path: root/Build/source/texk/web2c/mplibdir/mpmathbinary.w

% $Id$
%
% 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 MPFR based math}
\pdfoutput=1

@ Introduction.

@c 
#include <w2c/config.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <math.h>
#include "mpmathbinary.h" /* internal header */
#define ROUND(a) floor((a)+0.5)
@h

@ @c
@<Declarations@>;

@ @(mpmathbinary.h@>=
#ifndef MPMATHBINARY_H
#define  MPMATHBINARY_H 1
#include "mplib.h"
#include "mpmp.h" /* internal header */
#include <gmp.h>
#include <mpfr.h>

#ifdef HAVE_CONFIG_H
#define MP_STR_HELPER(x) #x
#define MP_STR(x) MP_STR_HELPER(x)
const char * const COMPILED_gmp_version  = MP_STR(__GNU_MP_VERSION) "." MP_STR( __GNU_MP_VERSION_MINOR) "." MP_STR(__GNU_MP_VERSION_PATCHLEVEL);
#else
const char * const COMPILED_gmp_version  = "unknown";
#endif

const char *COMPILED_MPFR_VERSION_STRING = MPFR_VERSION_STRING;
int COMPILED__GNU_MP_VERSION = __GNU_MP_VERSION ;
int COMPILED__GNU_MP_VERSION_MINOR = __GNU_MP_VERSION_MINOR ;
int COMPILED__GNU_MP_VERSION_PATCHLEVEL = __GNU_MP_VERSION_PATCHLEVEL ;

@<Internal library declarations@>;
#endif

@* Math initialization.

First, here are some very important constants.

@d ROUNDING MPFR_RNDN
@d E_STRING  "2.7182818284590452353602874713526624977572470936999595749669676277240766303535"
@d PI_STRING "3.1415926535897932384626433832795028841971693993751058209749445923078164062862"
@d fraction_multiplier 4096
@d angle_multiplier 16

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

@<Declarations@>=
#define DEBUG 0
static void mp_binary_scan_fractional_token (MP mp, int n);
static void mp_binary_scan_numeric_token (MP mp, int n);
static void mp_binary_ab_vs_cd (MP mp, mp_number *ret, mp_number a, mp_number b, mp_number c, mp_number d);
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_binary_crossing_point (MP mp, mp_number *ret, mp_number a, mp_number b, mp_number c);
static void mp_binary_number_modulo (mp_number *a, mp_number b);
static void mp_binary_print_number (MP mp, mp_number n);
static char * mp_binary_number_tostring (MP mp, mp_number n);
static void mp_binary_slow_add (MP mp, mp_number *ret, mp_number x_orig, mp_number y_orig);
static void mp_binary_square_rt (MP mp, mp_number *ret, mp_number x_orig);
static void mp_binary_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_binary_m_unif_rand (MP mp, mp_number *ret, mp_number x_orig);
static void mp_binary_m_norm_rand (MP mp, mp_number *ret);
static void mp_binary_m_exp (MP mp, mp_number *ret, mp_number x_orig);
static void mp_binary_m_log (MP mp, mp_number *ret, mp_number x_orig);
static void mp_binary_pyth_sub (MP mp, mp_number *r, mp_number a, mp_number b);
static void mp_binary_pyth_add (MP mp, mp_number *r, mp_number a, mp_number b);
static void mp_binary_n_arg (MP mp, mp_number *ret, mp_number x, mp_number y);
static void mp_binary_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_binary_from_int(mp_number *A, int B);
static void mp_set_binary_from_boolean(mp_number *A, int B);
static void mp_set_binary_from_scaled(mp_number *A, int B);
static void mp_set_binary_from_addition(mp_number *A, mp_number B, mp_number C);
static void mp_set_binary_from_substraction (mp_number *A, mp_number B, mp_number C);
static void mp_set_binary_from_div(mp_number *A, mp_number B, mp_number C);
static void mp_set_binary_from_mul(mp_number *A, mp_number B, mp_number C);
static void mp_set_binary_from_int_div(mp_number *A, mp_number B, int C);
static void mp_set_binary_from_int_mul(mp_number *A, mp_number B, int C);
static void mp_set_binary_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_binary_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_int(mp_number A);
static int mp_number_to_scaled(mp_number A);
static int mp_number_to_boolean(mp_number A);
static double mp_number_to_double(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_binary_fraction_to_round_scaled (mp_number *x);
static void mp_binary_number_make_scaled (MP mp, mp_number *r, mp_number p, mp_number q);
static void mp_binary_number_make_fraction (MP mp, mp_number *r, mp_number p, mp_number q);
static void mp_binary_number_take_fraction (MP mp, mp_number *r, mp_number p, mp_number q);
static void mp_binary_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_set_binary_from_double(mp_number *A, double B);
static void mp_free_binary_math (MP mp);
static void mp_binary_set_precision (MP mp);
static void mp_check_mpfr_t (MP mp, mpfr_t dec);
static int binary_number_check (mpfr_t dec);
static char * mp_binnumber_tostring (mpfr_t n);
static void init_binary_constants (void);
static void free_binary_constants (void);
static mpfr_prec_t precision_digits_to_bits(double i);
static double precision_bits_to_digits (mpfr_prec_t i);

@ We do not want special numbers as return values for functions, so:

@d mpfr_negative_p(a) (mpfr_sgn((a))<0)
@d mpfr_positive_p(a) (mpfr_sgn((a))>0)
@d checkZero(dec)  if (mpfr_zero_p(dec) && mpfr_negative_p(dec)) {
     mpfr_set_zero(dec,1);
   }

@c 
int binary_number_check (mpfr_t dec)
{
   int test = false;
   if (!mpfr_number_p(dec)) {
      test = true;
      if (mpfr_inf_p(dec)) {
        mpfr_set(dec, EL_GORDO_mpfr_t, ROUNDING);
        if (mpfr_negative_p(dec)) {
          mpfr_neg(dec, dec, ROUNDING);
        }
      } else { // Nan 
        mpfr_set_zero(dec,1); /* 1 == positive */
      }
   }
   checkZero(dec);
   return test;
}
void mp_check_mpfr_t (MP mp, mpfr_t dec)
{
  mp->arith_error = binary_number_check (dec);
}




@ Precision IO uses |double| because |MPFR_PREC_MAX| overflows int. 

@c
static double precision_bits;
mpfr_prec_t precision_digits_to_bits (double i)
{
  return i/log10(2);
}
double precision_bits_to_digits (mpfr_prec_t d)
{
  return d*log10(2);
}


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

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

@ 

@d unity 1
@d two 2
@d three 3
@d four 4
@d half_unit  0.5
@d three_quarter_unit 0.75
@d coef_bound ((7.0/3.0)*fraction_multiplier) /* |fraction| approximation to 7/3 */
@d fraction_threshold 0.04096 /* a |fraction| coefficient less than this is zeroed */
@d half_fraction_threshold (fraction_threshold/2) /* half of |fraction_threshold| */
@d scaled_threshold 0.000122 /* a |scaled| coefficient less than this is zeroed */
@d half_scaled_threshold (scaled_threshold/2) /* half of |scaled_threshold| */
@d near_zero_angle (0.0256*angle_multiplier)  /* an angle of about 0.0256 */
@d p_over_v_threshold 0x80000 /* TODO */
@d equation_threshold 0.001
@d tfm_warn_threshold 0.0625
@d warning_limit pow(2.0,52.0)  /* this is a large value that can just be expressed without loss of precision */
@d epsilon "1E-52"
@d epsilonf pow(2.0,-52.0)
@d EL_GORDO   "1E1000000" /* the largest value that \MP\ likes. */
@d one_third_EL_GORDO (EL_GORDO/3.0)

@<Declarations@>=
static mpfr_t zero;
static mpfr_t one;
static mpfr_t minusone;
static mpfr_t two_mpfr_t;
static mpfr_t three_mpfr_t;
static mpfr_t four_mpfr_t;
static mpfr_t fraction_multiplier_mpfr_t;
static mpfr_t angle_multiplier_mpfr_t;
static mpfr_t fraction_one_mpfr_t;
static mpfr_t fraction_one_plus_mpfr_t;
static mpfr_t PI_mpfr_t;
static mpfr_t epsilon_mpfr_t;
static mpfr_t EL_GORDO_mpfr_t;
static boolean initialized = false;


@ @c
void init_binary_constants (void) {
  if (!initialized) {
    mpfr_inits2 (precision_bits, one, minusone, zero, two_mpfr_t, three_mpfr_t, four_mpfr_t, fraction_multiplier_mpfr_t,
              fraction_one_mpfr_t, fraction_one_plus_mpfr_t,  angle_multiplier_mpfr_t, PI_mpfr_t, 
              epsilon_mpfr_t, EL_GORDO_mpfr_t, (mpfr_ptr) 0);
    mpfr_set_si (one, 1, ROUNDING);
    mpfr_set_si (minusone, -1, ROUNDING);
    mpfr_set_si (zero, 0, ROUNDING);
    mpfr_set_si (two_mpfr_t, two, ROUNDING);
    mpfr_set_si (three_mpfr_t, three, ROUNDING);
    mpfr_set_si (four_mpfr_t, four, ROUNDING);
    mpfr_set_si (fraction_multiplier_mpfr_t, fraction_multiplier, ROUNDING);
    mpfr_set_si (fraction_one_mpfr_t, fraction_one, ROUNDING);
    mpfr_set_si (fraction_one_plus_mpfr_t, (fraction_one+1), ROUNDING);
    mpfr_set_si (angle_multiplier_mpfr_t, angle_multiplier, ROUNDING);
    mpfr_set_str (PI_mpfr_t, PI_STRING, 10, ROUNDING);
    mpfr_set_str (epsilon_mpfr_t, epsilon, 10, ROUNDING);
    mpfr_set_str (EL_GORDO_mpfr_t, EL_GORDO, 10, ROUNDING);
    initialized = true;
  }
}
void free_binary_constants (void) {
  /* For sake of speed, we accept this memory leak. */
  /*mpfr_clears (one, minusone, zero, two_mpfr_t, three_mpfr_t, four_mpfr_t, fraction_multiplier_mpfr_t,*/
  /*            fraction_one_mpfr_t, fraction_one_plus_mpfr_t,  angle_multiplier_mpfr_t, PI_mpfr_t, */
  /*            epsilon_mpfr_t, EL_GORDO_mpfr_t, (mpfr_ptr) 0); */
  /*mpfr_free_cache ();*/
}

@ |precision_max| is limited to 1000, because the precision of already initialized 
|mpfr_t| numbers cannot be raised, only lowered. The value of 1000.0 is a tradeoff
between precision and allocation size / processing speed.

@d MAX_PRECISION 1000.0
@d DEF_PRECISION 34.0

@c
void * mp_initialize_binary_math (MP mp) {
  math_data *math = (math_data *)mp_xmalloc(mp,1,sizeof(math_data));
  precision_bits = precision_digits_to_bits(MAX_PRECISION);
  init_binary_constants();
  /* alloc */
  math->allocate = mp_new_number;
  math->free = mp_free_number;
  mp_new_number (mp, &math->precision_default, mp_scaled_type);
  mpfr_set_d(math->precision_default.data.num, DEF_PRECISION, ROUNDING);
  mp_new_number (mp, &math->precision_max, mp_scaled_type);
  mpfr_set_d(math->precision_max.data.num, MAX_PRECISION, ROUNDING);
  mp_new_number (mp, &math->precision_min, mp_scaled_type);
  /* really should be |precision_bits_to_digits(MPFR_PREC_MIN)| but that produces a horrible number */
  mpfr_set_d(math->precision_min.data.num, 1.0 , ROUNDING); 
  /* here are the constants for |scaled| objects */
  mp_new_number (mp, &math->epsilon_t, mp_scaled_type);
  mpfr_set (math->epsilon_t.data.num, epsilon_mpfr_t, ROUNDING);
  mp_new_number (mp, &math->inf_t, mp_scaled_type);
  mpfr_set (math->inf_t.data.num, EL_GORDO_mpfr_t, ROUNDING);
  mp_new_number (mp, &math->warning_limit_t, mp_scaled_type);
  mpfr_set_d (math->warning_limit_t.data.num, warning_limit, ROUNDING);
  mp_new_number (mp, &math->one_third_inf_t, mp_scaled_type);
  mpfr_div (math->one_third_inf_t.data.num, math->inf_t.data.num, three_mpfr_t, ROUNDING);
  mp_new_number (mp, &math->unity_t, mp_scaled_type);
  mpfr_set (math->unity_t.data.num, one, ROUNDING);
  mp_new_number (mp, &math->two_t, mp_scaled_type);
  mpfr_set_si(math->two_t.data.num, two, ROUNDING);
  mp_new_number (mp, &math->three_t, mp_scaled_type);
  mpfr_set_si(math->three_t.data.num, three, ROUNDING);
  mp_new_number (mp, &math->half_unit_t, mp_scaled_type);
  mpfr_set_d(math->half_unit_t.data.num, half_unit, ROUNDING);
  mp_new_number (mp, &math->three_quarter_unit_t, mp_scaled_type);
  mpfr_set_d (math->three_quarter_unit_t.data.num, three_quarter_unit, ROUNDING);
  mp_new_number (mp, &math->zero_t, mp_scaled_type);
  mpfr_set_zero (math->zero_t.data.num, 1);
  /* |fractions| */
  mp_new_number (mp, &math->arc_tol_k, mp_fraction_type);
  {
     mpfr_div_si (math->arc_tol_k.data.num, one, 4096, ROUNDING);
     /* quit when change in arc length estimate reaches this */
  }
  mp_new_number (mp, &math->fraction_one_t, mp_fraction_type);
  mpfr_set_si(math->fraction_one_t.data.num, fraction_one, ROUNDING);
  mp_new_number (mp, &math->fraction_half_t, mp_fraction_type);
  mpfr_set_si(math->fraction_half_t.data.num, fraction_half, ROUNDING);
  mp_new_number (mp, &math->fraction_three_t, mp_fraction_type);
  mpfr_set_si(math->fraction_three_t.data.num, fraction_three, ROUNDING);
  mp_new_number (mp, &math->fraction_four_t, mp_fraction_type);
  mpfr_set_si(math->fraction_four_t.data.num, fraction_four, ROUNDING);
  /* |angles| */
  mp_new_number (mp, &math->three_sixty_deg_t, mp_angle_type);
  mpfr_set_si(math->three_sixty_deg_t.data.num, 360  * angle_multiplier, ROUNDING);
  mp_new_number (mp, &math->one_eighty_deg_t, mp_angle_type);
  mpfr_set_si(math->one_eighty_deg_t.data.num, 180 * angle_multiplier, ROUNDING);
  /* various approximations */
  mp_new_number (mp, &math->one_k, mp_scaled_type);
  mpfr_set_d(math->one_k.data.num, 1.0/64, ROUNDING);
  mp_new_number (mp, &math->sqrt_8_e_k, mp_scaled_type); 
  {
    mpfr_set_d(math->sqrt_8_e_k.data.num, 112428.82793 / 65536.0, ROUNDING);
    /* $2^{16}\sqrt{8/e}\approx 112428.82793$ */
  }
  mp_new_number (mp, &math->twelve_ln_2_k, mp_fraction_type); 
  {
    mpfr_set_d(math->twelve_ln_2_k.data.num, 139548959.6165 / 65536.0, ROUNDING);
    /* $2^{24}\cdot12\ln2\approx139548959.6165$ */
  }
  mp_new_number (mp, &math->coef_bound_k, mp_fraction_type);
  mpfr_set_d(math->coef_bound_k.data.num,coef_bound, ROUNDING);
  mp_new_number (mp, &math->coef_bound_minus_1, mp_fraction_type);
  mpfr_set_d(math->coef_bound_minus_1.data.num,coef_bound - 1 / 65536.0, ROUNDING);
  mp_new_number (mp, &math->twelvebits_3, mp_scaled_type);
  {
    mpfr_set_d(math->twelvebits_3.data.num, 1365 / 65536.0, ROUNDING);
    /* $1365\approx 2^{12}/3$ */
  }
  mp_new_number (mp, &math->twentysixbits_sqrt2_t, mp_fraction_type);
  {
    mpfr_set_d(math->twentysixbits_sqrt2_t.data.num, 94906265.62 / 65536.0, ROUNDING);
    /* $2^{26}\sqrt2\approx94906265.62$ */
  }
  mp_new_number (mp, &math->twentyeightbits_d_t, mp_fraction_type);
  {
    mpfr_set_d(math->twentyeightbits_d_t.data.num, 35596754.69  / 65536.0, ROUNDING);
    /* $2^{28}d\approx35596754.69$ */
  }
  mp_new_number (mp, &math->twentysevenbits_sqrt2_d_t, mp_fraction_type);
  {
    mpfr_set_d(math->twentysevenbits_sqrt2_d_t.data.num, 25170706.63  / 65536.0, ROUNDING);
    /* $2^{27}\sqrt2\,d\approx25170706.63$ */
  }
  /* thresholds */
  mp_new_number (mp, &math->fraction_threshold_t, mp_fraction_type);
  mpfr_set_d(math->fraction_threshold_t.data.num, fraction_threshold, ROUNDING);
  mp_new_number (mp, &math->half_fraction_threshold_t, mp_fraction_type);
  mpfr_set_d(math->half_fraction_threshold_t.data.num, half_fraction_threshold, ROUNDING);
  mp_new_number (mp, &math->scaled_threshold_t, mp_scaled_type);
  mpfr_set_d(math->scaled_threshold_t.data.num, scaled_threshold, ROUNDING);
  mp_new_number (mp, &math->half_scaled_threshold_t, mp_scaled_type);
  mpfr_set_d(math->half_scaled_threshold_t.data.num, half_scaled_threshold, ROUNDING);
  mp_new_number (mp, &math->near_zero_angle_t, mp_angle_type);
  mpfr_set_d(math->near_zero_angle_t.data.num, near_zero_angle, ROUNDING);
  mp_new_number (mp, &math->p_over_v_threshold_t, mp_fraction_type);
  mpfr_set_d(math->p_over_v_threshold_t.data.num, p_over_v_threshold, ROUNDING);
  mp_new_number (mp, &math->equation_threshold_t, mp_scaled_type);
  mpfr_set_d(math->equation_threshold_t.data.num, equation_threshold, ROUNDING);
  mp_new_number (mp, &math->tfm_warn_threshold_t, mp_scaled_type);
  mpfr_set_d(math->tfm_warn_threshold_t.data.num, tfm_warn_threshold, ROUNDING);
  /* functions */
  math->from_int = mp_set_binary_from_int;
  math->from_boolean = mp_set_binary_from_boolean;
  math->from_scaled = mp_set_binary_from_scaled;
  math->from_double = mp_set_binary_from_double;
  math->from_addition  = mp_set_binary_from_addition;
  math->from_substraction  = mp_set_binary_from_substraction;
  math->from_oftheway  = mp_set_binary_from_of_the_way;
  math->from_div  = mp_set_binary_from_div;
  math->from_mul  = mp_set_binary_from_mul;
  math->from_int_div  = mp_set_binary_from_int_div;
  math->from_int_mul  = mp_set_binary_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_binary_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_boolean = mp_number_to_boolean;
  math->to_scaled = mp_number_to_scaled;
  math->to_double = mp_number_to_double;
  math->to_int = mp_number_to_int;
  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_binary_fraction_to_round_scaled;
  math->make_scaled = mp_binary_number_make_scaled;
  math->make_fraction = mp_binary_number_make_fraction;
  math->take_fraction = mp_binary_number_take_fraction;
  math->take_scaled = mp_binary_number_take_scaled;
  math->velocity = mp_binary_velocity;
  math->n_arg = mp_binary_n_arg;
  math->m_log = mp_binary_m_log;
  math->m_exp = mp_binary_m_exp;
  math->m_unif_rand = mp_binary_m_unif_rand;
  math->m_norm_rand = mp_binary_m_norm_rand;
  math->pyth_add = mp_binary_pyth_add;
  math->pyth_sub = mp_binary_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_binary_sin_cos;
  math->slow_add = mp_binary_slow_add;
  math->sqrt = mp_binary_square_rt;
  math->print = mp_binary_print_number;
  math->tostring = mp_binary_number_tostring;
  math->modulo = mp_binary_number_modulo;
  math->ab_vs_cd = mp_ab_vs_cd;
  math->crossing_point = mp_binary_crossing_point;
  math->scan_numeric = mp_binary_scan_numeric_token;
  math->scan_fractional = mp_binary_scan_fractional_token;
  math->free_math = mp_free_binary_math;
  math->set_precision = mp_binary_set_precision;  
  return (void *)math;
}

void mp_binary_set_precision (MP mp) {
  double d = mpfr_get_d(internal_value (mp_number_precision).data.num, ROUNDING);
  precision_bits = precision_digits_to_bits(d);
}

void mp_free_binary_math (MP mp) {
  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)->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)->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)->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_binary_constants();
  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.num = mp_xmalloc(mp,1,sizeof(mpfr_t));
  mpfr_init2 ((mpfr_ptr)(n->data.num), precision_bits);
  mpfr_set_zero((mpfr_ptr)(n->data.num),1); /* 1 == positive */
  n->type = t;
}

@ 

@c
void mp_free_number (MP mp, mp_number *n) {
  (void)mp;
  if (n->data.num) {
    mpfr_clear (n->data.num);
    n->data.num = NULL;
  }
  n->type = mp_nan_type;
}

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

@c 
void mp_set_binary_from_int(mp_number *A, int B) {
  mpfr_set_si(A->data.num,B, ROUNDING);
}
void mp_set_binary_from_boolean(mp_number *A, int B) {
  mpfr_set_si(A->data.num,B, ROUNDING);
}
void mp_set_binary_from_scaled(mp_number *A, int B) {
  mpfr_set_si(A->data.num, B, ROUNDING);
  mpfr_div_si(A->data.num, A->data.num, 65536, ROUNDING);
}
void mp_set_binary_from_double(mp_number *A, double B) {
  mpfr_set_d(A->data.num, B, ROUNDING);
}
void mp_set_binary_from_addition(mp_number *A, mp_number B, mp_number C) {
  mpfr_add(A->data.num,B.data.num,C.data.num, ROUNDING);
}
void mp_set_binary_from_substraction (mp_number *A, mp_number B, mp_number C) {
 mpfr_sub(A->data.num,B.data.num,C.data.num, ROUNDING);
}
void mp_set_binary_from_div(mp_number *A, mp_number B, mp_number C) {
  mpfr_div(A->data.num,B.data.num,C.data.num, ROUNDING);
}
void mp_set_binary_from_mul(mp_number *A, mp_number B, mp_number C) {
 mpfr_mul(A->data.num,B.data.num,C.data.num, ROUNDING);
}
void mp_set_binary_from_int_div(mp_number *A, mp_number B, int C) {
  mpfr_div_si(A->data.num,B.data.num,C, ROUNDING);
}
void mp_set_binary_from_int_mul(mp_number *A, mp_number B, int C) {
  mpfr_mul_si(A->data.num,B.data.num, C, ROUNDING);
}
void mp_set_binary_from_of_the_way(MP mp, mp_number *A, mp_number t, mp_number B, mp_number C) {
  mpfr_t c, r1;
  mpfr_init2(c, precision_bits);
  mpfr_init2(r1, precision_bits);
  mpfr_sub (c,B.data.num, C.data.num, ROUNDING);
  mp_binary_take_fraction(mp, r1, c, t.data.num);
  mpfr_sub (A->data.num, B.data.num, r1, ROUNDING);
  mpfr_clear(c);
  mpfr_clear(r1);
  mp_check_mpfr_t(mp, A->data.num);
}
void mp_number_negate(mp_number *A) {
  mpfr_neg (A->data.num, A->data.num, ROUNDING);
  checkZero((mpfr_ptr)A->data.num);
}
void mp_number_add(mp_number *A, mp_number B) {
  mpfr_add (A->data.num,A->data.num,B.data.num, ROUNDING);
}
void mp_number_substract(mp_number *A, mp_number B) {
  mpfr_sub (A->data.num,A->data.num,B.data.num, ROUNDING);
}
void mp_number_half(mp_number *A) {
  mpfr_div_si(A->data.num, A->data.num, 2, ROUNDING);
}
void mp_number_halfp(mp_number *A) {
  mpfr_div_si(A->data.num,A->data.num, 2, ROUNDING);
}
void mp_number_double(mp_number *A) {
  mpfr_mul_si(A->data.num,A->data.num, 2, ROUNDING);
}
void mp_number_add_scaled(mp_number *A, int B) { /* also for negative B */
  mpfr_add_d (A->data.num,A->data.num, B/65536.0, ROUNDING);
}
void mp_number_multiply_int(mp_number *A, int B) {
  mpfr_mul_si(A->data.num,A->data.num, B, ROUNDING);
}
void mp_number_divide_int(mp_number *A, int B) {
  mpfr_div_si(A->data.num,A->data.num, B, ROUNDING);
}
void mp_binary_abs(mp_number *A) {   
  mpfr_abs(A->data.num, A->data.num, ROUNDING);
}
void mp_number_clone(mp_number *A, mp_number B) {
  mpfr_prec_round (A->data.num, precision_bits, ROUNDING);
  mpfr_set(A->data.num, (mpfr_ptr)B.data.num, ROUNDING);
}
void mp_number_swap(mp_number *A, mp_number *B) {
  mpfr_swap(A->data.num, B->data.num);
}
void mp_number_fraction_to_scaled (mp_number *A) {
    A->type = mp_scaled_type;
    mpfr_div (A->data.num, A->data.num, fraction_multiplier_mpfr_t, ROUNDING);
}
void mp_number_angle_to_scaled (mp_number *A) {
    A->type = mp_scaled_type;
    mpfr_div (A->data.num, A->data.num, angle_multiplier_mpfr_t, ROUNDING);
}
void mp_number_scaled_to_fraction (mp_number *A) {
    A->type = mp_fraction_type;
    mpfr_mul (A->data.num, A->data.num, fraction_multiplier_mpfr_t, ROUNDING);
}
void mp_number_scaled_to_angle (mp_number *A) {
    A->type = mp_angle_type;
    mpfr_mul(A->data.num, A->data.num, angle_multiplier_mpfr_t, ROUNDING);
}


@* Query functions.

@ Convert a number to a scaled value. |decNumberToInt32| is not
able to make this conversion properly, so instead we are using
|decNumberToDouble| and a typecast. Bad!

@c
int mp_number_to_scaled(mp_number A) {
  double v = mpfr_get_d (A.data.num, ROUNDING);
  return (int)(v * 65536.0);
}

@ 

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

@c
int mp_number_to_int(mp_number A) {
  int32_t result = 0;
  if (mpfr_fits_sint_p(A.data.num, ROUNDING)) {
    result = mpfr_get_si(A.data.num, ROUNDING);
  }
  return result;
}
int mp_number_to_boolean(mp_number A) {
  int32_t result = 0;
  if (mpfr_fits_sint_p(A.data.num, ROUNDING)) {
    result = mpfr_get_si(A.data.num, ROUNDING);
  }
  return result;
}
double mp_number_to_double(mp_number A) {
  double res = 0.0;
  if (mpfr_number_p (A.data.num)) { 
    res = mpfr_get_d(A.data.num, ROUNDING);
  }
  return res;
}
int mp_number_odd(mp_number A) {
  return odd(mp_number_to_int(A));
}
int mp_number_equal(mp_number A, mp_number B) {
  return mpfr_equal_p(A.data.num,B.data.num);
}
int mp_number_greater(mp_number A, mp_number B) {
  return mpfr_greater_p(A.data.num,B.data.num);
}
int mp_number_less(mp_number A, mp_number B) {
  return mpfr_less_p(A.data.num,B.data.num);
}
int mp_number_nonequalabs(mp_number A, mp_number B) {
  return !(mpfr_cmpabs(A.data.num, B.data.num)==0);
}

@ 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.

@ 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.

@ Here is a procedure analogous to |print_int|.  The current version
is fairly stupid, and it is not round-trip safe, but this is good
enough for a beta test.

@c
char * mp_binnumber_tostring (mpfr_t n) {
  char *str = NULL, *buffer = NULL;
  mpfr_exp_t exp = 0;
  int neg = 0;
  if ((str = mpfr_get_str (NULL, &exp, 10, 0, n, ROUNDING))>0) {
    int numprecdigits = precision_bits_to_digits(precision_bits);
    if (*str == '-') {
      neg = 1;
    }
    while (strlen(str)>0 && *(str+strlen(str)-1) == '0' ) {
      *(str+strlen(str)-1) = '\0'; /* get rid of trailing zeroes */
    }
    buffer = malloc(strlen(str)+13+numprecdigits+1); 
    /* the buffer should also fit at least strlen("E+\%d", exp) or (numprecdigits-2) worth of zeroes, 
     * because with numprecdigits == 33, the str for "1E32" will be "1", and needing 32 extra zeroes,
     * and the decimal dot. To avoid miscalculations by myself, it is safer to add these
     * three together.
     */
    if (buffer) {
      int i = 0, j = 0;
      if (neg) {
        buffer[i++] = '-';
	j = 1;
      }
      if (strlen(str+j) == 0) {
         buffer[i++] = '0';
      } else {
         /* non-zero */
         if (exp<=numprecdigits && exp > -6) {
           if (exp>0) {
             buffer[i++] = str[j++];
             while (--exp>0) {
	        buffer[i++] = (str[j] ? str[j++] : '0');
             }
             if (str[j]) {
	        buffer[i++] = '.';
                while (str[j]) {
   	          buffer[i++] = str[j++];
                }
             }
           } else {
             int absexp;
             buffer[i++] = '0';
             buffer[i++] = '.';
             absexp = -exp;
             while (absexp-- > 0) {
               buffer[i++] = '0';
             }
             while (str[j]) {
               buffer[i++] = str[j++];
             }
           }
         } else {
           buffer[i++] = str[j++];
           if (str[j]) {
              buffer[i++] = '.';
              while (str[j]) {
                buffer[i++] = str[j++];
              }
           }
	   {
	     char msg[256];
             int k = 0;
             mp_snprintf (msg, 256, "%s%d", (exp>0?"+":""), (int)(exp>0 ? (exp-1) : (exp-1)));
             buffer[i++] = 'E';
             while (msg[k]) {
                buffer[i++] = msg[k++];
             }
           }
         }
      }
      buffer[i++] = '\0';
    }
    mpfr_free_str(str);
  }
  return buffer;
}
char * mp_binary_number_tostring (MP mp, mp_number n) {
  return mp_binnumber_tostring(n.data.num);
}


@ @c
void mp_binary_print_number (MP mp, mp_number n) {
  char *str = mp_binary_number_tostring(mp, n);
  mp_print (mp, str);
  free (str);
}




@ 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_binary_slow_add (MP mp, mp_number *ret, mp_number A, mp_number B) {
  mpfr_add(ret->data.num,A.data.num,B.data.num, ROUNDING);
}

@ 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@>

@c
void mp_binary_make_fraction (MP mp, mpfr_t ret, mpfr_t p, mpfr_t q) {
  mpfr_div (ret, p, q, ROUNDING);
  mp_check_mpfr_t(mp, ret);
  mpfr_mul (ret, ret, fraction_multiplier_mpfr_t, ROUNDING);
}
void mp_binary_number_make_fraction (MP mp, mp_number *ret, mp_number p, mp_number q) {
  mp_binary_make_fraction (mp, ret->data.num, p.data.num, q.data.num);
}

@ @<Declarations@>=
void mp_binary_make_fraction (MP mp, mpfr_t ret, mpfr_t p, mpfr_t q);

@ 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@>

@c
void mp_binary_take_fraction (MP mp, mpfr_t ret, mpfr_t p, mpfr_t q) {
  mpfr_mul(ret, p, q, ROUNDING);
  mpfr_div(ret, ret, fraction_multiplier_mpfr_t, ROUNDING);
}
void mp_binary_number_take_fraction (MP mp, mp_number *ret, mp_number p, mp_number q) {
  mp_binary_take_fraction (mp, ret->data.num, p.data.num, q.data.num);
}

@ @<Declarations@>=
void mp_binary_take_fraction (MP mp, mpfr_t ret, mpfr_t p, mpfr_t q);

@ 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@>

@c
void mp_binary_number_take_scaled (MP mp, mp_number *ret, mp_number p_orig, mp_number q_orig) {
  mpfr_mul(ret->data.num, p_orig.data.num, q_orig.data.num, ROUNDING);
}


@ 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.)

@c
void mp_binary_number_make_scaled (MP mp, mp_number *ret, mp_number p_orig, mp_number q_orig) {
  mpfr_div(ret->data.num, p_orig.data.num, q_orig.data.num, ROUNDING);
  mp_check_mpfr_t(mp, ret->data.num);
}

@ 
@d halfp(A) (integer)((unsigned)(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) mpfr_set((mpfr_ptr)(mp->cur_mod_->data.n.data.num),A, ROUNDING)

@<Declarations...@>=
static void mp_wrapup_numeric_token(MP mp, unsigned char *start, unsigned char *stop);

@ The check of the precision is based on the article "27 Bits are not enough for 8-Digit accuracy" 
@ by Bennet Goldberg  which roughly says that
@ given $p$ digits in base 10 and $q$ digits in base 2, 
@ conversion from base 10 round-trip through base 2 if and only if $10^p < 2^{q-1}$.
@ In our case  $p/\log_{10}2 + 1 < q$, or $q\geq a$
@ where $q$ is the current precision in bits and $a=\left\lceil p/\log_{10}2 + 1\right\rceil$. 
@ Therefore if $a>q$ the required precision could be too high and we emit a warning.
@d too_precise(a) (a>precision_bits)
@c
void mp_wrapup_numeric_token(MP mp, unsigned char *start, unsigned char *stop) {
  int invalid = 0;
  mpfr_t result;
  size_t l = stop-start+1;
  unsigned long lp, lpbit;
  char *buf = mp_xmalloc(mp, l+1, 1);
  char *bufp = buf; 
  buf[l] = '\0';
  mpfr_init2(result, precision_bits);
  (void)strncpy(buf,(const char *)start, l);
  invalid = mpfr_set_str(result,buf, 10, ROUNDING);
  /*|fprintf(stdout,"scan of [%s] produced %s, ", buf, mp_binnumber_tostring(result));|*/
  lp = (unsigned long) l;
  /* strip leading - or + or 0 or .*/
  if ( (*bufp=='-') || (*bufp=='+') || (*bufp=='0') || (*bufp=='.') ) { lp--; bufp++;}
  /* strip also . */
  lp = strchr(bufp,'.') ? lp-1: lp;
  /* strip also trailing 0s */ 
  bufp = buf+l-1;
  while(*bufp == '0') {bufp--; lp=( ((lp==0)||(lp==1))?1:lp-1);}
  /* at least one digit, even if the number is  0 */
  lp = lp>0? lp: 1;
  /* bits needed for buf */
  lpbit = (unsigned long)ceil(lp/log10(2)+1);
  free(buf);
  bufp = NULL;
  if (invalid == 0) {
    set_cur_mod(result);
   /* |fprintf(stdout,"mod=%s\n", mp_binary_number_tostring(mp,mp->cur_mod_->data.n));|*/
    if (too_precise(lpbit)) {
       if (mpfr_positive_p((mpfr_ptr)(internal_value (mp_warning_check).data.num)) &&
          (mp->scanner_status != tex_flushing)) {
        char msg[256];
        const char *hlp[] = {"Continue and I'll try to cope",
               "with that value; but it might be dangerous.",
               "(Set warningcheck:=0 to suppress this message.)",
               NULL };
        mp_snprintf (msg, 256, "Required precision is too high (%d vs. numberprecision = %f, required precision=%d bits vs internal precision=%f bits)", (unsigned int)lp,mpfr_get_d(internal_value (mp_number_precision).data.num, ROUNDING),(int)lpbit,precision_bits);
@.Number is too large@>;
        mp_error (mp, msg, hlp, true);
      }
    }
  } else if (mp->scanner_status != tex_flushing) {
    const char *hlp[] = {"I could not handle this number specification",
                         "probably because it is out of range. Error:",
                         "",
                          NULL };   
    hlp[2] = strerror(errno);
    mp_error (mp, "Enormous number has been reduced.", hlp, false);
@.Enormous number...@>;
    set_cur_mod((mpfr_ptr)(((math_data *)(mp->math))->inf_t.data.num));
  }
  set_cur_cmd((mp_variable_type)mp_numeric_token);
  mpfr_clear(result);
}

@ @c
static void find_exponent (MP mp)  {
  if (mp->buffer[mp->cur_input.loc_field] == 'e' || 
      mp->buffer[mp->cur_input.loc_field] == 'E') {
     mp->cur_input.loc_field++;
     if (!(mp->buffer[mp->cur_input.loc_field] == '+' || 
        mp->buffer[mp->cur_input.loc_field] == '-' ||
	mp->char_class[mp->buffer[mp->cur_input.loc_field]] == digit_class)) {
       mp->cur_input.loc_field--;
       return;
     }     
     if (mp->buffer[mp->cur_input.loc_field] == '+' || 
        mp->buffer[mp->cur_input.loc_field] == '-') {
        mp->cur_input.loc_field++;
     }
     while (mp->char_class[mp->buffer[mp->cur_input.loc_field]] == digit_class) {
       mp->cur_input.loc_field++;
     }
  }
}
void mp_binary_scan_fractional_token (MP mp, int n) { /* n: scaled */
  unsigned char *start = &mp->buffer[mp->cur_input.loc_field -1];
  unsigned char *stop;
  while (mp->char_class[mp->buffer[mp->cur_input.loc_field]] == digit_class) {
     mp->cur_input.loc_field++;
  }
  find_exponent(mp);
  stop = &mp->buffer[mp->cur_input.loc_field-1];
  mp_wrapup_numeric_token (mp, start, stop);
}


@ We just have to collect bytes.

@c
void mp_binary_scan_numeric_token (MP mp, int n) { /* n: scaled */
  unsigned char *start = &mp->buffer[mp->cur_input.loc_field -1];
  unsigned char *stop;
  while (mp->char_class[mp->buffer[mp->cur_input.loc_field]] == digit_class) {
     mp->cur_input.loc_field++;
  }
  if (mp->buffer[mp->cur_input.loc_field] == '.' && 
      mp->buffer[mp->cur_input.loc_field+1] != '.') {
     mp->cur_input.loc_field++;
     while (mp->char_class[mp->buffer[mp->cur_input.loc_field]] == digit_class) {
       mp->cur_input.loc_field++;
     }
  } 
  find_exponent(mp);
  stop = &mp->buffer[mp->cur_input.loc_field-1];
  mp_wrapup_numeric_token (mp, start, stop);
}

@ 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 (fraction_multiplier/2)
@d fraction_one (1*fraction_multiplier)
@d fraction_two (2*fraction_multiplier)
@d fraction_three (3*fraction_multiplier)
@d fraction_four (4*fraction_multiplier)

@ 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.  

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_binary_velocity (MP mp, mp_number *ret, mp_number st, mp_number ct, mp_number sf,
                  mp_number cf, mp_number t) {
  mpfr_t acc, num, denom;      /* registers for intermediate calculations */
  mpfr_t r1, r2;
  mpfr_t arg1, arg2;
  mpfr_t i16, fone, fhalf, ftwo, sqrtfive;
  mpfr_inits2 (precision_bits, acc, num, denom, r1, r2, arg1, arg2, i16, fone, fhalf, ftwo, sqrtfive, (mpfr_ptr)0);
  mpfr_set_si(i16, 16, ROUNDING);
  mpfr_set_si(fone, fraction_one, ROUNDING);
  mpfr_set_si(fhalf, fraction_half, ROUNDING);
  mpfr_set_si(ftwo, fraction_two, ROUNDING);
  mpfr_set_si(sqrtfive, 5, ROUNDING);
  mpfr_sqrt (sqrtfive, sqrtfive, ROUNDING);          
  mpfr_div (arg1,sf.data.num, i16, ROUNDING); // arg1 = sf / 16
  mpfr_sub (arg1,st.data.num, arg1, ROUNDING); // arg1 = st - arg1
  mpfr_div (arg2,st.data.num, i16, ROUNDING); // arg2 = st / 16
  mpfr_sub (arg2,sf.data.num, arg2, ROUNDING); // arg2 = sf - arg2
  mp_binary_take_fraction (mp, acc, arg1, arg2); // acc = (arg1 * arg2) / fmul

  mpfr_set (arg1, acc, ROUNDING);
  mpfr_sub (arg2, ct.data.num, cf.data.num, ROUNDING); // arg2 = ct - cf
  mp_binary_take_fraction (mp, acc, arg1, arg2); // acc = (arg1 * arg2 ) / fmul

  mpfr_sqrt(arg1, two_mpfr_t, ROUNDING); // arg1 = sqrt(2)
  mpfr_mul(arg1, arg1, fone, ROUNDING);     // arg1 = arg1 * fmul
  mp_binary_take_fraction (mp, r1, acc, arg1);  // r1 = (acc * arg1) / fmul
  mpfr_add(num, ftwo, r1, ROUNDING);             // num = ftwo + r1
  
  mpfr_sub(arg1,sqrtfive, one, ROUNDING);   // arg1 = sqrt(5) - 1
  mpfr_mul(arg1,arg1,fhalf, ROUNDING);      // arg1 = arg1 * fmul/2
  mpfr_mul(arg1,arg1,three_mpfr_t, ROUNDING); // arg1 = arg1 * 3

  mpfr_sub(arg2,three_mpfr_t, sqrtfive, ROUNDING); // arg2 = 3 - sqrt(5)
  mpfr_mul(arg2,arg2,fhalf, ROUNDING);            // arg2 = arg2 * fmul/2
  mpfr_mul(arg2,arg2,three_mpfr_t, ROUNDING);  // arg2 = arg2 * 3
  mp_binary_take_fraction (mp, r1, ct.data.num, arg1) ; // r1 = (ct * arg1) / fmul
  mp_binary_take_fraction (mp, r2, cf.data.num, arg2);  // r2 = (cf * arg2) / fmul

  mpfr_set_si(denom, fraction_three, ROUNDING);  // denom = 3fmul
  mpfr_add(denom, denom, r1, ROUNDING);     // denom = denom + r1
  mpfr_add(denom, denom, r2, ROUNDING);     // denom = denom + r2

  if (!mpfr_equal_p(t.data.num, one)) {                 // t != 1
    mpfr_div(num, num, t.data.num, ROUNDING); // num = num / t
  }
  mpfr_set(r2, num, ROUNDING);                        // r2 = num / 4
  mpfr_div(r2, r2, four_mpfr_t, ROUNDING);
  if (mpfr_less_p(denom,r2)) { // num/4 >= denom => denom < num/4
    mpfr_set_si(ret->data.num,fraction_four, ROUNDING);
  } else {
    mp_binary_make_fraction (mp, ret->data.num, num, denom);
  }
  mpfr_clears (acc, num, denom, r1, r2, arg1, arg2, i16, fone, fhalf, ftwo, sqrtfive, (mpfr_ptr)0);
  mp_check_mpfr_t(mp, ret->data.num);
}


@ 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
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) {
  mpfr_t q, r, test; /* temporary registers */
  mpfr_t a, b, c, d;
  int cmp = 0;
  (void)mp;
  mpfr_inits2(precision_bits, q,r,test,a,b,c,d,(mpfr_ptr)0);
  mpfr_set(a, (mpfr_ptr)a_orig.data.num, ROUNDING);
  mpfr_set(b, (mpfr_ptr )b_orig.data.num, ROUNDING);
  mpfr_set(c, (mpfr_ptr )c_orig.data.num, ROUNDING);
  mpfr_set(d, (mpfr_ptr )d_orig.data.num, ROUNDING);
  
  mpfr_mul(q,a,b,ROUNDING);
  mpfr_mul(r,c,d,ROUNDING);
  cmp = mpfr_cmp(q,r);
  if (cmp==0) {
    mpfr_set(ret->data.num, zero, ROUNDING);
    goto RETURN;
  }
  if (cmp>0) {
    mpfr_set(ret->data.num, one, ROUNDING);
    goto RETURN;
  }
  if (cmp<0) {
    mpfr_set(ret->data.num, minusone, ROUNDING);
    goto RETURN;
  }

  /*TODO: remove this part of the code until RETURN */
  @<Reduce to the case that |a,c>=0|, |b,d>0|@>;
  while (1) {
    mpfr_div(q,a,d, ROUNDING);
    mpfr_div(r,c,b, ROUNDING);
    cmp = mpfr_cmp(q,r);
    if (cmp) {
      if (cmp>1) {
         mpfr_set(ret->data.num, one, ROUNDING);
      } else {
         mpfr_set(ret->data.num, minusone, ROUNDING);
      }
      goto RETURN;
    }
    mpfr_remainder(q,a,d, ROUNDING);
    mpfr_remainder(r,c,b, ROUNDING);
    if (mpfr_zero_p(r)) {
      if (mpfr_zero_p(q)) {
         mpfr_set(ret->data.num, zero, ROUNDING);
      } else {
         mpfr_set(ret->data.num, one, ROUNDING);
      }
      goto RETURN;
    }
    if (mpfr_zero_p(q)) {
      mpfr_set(ret->data.num, minusone, ROUNDING);
      goto RETURN;
    }
    mpfr_set(a,b, ROUNDING);
    mpfr_set(b,q, ROUNDING);
    mpfr_set(c,d, ROUNDING);
    mpfr_set(d,r, ROUNDING);
  }                             /* now |a>d>0| and |c>b>0| */
RETURN:
#if DEBUG
  fprintf(stdout, "\n%f = ab_vs_cd(%f,%f,%f,%f)", mp_number_to_double(*ret), 
mp_number_to_double(a_orig),mp_number_to_double(b_orig),
mp_number_to_double(c_orig),mp_number_to_double(d_orig));
#endif
  mp_check_mpfr_t(mp, ret->data.num);
  mpfr_clears(q,r,test,a,b,c,d,(mpfr_ptr)0);
  return;
}


@ @<Reduce to the case that |a...@>=
if (mpfr_negative_p(a)) {
  mpfr_neg(a, a, ROUNDING);
  mpfr_neg(b, b, ROUNDING);
}
if (mpfr_negative_p(c)) {
  mpfr_neg(c, c, ROUNDING);
  mpfr_neg(d, d, ROUNDING);
}
if (!mpfr_positive_p(d)) {
  if (!mpfr_negative_p(b)) {
    if ((mpfr_zero_p(a) || mpfr_zero_p(b)) && (mpfr_zero_p(c) || mpfr_zero_p(d))) 
         mpfr_set(ret->data.num, zero, ROUNDING);
    else
         mpfr_set(ret->data.num, one, ROUNDING);
    goto RETURN;
  }
  if (mpfr_zero_p(d)) {
    if (mpfr_zero_p(a)) 
         mpfr_set(ret->data.num, zero, ROUNDING);
    else
         mpfr_set(ret->data.num, minusone, ROUNDING);
    goto RETURN;
  }
  mpfr_set(q, a, ROUNDING);
  mpfr_set(a, c, ROUNDING);
  mpfr_set(c, q, ROUNDING);
  mpfr_neg(q, b, ROUNDING);
  mpfr_neg(b, d, ROUNDING);
  mpfr_set(d, q, ROUNDING);
} else if (!mpfr_positive_p(b)) {
  if (mpfr_negative_p(b) && mpfr_positive_p(a)) {
    mpfr_set(ret->data.num, minusone, ROUNDING);
    goto RETURN;
  }
  if (mpfr_zero_p(c)) 
    mpfr_set(ret->data.num, zero, ROUNDING);
  else
    mpfr_set(ret->data.num, minusone, ROUNDING);
  goto 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   { mpfr_set(ret->data.num, fraction_one_plus_mpfr_t, ROUNDING); goto RETURN; }
@d one_crossing  { mpfr_set(ret->data.num, fraction_one_mpfr_t, ROUNDING); goto RETURN; }
@d zero_crossing { mpfr_set(ret->data.num, zero, ROUNDING); goto RETURN; }

@c
static void mp_binary_crossing_point (MP mp, mp_number *ret, mp_number aa, mp_number bb, mp_number cc) {
  mpfr_t a,b,c;
  double d;    /* recursive counter */
  mpfr_t x, xx, x0, x1, x2;    /* temporary registers for bisection */
  mpfr_t scratch;
  mpfr_inits2 (precision_bits, a,b,c, x,xx,x0,x1,x2, scratch,(mpfr_ptr)0);
  mpfr_set(a, (mpfr_ptr )aa.data.num, ROUNDING);
  mpfr_set(b, (mpfr_ptr )bb.data.num, ROUNDING);
  mpfr_set(c, (mpfr_ptr )cc.data.num, ROUNDING);
  if (mpfr_negative_p(a))
    zero_crossing;
  if (!mpfr_negative_p(c)) {
    if (!mpfr_negative_p(b)) {
      if (mpfr_positive_p(c)) {
        no_crossing;
      } else if (mpfr_zero_p(a) && mpfr_zero_p(b)) {
        no_crossing;
      } else {
        one_crossing;
      }
    }
    if (mpfr_zero_p(a))
      zero_crossing;
  } else if (mpfr_zero_p(a)) {
    if (!mpfr_positive_p(b))
      zero_crossing;
  }

  /* Use bisection to find the crossing point... */
  d = epsilonf;
  mpfr_set(x0, a, ROUNDING);
  mpfr_sub(x1,a, b, ROUNDING);
  mpfr_sub(x2,b, c, ROUNDING);
  do {
    /* not sure why the error correction has to be >= 1E-12 */
    mpfr_add(x, x1, x2, ROUNDING);
    mpfr_div(x, x, two_mpfr_t, ROUNDING);
    mpfr_add_d (x, x, 1E-12, ROUNDING);
    mpfr_sub(scratch, x1, x0, ROUNDING);
    if (mpfr_greater_p(scratch, x0)) {
      mpfr_set(x2, x, ROUNDING);
      mpfr_add(x0, x0, x0, ROUNDING);
      d += d;
    } else {
      mpfr_add(xx, scratch, x, ROUNDING);
      if (mpfr_greater_p(xx,x0)) {
        mpfr_set(x2,x, ROUNDING);
        mpfr_add(x0, x0, x0, ROUNDING);
        d += d;
      } else {
        mpfr_sub(x0, x0, xx, ROUNDING);
        if (!mpfr_greater_p(x,x0)) {
          mpfr_add(scratch, x, x2, ROUNDING);
          if (!mpfr_greater_p(scratch, x0))
            no_crossing;
        }
        mpfr_set(x1,x, ROUNDING);
        d = d + d + epsilonf;
      }
    }
  } while (d < fraction_one);
  mpfr_set_d(scratch, d, ROUNDING);
  mpfr_sub(ret->data.num,scratch, fraction_one_mpfr_t, ROUNDING);
RETURN:
#if DEBUG
  fprintf(stdout, "\n%f = crossing_point(%f,%f,%f)", mp_number_to_double(*ret), 
mp_number_to_double(aa),mp_number_to_double(bb),mp_number_to_double(cc));
#endif
  mpfr_clears (a,b,c, x,xx,x0,x1,x2, scratch, (mpfr_ptr)0);
  mp_check_mpfr_t(mp, ret->data.num);
  return;
}
 

@ 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) {
  double xx = mp_number_to_double(x_orig);
  int x = (int)ROUND(xx);
  return x;
}

@ |number_floor| floors a number

@c
void mp_number_floor (mp_number *i) {
  mpfr_rint_floor(i->data.num, i->data.num, MPFR_RNDD);
}

@ |fraction_to_scaled| rounds a |fraction| and converts it to |scaled|
@c
void mp_binary_fraction_to_round_scaled (mp_number *x_orig) {
  x_orig->type = mp_scaled_type;
  mpfr_div(x_orig->data.num, x_orig->data.num, fraction_multiplier_mpfr_t, ROUNDING);
}



@* 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.

@ 

@c
void mp_binary_square_rt (MP mp, mp_number *ret, mp_number x_orig) { /* return, x: scaled */
  if (!mpfr_positive_p((mpfr_ptr)x_orig.data.num)) {
    @<Handle square root of zero or negative argument@>;
  } else {
    mpfr_sqrt(ret->data.num, x_orig.data.num, ROUNDING);
  }
  mp_check_mpfr_t(mp, ret->data.num);
}


@ @<Handle square root of zero...@>=
{  
  if (mpfr_negative_p((mpfr_ptr)x_orig.data.num)) {
    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 *xstr = mp_binary_number_tostring (mp, x_orig);
    mp_snprintf(msg, 256, "Square root of %s has been replaced by 0", xstr);
    free(xstr);
@.Square root...replaced by 0@>;
    mp_error (mp, msg, hlp, true);
  }
  mpfr_set_zero(ret->data.num,1); /* 1 == positive */
  return;
}


@ Pythagorean addition $\psqrt{a^2+b^2}$ is implemented by a quick hack

@c
void mp_binary_pyth_add (MP mp, mp_number *ret, mp_number a_orig, mp_number b_orig) {
  mpfr_t a, b, asq, bsq;
  mpfr_inits2(precision_bits, a,b, asq, bsq, (mpfr_ptr)0);  
  mpfr_set(a, (mpfr_ptr)a_orig.data.num, ROUNDING);
  mpfr_set(b, (mpfr_ptr)b_orig.data.num, ROUNDING);
  mpfr_mul(asq, a, a, ROUNDING);
  mpfr_mul(bsq, b, b, ROUNDING);
  mpfr_add(a, asq, bsq, ROUNDING);
  mpfr_sqrt(ret->data.num, a, ROUNDING);
  mp_check_mpfr_t(mp, ret->data.num);
  mpfr_clears(a,b, asq, bsq, (mpfr_ptr)0);  
}

@ Here is a similar algorithm for $\psqrt{a^2-b^2}$. Same quick hack, also.

@c
void mp_binary_pyth_sub (MP mp, mp_number *ret, mp_number a_orig, mp_number b_orig) {
  mpfr_t a, b, asq, bsq;
  mpfr_inits2(precision_bits, a,b, asq, bsq, (mpfr_ptr)0);  
  mpfr_set(a, (mpfr_ptr)a_orig.data.num, ROUNDING);
  mpfr_set(b, (mpfr_ptr)b_orig.data.num, ROUNDING);
  if (!mpfr_greater_p(a,b)) {
    @<Handle erroneous |pyth_sub| and set |a:=0|@>;
  } else {
    mpfr_mul(asq, a, a, ROUNDING);
    mpfr_mul(bsq, b, b, ROUNDING);
    mpfr_sub(a, asq, bsq, ROUNDING);
    mpfr_sqrt(a, a, ROUNDING);
  }
  mpfr_set(ret->data.num, a, ROUNDING);
  mp_check_mpfr_t(mp, ret->data.num);
}


@ @<Handle erroneous |pyth_sub| and set |a:=0|@>=
{
  if (mpfr_less_p(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 = mp_binary_number_tostring (mp, a_orig);
    char *bstr = mp_binary_number_tostring (mp, b_orig);
    mp_snprintf (msg, 256, "Pythagorean subtraction %s+-+%s has been replaced by 0", astr, bstr);
    free(astr);
    free(bstr);
@.Pythagorean...@>;
    mp_error (mp, msg, hlp, true);
  }
  mpfr_set_zero(a,1); /* 1 == positive */
}


@ Here is the routine that calculates $2^8$ times the natural logarithm
of a |scaled| quantity; 

@c
void mp_binary_m_log (MP mp, mp_number *ret, mp_number x_orig) {
  if (!mpfr_positive_p((mpfr_ptr)x_orig.data.num)) {
    @<Handle non-positive logarithm@>;
  } else {
    mpfr_log(ret->data.num, x_orig.data.num, ROUNDING);
    mp_check_mpfr_t(mp, ret->data.num);
    mpfr_mul_si(ret->data.num, ret->data.num, 256, ROUNDING);
  }
  mp_check_mpfr_t(mp, ret->data.num);
}

@ @<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 };
  char *xstr = mp_binary_number_tostring (mp, x_orig);
  mp_snprintf (msg, 256, "Logarithm of %s has been replaced by 0", xstr);
  free (xstr);
@.Logarithm...replaced by 0@>;
  mp_error (mp, msg, hlp, true);
  mpfr_set_zero(ret->data.num,1); /* 1 == positive */
}


@ Conversely, the exponential routine calculates $\exp(x/2^8)$,
when |x| is |scaled|. 

@c
void mp_binary_m_exp (MP mp, mp_number *ret, mp_number x_orig) {
  mpfr_t temp;
  mpfr_init2(temp, precision_bits);
  mpfr_div_si(temp, x_orig.data.num, 256, ROUNDING);
  mpfr_exp(ret->data.num, temp, ROUNDING);
  mp_check_mpfr_t(mp, ret->data.num);
  mpfr_clear (temp);
}


@ Given integers |x| and |y|, not both zero, the |n_arg| function
returns the |angle| whose tangent points in the direction $(x,y)$.

@c
void mp_binary_n_arg (MP mp, mp_number *ret, mp_number x_orig, mp_number y_orig) {
  if (mpfr_zero_p((mpfr_ptr )x_orig.data.num) && mpfr_zero_p((mpfr_ptr )y_orig.data.num)) {
    @<Handle undefined arg@>;
  } else {
    mpfr_t atan2val, oneeighty_angle;
    mpfr_init2(atan2val, precision_bits);
    mpfr_init2(oneeighty_angle, precision_bits);
    ret->type = mp_angle_type;
    mpfr_set_si(oneeighty_angle, 180 * angle_multiplier, ROUNDING);
    mpfr_div(oneeighty_angle, oneeighty_angle, PI_mpfr_t, ROUNDING);
    checkZero((mpfr_ptr)y_orig.data.num);
    checkZero((mpfr_ptr)x_orig.data.num);
    mpfr_atan2(atan2val, y_orig.data.num, x_orig.data.num, ROUNDING);
    mpfr_mul(ret->data.num, atan2val, oneeighty_angle, ROUNDING);
    checkZero((mpfr_ptr)ret->data.num);
    mpfr_clear(atan2val);
    mpfr_clear(oneeighty_angle);
  }
  mp_check_mpfr_t(mp, ret->data.num);
}


@ @<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@>;
  mpfr_set_zero(ret->data.num,1); /* 1 == positive */
}


@ 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|.

@ Calculate sines and cosines.

@c
void mp_binary_sin_cos (MP mp, mp_number z_orig, mp_number *n_cos, mp_number *n_sin) {
  mpfr_t rad;
  mpfr_t one_eighty;
  mpfr_init2(rad, precision_bits);
  mpfr_init2(one_eighty, precision_bits);
  mpfr_set_si(one_eighty, 180 * 16, ROUNDING);
  mpfr_mul (rad, z_orig.data.num, PI_mpfr_t, ROUNDING);
  mpfr_div (rad, rad, one_eighty, ROUNDING);

  mpfr_sin (n_sin->data.num, rad, ROUNDING);
  mpfr_cos (n_cos->data.num, rad, ROUNDING);

  mpfr_mul (n_cos->data.num,n_cos->data.num, fraction_multiplier_mpfr_t, ROUNDING);
  mpfr_mul (n_sin->data.num,n_sin->data.num, fraction_multiplier_mpfr_t, ROUNDING);
  mp_check_mpfr_t(mp, n_cos->data.num);
  mp_check_mpfr_t(mp, n_sin->data.num);
  mpfr_clear (rad);
  mpfr_clear (one_eighty);
}

@ This is the http://www-cs-faculty.stanford.edu/~uno/programs/rng.c
with  small cosmetic modifications.

@c
#define KK 100                     /* the long lag  */
#define LL  37                     /* the short lag */
#define MM (1L<<30)                /* the modulus   */
#define mod_diff(x,y) (((x)-(y))&(MM-1)) /* subtraction mod MM */
/* */ 
static long ran_x[KK];                    /* the generator state */
/* */ 
static void ran_array(long aa[],int n) /* put n new random numbers in aa */
  /* long aa[]    destination */
  /* int n       array length (must be at least KK) */
{
  register int i,j;
  for (j=0;j<KK;j++) aa[j]=ran_x[j];
  for (;j<n;j++) aa[j]=mod_diff(aa[j-KK],aa[j-LL]);
  for (i=0;i<LL;i++,j++) ran_x[i]=mod_diff(aa[j-KK],aa[j-LL]);
  for (;i<KK;i++,j++) ran_x[i]=mod_diff(aa[j-KK],ran_x[i-LL]);
}
/* */ 
/* the following routines are from exercise 3.6--15 */
/* after calling |ran_start|, get new randoms by, e.g., |x=ran_arr_next()| */
/* */ 
#define QUALITY 1009 /* recommended quality level for high-res use */
static long ran_arr_buf[QUALITY];
static long ran_arr_dummy=-1, ran_arr_started=-1;
static long *ran_arr_ptr=&ran_arr_dummy; /* the next random number, or -1 */
/* */ 
#define TT  70   /* guaranteed separation between streams */
#define is_odd(x)  ((x)&1)          /* units bit of x */
/* */ 
static void ran_start(long seed) /* do this before using |ran_array| */
  /* long seed             selector for different streams */
{
  register int t,j;
  long x[KK+KK-1];              /* the preparation buffer */
  register long ss=(seed+2)&(MM-2);
  for (j=0;j<KK;j++) {
    x[j]=ss;                      /* bootstrap the buffer */
    ss<<=1; if (ss>=MM) ss-=MM-2; /* cyclic shift 29 bits */
  }
  x[1]++;              /* make x[1] (and only x[1]) odd */
  for (ss=seed&(MM-1),t=TT-1; t; ) {       
    for (j=KK-1;j>0;j--) x[j+j]=x[j], x[j+j-1]=0; /* "square" */
    for (j=KK+KK-2;j>=KK;j--)
      x[j-(KK-LL)]=mod_diff(x[j-(KK-LL)],x[j]),
      x[j-KK]=mod_diff(x[j-KK],x[j]);
    if (is_odd(ss)) {              /* "multiply by z" */
      for (j=KK;j>0;j--)  x[j]=x[j-1];
      x[0]=x[KK];            /* shift the buffer cyclically */
      x[LL]=mod_diff(x[LL],x[KK]);
    }
    if (ss) ss>>=1; else t--;
  }
  for (j=0;j<LL;j++) ran_x[j+KK-LL]=x[j];
  for (;j<KK;j++) ran_x[j-LL]=x[j];
  for (j=0;j<10;j++) ran_array(x,KK+KK-1); /* warm things up */
  ran_arr_ptr=&ran_arr_started;
}
/* */ 
#define ran_arr_next() (*ran_arr_ptr>=0? *ran_arr_ptr++: ran_arr_cycle())
static long ran_arr_cycle(void)
{
  if (ran_arr_ptr==&ran_arr_dummy)
    ran_start(314159L); /* the user forgot to initialize */
  ran_array(ran_arr_buf,QUALITY);
  ran_arr_buf[KK]=-1;
  ran_arr_ptr=ran_arr_buf+1;
  return ran_arr_buf[0];
}




@ 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;
    mpfr_set_si(mp->randoms[(i * 21) % 55].data.num, j, ROUNDING);
  }
  mp_new_randoms (mp);
  mp_new_randoms (mp);
  mp_new_randoms (mp);          /* ``warm up'' the array */
  
  ran_start ((unsigned long)seed);  

}

@ @c
void mp_binary_number_modulo (mp_number *a, mp_number b) {
   mpfr_remainder (a->data.num, a->data.num, b.data.num, ROUNDING);
}

@ To consume a random  integer for the uniform generator, the program below will say `|next_unif_random|'.

@c 
static void mp_next_unif_random (MP mp, mp_number *ret) { 
  mp_number rop;
  unsigned long int op;
  float flt_op ;  
  (void)mp;
  mp_new_number (mp, &rop, mp_scaled_type);
  op = (unsigned)ran_arr_next();
  flt_op = op/(MM*1.0);
  mpfr_set_d ((mpfr_ptr)(rop.data.num), flt_op,ROUNDING);
  mp_number_clone (ret, rop);
  free_number (rop);
}



@ 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_binary_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;
  char *r ;mpfr_exp_t e;
  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_binary_abs (&abs_x);
  mp_next_unif_random(mp, &u);
  mpfr_mul (y.data.num, abs_x.data.num, u.data.num, ROUNDING);
  free_number (u);
  if (mp_number_equal(y, abs_x)) {
    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);
  }
  r = mpfr_get_str(NULL,    /* |char *str|,         */     
                  &e,       /* |mpfr_exp_t *expptr|,*/
                  10,       /* |int b|,             */
                  0,        /* |size_t n|,          */
                  ret->data.num, /* |mpfr_t op|,    */      
                  ROUNDING       /* |mpfr_rnd_t rnd|*/
                  );
  mpfr_free_str(r);
  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_binary_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_binary_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_binary_abs (&abs_x);
    } while (!mp_number_less(abs_x, u));
    mp_binary_number_make_fraction (mp, &r, xa, u);
    mp_number_clone (&xa, r);
    mp_binary_m_log (mp,&la, u);
    mp_set_binary_from_substraction(&la, ((math_data *)mp->math)->twelve_ln_2_k, la);
    mp_binary_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);
}



@ The following subroutine is used only in |norm_rand| and tests  if $ab$ is
greater than, equal to, or less than~$cd$.
The result is $+1$, 0, or~$-1$ in the three respective cases.

@c
static void mp_binary_ab_vs_cd (MP mp, mp_number *ret, mp_number a_orig, mp_number b_orig, mp_number c_orig, mp_number d_orig) {
  mpfr_t a, b, c, d;
  mpfr_t ab, cd;

  int cmp = 0;
  (void)mp;
  mpfr_inits2(precision_bits, a,b,c,d,ab,cd,(mpfr_ptr)0);
  mpfr_set(a, (mpfr_ptr )a_orig.data.num, ROUNDING);
  mpfr_set(b, (mpfr_ptr )b_orig.data.num, ROUNDING);
  mpfr_set(c, (mpfr_ptr )c_orig.data.num, ROUNDING);
  mpfr_set(d, (mpfr_ptr )d_orig.data.num, ROUNDING);

  mpfr_mul(ab,a,b, ROUNDING);
  mpfr_mul(cd,c,d, ROUNDING);
 
  mpfr_set(ret->data.num, zero, ROUNDING);
  cmp = mpfr_cmp(ab,cd);
  if (cmp) {
   if (cmp>0) 
     mpfr_set(ret->data.num, one, ROUNDING);
   else 
     mpfr_set(ret->data.num, minusone, ROUNDING);
  }
  mp_check_mpfr_t(mp, ret->data.num);
  mpfr_clears(a,b,c,d,ab,cd,(mpfr_ptr)0);
  return;
}