% $Id $
%
% Copyright 2008-2010 Taco Hoekwater.
%
% This program is free software: you can redistribute it and/or modify
% it under the terms of the GNU Lesser General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
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% This program 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.
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% along with this program.  If not, see <http://www.gnu.org/licenses/>.
%
% TeX is a trademark of the American Mathematical Society.
% METAFONT is a trademark of Addison-Wesley Publishing Company.
% PostScript is a trademark of Adobe Systems Incorporated.

% Here is TeX material that gets inserted after \input webmac

\font\tenlogo=logo10 % font used for the METAFONT logo
\font\logos=logosl10
\def\MF{{\tenlogo META}\-{\tenlogo FONT}}
\def\MP{{\tenlogo META}\-{\tenlogo POST}}

\def\title{Reading TEX metrics files}
\pdfoutput=1

@ Introduction.

@
@d hlp1(A) mp->help_line[0]=A; }
@d hlp2(A,B) mp->help_line[1]=A; hlp1(B)
@d help1  { mp->help_ptr=1; hlp1 /* use this with one help line */
@d help2  { mp->help_ptr=2; hlp2 /* use this with two help lines */

@ @c 
#include <w2c/config.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include "mplib.h"
#include "mpmp.h" /* internal header */
#include "mpmath.h" /* internal header */
@h

@ @c
@<Declarations@>;

@ @(mpmath.h@>=
@<Types@>;
@<Internal library declarations@>;

@ Currently empty
@<Declarations@>=

@* Math initialization.

@<Types@>=
typedef struct math_data {
  scaled max_scaled_;
  scaled one_third_max_scaled_;
  scaled unity_;
  scaled two_;
  scaled three_;
  scaled half_unit_;
  scaled three_quarter_unit_;
  fraction fraction_one_;
  fraction fraction_half_;
  fraction fraction_two_;
  fraction fraction_three_;
  fraction fraction_four_;
  angle ninety_deg_;
  angle one_eighty_deg_;
  angle three_sixty_deg_;
} math_data;

@ @<Internal library declarations@>=
void * mp_initialize_math (MP mp);
void mp_free_math (MP mp);

@ @c
void * mp_initialize_math (MP mp) {
  math_data *math = (math_data *)mp_xmalloc(mp,1,sizeof(math_data));
  /* here are the constants for |scaled| objects */
  math->max_scaled_ = EL_GORDO;
  math->one_third_max_scaled_ = one_third_EL_GORDO;
  math->unity_ = unity;
  math->two_  = two;
  math->three_ = three;
  math->half_unit_ = half_unit;
  math->three_quarter_unit_ = three_quarter_unit;
  /* |fractions| */
  math->fraction_one_   = fraction_one;
  math->fraction_half_  = fraction_half;
  math->fraction_two_   = fraction_two;
  math->fraction_three_ = fraction_three;
  math->fraction_four_  = fraction_four;
  /* |angles| */
  math->ninety_deg_ = ninety_deg;
  math->one_eighty_deg_ = one_eighty_deg;
  math->three_sixty_deg_ = three_sixty_deg;
  return (void *)math;
}

void mp_free_math (MP mp) {
  free(mp->math);
}

@ Fixed-point arithmetic is done on {\sl scaled integers\/} that are multiples
of $2^{-16}$. In other words, a binary point is assumed to be sixteen bit
positions from the right end of a binary computer word.

@d unity   0x10000 /* $2^{16}$, represents 1.00000 */
@d two (2*unity) /* $2^{17}$, represents 2.00000 */
@d three (3*unity) /* $2^{17}+2^{16}$, represents 3.00000 */
@d half_unit   (unity/2) /* $2^{15}$, represents 0.50000 */
@d three_quarter_unit (3*(unity/4)) /* $3\cdot2^{14}$, represents 0.75000 */

@d EL_GORDO   0x7fffffff /* $2^{31}-1$, the largest value that \MP\ likes */
@d one_third_EL_GORDO 05252525252

@ One of \MP's most common operations is the calculation of
$\lfloor{a+b\over2}\rfloor$,
the midpoint of two given integers |a| and~|b|. The most decent way to do
this is to write `|(a+b)/2|'; but on many machines it is more efficient 
to calculate `|(a+b)>>1|'.

Therefore the midpoint operation will always be denoted by `|half(a+b)|'
in this program. If \MP\ is being implemented with languages that permit
binary shifting, the |half| macro should be changed to make this operation
as efficient as possible.  Since some systems have shift operators that can
only be trusted to work on positive numbers, there is also a macro |halfp|
that is used only when the quantity being halved is known to be positive
or zero.

@<Internal library declarations@>=
#define half(A) ((A) / 2)
#define halfp(A) (integer)((unsigned)(A) >> 1)

@ Todo: Here are some compilation tricks for problems to be sorted out later

@<Internal library declarations@>=
#define integer_as_fraction(A) (fraction)(A)


@ Here is a procedure analogous to |print_int|. If the output
of this procedure is subsequently read by \MP\ and converted by the
|round_decimals| routine above, it turns out that the original value will
be reproduced exactly. A decimal point is printed only if the value is
not an integer. If there is more than one way to print the result with
the optimum number of digits following the decimal point, the closest
possible value is given.

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.

@<Internal library declarations@>=
void mp_print_scaled (MP mp, scaled s);

@ @c
void mp_print_scaled (MP mp, scaled s) {                               /* prints scaled real, rounded to five  digits */
  scaled delta; /* amount of allowable inaccuracy */
  if (s < 0) {
    mp_print_char (mp, xord ('-'));
    s = -s;                 /* print the sign, if negative */
  }
  mp_print_int (mp, s / unity); /* print the integer part */
  s = 10 * (s % unity) + 5;
  if (s != 5) {
    delta = 10;
    mp_print_char (mp, xord ('.'));
    do {
      if (delta > unity)
        s = s + 0100000 - (delta / 2);  /* round the final digit */
      mp_print_char (mp, xord ('0' + (s / unity)));
      s = 10 * (s % unity);
      delta = delta * 10;
    } while (s > delta);
  }
}

@ Addition is not always checked to make sure that it doesn't overflow,
but in places where overflow isn't too unlikely the |slow_add| routine
is used.

@<Internal library declarations@>=
integer mp_slow_add (MP mp, integer x, integer y);

@ @c
integer mp_slow_add (MP mp, integer x, integer y) {
  if (x >= 0) {
    if (y <= EL_GORDO - x) {
      return x + y;
    } else {
      mp->arith_error = true;
      return EL_GORDO;
    }
  } else if (-y <= EL_GORDO + x) {
    return x + y;
  } else {
    mp->arith_error = true;
    return -EL_GORDO;
  }
}

@ The |make_fraction| 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_fraction(t,t)=fraction| is valid;
and it's also possible to use the subroutine ``backwards,'' using
the relation |make_fraction(t,fraction)=t| between scaled types.

If the result would have magnitude $2^{31}$ or more, |make_fraction|
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 int-eger 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\pct! 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.
@^inner loop@>

@<Internal library declarations@>=
fraction mp_make_fraction (MP mp, integer p, integer q);

@ We need these preprocessor values

@d TWEXP31  2147483648.0
@d TWEXP28  268435456.0
@d TWEXP16 65536.0
@d TWEXP_16 (1.0/65536.0)
@d TWEXP_28 (1.0/268435456.0)


@c
fraction mp_make_fraction (MP mp, integer p, integer q) {
  fraction i;
  if (q == 0)
    mp_confusion (mp, "/");
@:this can't happen /}{\quad \./@> {
    register double d;
    d = TWEXP28 * (double) p / (double) q;
    if ((p ^ q) >= 0) {
      d += 0.5;
      if (d >= TWEXP31) {
        mp->arith_error = true;
        return EL_GORDO;
      }
      i = (integer) d;
      if (d == (double) i && (((q > 0 ? -q : q) & 077777)
                              * (((i & 037777) << 1) - 1) & 04000) != 0)
        --i;
    } else {
      d -= 0.5;
      if (d <= -TWEXP31) {
        mp->arith_error = true;
        return -EL_GORDO;
      }
      i = (integer) d;
      if (d == (double) i && (((q > 0 ? q : -q) & 077777)
                              * (((i & 037777) << 1) + 1) & 04000) != 0)
        ++i;
    }
  }
  return i;
}


@ The dual of |make_fraction| is |take_fraction|, which multiplies a
given integer~|q| by a fraction~|f|. When the operands are positive, it
computes $p=\lfloor qf/2^{28}+{1\over2}\rfloor$, a symmetric function
of |q| and~|f|.

This routine is even more ``inner loopy'' than |make_fraction|;
the present implementation consumes almost 20\pct! of \MP's computation
time during typical jobs, so a machine-language substitute is advisable.
@^inner loop@> @^system dependencies@>

@<Internal library declarations@>=
integer mp_take_fraction (MP mp, integer q, fraction f);

@ @c
integer mp_take_fraction (MP mp, integer p, fraction q) {
  register double d;
  register integer i;
  d = (double) p *(double) q *TWEXP_28;
  if ((p ^ q) >= 0) {
    d += 0.5;
    if (d >= TWEXP31) {
      if (d != TWEXP31 || (((p & 077777) * (q & 077777)) & 040000) == 0)
        mp->arith_error = true;
      return EL_GORDO;
    }
    i = (integer) d;
    if (d == (double) i && (((p & 077777) * (q & 077777)) & 040000) != 0)
      --i;
  } else {
    d -= 0.5;
    if (d <= -TWEXP31) {
      if (d != -TWEXP31 || ((-(p & 077777) * (q & 077777)) & 040000) == 0)
        mp->arith_error = true;
      return -EL_GORDO;
    }
    i = (integer) d;
    if (d == (double) i && ((-(p & 077777) * (q & 077777)) & 040000) != 0)
      ++i;
  }
  return i;
}


@ When we want to multiply something by a |scaled| quantity, we use a scheme
analogous to |take_fraction| but with a different scaling.
Given positive operands, |take_scaled|
computes the quantity $p=\lfloor qf/2^{16}+{1\over2}\rfloor$.

Once again it is a good idea to use a machine-language replacement if
possible; otherwise |take_scaled| will use more than 2\pct! of the running time
when the Computer Modern fonts are being generated.
@^inner loop@>

@<Internal library declarations@>=
integer mp_take_scaled (MP mp, integer q, scaled f);

@ @c
integer mp_take_scaled (MP mp, integer p, scaled q) {
  register double d;
  register integer i;
  d = (double) p *(double) q *TWEXP_16;
  if ((p ^ q) >= 0) {
    d += 0.5;
    if (d >= TWEXP31) {
      if (d != TWEXP31 || (((p & 077777) * (q & 077777)) & 040000) == 0)
        mp->arith_error = true;
      return EL_GORDO;
    }
    i = (integer) d;
    if (d == (double) i && (((p & 077777) * (q & 077777)) & 040000) != 0)
      --i;
  } else {
    d -= 0.5;
    if (d <= -TWEXP31) {
      if (d != -TWEXP31 || ((-(p & 077777) * (q & 077777)) & 040000) == 0)
        mp->arith_error = true;
      return -EL_GORDO;
    }
    i = (integer) d;
    if (d == (double) i && ((-(p & 077777) * (q & 077777)) & 040000) != 0)
      ++i;
  }
  return i;
}


@ For completeness, there's also |make_scaled|, which computes a
quotient as a |scaled| number instead of as a |fraction|.
In other words, the result is $\lfloor2^{16}p/q+{1\over2}\rfloor$, if the
operands are positive. \ (This procedure is not used especially often,
so it is not part of \MP's inner loop.)

@<Internal library ...@>=
scaled mp_make_scaled (MP mp, integer p, integer q);

@ @c
scaled mp_make_scaled (MP mp, integer p, integer q) {
  register integer i;
  if (q == 0)
    mp_confusion (mp, "/");
@:this can't happen /}{\quad \./@> {
    register double d;
    d = TWEXP16 * (double) p / (double) q;
    if ((p ^ q) >= 0) {
      d += 0.5;
      if (d >= TWEXP31) {
        mp->arith_error = true;
        return EL_GORDO;
      }
      i = (integer) d;
      if (d == (double) i && (((q > 0 ? -q : q) & 077777)
                              * (((i & 037777) << 1) - 1) & 04000) != 0)
        --i;
    } else {
      d -= 0.5;
      if (d <= -TWEXP31) {
        mp->arith_error = true;
        return -EL_GORDO;
      }
      i = (integer) d;
      if (d == (double) i && (((q > 0 ? q : -q) & 077777)
                              * (((i & 037777) << 1) + 1) & 04000) != 0)
        ++i;
    }
  }
  return i;
}

@ The following function divides |s| by |m|. |dd| is number of decimal digits.

@<Internal library ...@>=
scaled mp_divide_scaled (MP mp, scaled s, scaled m, integer dd);

@ @c
scaled mp_divide_scaled (MP mp, scaled s, scaled m, integer dd) {
  scaled q, r;
  integer sign, i;
  sign = 1;
  if (s < 0) {
    sign = -sign;
    s = -s;
  }
  if (m < 0) {
    sign = -sign;
    m = -m;
  }
  if (m == 0)
    mp_confusion (mp, "arithmetic: divided by zero");
  else if (m >= (EL_GORDO / 10))
    mp_confusion (mp, "arithmetic: number too big");
  q = s / m;
  r = s % m;
  for (i = 1; i <= dd; i++) {
    q = 10 * q + (10 * r) / m;
    r = (10 * r) % m;
  }
  if (2 * r >= m) {
    q++;
    r = r - m;
  }
  mp->scaled_out = sign * (s - (r / mp->ten_pow[dd]));
  return (sign * q);
}


@ 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|.

@<Internal library declarations@>=
scaled mp_round_decimals (MP mp, unsigned char *b, quarterword k);

@ @c
scaled mp_round_decimals (MP mp, unsigned char *b, quarterword k) {
  /* converts a decimal fraction */
  unsigned a = 0;       /* the accumulator */
  int l = 0;
  for ( l = k-1; l >= 0; l-- ) {
    if (l<16)    /* digits for |k>=17| cannot affect the result */
      a = (a + (unsigned) (*(b+l) - '0') * two) / 10;
  }
  return (scaled) halfp (a + 1);
}

@ The |scaled| quantities in \MP\ programs are generally supposed to be
less than $2^{12}$ in absolute value, so \MP\ does much of its internal
arithmetic with 28~significant bits of precision. A |fraction| denotes
a scaled integer whose binary point is assumed to be 28 bit positions
from the right.

@d fraction_half 01000000000 /* $2^{27}$, represents 0.50000000 */
@d fraction_one 02000000000 /* $2^{28}$, represents 1.00000000 */
@d fraction_two 04000000000 /* $2^{29}$, represents 2.00000000 */
@d fraction_three 06000000000 /* $3\cdot2^{28}$, represents 3.00000000 */
@d fraction_four 010000000000 /* $2^{30}$, represents 4.00000000 */

@ Here is a typical example of how the routines above can be used.
It computes the function
$${1\over3\tau}f(\theta,\phi)=
{\tau^{-1}\bigl(2+\sqrt2\,(\sin\theta-{1\over16}\sin\phi)
 (\sin\phi-{1\over16}\sin\theta)(\cos\theta-\cos\phi)\bigr)\over
3\,\bigl(1+{1\over2}(\sqrt5-1)\cos\theta+{1\over2}(3-\sqrt5\,)\cos\phi\bigr)},$$
where $\tau$ is a |scaled| ``tension'' parameter. This is \MP's magic
fudge factor for placing the first control point of a curve that starts
at an angle $\theta$ and ends at an angle $\phi$ from the straight path.
(Actually, if the stated quantity exceeds 4, \MP\ reduces it to~4.)

The trigonometric quantity to be multiplied by $\sqrt2$ is less than $\sqrt2$.
(It's a sum of eight terms whose absolute values can be bounded using
relations such as $\sin\theta\cos\theta\L{1\over2}$.) Thus the numerator
is positive; and since the tension $\tau$ is constrained to be at least
$3\over4$, the numerator is less than $16\over3$. The denominator is
nonnegative and at most~6.  Hence the fixed-point calculations below
are guaranteed to stay within the bounds of a 32-bit computer word.

The angles $\theta$ and $\phi$ are given implicitly in terms of |fraction|
arguments |st|, |ct|, |sf|, and |cf|, representing $\sin\theta$, $\cos\theta$,
$\sin\phi$, and $\cos\phi$, respectively.

@<Internal library declarations@>=
fraction mp_velocity (MP mp, fraction st, fraction ct, fraction sf,
	                     fraction cf, scaled t);

@ @c
fraction mp_velocity (MP mp, fraction st, fraction ct, fraction sf,
	                     fraction cf, scaled t) {
  integer acc, num, denom;      /* registers for intermediate calculations */
  acc = mp_take_fraction (mp, st - (sf / 16), sf - (st / 16));
  acc = mp_take_fraction (mp, acc, ct - cf);
  num = fraction_two + mp_take_fraction (mp, acc, 379625062);
  /* $2^{28}\sqrt2\approx379625062.497$ */
  denom =
    fraction_three + mp_take_fraction (mp, ct,
                                       497706707) + mp_take_fraction (mp, cf,
                                                                      307599661);
  /* $3\cdot2^{27}\cdot(\sqrt5-1)\approx497706706.78$ and
     $3\cdot2^{27}\cdot(3-\sqrt5\,)\approx307599661.22$ */
  if (t != unity)
    num = mp_make_scaled (mp, num, t);
  /* |make_scaled(fraction,scaled)=fraction| */
  if (num / 4 >= denom)
    return fraction_four;
  else
    return mp_make_fraction (mp, num, denom);
}


@ 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.

@<Internal library declarations@>=
integer mp_ab_vs_cd (MP mp, integer a, integer b, integer c, integer d);

@ @c
integer mp_ab_vs_cd (MP mp, integer a, integer b, integer c, integer d) {
  integer q, r; /* temporary registers */
  (void)mp;
  @<Reduce to the case that |a,c>=0|, |b,d>0|@>;
  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 ? 1 : 0);
    if (q == 0)
      return -1;
    a = b;
    b = q;
    c = d;
    d = r;
  }                             /* now |a>d>0| and |c>b>0| */
}


@ @<Reduce to the case that |a...@>=
if (a < 0) {
  a = -a;
  b = -b;
};
if (c < 0) {
  c = -c;
  d = -d;
};
if (d <= 0) {
  if (b >= 0) {
    if ((a == 0 || b == 0) && (c == 0 || d == 0))
      return 0;
    else
      return 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)
    if (a > 0)
      return -1;
  return (c == 0 ? 0 : -1);
}

@ We conclude this set of elementary routines with some simple rounding
and truncation operations.

@ |floor_scaled| floors a |scaled|
@<Internal library declarations@>=
#define mp_floor_scaled(M,i) ((i)&(-65536))

@ |round_unscaled| rounds a |scaled| and converts it to |int|
@<Internal library declarations@>=
#define mp_round_unscaled(M,x) (x>=0100000 ? 1+((x-0100000) / 0200000) \
  : ( x>=-0100000 ? 0 : -(1+((-(x+1)-0100000) / 0200000))))

@ |round_fraction| rounds a |fraction| and converts it to |scaled|
@<Internal library declarations@>=
#define mp_round_fraction(M,x) (x>=2048 ? 1+((x-2048) / 4096) \
  : ( x>=-2048 ? 0 : -(1+((-(x+1)-2048) / 4096))))



@* Algebraic and transcendental functions.
\MP\ computes all of the necessary special functions from scratch, without
relying on |real| arithmetic or system subroutines for sines, cosines, etc.

@ To get the square root of a |scaled| number |x|, we want to calculate
$s=\lfloor 2^8\!\sqrt x +{1\over2}\rfloor$. If $x>0$, this is the unique
integer such that $2^{16}x-s\L s^2<2^{16}x+s$. The following subroutine
determines $s$ by an iterative method that maintains the invariant
relations $x=2^{46-2k}x_0\bmod 2^{30}$, $0<y=\lfloor 2^{16-2k}x_0\rfloor
-s^2+s\L q=2s$, where $x_0$ is the initial value of $x$. The value of~$y$
might, however, be zero at the start of the first iteration.

@<Internal library declarations@>=
scaled mp_square_rt (MP mp, scaled x);

@ @c
scaled mp_square_rt (MP mp, scaled x) {
  quarterword k;        /* iteration control counter */
  integer y;    /* register for intermediate calculations */
  integer q;    /* register for intermediate calculations */
  if (x <= 0) {
    @<Handle square root of zero or negative argument@>;
  } else {
    k = 23;
    q = 2;
    while (x < fraction_two) {  /* i.e., |while x<@t$2^{29}$@>|\unskip */
      k--;
      x = x + x + x + x;
    }
    if (x < fraction_four)
      y = 0;
    else {
      x = x - fraction_four;
      y = 1;
    };
    do {
      @<Decrease |k| by 1, maintaining the invariant
      relations between |x|, |y|, and~|q|@>;
    } while (k != 0);
    return (scaled) (halfp (q));
  }
}


@ @<Handle square root of zero...@>=
{
  if (x < 0) {
    mp_print_err (mp, "Square root of ");
@.Square root...replaced by 0@>;
    mp_print_scaled (mp, x);
    mp_print (mp, " has been replaced by 0");
    help2 ("Since I don't take square roots of negative numbers,",
           "I'm zeroing this one. Proceed, with fingers crossed.");
    mp_error (mp);
  };
  return 0;
}


@ @<Decrease |k| by 1, maintaining...@>=
x += x;
y += y;
if (x >= fraction_four) {       /* note that |fraction_four=@t$2^{30}$@>| */
  x = x - fraction_four;
  y++;
};
x += x;
y = y + y - q;
q += q;
if (x >= fraction_four) {
  x = x - fraction_four;
  y++;
};
if (y > (int) q) {
  y -= q;
  q += 2;
} else if (y <= 0) {
  q -= 2;
  y += q;
};
k--

@ Pythagorean addition $\psqrt{a^2+b^2}$ is implemented by an elegant
iterative scheme due to Cleve Moler and Donald Morrison [{\sl IBM Journal
@^Moler, Cleve Barry@>
@^Morrison, Donald Ross@>
of Research and Development\/ \bf27} (1983), 577--581]. It modifies |a| and~|b|
in such a way that their Pythagorean sum remains invariant, while the
smaller argument decreases.

@<Internal library ...@>=
integer mp_pyth_add (MP mp, integer a, integer b);


@ @c
integer mp_pyth_add (MP mp, integer a, integer b) {
  fraction r;   /* register used to transform |a| and |b| */
  boolean big;  /* is the result dangerously near $2^{31}$? */
  a = abs (a);
  b = abs (b);
  if (a < b) {
    r = b;
    b = a;
    a = r;
  };                            /* now |0<=b<=a| */
  if (b > 0) {
    if (a < fraction_two) {
      big = false;
    } else {
      a = a / 4;
      b = b / 4;
      big = true;
    };                          /* we reduced the precision to avoid arithmetic overflow */
    @<Replace |a| by an approximation to $\psqrt{a^2+b^2}$@>;
    if (big) {
      if (a < fraction_two) {
        a = a + a + a + a;
      } else {
        mp->arith_error = true;
        a = EL_GORDO;
      };
    }
  }
  return a;
}


@ The key idea here is to reflect the vector $(a,b)$ about the
line through $(a,b/2)$.

@<Replace |a| by an approximation to $\psqrt{a^2+b^2}$@>=
while (1) {
  r = mp_make_fraction (mp, b, a);
  r = mp_take_fraction (mp, r, r);      /* now $r\approx b^2/a^2$ */
  if (r == 0)
    break;
  r = mp_make_fraction (mp, r, fraction_four + r);
  a = a + mp_take_fraction (mp, a + a, r);
  b = mp_take_fraction (mp, b, r);
}


@ Here is a similar algorithm for $\psqrt{a^2-b^2}$.
It converges slowly when $b$ is near $a$, but otherwise it works fine.

@<Internal library declarations@>=
integer mp_pyth_sub (MP mp, integer a, integer b);

@ @c
integer mp_pyth_sub (MP mp, integer a, integer b) {
  fraction r;   /* register used to transform |a| and |b| */
  boolean big;  /* is the input dangerously near $2^{31}$? */
  a = abs (a);
  b = abs (b);
  if (a <= b) {
    @<Handle erroneous |pyth_sub| and set |a:=0|@>;
  } else {
    if (a < fraction_four) {
      big = false;
    } else {
      a = (integer) halfp (a);
      b = (integer) halfp (b);
      big = true;
    }
    @<Replace |a| by an approximation to $\psqrt{a^2-b^2}$@>;
    if (big)
      a *= 2;
  }
  return a;
}


@ @<Replace |a| by an approximation to $\psqrt{a^2-b^2}$@>=
while (1) {
  r = mp_make_fraction (mp, b, a);
  r = mp_take_fraction (mp, r, r);      /* now $r\approx b^2/a^2$ */
  if (r == 0)
    break;
  r = mp_make_fraction (mp, r, fraction_four - r);
  a = a - mp_take_fraction (mp, a + a, r);
  b = mp_take_fraction (mp, b, r);
}


@ @<Handle erroneous |pyth_sub| and set |a:=0|@>=
{
  if (a < b) {
    mp_print_err (mp, "Pythagorean subtraction ");
    mp_print_scaled (mp, a);
    mp_print (mp, "+-+");
    mp_print_scaled (mp, b);
    mp_print (mp, " has been replaced by 0");
@.Pythagorean...@>;
    help2 ("Since I don't take square roots of negative numbers,",
           "I'm zeroing this one. Proceed, with fingers crossed.");
    mp_error (mp);
  }
  a = 0;
}


@ 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.

@d two_to_the(A) (1<<(unsigned)(A))

@<Declarations@>=
static const integer spec_log[29] = { 0,        /* special logarithms */
  93032640, 38612034, 17922280, 8662214, 4261238, 2113709,
  1052693, 525315, 262400, 131136, 65552, 32772, 16385,
  8192, 4096, 2048, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1, 1
};


@ Here is the routine that calculates $2^8$ times the natural logarithm
of a |scaled| quantity; it is an integer approximation to $2^{24}\ln(x/2^{16})$,
when |x| is a given positive integer.

The method is based on exercise 1.2.2--25 in {\sl The Art of Computer
Programming\/}: During the main iteration we have $1\L 2^{-30}x<1/(1-2^{1-k})$,
and the logarithm of $2^{30}x$ remains to be added to an accumulator
register called~$y$. Three auxiliary bits of accuracy are retained in~$y$
during the calculation, and sixteen auxiliary bits to extend |y| are
kept in~|z| during the initial argument reduction. (We add
$100\cdot2^{16}=6553600$ to~|z| and subtract 100 from~|y| so that |z| will
not become negative; also, the actual amount subtracted from~|y| is~96,
not~100, because we want to add~4 for rounding before the final division by~8.)

@<Internal library declarations@>=
scaled mp_m_log (MP mp, scaled x);

@ @c
scaled mp_m_log (MP mp, scaled x) {
  integer y, z; /* auxiliary registers */
  integer k;    /* iteration counter */
  if (x <= 0) {
    @<Handle non-positive logarithm@>;
  } else {
    y = 1302456956 + 4 - 100;   /* $14\times2^{27}\ln2\approx1302456956.421063$ */
    z = 27595 + 6553600;        /* and $2^{16}\times .421063\approx 27595$ */
    while (x < fraction_four) {
      x = 2*x;
      y -= 93032639;
      z -= 48782;
    }                           /* $2^{27}\ln2\approx 93032639.74436163$ and $2^{16}\times.74436163\approx 48782$ */
    y = y + (z / unity);
    k = 2;
    while (x > fraction_four + 4) {
      @<Increase |k| until |x| can be multiplied by a
        factor of $2^{-k}$, and adjust $y$ accordingly@>;
    }
    return (y / 8);
  }
}


@ @<Increase |k| until |x| can...@>=
{
  z = ((x - 1) / two_to_the (k)) + 1;   /* $z=\lceil x/2^k\rceil$ */
  while (x < fraction_four + z) {
    z = halfp (z + 1);
    k++;
  };
  y += spec_log[k];
  x -= z;
}


@ @<Handle non-positive logarithm@>=
{
  mp_print_err (mp, "Logarithm of ");
@.Logarithm...replaced by 0@>;
  mp_print_scaled (mp, x);
  mp_print (mp, " 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.");
  mp_error (mp);
  return 0;
}


@ Conversely, the exponential routine calculates $\exp(x/2^8)$,
when |x| is |scaled|. The result is an integer approximation to
$2^{16}\exp(x/2^{24})$, when |x| is regarded as an integer.

@<Internal library declarations@>=
scaled mp_m_exp (MP mp, scaled x);

@ @c
scaled mp_m_exp (MP mp, scaled x) {
  quarterword k;        /* loop control index */
  integer y, z; /* auxiliary registers */
  if (x > 174436200) {
    /* $2^{24}\ln((2^{31}-1)/2^{16})\approx 174436199.51$ */
    mp->arith_error = true;
    return EL_GORDO;
  } else if (x < -197694359) {
    /* $2^{24}\ln(2^{-1}/2^{16})\approx-197694359.45$ */
    return 0;
  } else {
    if (x <= 0) {
      z = -8 * x;
      y = 04000000;             /* $y=2^{20}$ */
    } else {
      if (x <= 127919879) {
        z = 1023359037 - 8 * x;
        /* $2^{27}\ln((2^{31}-1)/2^{20})\approx 1023359037.125$ */
      } else {
        z = 8 * (174436200 - x);        /* |z| is always nonnegative */
      }
      y = EL_GORDO;
    };
    @<Multiply |y| by $\exp(-z/2^{27})$@>;
    if (x <= 127919879)
      return ((y + 8) / 16);
    else
      return y;
  }
}


@ The idea here is that subtracting |spec_log[k]| from |z| corresponds
to multiplying |y| by $1-2^{-k}$.

A subtle point (which had to be checked) was that if $x=127919879$, the
value of~|y| will decrease so that |y+8| doesn't overflow. In fact,
$z$ will be 5 in this case, and |y| will decrease by~64 when |k=25|
and by~16 when |k=27|.

@<Multiply |y| by...@>=
k = 1;
while (z > 0) {
  while (z >= spec_log[k]) {
    z -= spec_log[k];
    y = y - 1 - ((y - two_to_the (k - 1)) / two_to_the (k));
  }
  k++;
}

@ The trigonometric subroutines use an auxiliary table such that
|spec_atan[k]| contains an approximation to the |angle| whose tangent
is~$1/2^k$. $\arctan2^{-k}$ times $2^{20}\cdot180/\pi$ 

@<Declarations@>=
static const angle spec_atan[27] = { 0, 27855475, 14718068, 7471121, 3750058,
  1876857, 938658, 469357, 234682, 117342, 58671, 29335, 14668, 7334, 3667,
  1833, 917, 458, 229, 115, 57, 29, 14, 7, 4, 2, 1
};


@ Given integers |x| and |y|, not both zero, the |n_arg| function
returns the |angle| whose tangent points in the direction $(x,y)$.
This subroutine first determines the correct octant, then solves the
problem for |0<=y<=x|, then converts the result appropriately to
return an answer in the range |-one_eighty_deg<=@t$\theta$@><=one_eighty_deg|.
(The answer is |+one_eighty_deg| if |y=0| and |x<0|, but an answer of
|-one_eighty_deg| is possible if, for example, |y=-1| and $x=-2^{30}$.)

The octants are represented in a ``Gray code,'' since that turns out
to be computationally simplest.

@d negate_x 1
@d negate_y 2
@d switch_x_and_y 4
@d first_octant 1
@d second_octant (first_octant+switch_x_and_y)
@d third_octant (first_octant+switch_x_and_y+negate_x)
@d fourth_octant (first_octant+negate_x)
@d fifth_octant (first_octant+negate_x+negate_y)
@d sixth_octant (first_octant+switch_x_and_y+negate_x+negate_y)
@d seventh_octant (first_octant+switch_x_and_y+negate_y)
@d eighth_octant (first_octant+negate_y)

@<Internal library declarations@>=
angle mp_n_arg (MP mp, integer x, integer y);

@ @c
angle mp_n_arg (MP mp, integer x, integer y) {
  angle z;      /* auxiliary register */
  integer t;    /* temporary storage */
  quarterword k;        /* loop counter */
  int octant;   /* octant code */
  if (x >= 0) {
    octant = first_octant;
  } else {
    x = -x;
    octant = first_octant + negate_x;
  }
  if (y < 0) {
    y = -y;
    octant = octant + negate_y;
  }
  if (x < y) {
    t = y;
    y = x;
    x = t;
    octant = octant + switch_x_and_y;
  }
  if (x == 0) {
    @<Handle undefined arg@>;
  } else {
    @<Set variable |z| to the arg of $(x,y)$@>;
    @<Return an appropriate answer based on |z| and |octant|@>;
  }
}


@ @<Handle undefined arg@>=
{
  mp_print_err (mp, "angle(0,0) is taken as zero");
@.angle(0,0)...zero@>;
  help2 ("The `angle' between two identical points is undefined.",
         "I'm zeroing this one. Proceed, with fingers crossed.");
  mp_error (mp);
  return 0;
}


@ @<Return an appropriate answer...@>=
switch (octant) {
case first_octant:
  return z;
case second_octant:
  return (ninety_deg - z);
case third_octant:
  return (ninety_deg + z);
case fourth_octant:
  return (one_eighty_deg - z);
case fifth_octant:
  return (z - one_eighty_deg);
case sixth_octant:
  return (-z - ninety_deg);
case seventh_octant:
  return (z - ninety_deg);
case eighth_octant:
  return (-z);
};                              /* there are no other cases */
return 0

@ At this point we have |x>=y>=0|, and |x>0|. The numbers are scaled up
or down until $2^{28}\L x<2^{29}$, so that accurate fixed-point calculations
will be made.

@<Set variable |z| to the arg...@>=
while (x >= fraction_two) {
  x = halfp (x);
  y = halfp (y);
}
z = 0;
if (y > 0) {
  while (x < fraction_one) {
    x += x;
    y += y;
  };
  @<Increase |z| to the arg of $(x,y)$@>;
}

@ During the calculations of this section, variables |x| and~|y|
represent actual coordinates $(x,2^{-k}y)$. We will maintain the
condition |x>=y|, so that the tangent will be at most $2^{-k}$.
If $x<2y$, the tangent is greater than $2^{-k-1}$. The transformation
$(a,b)\mapsto(a+b\tan\phi,b-a\tan\phi)$ replaces $(a,b)$ by
coordinates whose angle has decreased by~$\phi$; in the special case
$a=x$, $b=2^{-k}y$, and $\tan\phi=2^{-k-1}$, this operation reduces
to the particularly simple iteration shown here. [Cf.~John E. Meggitt,
@^Meggitt, John E.@>
{\sl IBM Journal of Research and Development\/ \bf6} (1962), 210--226.]

The initial value of |x| will be multiplied by at most
$(1+{1\over2})(1+{1\over8})(1+{1\over32})\cdots\approx 1.7584$; hence
there is no chance of integer overflow.

@<Increase |z|...@>=
k = 0;
do {
  y += y;
  k++;
  if (y > x) {
    z = z + spec_atan[k];
    t = x;
    x = x + (y / two_to_the (k + k));
    y = y - t;
  };
} while (k != 15);
do {
  y += y;
  k++;
  if (y > x) {
    z = z + spec_atan[k];
    y = y - x;
  };
} while (k != 26)

@ Conversely, the |n_sin_cos| routine takes an |angle| and produces the sine
and cosine of that angle. The results of this routine are
stored in global integer variables |n_sin| and |n_cos|.

@ Given an integer |z| that is $2^{20}$ times an angle $\theta$ in degrees,
the purpose of |n_sin_cos(z)| is to set
|x=@t$r\cos\theta$@>| and |y=@t$r\sin\theta$@>| (approximately),
for some rather large number~|r|. The maximum of |x| and |y|
will be between $2^{28}$ and $2^{30}$, so that there will be hardly
any loss of accuracy. Then |x| and~|y| are divided by~|r|.

@d forty_five_deg 0264000000 /* $45\cdot2^{20}$, represents $45^\circ$ */
@d ninety_deg 0550000000 /* $90\cdot2^{20}$, represents $90^\circ$ */
@d one_eighty_deg 01320000000 /* $180\cdot2^{20}$, represents $180^\circ$ */
@d three_sixty_deg 02640000000 /* $360\cdot2^{20}$, represents $360^\circ$ */

@d odd(A)   ((A)%2==1)

@<Internal library declarations@>=
void mp_n_sin_cos (MP mp, angle z, fraction *n_cos, fraction *n_sin);

@ Compute a multiple of the sine and cosine

@c
void mp_n_sin_cos (MP mp, angle z, fraction *n_cos, fraction *n_sin) {
  quarterword k;        /* loop control variable */
  int q;        /* specifies the quadrant */
  fraction r;   /* magnitude of |(x,y)| */
  integer x, y, t;      /* temporary registers */
  while (z < 0)
    z = z + three_sixty_deg;
  z = z % three_sixty_deg;      /* now |0<=z<three_sixty_deg| */
  q = z / forty_five_deg;
  z = z % forty_five_deg;
  x = fraction_one;
  y = x;
  if (!odd (q))
    z = forty_five_deg - z;
  @<Subtract angle |z| from |(x,y)|@>;
  @<Convert |(x,y)| to the octant determined by~|q|@>;
  r = mp_pyth_add (mp, x, y);
  *n_cos = mp_make_fraction (mp, x, r);
  *n_sin = mp_make_fraction (mp, y, r);
}


@ In this case the octants are numbered sequentially.

@<Convert |(x,...@>=
switch (q) {
case 0:
  break;
case 1:
  t = x;
  x = y;
  y = t;
  break;
case 2:
  t = x;
  x = -y;
  y = t;
  break;
case 3:
  x = -x;
  break;
case 4:
  x = -x;
  y = -y;
  break;
case 5:
  t = x;
  x = -y;
  y = -t;
  break;
case 6:
  t = x;
  x = y;
  y = -t;
  break;
case 7:
  y = -y;
  break;
}                               /* there are no other cases */


@ The main iteration of |n_sin_cos| is similar to that of |n_arg| but
applied in reverse. The values of |spec_atan[k]| decrease slowly enough
that this loop is guaranteed to terminate before the (nonexistent) value
|spec_atan[27]| would be required.

@<Subtract angle |z|...@>=
k = 1;
while (z > 0) {
  if (z >= spec_atan[k]) {
    z = z - spec_atan[k];
    t = x;
    x = t + y / two_to_the (k);
    y = y - t / two_to_the (k);
  }
  k++;
}
if (y < 0)
  y = 0                         /* this precaution may never be needed */