tex Reduce to the case that |f>=0| and |q>0|. */
    if (f < 0) {
        negate(f);
        negative = true;
    }
    if (q < 0) {
        negate(q);
        negative = !negative;
    }
    if (f < fraction_one) {
        n = 0;
    } else {
        n = f / fraction_one;
        f = f % fraction_one;
        if (q <= el_gordo / n) {
            n = n * q;
        } else {
            arith_error = true;
            n = el_gordo;
        }
    }
    f = f + fraction_one;
    /*tex

        Compute $p=\lfloor qf/2^{28}+{1\over2}\rfloor-q$. The invariant relations
        in this case are (i)~$\lfloor(qf+p)/2^k\rfloor =\lfloor
        qf_0/2^{28}+{1\over2}\rfloor$, where $k$ is an integer and $f_0$ is the
        original value of~$f$; (ii)~$2^k\L f<2^{k+1}$.

        Here |p| becomes $2^{27}$; the invariants hold now with $k=28$:

    */
    p = fraction_half;
    if (q < fraction_four) {
        do {
            if (odd(f))
                p = halfp(p + q);
            else
                p = halfp(p);
            f = halfp(f);
        } while (f != 1);
    } else {
        do {
            if (odd(f))
                p = p + halfp(q - p);
            else
                p = halfp(p);
            f = halfp(f);
        } while (f != 1);
    }
    be_careful = n - el_gordo;
    if (be_careful + p > 0) {
        arith_error = true;
        n = el_gordo - p;
    }
    if (negative)
        return (-(n + p));
    else
        return (n + p);
}

/*tex

The subroutines for logarithm and exponential involve two tables. The first is
simple: |two_to_the[k]| equals $2^k$. The second involves a bit more calculation,
which the author claims to have done correctly: |spec_log[k]| is $2^{27}$ times
$\ln\bigl(1/(1-2^{-k})\bigr)= 2^{-k}+{1\over2}2^{-2k}+{1\over3}2^{-3k}+\cdots\,$,
rounded to the nearest integer.

*/

/*tex The powers of two: */

static int two_to_the[31];

/*tex Special logarithms: */

static int spec_log[29];

void initialize_arithmetic(void)
{
    int k;
    two_to_the[0] = 1;
    for (k = 1; k <= 30; k++) {
        two_to_the[k] = 2 * two_to_the[k - 1];
    }
    spec_log [1] = 93032640;
    spec_log [2] = 38612034;
    spec_log [3] = 17922280;
    spec_log [4] =  8662214;
    spec_log [5] =  4261238;
    spec_log [6] =  2113709;
    spec_log [7] =  1052693;
    spec_log [8] =   525315;
    spec_log [9] =   262400;
    spec_log[10] =   131136;
    spec_log[11] =    65552;
    spec_log[12] =    32772;
    spec_log[13] =    16385;
    for (k = 14; k <= 27; k++) {
        spec_log[k] = two_to_the[27 - k];
    }
    spec_log[28] = 1;
}

static int m_log(int x)
{
    /*tex Auxiliary registers: */
    int y, z;
    /*tex Iteration counter: */
    int k;
    if (x <= 0) {
        /*tex Handle non-positive logarithm. */
        print_err("Logarithm of ");
        print_scaled(x);
        tprint(" has been replaced by 0");
        help2(
            "Since I don't take logs of non-positive numbers,",
            "I'm zeroing this one. Proceed, with fingers crossed."
        );
        error();
        return 0;
    } else {
        /*tex $14\times2^{27}\ln2\approx1302456956.421063$ */
        y = 1302456956 + 4 - 100;
        /*tex $2^{16}\times .421063\approx 27595$ */
        z = 27595 + 6553600;
        while (x < fraction_four) {
            x += x;
            /*tex $2^{27}\ln2\approx 93032639.74436163$ */
            y = y - 93032639;
            /*tex $2^{16}\times.74436163\approx 48782$ */
            z = z - 48782;
        }

        y = y + (z / unity);
        k = 2;
        while (x > fraction_four + 4) {
            /*tex
                Increase |k| until |x| can be multiplied by a factor of $2^{-k}$,
                and adjust $y$ accordingly. Here $z=\lceil x/2^k\rceil$.
            */
            z = ((x - 1) / two_to_the[k]) + 1;
            while (x < fraction_four + z) {
                z = halfp(z + 1);
                k = k + 1;
            }
            y = y + spec_log[k];
            x = x - z;
        }
        return (y / 8);
    }
}

/*tex

The following somewhat different subroutine tests rigorously if $ab$ is greater
than, equal to, or less than~$cd$, given integers $(a,b,c,d)$. In most cases a
quick decision is reached. The result is $+1$, 0, or~$-1$ in the three respective
cases.

*/

static int ab_vs_cd(int a, int b, int c, int d)
{
    int q, r;
    /*tex Reduce to the case that |a,c>=0| and |b,d>0|. */
    if (a < 0) {
        negate(a);
        negate(b);
    }
    if (c < 0) {
        negate(c);
        negate(d);
    }
    if (d <= 0) {
        if (b >= 0)
            return (((a == 0 || b == 0) && (c == 0 || d == 0)) ? 0 : 1);
        if (d == 0)
            return (a == 0 ? 0 : -1);
        q = a;
        a = c;
        c = q;
        q = -b;
        b = -d;
        d = q;
    } else if (b <= 0) {
        if (b < 0 && a > 0)
            return -1;
        return (c == 0 ? 0 : -1);
    }
    while (1) {
        q = a / d;
        r = c / b;
        if (q != r)
            return (q > r ? 1 : -1);
        q = a % d;
        r = c % b;
        if (r == 0)
            return (q == 0 ? 0 : 1);
        if (q == 0)
            return -1;
        a = b;
        b = q;
        c = d;
        d = r;
        /*tex Now |a>d>0| and |c>b>0|. */
    }
}

/*tex

To consume a random integer, the program below will say `|next_random|' and then
it will fetch |randoms[j_random]|.

*/

#define next_random() do { \
    if (j_random==0) \
        new_randoms(); \
    else \
        decr(j_random); \
} while (0)

static void new_randoms(void)
{
    /*tex The index into |randoms|. */
    int k;
    /*tex The accumulator. */
    int x;
    for (k = 0; k <= 23; k++) {
        x = randoms[k] - randoms[k + 31];
        if (x < 0)
            x = x + fraction_one;
        randoms[k] = x;
    }
    for (k = 24; k <= 54; k++) {
        x = randoms[k] - randoms[k - 24];
        if (x < 0)
            x = x + fraction_one;
        randoms[k] = x;
    }
    j_random = 54;
}

/*tex

To initialize the |randoms| table, we call the following routine.

*/

void init_randoms(int seed)
{
    /*tex Three more or less random integers. */
    int j, jj, k;
    /*tex The index into |randoms|. */
    int i;
    j = abs(seed);
    while (j >= fraction_one)
        j = halfp(j);
    k = 1;
    for (i = 0; i <= 54; i++) {
        jj = k;
        k = j - k;
        j = jj;
        if (k < 0)
            k = k + fraction_one;
        randoms[(i * 21) % 55] = j;
    }
    /*tex We ``warm up'' the array. */
    new_randoms();
    new_randoms();
    new_randoms();
}

/*tex

To produce a uniform random number in the range |0<=u<x| or |0>=u>x| or |0=u=x|,
given a |scaled| value~|x|, we proceed as shown here.

Note that the call of |take_frac| will produce the values 0 and~|x| with about
half the probability that it will produce any other particular values between 0
and~|x|, because it rounds its answers.

*/

int unif_rand(int x)
{
    int y;
    next_random();
    y = take_frac(abs(x), randoms[j_random]);
    if (y == abs(x))
        return 0;
    else if (x > 0)
        return y;
    else
        return -y;
}

/*tex

Finally, a normal deviate with mean zero and unit standard deviation can readily
be obtained with the ratio method (Algorithm 3.4.1R in {\sl The Art of Computer
Programming\/}.

*/

int norm_rand(void)
{
    /*tex What the book would call $2^{16}X$, $2^{28}U$, and $-2^{24}\ln U$. */
    int x, u, l;
    do {
        do {
            next_random();
            x = take_frac(112429, randoms[j_random] - fraction_half);
            /*tex Which is $2^{16}\sqrt{8/e}\approx 112428.82793$. */
            next_random();
            u = randoms[j_random];
        } while (abs(x) >= u);
        x = make_frac(x, u);
        /*tex More fuzzyness: $2^{24}\cdot12\ln2\approx139548959.6165$. */
        l = 139548960 - m_log(u);
    } while (ab_vs_cd(1024, l, x, x) < 0);
    return x;
}

/*tex

This function could also be expressed as a macro, but it is a useful breakpoint
for debugging.

*/

int fix_int(int val, int min, int max)
{
    return (val < min ? min : (val > max ? max : val));
}