% $Id$
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% This file is part of MetaPost;
% the MetaPost program is in the public domain.
% See the <Show version...> code in mpost.w for more info.

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\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\pct!{{\char`\%}} % percent sign in ordinary text
\def\psqrt#1{\sqrt{\mathstrut#1}}

\def\title{Math support functions for 32-bit integer math}
\pdfoutput=1

@ Introduction.

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

@ @c
@<Declarations@>;

@ @(mpmath.h@>=
#ifndef MPMATH_H
#define MPMATH_H 1
#include "mplib.h"
#include "mpmp.h" /* internal header */
@<Internal library declarations@>;
#endif

@* Math initialization.

@ Here are the functions that are static as they are not used elsewhere

@<Declarations@>=
static void mp_scan_fractional_token (MP mp, int n);
static void mp_scan_numeric_token (MP mp, int n);
static void mp_ab_vs_cd (MP mp, mp_number *ret, mp_number a, mp_number b, mp_number c, mp_number d);
static void mp_crossing_point (MP mp, mp_number *ret, mp_number a, mp_number b, mp_number c);
static void mp_number_modulo (mp_number *a, mp_number b);
static void mp_print_number (MP mp, mp_number n);
static char * mp_number_tostring (MP mp, mp_number n);
static void mp_slow_add (MP mp, mp_number *ret, mp_number x_orig, mp_number y_orig);
static void mp_square_rt (MP mp, mp_number *ret, mp_number x_orig);
static void mp_n_sin_cos (MP mp, mp_number z_orig, mp_number *n_cos, mp_number *n_sin);
static void mp_init_randoms (MP mp, int seed);
static void mp_number_angle_to_scaled (mp_number *A);
static void mp_number_fraction_to_scaled (mp_number *A);
static void mp_number_scaled_to_fraction (mp_number *A);
static void mp_number_scaled_to_angle (mp_number *A);
static void mp_m_unif_rand (MP mp, mp_number *ret, mp_number x_orig);
static void mp_m_norm_rand (MP mp, mp_number *ret);
static void mp_m_exp (MP mp, mp_number *ret, mp_number x_orig);
static void mp_m_log (MP mp, mp_number *ret, mp_number x_orig);
static void mp_pyth_sub (MP mp, mp_number *r, mp_number a, mp_number b);
static void mp_n_arg (MP mp, mp_number *ret, mp_number x, mp_number y);
static void mp_velocity (MP mp, mp_number *ret, mp_number st, mp_number ct, mp_number sf,  mp_number cf, mp_number t);
static void mp_set_number_from_int(mp_number *A, int B);
static void mp_set_number_from_boolean(mp_number *A, int B);
static void mp_set_number_from_scaled(mp_number *A, int B);
static void mp_set_number_from_boolean(mp_number *A, int B);
static void mp_set_number_from_addition(mp_number *A, mp_number B, mp_number C);
static void mp_set_number_from_substraction (mp_number *A, mp_number B, mp_number C);
static void mp_set_number_from_div(mp_number *A, mp_number B, mp_number C);
static void mp_set_number_from_mul(mp_number *A, mp_number B, mp_number C);
static void mp_set_number_from_int_div(mp_number *A, mp_number B, int C);
static void mp_set_number_from_int_mul(mp_number *A, mp_number B, int C);
static void mp_set_number_from_of_the_way(MP mp, mp_number *A, mp_number t, mp_number B, mp_number C);
static void mp_number_negate(mp_number *A);
static void mp_number_add(mp_number *A, mp_number B);
static void mp_number_substract(mp_number *A, mp_number B);
static void mp_number_half(mp_number *A);
static void mp_number_halfp(mp_number *A);
static void mp_number_double(mp_number *A);
static void mp_number_add_scaled(mp_number *A, int B); /* also for negative B */
static void mp_number_multiply_int(mp_number *A, int B);
static void mp_number_divide_int(mp_number *A, int B);
static void mp_number_abs(mp_number *A);   
static void mp_number_clone(mp_number *A, mp_number B);
static void mp_number_swap(mp_number *A, mp_number *B);
static int mp_round_unscaled(mp_number x_orig);
static int mp_number_to_scaled(mp_number A);
static int mp_number_to_boolean(mp_number A);
static int mp_number_to_int(mp_number A);
static int mp_number_odd(mp_number A);
static int mp_number_equal(mp_number A, mp_number B);
static int mp_number_greater(mp_number A, mp_number B);
static int mp_number_less(mp_number A, mp_number B);
static int mp_number_nonequalabs(mp_number A, mp_number B);
static void mp_number_floor (mp_number *i);
static void mp_fraction_to_round_scaled (mp_number *x);
static void mp_number_make_scaled (MP mp, mp_number *r, mp_number p, mp_number q);
static void mp_number_make_fraction (MP mp, mp_number *r, mp_number p, mp_number q);
static void mp_number_take_fraction (MP mp, mp_number *r, mp_number p, mp_number q);
static void mp_number_take_scaled (MP mp, mp_number *r, mp_number p, mp_number q);
static void mp_new_number (MP mp, mp_number *n, mp_number_type t) ;
static void mp_free_number (MP mp, mp_number *n) ;
static void mp_free_scaled_math (MP mp);
static void mp_scaled_set_precision (MP mp);

@ And these are the ones that {\it are} used elsewhere

@<Internal library declarations@>=
void * mp_initialize_scaled_math (MP mp);
void mp_set_number_from_double(mp_number *A, double B);
void mp_pyth_add (MP mp, mp_number *r, mp_number a, mp_number b);
double mp_number_to_double(mp_number A);

@ 

@d coef_bound 04525252525 /* |fraction| approximation to 7/3 */
@d fraction_threshold 2685 /* a |fraction| coefficient less than this is zeroed */
@d half_fraction_threshold 1342 /* half of |fraction_threshold| */
@d scaled_threshold 8 /* a |scaled| coefficient less than this is zeroed */
@d half_scaled_threshold 4 /* half of |scaled_threshold| */
@d near_zero_angle 26844
@d p_over_v_threshold 0x80000
@d equation_threshold 64
@d tfm_warn_threshold 4096


@c
void * mp_initialize_scaled_math (MP mp) {
  math_data *math = (math_data *)mp_xmalloc(mp,1,sizeof(math_data));
  /* alloc */
  math->allocate = mp_new_number;
  math->free = mp_free_number;
  mp_new_number (mp, &math->precision_default, mp_scaled_type);
  math->precision_default.data.val  = unity * 10;
  mp_new_number (mp, &math->precision_max, mp_scaled_type);
  math->precision_max.data.val  = unity * 10;
  mp_new_number (mp, &math->precision_min, mp_scaled_type);
  math->precision_min.data.val  = unity * 10;
  /* here are the constants for |scaled| objects */
  mp_new_number (mp, &math->epsilon_t, mp_scaled_type);
  math->epsilon_t.data.val  = 1;
  mp_new_number (mp, &math->inf_t, mp_scaled_type);
  math->inf_t.data.val  = EL_GORDO;
  mp_new_number (mp, &math->warning_limit_t, mp_scaled_type);
  math->warning_limit_t.data.val  = fraction_one;
  mp_new_number (mp, &math->one_third_inf_t, mp_scaled_type);
  math->one_third_inf_t.data.val = one_third_EL_GORDO;
  mp_new_number (mp, &math->unity_t, mp_scaled_type);
  math->unity_t.data.val = unity;
  mp_new_number (mp, &math->two_t, mp_scaled_type);
  math->two_t.data.val = two;
  mp_new_number (mp, &math->three_t, mp_scaled_type);
  math->three_t.data.val = three;
  mp_new_number (mp, &math->half_unit_t, mp_scaled_type);
  math->half_unit_t.data.val = half_unit;
  mp_new_number (mp, &math->three_quarter_unit_t, mp_scaled_type);
  math->three_quarter_unit_t.data.val = three_quarter_unit;
  mp_new_number (mp, &math->zero_t, mp_scaled_type);
  /* |fractions| */
  mp_new_number (mp, &math->arc_tol_k, mp_fraction_type);
  math->arc_tol_k.data.val = (unity/4096);  /* quit when change in arc length estimate reaches this */
  mp_new_number (mp, &math->fraction_one_t, mp_fraction_type);
  math->fraction_one_t.data.val = fraction_one;
  mp_new_number (mp, &math->fraction_half_t, mp_fraction_type);
  math->fraction_half_t.data.val = fraction_half;
  mp_new_number (mp, &math->fraction_three_t, mp_fraction_type);
  math->fraction_three_t.data.val = fraction_three;
  mp_new_number (mp, &math->fraction_four_t, mp_fraction_type);
  math->fraction_four_t.data.val = fraction_four;
  /* |angles| */
  mp_new_number (mp, &math->three_sixty_deg_t, mp_angle_type);
  math->three_sixty_deg_t.data.val = three_sixty_deg;
  mp_new_number (mp, &math->one_eighty_deg_t, mp_angle_type);
  math->one_eighty_deg_t.data.val = one_eighty_deg;
  /* various approximations */
  mp_new_number (mp, &math->one_k, mp_scaled_type);
  math->one_k.data.val = 1024;
  mp_new_number (mp, &math->sqrt_8_e_k, mp_scaled_type); 
  math->sqrt_8_e_k.data.val = 112429; /* $2^{16}\sqrt{8/e}\approx 112428.82793$ */
  mp_new_number (mp, &math->twelve_ln_2_k, mp_fraction_type); 
  math->twelve_ln_2_k.data.val = 139548960; /* $2^{24}\cdot12\ln2\approx139548959.6165$ */
  mp_new_number (mp, &math->coef_bound_k, mp_fraction_type);
  math->coef_bound_k.data.val = coef_bound;
  mp_new_number (mp, &math->coef_bound_minus_1, mp_fraction_type);
  math->coef_bound_minus_1.data.val = coef_bound - 1;
  mp_new_number (mp, &math->twelvebits_3, mp_scaled_type);
  math->twelvebits_3.data.val = 1365;  /* $1365\approx 2^{12}/3$ */
  mp_new_number (mp, &math->twentysixbits_sqrt2_t, mp_fraction_type);
  math->twentysixbits_sqrt2_t.data.val = 94906266;      /* $2^{26}\sqrt2\approx94906265.62$ */
  mp_new_number (mp, &math->twentyeightbits_d_t, mp_fraction_type);
  math->twentyeightbits_d_t.data.val = 35596755;        /* $2^{28}d\approx35596754.69$ */
  mp_new_number (mp, &math->twentysevenbits_sqrt2_d_t, mp_fraction_type);
  math->twentysevenbits_sqrt2_d_t.data.val = 25170707;  /* $2^{27}\sqrt2\,d\approx25170706.63$ */
  /* thresholds */
  mp_new_number (mp, &math->fraction_threshold_t, mp_fraction_type);
  math->fraction_threshold_t.data.val = fraction_threshold;
  mp_new_number (mp, &math->half_fraction_threshold_t, mp_fraction_type);
  math->half_fraction_threshold_t.data.val = half_fraction_threshold;
  mp_new_number (mp, &math->scaled_threshold_t, mp_scaled_type);
  math->scaled_threshold_t.data.val = scaled_threshold;
  mp_new_number (mp, &math->half_scaled_threshold_t, mp_scaled_type);
  math->half_scaled_threshold_t.data.val = half_scaled_threshold;
  mp_new_number (mp, &math->near_zero_angle_t, mp_angle_type);
  math->near_zero_angle_t.data.val = near_zero_angle;
  mp_new_number (mp, &math->p_over_v_threshold_t, mp_fraction_type);
  math->p_over_v_threshold_t.data.val = p_over_v_threshold;
  mp_new_number (mp, &math->equation_threshold_t, mp_scaled_type);
  math->equation_threshold_t.data.val = equation_threshold;
  mp_new_number (mp, &math->tfm_warn_threshold_t, mp_scaled_type);
  math->tfm_warn_threshold_t.data.val = tfm_warn_threshold;
  /* functions */
  math->from_int = mp_set_number_from_int;
  math->from_boolean = mp_set_number_from_boolean;
  math->from_scaled = mp_set_number_from_scaled;
  math->from_double = mp_set_number_from_double;
  math->from_addition  = mp_set_number_from_addition;
  math->from_substraction  = mp_set_number_from_substraction;
  math->from_oftheway  = mp_set_number_from_of_the_way;
  math->from_div  = mp_set_number_from_div;
  math->from_mul  = mp_set_number_from_mul;
  math->from_int_div  = mp_set_number_from_int_div;
  math->from_int_mul  = mp_set_number_from_int_mul;
  math->negate = mp_number_negate;
  math->add  = mp_number_add;
  math->substract = mp_number_substract;
  math->half = mp_number_half;
  math->halfp = mp_number_halfp;
  math->do_double = mp_number_double;
  math->abs = mp_number_abs;
  math->clone = mp_number_clone;
  math->swap = mp_number_swap;
  math->add_scaled = mp_number_add_scaled;
  math->multiply_int = mp_number_multiply_int;
  math->divide_int = mp_number_divide_int;
  math->to_int = mp_number_to_int;
  math->to_boolean = mp_number_to_boolean;
  math->to_scaled = mp_number_to_scaled;
  math->to_double = mp_number_to_double;
  math->odd = mp_number_odd;
  math->equal = mp_number_equal;
  math->less = mp_number_less;
  math->greater = mp_number_greater;
  math->nonequalabs = mp_number_nonequalabs;
  math->round_unscaled = mp_round_unscaled;
  math->floor_scaled = mp_number_floor;
  math->fraction_to_round_scaled = mp_fraction_to_round_scaled;
  math->make_scaled = mp_number_make_scaled;
  math->make_fraction = mp_number_make_fraction;
  math->take_fraction = mp_number_take_fraction;
  math->take_scaled = mp_number_take_scaled;
  math->velocity = mp_velocity;
  math->n_arg = mp_n_arg;
  math->m_log = mp_m_log;
  math->m_exp = mp_m_exp;
  math->m_unif_rand = mp_m_unif_rand;
  math->m_norm_rand = mp_m_norm_rand;
  math->pyth_add = mp_pyth_add;
  math->pyth_sub = mp_pyth_sub;
  math->fraction_to_scaled = mp_number_fraction_to_scaled;
  math->scaled_to_fraction = mp_number_scaled_to_fraction;
  math->scaled_to_angle = mp_number_scaled_to_angle;
  math->angle_to_scaled = mp_number_angle_to_scaled;
  math->init_randoms = mp_init_randoms;
  math->sin_cos = mp_n_sin_cos;
  math->slow_add = mp_slow_add;
  math->sqrt = mp_square_rt;
  math->print = mp_print_number;
  math->tostring = mp_number_tostring;
  math->modulo = mp_number_modulo;
  math->ab_vs_cd = mp_ab_vs_cd;
  math->crossing_point = mp_crossing_point;
  math->scan_numeric = mp_scan_numeric_token;
  math->scan_fractional = mp_scan_fractional_token;
  math->free_math = mp_free_scaled_math;
  math->set_precision = mp_scaled_set_precision;
  return (void *)math;
}

void mp_scaled_set_precision (MP mp) {
}

void mp_free_scaled_math (MP mp) {
  free_number (((math_data *)mp->math)->epsilon_t);
  free_number (((math_data *)mp->math)->inf_t);
  free_number (((math_data *)mp->math)->arc_tol_k);
  free_number (((math_data *)mp->math)->three_sixty_deg_t);
  free_number (((math_data *)mp->math)->one_eighty_deg_t);
  free_number (((math_data *)mp->math)->fraction_one_t);
  free_number (((math_data *)mp->math)->fraction_half_t);
  free_number (((math_data *)mp->math)->fraction_three_t);
  free_number (((math_data *)mp->math)->fraction_four_t);
  free_number (((math_data *)mp->math)->zero_t);
  free_number (((math_data *)mp->math)->half_unit_t);
  free_number (((math_data *)mp->math)->three_quarter_unit_t);
  free_number (((math_data *)mp->math)->unity_t);
  free_number (((math_data *)mp->math)->two_t);
  free_number (((math_data *)mp->math)->three_t);
  free_number (((math_data *)mp->math)->one_third_inf_t);
  free_number (((math_data *)mp->math)->warning_limit_t);
  free_number (((math_data *)mp->math)->one_k);
  free_number (((math_data *)mp->math)->sqrt_8_e_k);
  free_number (((math_data *)mp->math)->twelve_ln_2_k);
  free_number (((math_data *)mp->math)->coef_bound_k);
  free_number (((math_data *)mp->math)->coef_bound_minus_1);
  free_number (((math_data *)mp->math)->twelvebits_3);
  free_number (((math_data *)mp->math)->twentysixbits_sqrt2_t);
  free_number (((math_data *)mp->math)->twentyeightbits_d_t);
  free_number (((math_data *)mp->math)->twentysevenbits_sqrt2_d_t);
  free_number (((math_data *)mp->math)->fraction_threshold_t);
  free_number (((math_data *)mp->math)->half_fraction_threshold_t);
  free_number (((math_data *)mp->math)->scaled_threshold_t);
  free_number (((math_data *)mp->math)->half_scaled_threshold_t);
  free_number (((math_data *)mp->math)->near_zero_angle_t);
  free_number (((math_data *)mp->math)->p_over_v_threshold_t);
  free_number (((math_data *)mp->math)->equation_threshold_t);
  free_number (((math_data *)mp->math)->tfm_warn_threshold_t);
  free(mp->math);
}

@ Creating an destroying |mp_number| objects

@ @c
void mp_new_number (MP mp, mp_number *n, mp_number_type t) {
  (void)mp;
  n->data.val = 0;
  n->type = t;
}

@ 
@c
void mp_free_number (MP mp, mp_number *n) {
  (void)mp;
  n->type = mp_nan_type;
}

@ Here are the low-level functions on |mp_number| items, setters first.

@c 
void mp_set_number_from_int(mp_number *A, int B) {
  A->data.val = B;
}
void mp_set_number_from_boolean(mp_number *A, int B) {
  A->data.val = B;
}
void mp_set_number_from_scaled(mp_number *A, int B) {
  A->data.val = B;
}
void mp_set_number_from_double(mp_number *A, double B) {
  A->data.val = (int)(B*65536.0);
}
void mp_set_number_from_addition(mp_number *A, mp_number B, mp_number C) {
  A->data.val = B.data.val+C.data.val;
}
void mp_set_number_from_substraction (mp_number *A, mp_number B, mp_number C) {
 A->data.val = B.data.val-C.data.val;
}
void mp_set_number_from_div(mp_number *A, mp_number B, mp_number C) {
  A->data.val = B.data.val / C.data.val;
}
void mp_set_number_from_mul(mp_number *A, mp_number B, mp_number C) {
  A->data.val = B.data.val * C.data.val;
}
void mp_set_number_from_int_div(mp_number *A, mp_number B, int C) {
  A->data.val = B.data.val / C;
}
void mp_set_number_from_int_mul(mp_number *A, mp_number B, int C) {
  A->data.val = B.data.val * C;
}
void mp_set_number_from_of_the_way(MP mp, mp_number *A, mp_number t, mp_number B, mp_number C) {
  A->data.val = B.data.val - mp_take_fraction(mp, (B.data.val - C.data.val), t.data.val);
}
void mp_number_negate(mp_number *A) {
  A->data.val = -A->data.val;
}
void mp_number_add(mp_number *A, mp_number B) {
  A->data.val = A->data.val + B.data.val;
}
void mp_number_substract(mp_number *A, mp_number B) {
  A->data.val = A->data.val - B.data.val;
}
void mp_number_half(mp_number *A) {
  A->data.val = A->data.val/2;
}
void mp_number_halfp(mp_number *A) {
  A->data.val = (A->data.val>>1);
}
void mp_number_double(mp_number *A) {
  A->data.val = A->data.val + A->data.val;
}
void mp_number_add_scaled(mp_number *A, int B) { /* also for negative B */
  A->data.val = A->data.val + B;
}
void mp_number_multiply_int(mp_number *A, int B) {
  A->data.val = B * A->data.val;
}
void mp_number_divide_int(mp_number *A, int B) {
  A->data.val = A->data.val / B;
}
void mp_number_abs(mp_number *A) {   
  A->data.val = abs(A->data.val);
}
void mp_number_clone(mp_number *A, mp_number B) {
  A->data.val = B.data.val;
}
void mp_number_swap(mp_number *A, mp_number *B) {
  int swap_tmp = A->data.val;
  A->data.val = B->data.val;
  B->data.val = swap_tmp;
}
void mp_number_fraction_to_scaled (mp_number *A) {
    A->type = mp_scaled_type;
    A->data.val = A->data.val / 4096;
}
void mp_number_angle_to_scaled (mp_number *A) {
    A->type = mp_scaled_type;
    if (A->data.val >= 0) {
      A->data.val = (A->data.val + 8) / 16;
    } else {
      A->data.val = -((-A->data.val + 8) / 16);
    }
}
void mp_number_scaled_to_fraction (mp_number *A) {
    A->type = mp_fraction_type;
    A->data.val = A->data.val * 4096;
}
void mp_number_scaled_to_angle (mp_number *A) {
    A->type = mp_angle_type;
    A->data.val = A->data.val * 16;
}


@ Query functions

@c
int mp_number_to_int(mp_number A) {
  return A.data.val;
}
int mp_number_to_scaled(mp_number A) {
  return A.data.val;
}
int mp_number_to_boolean(mp_number A) {
  return A.data.val;
}
double mp_number_to_double(mp_number A) {
  return (A.data.val/65536.0);
}
int mp_number_odd(mp_number A) {
  return odd(A.data.val);
}
int mp_number_equal(mp_number A, mp_number B) {
  return (A.data.val==B.data.val);
}
int mp_number_greater(mp_number A, mp_number B) {
  return (A.data.val>B.data.val);
}
int mp_number_less(mp_number A, mp_number B) {
  return (A.data.val<B.data.val);
}
int mp_number_nonequalabs(mp_number A, mp_number B) {
  return (!(abs(A.data.val)==abs(B.data.val)));
}

@ 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.

@d halfp(A) (integer)((unsigned)(A) >> 1)

@ 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.

@<Declarations@>=
static void mp_print_scaled (MP mp, int s); /* scaled */
static char *mp_string_scaled (MP mp, int s);

@ @c
static void mp_print_scaled (MP mp, int s) {  /* s=scaled prints scaled real, rounded to five  digits */
  int delta; /* amount of allowable inaccuracy, scaled */
  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);
  }
}

static  char *mp_string_scaled (MP mp, int s) {    /* s=scaled prints scaled real, rounded to five  digits */
  static char scaled_string[32];
  int delta; /* amount of allowable inaccuracy, scaled */
  int i = 0;
  if (s < 0) {
    scaled_string[i++] = xord ('-');
    s = -s;                 /* print the sign, if negative */
  }
  /* print the integer part */
  mp_snprintf ((scaled_string+i), 12, "%d", (int) (s / unity));
  while (*(scaled_string+i)) i++;

  s = 10 * (s % unity) + 5;
  if (s != 5) {
    delta = 10;
    scaled_string[i++] =  xord ('.');
    do {
      if (delta > unity)
        s = s + 0100000 - (delta / 2);  /* round the final digit */
      scaled_string[i++] = xord ('0' + (s / unity));
      s = 10 * (s % unity);
      delta = delta * 10;
    } while (s > delta);
  }
  scaled_string[i] = '\0';
  return scaled_string;
}

@ 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.

@c
void mp_slow_add (MP mp, mp_number *ret, mp_number x_orig, mp_number y_orig) {
  integer x, y;
  x = x_orig.data.val;
  y = y_orig.data.val;
  if (x >= 0) {
    if (y <= EL_GORDO - x) {
      ret->data.val = x + y;
    } else {
      mp->arith_error = true;
      ret->data.val =  EL_GORDO;
    }
  } else if (-y <= EL_GORDO + x) {
    ret->data.val = x + y;
  } else {
    mp->arith_error = true;
    ret->data.val =  -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@>

@ 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
static integer mp_make_fraction (MP mp, integer p, integer q) {
  integer 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;
        i = EL_GORDO;
        goto RETURN;
      }
      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;
        i = -EL_GORDO;
        goto RETURN;
      }
      i = (integer) d;
      if (d == (double) i && (((q > 0 ? q : -q) & 077777)
                              * (((i & 037777) << 1) + 1) & 04000) != 0)
        ++i;
    }
  }
RETURN:
  return i;
}
void mp_number_make_fraction (MP mp, mp_number *ret, mp_number p, mp_number q) {
  ret->data.val = mp_make_fraction (mp, p.data.val, q.data.val);
}


@ 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@>=
/* still in use by tfmin.w */
integer mp_take_fraction (MP mp, integer q, int f);

@ @c
integer mp_take_fraction (MP mp, integer p, int q) { /* q = fraction */
  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;
}
void mp_number_take_fraction (MP mp, mp_number *ret, mp_number p_orig, mp_number q_orig) {
  ret->data.val = mp_take_fraction (mp, p_orig.data.val, q_orig.data.val);
}


@ 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@>

@<Declarations@>=
static integer mp_take_scaled (MP mp, integer q, int f);

@ @c
static integer mp_take_scaled (MP mp, integer p, int q) { /* q = scaled */
  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;
}
void mp_number_take_scaled (MP mp, mp_number *ret, mp_number p_orig, mp_number q_orig) {
  ret->data.val = mp_take_scaled (mp, p_orig.data.val, q_orig.data.val);
}


@ 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 ...@>=
/* still in use by svgout.w */
int mp_make_scaled (MP mp, integer p, integer q);

@ @c
int mp_make_scaled (MP mp, integer p, integer q) { /* return scaled */
  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;
}
void mp_number_make_scaled (MP mp, mp_number *ret, mp_number p_orig, mp_number q_orig) {
  ret->data.val = mp_make_scaled (mp, p_orig.data.val, q_orig.data.val);
}

@ 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|.

@<Declarations@>=
static int mp_round_decimals (MP mp, unsigned char *b, quarterword k);

@ @c
static int mp_round_decimals (MP mp, unsigned char *b, quarterword k) { /* return: scaled */
  /* converts a decimal fraction */
  unsigned a = 0;       /* the accumulator */
  int l = 0;
  (void)mp; /* Will be needed later */
  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 (int) halfp (a + 1);
}

@* Scanning numbers in the input.

The definitions below are temporarily here.

@d set_cur_cmd(A) mp->cur_mod_->type=(A)
@d set_cur_mod(A) mp->cur_mod_->data.n.data.val=(A)

@<Declarations...@>=
static void mp_wrapup_numeric_token(MP mp, int n, int f);

@ @c
static void mp_wrapup_numeric_token(MP mp, int n, int f) { /* n,f: scaled */
  int mod ; /* scaled */
  if (n < 32768) {
    mod = (n * unity + f);
    set_cur_mod(mod);
    if (mod >= fraction_one) {
      if (internal_value (mp_warning_check).data.val > 0 &&
          (mp->scanner_status != tex_flushing)) {
        char msg[256];
        const char *hlp[] = {"It is at least 4096. Continue and I'll try to cope",
               "with that big value; but it might be dangerous.",
               "(Set warningcheck:=0 to suppress this message.)",
               NULL };
        mp_snprintf (msg, 256, "Number is too large (%s)", mp_string_scaled(mp,mod));
@.Number is too large@>;
        mp_error (mp, msg, hlp, true);
      }
    }
  } else if (mp->scanner_status != tex_flushing) {
    const char *hlp[] = {"I can\'t handle numbers bigger than 32767.99998;",
         "so I've changed your constant to that maximum amount.", 
         NULL };
    mp_error (mp, "Enormous number has been reduced", hlp, false);
@.Enormous number...@>;
    set_cur_mod(EL_GORDO);
  }
  set_cur_cmd((mp_variable_type)mp_numeric_token);
}

@ @c
void mp_scan_fractional_token (MP mp, int n) { /* n: scaled */
  int f; /* scaled */
  int k = 0;
  do {
    k++;
    mp->cur_input.loc_field++;
  } while (mp->char_class[mp->buffer[mp->cur_input.loc_field]] == digit_class);
  f = mp_round_decimals (mp, (unsigned char *)(mp->buffer+mp->cur_input.loc_field-k), (quarterword) k);
  if (f == unity) {
    n++;
    f = 0;
  }
  mp_wrapup_numeric_token(mp, n, f);
}


@ @c
void mp_scan_numeric_token (MP mp, int n) { /* n: scaled */
  while (mp->char_class[mp->buffer[mp->cur_input.loc_field]] == digit_class) {
    if (n < 32768)
      n = 10 * n + mp->buffer[mp->cur_input.loc_field] - '0';
    mp->cur_input.loc_field++;
  }
  if (!(mp->buffer[mp->cur_input.loc_field] == '.' &&
        mp->char_class[mp->buffer[mp->cur_input.loc_field + 1]] == digit_class)) {
    mp_wrapup_numeric_token(mp, n, 0);
  } else {
    mp->cur_input.loc_field++;
    mp_scan_fractional_token(mp, n);
  }
}

@ 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.

@c
void mp_velocity (MP mp, mp_number *ret, mp_number st, mp_number ct, mp_number sf,
                  mp_number cf, mp_number t) {
  integer acc, num, denom;      /* registers for intermediate calculations */
  acc = mp_take_fraction (mp, st.data.val - (sf.data.val / 16), sf.data.val - (st.data.val / 16));
  acc = mp_take_fraction (mp, acc, ct.data.val - cf.data.val);
  num = fraction_two + mp_take_fraction (mp, acc, 379625062);
  /* $2^{28}\sqrt2\approx379625062.497$ */
  denom =
    fraction_three + mp_take_fraction (mp, ct.data.val,
                                       497706707) + mp_take_fraction (mp, cf.data.val,
                                                                      307599661);
  /* $3\cdot2^{27}\cdot(\sqrt5-1)\approx497706706.78$ and
     $3\cdot2^{27}\cdot(3-\sqrt5\,)\approx307599661.22$ */
  if (t.data.val != unity)
    num = mp_make_scaled (mp, num, t.data.val); /* |make_scaled(fraction,scaled)=fraction| */
  if (num / 4 >= denom) {
    ret->data.val = fraction_four;
  } else {
    ret->data.val = mp_make_fraction (mp, num, denom);
  }
/*  |printf ("num,denom=%f,%f -=> %f\n", num/65536.0, denom/65536.0, ret.data.val/65536.0);|*/
}


@ 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.

@c
static void mp_ab_vs_cd (MP mp, mp_number *ret, mp_number a_orig, mp_number b_orig, mp_number c_orig, mp_number d_orig) {
  integer q, r; /* temporary registers */
  integer a, b, c, d;
  (void)mp;
  a = a_orig.data.val;
  b = b_orig.data.val;
  c = c_orig.data.val;
  d = d_orig.data.val;
  @<Reduce to the case that |a,c>=0|, |b,d>0|@>;
  while (1) {
    q = a / d;
    r = c / b;
    if (q != r) {
      ret->data.val = (q > r ? 1 : -1);
      return;
    }
    q = a % d;
    r = c % b;
    if (r == 0) {
      ret->data.val = (q ? 1 : 0);
      return;
    }
    if (q == 0) {
      ret->data.val = -1;
      return;
    }
    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)) 
      ret->data.val = 0;
    else
      ret->data.val = 1;
    return;
  }
  if (d == 0) {
    ret->data.val = (a == 0 ? 0 : -1);
    return;
  }
  q = a;
  a = c;
  c = q;
  q = -b;
  b = -d;
  d = q;
} else if (b <= 0) {
  if (b < 0 && a > 0) {
    ret->data.val  = -1;
    return;
  }
  ret->data.val = (c == 0 ? 0 : -1);
  return;
}

@ Now here's a subroutine that's handy for all sorts of path computations:
Given a quadratic polynomial $B(a,b,c;t)$, the |crossing_point| function
returns the unique |fraction| value |t| between 0 and~1 at which
$B(a,b,c;t)$ changes from positive to negative, or returns
|t=fraction_one+1| if no such value exists. If |a<0| (so that $B(a,b,c;t)$
is already negative at |t=0|), |crossing_point| returns the value zero.

The general bisection method is quite simple when $n=2$, hence
|crossing_point| does not take much time. At each stage in the
recursion we have a subinterval defined by |l| and~|j| such that
$B(a,b,c;2^{-l}(j+t))=B(x_0,x_1,x_2;t)$, and we want to ``zero in'' on
the subinterval where $x_0\G0$ and $\min(x_1,x_2)<0$.

It is convenient for purposes of calculation to combine the values
of |l| and~|j| in a single variable $d=2^l+j$, because the operation
of bisection then corresponds simply to doubling $d$ and possibly
adding~1. Furthermore it proves to be convenient to modify
our previous conventions for bisection slightly, maintaining the
variables $X_0=2^lx_0$, $X_1=2^l(x_0-x_1)$, and $X_2=2^l(x_1-x_2)$.
With these variables the conditions $x_0\ge0$ and $\min(x_1,x_2)<0$ are
equivalent to $\max(X_1,X_1+X_2)>X_0\ge0$.

The following code maintains the invariant relations
$0\L|x0|<\max(|x1|,|x1|+|x2|)$,
$\vert|x1|\vert<2^{30}$, $\vert|x2|\vert<2^{30}$;
it has been constructed in such a way that no arithmetic overflow
will occur if the inputs satisfy
$a<2^{30}$, $\vert a-b\vert<2^{30}$, and $\vert b-c\vert<2^{30}$.

@d no_crossing   { ret->data.val = fraction_one + 1; return; }
@d one_crossing  { ret->data.val = fraction_one; return; }
@d zero_crossing { ret->data.val = 0; return; }

@c
static void mp_crossing_point (MP mp, mp_number *ret, mp_number aa, mp_number bb, mp_number cc) {
  integer a,b,c;
  integer d;    /* recursive counter */
  integer x, xx, x0, x1, x2;    /* temporary registers for bisection */
  a = aa.data.val;
  b = bb.data.val;
  c = cc.data.val;
  if (a < 0)
    zero_crossing;
  if (c >= 0) {
    if (b >= 0) {
      if (c > 0) {
        no_crossing;
      } else if ((a == 0) && (b == 0)) {
        no_crossing;
      } else {
        one_crossing;
      }
    }
    if (a == 0)
      zero_crossing;
  } else if (a == 0) {
    if (b <= 0)
      zero_crossing;
  }

  /* Use bisection to find the crossing point... */
  d = 1;
  x0 = a;
  x1 = a - b;
  x2 = b - c;
  do {
    x = (x1 + x2) / 2;
    if (x1 - x0 > x0) {
      x2 = x;
      x0 += x0;
      d += d;
    } else {
      xx = x1 + x - x0;
      if (xx > x0) {
        x2 = x;
        x0 += x0;
        d += d;
      } else {
        x0 = x0 - xx;
        if (x <= x0) {
          if (x + x2 <= x0)
            no_crossing;
        }
        x1 = x;
        d = d + d + 1;
      }
    }
  } while (d < fraction_one);
  ret->data.val = (d - fraction_one); 
}
 

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


@ |round_unscaled| rounds a |scaled| and converts it to |int|
@c
int mp_round_unscaled(mp_number x_orig) {
  int x = x_orig.data.val;
  if (x >= 32768) {
    return 1+((x-32768) / 65536);
  } else if ( x>=-32768) {
    return 0;
  } else {
    return  -(1+((-(x+1)-32768) / 65536));
  }
}

@ |number_floor| floors a |scaled|

@c
void mp_number_floor (mp_number *i) {
  i->data.val = i->data.val&-65536;
}

@ |fraction_to_scaled| rounds a |fraction| and converts it to |scaled|
@c
void mp_fraction_to_round_scaled (mp_number *x_orig) {
  int x = x_orig->data.val;
  x_orig->type = mp_scaled_type;
  x_orig->data.val = (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.

@c
void mp_square_rt (MP mp, mp_number *ret, mp_number x_orig) { /* return, x: scaled */
  integer x;
  quarterword k;        /* iteration control counter */
  integer y;    /* register for intermediate calculations */
  integer q;    /* register for intermediate calculations */
  x = x_orig.data.val;
  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);
    ret->data.val = (int) (halfp (q));
  }
}


@ @<Handle square root of zero...@>=
{  
  if (x < 0) {
    char msg[256];
    const char *hlp[] = {
           "Since I don't take square roots of negative numbers,",
           "I'm zeroing this one. Proceed, with fingers crossed.",
           NULL };
    mp_snprintf(msg, 256, "Square root of %s has been replaced by 0", mp_string_scaled (mp, x));
@.Square root...replaced by 0@>;
    mp_error (mp, msg, hlp, true);
  }
  ret->data.val = 0;
  return;
}


@ @<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.

@c
void mp_pyth_add (MP mp, mp_number *ret, mp_number a_orig, mp_number b_orig) {
  int a, b; /* a,b : scaled */
  int r;   /* register used to transform |a| and |b|, fraction */
  boolean big;  /* is the result dangerously near $2^{31}$? */
  a = abs (a_orig.data.val);
  b = abs (b_orig.data.val);
  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;
      };
    }
  }
  ret->data.val = 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.

@c
void mp_pyth_sub (MP mp, mp_number *ret, mp_number a_orig, mp_number b_orig) {
  int a, b; /* a,b: scaled */
  int r;   /* register used to transform |a| and |b|, fraction */
  boolean big;  /* is the result dangerously near $2^{31}$? */
  a = abs (a_orig.data.val);
  b = abs (b_orig.data.val);
  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;
  }
  ret->data.val = 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) {
    char msg[256];
    const char *hlp[] = {
         "Since I don't take square roots of negative numbers,",
         "I'm zeroing this one. Proceed, with fingers crossed.",
         NULL };
    char *astr = strdup(mp_string_scaled (mp, a));
    assert (astr);
    mp_snprintf (msg, 256, "Pythagorean subtraction %s+-+%s has been replaced by 0", astr, mp_string_scaled (mp, b));
    free(astr);
@.Pythagorean...@>;
    mp_error (mp, msg, hlp, true);
  }
  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.)

@c
void mp_m_log (MP mp, mp_number *ret, mp_number x_orig) { /* return, x: scaled */
  int x;
  integer y, z; /* auxiliary registers */
  integer k;    /* iteration counter */
  x = x_orig.data.val;
  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@>;
    }
    ret->data.val = (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@>=
{
  char msg[256];
  const char *hlp[] = { 
         "Since I don't take logs of non-positive numbers,",
         "I'm zeroing this one. Proceed, with fingers crossed.",
          NULL };
  mp_snprintf (msg, 256, "Logarithm of %s has been replaced by 0", mp_string_scaled (mp, x));
@.Logarithm...replaced by 0@>;
  mp_error (mp, msg, hlp, true);
  ret->data.val = 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.

@c
void mp_m_exp (MP mp, mp_number *ret, mp_number x_orig) {
  quarterword k;        /* loop control index */
  integer y, z; /* auxiliary registers */
  int x;
  x = x_orig.data.val;
  if (x > 174436200) {
    /* $2^{24}\ln((2^{31}-1)/2^{16})\approx 174436199.51$ */
    mp->arith_error = true;
    ret->data.val = EL_GORDO;
  } else if (x < -197694359) {
    /* $2^{24}\ln(2^{-1}/2^{16})\approx-197694359.45$ */
    ret->data.val = 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)
      ret->data.val = ((y + 8) / 16);
    else
      ret->data.val = 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 int 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)

@c
void mp_n_arg (MP mp, mp_number *ret, mp_number x_orig, mp_number y_orig) {
  integer z;      /* auxiliary register */
  integer t;    /* temporary storage */
  quarterword k;        /* loop counter */
  int octant;   /* octant code */
  integer x, y;
  x = x_orig.data.val;
  y = y_orig.data.val;
  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 {
    ret->type = mp_angle_type;
    @<Set variable |z| to the arg of $(x,y)$@>;
    @<Return an appropriate answer based on |z| and |octant|@>;
  }
}


@ @<Handle undefined arg@>=
{
  const char *hlp[] = {
         "The `angle' between two identical points is undefined.",
         "I'm zeroing this one. Proceed, with fingers crossed.",
         NULL };
  mp_error (mp, "angle(0,0) is taken as zero", hlp, true);
@.angle(0,0)...zero@>;
  ret->data.val = 0;
}


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


@ 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)   (abs(A)%2==1)

@ Compute a multiple of the sine and cosine

@c
void mp_n_sin_cos (MP mp, mp_number z_orig, mp_number *n_cos, mp_number *n_sin) {
  quarterword k;        /* loop control variable */
  int q;        /* specifies the quadrant */
  integer x, y, t;      /* temporary registers */
  int z; /* scaled */
  mp_number x_n, y_n, ret;
  new_number (ret);
  new_number (x_n);
  new_number (y_n);
  z = z_orig.data.val;
  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|@>;
  x_n.data.val = x;
  y_n.data.val = y;
  mp_pyth_add (mp, &ret, x_n, y_n);
  n_cos->data.val = mp_make_fraction (mp, x, ret.data.val);
  n_sin->data.val = mp_make_fraction (mp, y, ret.data.val);
  free_number(ret);
  free_number(x_n);
  free_number(y_n);
}


@ 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 */
    

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

@c
void mp_init_randoms (MP mp, int seed) {
  int j, jj, k;    /* more or less random integers */
  int i;        /* index into |randoms| */
  j =  abs (seed);
  while (j >= fraction_one) {
    j = j/2;
  }
  k = 1;
  for (i = 0; i <= 54; i++) {
    jj = k;
    k = j - k;
    j = jj;
    if (k<0)
      k += fraction_one;
    mp->randoms[(i * 21) % 55].data.val = j;
  }
  mp_new_randoms (mp);
  mp_new_randoms (mp);
  mp_new_randoms (mp);          /* ``warm up'' the array */
}


@ @c
void mp_print_number (MP mp, mp_number n) {
  mp_print_scaled (mp, n.data.val);
}


@ @c
char * mp_number_tostring (MP mp, mp_number n) {
   return mp_string_scaled(mp, n.data.val);
}

@ @c
void mp_number_modulo (mp_number *a, mp_number b) {
   a->data.val = a->data.val % b.data.val;
}





@ To consume a random fraction, the program below will say `|next_random|'.

@c 
static void mp_next_random (MP mp, mp_number *ret) { 
  if ( mp->j_random==0 ) 
    mp_new_randoms(mp);
  else 
    mp->j_random = mp->j_random-1;
  mp_number_clone (ret, mp->randoms[mp->j_random]);
}


@ 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_fraction| 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.

@c
static void mp_m_unif_rand (MP mp, mp_number *ret, mp_number x_orig) {
  mp_number y;     /* trial value */
  mp_number x, abs_x;
  mp_number u;
  new_fraction (y);
  new_number (x);
  new_number (abs_x);
  new_number (u);
  mp_number_clone (&x, x_orig);
  mp_number_clone (&abs_x, x);
  mp_number_abs (&abs_x);
  mp_next_random(mp, &u);
  /*|take_fraction (y, abs_x, u);|*/
  mp_number_take_fraction (mp,&y, abs_x,u);
  free_number (u);
  if (mp_number_equal(y, abs_x)) {
    /*|set_number_to_zero(*ret);|*/
    mp_number_clone (ret, ((math_data *)mp->math)->zero_t);
  } else if (mp_number_greater(x, ((math_data *)mp->math)->zero_t)) {
    mp_number_clone (ret, y);
  } else {
    mp_number_clone (ret, y);
    mp_number_negate (ret);
  }
  free_number (abs_x);
  free_number (x);
  free_number (y);
}




@ 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\/}).

@c
static void mp_m_norm_rand (MP mp, mp_number *ret) {
  mp_number ab_vs_cd; 
  mp_number abs_x;
  mp_number u;
  mp_number r;
  mp_number la, xa;
  new_number (ab_vs_cd);
  new_number (la);
  new_number (xa);
  new_number (abs_x);
  new_number (u);
  new_number (r);
  do {
    do {
      mp_number v;
      new_number (v);
      mp_next_random(mp, &v);
      mp_number_substract (&v, ((math_data *)mp->math)->fraction_half_t); 
      mp_number_take_fraction (mp,&xa, ((math_data *)mp->math)->sqrt_8_e_k, v); 
      free_number (v);
      mp_next_random(mp, &u);
      mp_number_clone (&abs_x, xa);
      mp_number_abs (&abs_x);
    } while (!mp_number_less(abs_x, u));
    mp_number_make_fraction (mp, &r, xa, u);
    mp_number_clone (&xa, r);
    mp_m_log (mp,&la, u);
    mp_set_number_from_substraction(&la, ((math_data *)mp->math)->twelve_ln_2_k, la);
    mp_ab_vs_cd (mp,&ab_vs_cd, ((math_data *)mp->math)->one_k, la, xa, xa);
  } while (mp_number_less(ab_vs_cd,((math_data *)mp->math)->zero_t));
  mp_number_clone (ret, xa);
  free_number (ab_vs_cd);
  free_number (r);
  free_number (abs_x);
  free_number (la);
  free_number (xa);
  free_number (u);
}