scheme48/c/bignum.c

/* -*-C-*-

$Id: bignum.c,v 9.41 1994/02/09 00:53:27 cph Exp $

Copyright (c) 1989-94 Massachusetts Institute of Technology

This material was developed by the Scheme project at the Massachusetts
Institute of Technology, Department of Electrical Engineering and
Computer Science.  Permission to copy and modify this software, to
redistribute either the original software or a modified version, and
to use this software for any purpose is granted, subject to the
following restrictions and understandings.

1. Any copy made of this software must include this copyright notice
in full.

2. Users of this software agree to make their best efforts (a) to
return to the MIT Scheme project any improvements or extensions that
they make, so that these may be included in future releases; and (b)
to inform MIT of noteworthy uses of this software.

3. All materials developed as a consequence of the use of this
software shall duly acknowledge such use, in accordance with the usual
standards of acknowledging credit in academic research.

4. MIT has made no warrantee or representation that the operation of
this software will be error-free, and MIT is under no obligation to
provide any services, by way of maintenance, update, or otherwise.

5. In conjunction with products arising from the use of this material,
there shall be no use of the name of the Massachusetts Institute of
Technology nor of any adaptation thereof in any advertising,
promotional, or sales literature without prior written consent from
MIT in each case. */

/* Changes for Scheme 48:
 *  - Converted to ANSI.
 *  - Added bitwise operations.
 *  - Added s48_ to the beginning of all externally visible names.
 *  - Cached the bignum representations of -1, 0, and 1.
 */

#include "bignum.h"
#include "bignumint.h"
#include <limits.h>
#include "scheme48.h"	/* for S48_GC_PROTECT_GLOBAL */
#include <stdio.h>
#include <stdlib.h>	/* abort */
#include <math.h>

/* Forward references */
static int bignum_equal_p_unsigned(bignum_type, bignum_type);
static enum bignum_comparison bignum_compare_unsigned(bignum_type, bignum_type);
static bignum_type bignum_add_unsigned(bignum_type, bignum_type, int);
static bignum_type bignum_subtract_unsigned(bignum_type, bignum_type);
static bignum_type bignum_multiply_unsigned(bignum_type, bignum_type, int);
static bignum_type bignum_multiply_unsigned_small_factor
  (bignum_type, bignum_digit_type, int);
static void bignum_destructive_scale_up(bignum_type, bignum_digit_type);
static void bignum_destructive_add(bignum_type, bignum_digit_type);
static void bignum_divide_unsigned_large_denominator
  (bignum_type, bignum_type, bignum_type *, bignum_type *, int, int);
static void bignum_destructive_normalization(bignum_type, bignum_type, int);
static void bignum_destructive_unnormalization(bignum_type, int);
static void bignum_divide_unsigned_normalized(bignum_type, bignum_type, bignum_type);
static bignum_digit_type bignum_divide_subtract
  (bignum_digit_type *, bignum_digit_type *, bignum_digit_type,
   bignum_digit_type *);
static void bignum_divide_unsigned_medium_denominator
  (bignum_type, bignum_digit_type, bignum_type *, bignum_type *, int, int);
static bignum_digit_type bignum_digit_divide
  (bignum_digit_type, bignum_digit_type, bignum_digit_type, bignum_digit_type *);
static bignum_digit_type bignum_digit_divide_subtract
  (bignum_digit_type, bignum_digit_type, bignum_digit_type, bignum_digit_type *);
static void bignum_divide_unsigned_small_denominator
  (bignum_type, bignum_digit_type, bignum_type *, bignum_type *, int, int);
static bignum_digit_type bignum_destructive_scale_down
  (bignum_type, bignum_digit_type);
static bignum_type bignum_remainder_unsigned_small_denominator
  (bignum_type, bignum_digit_type, int);
static bignum_type bignum_digit_to_bignum(bignum_digit_type, int);
static bignum_type bignum_allocate(bignum_length_type, int);
static bignum_type bignum_allocate_zeroed(bignum_length_type, int);
static bignum_type bignum_shorten_length(bignum_type, bignum_length_type);
static bignum_type bignum_trim(bignum_type);
static bignum_type bignum_copy(bignum_type);
static bignum_type bignum_new_sign(bignum_type, int);
static bignum_type bignum_maybe_new_sign(bignum_type, int);
static void bignum_destructive_copy(bignum_type, bignum_type);
/* Unused
static void bignum_destructive_zero(bignum_type);
*/

/* Added for bitwise operations. */
static bignum_type bignum_magnitude_ash(bignum_type arg1, long n);
static bignum_type bignum_pospos_bitwise_op(int op, bignum_type, bignum_type);
static bignum_type bignum_posneg_bitwise_op(int op, bignum_type, bignum_type);
static bignum_type bignum_negneg_bitwise_op(int op, bignum_type, bignum_type);
static void        bignum_negate_magnitude(bignum_type);
static long        bignum_unsigned_logcount(bignum_type arg);
static int         bignum_unsigned_logbitp(int shift, bignum_type bignum);

static s48_value s48_bignum_zero    = (s48_value) NULL;
static s48_value s48_bignum_pos_one = (s48_value) NULL;
static s48_value s48_bignum_neg_one = (s48_value) NULL;

/* Exports */

/*
 * We have to allocate the cached constants slightly differently because
 * they have to be registered with the GC, which requires that we have
 * tagged pointers to them.
 */

void
s48_bignum_make_cached_constants(void)
{
  if (s48_bignum_zero == (s48_value) NULL) {
    bignum_type temp;
    
    s48_bignum_zero = (s48_value) BIGNUM_ALLOCATE_TAGGED(0);
    temp = S48_ADDRESS_AFTER_HEADER(s48_bignum_zero, long);
    BIGNUM_SET_HEADER (temp, 0, 0);
    S48_GC_PROTECT_GLOBAL(s48_bignum_zero);
    
    s48_bignum_pos_one = (s48_value) BIGNUM_ALLOCATE_TAGGED(1);
    temp = S48_ADDRESS_AFTER_HEADER(s48_bignum_pos_one, long);
    BIGNUM_SET_HEADER (temp, 1, 0);
    (BIGNUM_REF (temp, 0)) = 1;
    S48_GC_PROTECT_GLOBAL(s48_bignum_pos_one);
    
    s48_bignum_neg_one = (s48_value) BIGNUM_ALLOCATE_TAGGED(1);
    temp = S48_ADDRESS_AFTER_HEADER(s48_bignum_neg_one, long);
    BIGNUM_SET_HEADER (temp, 1, 1);
    (BIGNUM_REF (temp, 0)) = 1;
    S48_GC_PROTECT_GLOBAL(s48_bignum_neg_one);
  }
}

int
s48_bignum_equal_p(bignum_type x, bignum_type y)
{
  return
    ((BIGNUM_ZERO_P (x))
     ? (BIGNUM_ZERO_P (y))
     : ((! (BIGNUM_ZERO_P (y)))
	&& ((BIGNUM_NEGATIVE_P (x))
	    ? (BIGNUM_NEGATIVE_P (y))
	    : (! (BIGNUM_NEGATIVE_P (y))))
	&& (bignum_equal_p_unsigned (x, y))));
}

enum bignum_comparison
s48_bignum_test(bignum_type bignum)
{
  return
    ((BIGNUM_ZERO_P (bignum))
     ? bignum_comparison_equal
     : (BIGNUM_NEGATIVE_P (bignum))
     ? bignum_comparison_less
     : bignum_comparison_greater);
}

enum bignum_comparison
s48_bignum_compare(bignum_type x, bignum_type y)
{
  return
    ((BIGNUM_ZERO_P (x))
     ? ((BIGNUM_ZERO_P (y))
	? bignum_comparison_equal
	: (BIGNUM_NEGATIVE_P (y))
	? bignum_comparison_greater
	: bignum_comparison_less)
     : (BIGNUM_ZERO_P (y))
     ? ((BIGNUM_NEGATIVE_P (x))
	? bignum_comparison_less
	: bignum_comparison_greater)
     : (BIGNUM_NEGATIVE_P (x))
     ? ((BIGNUM_NEGATIVE_P (y))
	? (bignum_compare_unsigned (y, x))
	: (bignum_comparison_less))
     : ((BIGNUM_NEGATIVE_P (y))
	? (bignum_comparison_greater)
	: (bignum_compare_unsigned (x, y))));
}

bignum_type
s48_bignum_add(bignum_type x, bignum_type y)
{
  return
    ((BIGNUM_ZERO_P (x))
     ? (BIGNUM_MAYBE_COPY (y))
     : (BIGNUM_ZERO_P (y))
     ? (BIGNUM_MAYBE_COPY (x))
     : ((BIGNUM_NEGATIVE_P (x))
	? ((BIGNUM_NEGATIVE_P (y))
	   ? (bignum_add_unsigned (x, y, 1))
	   : (bignum_subtract_unsigned (y, x)))
	: ((BIGNUM_NEGATIVE_P (y))
	   ? (bignum_subtract_unsigned (x, y))
	   : (bignum_add_unsigned (x, y, 0)))));
}

bignum_type
s48_bignum_subtract(bignum_type x, bignum_type y)
{
  return
    ((BIGNUM_ZERO_P (x))
     ? ((BIGNUM_ZERO_P (y))
	? (BIGNUM_MAYBE_COPY (y))
	: (bignum_new_sign (y, (! (BIGNUM_NEGATIVE_P (y))))))
     : ((BIGNUM_ZERO_P (y))
	? (BIGNUM_MAYBE_COPY (x))
	: ((BIGNUM_NEGATIVE_P (x))
	   ? ((BIGNUM_NEGATIVE_P (y))
	      ? (bignum_subtract_unsigned (y, x))
	      : (bignum_add_unsigned (x, y, 1)))
	   : ((BIGNUM_NEGATIVE_P (y))
	      ? (bignum_add_unsigned (x, y, 0))
	      : (bignum_subtract_unsigned (x, y))))));
}

bignum_type
s48_bignum_negate(bignum_type x)
{
  return
    ((BIGNUM_ZERO_P (x))
     ? (BIGNUM_MAYBE_COPY (x))
     : (bignum_new_sign (x, (! (BIGNUM_NEGATIVE_P (x))))));
}

bignum_type
s48_bignum_multiply(bignum_type x, bignum_type y)
{
  bignum_length_type x_length = (BIGNUM_LENGTH (x));
  bignum_length_type y_length = (BIGNUM_LENGTH (y));
  int negative_p =
    ((BIGNUM_NEGATIVE_P (x))
     ? (! (BIGNUM_NEGATIVE_P (y)))
     : (BIGNUM_NEGATIVE_P (y)));
  if (BIGNUM_ZERO_P (x))
    return (BIGNUM_MAYBE_COPY (x));
  if (BIGNUM_ZERO_P (y))
    return (BIGNUM_MAYBE_COPY (y));
  if (x_length == 1)
    {
      bignum_digit_type digit = (BIGNUM_REF (x, 0));
      if (digit == 1)
	return (bignum_maybe_new_sign (y, negative_p));
      if (digit < BIGNUM_RADIX_ROOT)
	return (bignum_multiply_unsigned_small_factor (y, digit, negative_p));
    }
  if (y_length == 1)
    {
      bignum_digit_type digit = (BIGNUM_REF (y, 0));
      if (digit == 1)
	return (bignum_maybe_new_sign (x, negative_p));
      if (digit < BIGNUM_RADIX_ROOT)
	return (bignum_multiply_unsigned_small_factor (x, digit, negative_p));
    }
  return (bignum_multiply_unsigned (x, y, negative_p));
}

static int
bignum_divide(bignum_type numerator, bignum_type denominator,
		  bignum_type * quotient, bignum_type * remainder)
{
  if (BIGNUM_ZERO_P (denominator))
    return (1);
  if (BIGNUM_ZERO_P (numerator))
    {
      (*quotient) = (BIGNUM_MAYBE_COPY (numerator));
      (*remainder) = (BIGNUM_MAYBE_COPY (numerator));
    }
  else
    {
      int r_negative_p = (BIGNUM_NEGATIVE_P (numerator));
      int q_negative_p =
	((BIGNUM_NEGATIVE_P (denominator)) ? (! r_negative_p) : r_negative_p);
      switch (bignum_compare_unsigned (numerator, denominator))
	{
	case bignum_comparison_equal:
	  {
	    (*quotient) = (BIGNUM_ONE (q_negative_p));
	    (*remainder) = (BIGNUM_ZERO ());
	    break;
	  }
	case bignum_comparison_less:
	  {
	    (*quotient) = (BIGNUM_ZERO ());
	    (*remainder) = (BIGNUM_MAYBE_COPY (numerator));
	    break;
	  }
	case bignum_comparison_greater:
	  {
	    if ((BIGNUM_LENGTH (denominator)) == 1)
	      {
		bignum_digit_type digit = (BIGNUM_REF (denominator, 0));
		if (digit == 1)
		  {
		    (*quotient) =
		      (bignum_maybe_new_sign (numerator, q_negative_p));
		    (*remainder) = (BIGNUM_ZERO ());
		    break;
		  }
		else if (digit < BIGNUM_RADIX_ROOT)
		  {
		    bignum_divide_unsigned_small_denominator
		      (numerator, digit,
		       quotient, remainder,
		       q_negative_p, r_negative_p);
		    break;
		  }
		else
		  {
		    bignum_divide_unsigned_medium_denominator
		      (numerator, digit,
		       quotient, remainder,
		       q_negative_p, r_negative_p);
		    break;
		  }
	      }
	    bignum_divide_unsigned_large_denominator
	      (numerator, denominator,
	       quotient, remainder,
	       q_negative_p, r_negative_p);
	    break;
	  }
	}
    }
  return (0);
}

int
s48_bignum_divide(bignum_type numerator, bignum_type denominator,
		  void* quotient, void * remainder)
{
  return bignum_divide(numerator, denominator,
		       (bignum_type *)quotient, (bignum_type *)remainder);
}

bignum_type
s48_bignum_quotient(bignum_type numerator, bignum_type denominator)
{
  if (BIGNUM_ZERO_P (denominator))
    return (BIGNUM_OUT_OF_BAND);
  if (BIGNUM_ZERO_P (numerator))
    return (BIGNUM_MAYBE_COPY (numerator));
  {
    int q_negative_p =
      ((BIGNUM_NEGATIVE_P (denominator))
       ? (! (BIGNUM_NEGATIVE_P (numerator)))
       : (BIGNUM_NEGATIVE_P (numerator)));
    switch (bignum_compare_unsigned (numerator, denominator))
      {
      case bignum_comparison_equal:
	return (BIGNUM_ONE (q_negative_p));
      case bignum_comparison_less:
	return (BIGNUM_ZERO ());
      case bignum_comparison_greater:
      default:					/* to appease gcc -Wall */
	{
	  bignum_type quotient;
	  if ((BIGNUM_LENGTH (denominator)) == 1)
	    {
	      bignum_digit_type digit = (BIGNUM_REF (denominator, 0));
	      if (digit == 1)
		return (bignum_maybe_new_sign (numerator, q_negative_p));
	      if (digit < BIGNUM_RADIX_ROOT)
		bignum_divide_unsigned_small_denominator
		  (numerator, digit,
		   (&quotient), ((bignum_type *) 0),
		   q_negative_p, 0);
	      else
		bignum_divide_unsigned_medium_denominator
		  (numerator, digit,
		   (&quotient), ((bignum_type *) 0),
		   q_negative_p, 0);
	    }
	  else
	    bignum_divide_unsigned_large_denominator
	      (numerator, denominator,
	       (&quotient), ((bignum_type *) 0),
	       q_negative_p, 0);
	  return (quotient);
	}
      }
  }
}

bignum_type
s48_bignum_remainder(bignum_type numerator, bignum_type denominator)
{
  if (BIGNUM_ZERO_P (denominator))
    return (BIGNUM_OUT_OF_BAND);
  if (BIGNUM_ZERO_P (numerator))
    return (BIGNUM_MAYBE_COPY (numerator));
  switch (bignum_compare_unsigned (numerator, denominator))
    {
    case bignum_comparison_equal:
      return (BIGNUM_ZERO ());
    case bignum_comparison_less:
      return (BIGNUM_MAYBE_COPY (numerator));
    case bignum_comparison_greater:
    default:					/* to appease gcc -Wall */
      {
	bignum_type remainder;
	if ((BIGNUM_LENGTH (denominator)) == 1)
	  {
	    bignum_digit_type digit = (BIGNUM_REF (denominator, 0));
	    if (digit == 1)
	      return (BIGNUM_ZERO ());
	    if (digit < BIGNUM_RADIX_ROOT)
	      return
		(bignum_remainder_unsigned_small_denominator
		 (numerator, digit, (BIGNUM_NEGATIVE_P (numerator))));
	    bignum_divide_unsigned_medium_denominator
	      (numerator, digit,
	       ((bignum_type *) 0), (&remainder),
	       0, (BIGNUM_NEGATIVE_P (numerator)));
	  }
	else
	  bignum_divide_unsigned_large_denominator
	    (numerator, denominator,
	     ((bignum_type *) 0), (&remainder),
	     0, (BIGNUM_NEGATIVE_P (numerator)));
	return (remainder);
      }
    }
}

/* These procedures depend on the non-portable type `unsigned long'.
   If your compiler doesn't support this type, either define the
   switch `BIGNUM_NO_ULONG' to disable them (in "bignum.h"), or write
   new versions that don't use this type. */

#ifndef BIGNUM_NO_ULONG

bignum_type
s48_long_to_bignum(long n)
{
  int negative_p;
  bignum_digit_type result_digits [BIGNUM_DIGITS_FOR_LONG];
  bignum_digit_type * end_digits = result_digits;
  /* Special cases win when these small constants are cached. */
  if (n == 0) return (BIGNUM_ZERO ());
  if (n == 1) return (BIGNUM_ONE (0));
  if (n == -1) return (BIGNUM_ONE (1));
  {
    unsigned long accumulator = ((negative_p = (n < 0)) ? (-n) : n);
    do
      {
	(*end_digits++) = (accumulator & BIGNUM_DIGIT_MASK);
	accumulator >>= BIGNUM_DIGIT_LENGTH;
      }
    while (accumulator != 0);
  }
  {
    bignum_type result =
      (bignum_allocate ((end_digits - result_digits), negative_p));
    bignum_digit_type * scan_digits = result_digits;
    bignum_digit_type * scan_result = (BIGNUM_START_PTR (result));
    while (scan_digits < end_digits)
      (*scan_result++) = (*scan_digits++);
    return (result);
  }
}

long
s48_bignum_to_long(bignum_type bignum)
{
  if (BIGNUM_ZERO_P (bignum))
    return (0);
  {
    unsigned long accumulator = 0;
    bignum_digit_type * start = (BIGNUM_START_PTR (bignum));
    bignum_digit_type * scan = (start + (BIGNUM_LENGTH (bignum)));
    while (start < scan)
      accumulator = ((accumulator << BIGNUM_DIGIT_LENGTH) + (*--scan));
    return ((BIGNUM_NEGATIVE_P (bignum)) ? (-((long)accumulator)) : accumulator);
  }
}

bignum_type
ulong_to_bignum(unsigned long n)
{
  bignum_digit_type result_digits [BIGNUM_DIGITS_FOR_LONG];
  bignum_digit_type * end_digits = result_digits;
  /* Special cases win when these small constants are cached. */
  if (n == 0) return (BIGNUM_ZERO ());
  if (n == 1) return (BIGNUM_ONE (0));
  {
    unsigned long accumulator = n;
    do
      {
	(*end_digits++) = (accumulator & BIGNUM_DIGIT_MASK);
	accumulator >>= BIGNUM_DIGIT_LENGTH;
      }
    while (accumulator != 0);
  }
  {
    bignum_type result =
      (bignum_allocate ((end_digits - result_digits), 0));
    bignum_digit_type * scan_digits = result_digits;
    bignum_digit_type * scan_result = (BIGNUM_START_PTR (result));
    while (scan_digits < end_digits)
      (*scan_result++) = (*scan_digits++);
    return (result);
  }
}

unsigned long
s48_bignum_to_ulong(bignum_type bignum)
{
  if (BIGNUM_ZERO_P (bignum))
    return (0);
  {
    unsigned long accumulator = 0;
    bignum_digit_type * start = (BIGNUM_START_PTR (bignum));
    bignum_digit_type * scan = (start + (BIGNUM_LENGTH (bignum)));
    while (start < scan)
      accumulator = ((accumulator << BIGNUM_DIGIT_LENGTH) + (*--scan));
    return (accumulator);
  }
}

#endif /* not BIGNUM_NO_ULONG */

#define DTB_WRITE_DIGIT(factor)						\
{									\
  significand *= (factor);						\
  digit = ((bignum_digit_type) significand);				\
  (*--scan) = digit;							\
  significand -= ((double) digit);					\
}

bignum_type
s48_double_to_bignum(double x)
{
  int exponent;
  double significand = (frexp (x, (&exponent)));
  if (exponent <= 0) return (BIGNUM_ZERO ());
  if (exponent == 1) return (BIGNUM_ONE (x < 0));
  if (significand < 0) significand = (-significand);
  {
    bignum_length_type length = (BIGNUM_BITS_TO_DIGITS (exponent));
    bignum_type result = (bignum_allocate (length, (x < 0)));
    bignum_digit_type * start = (BIGNUM_START_PTR (result));
    bignum_digit_type * scan = (start + length);
    bignum_digit_type digit;
    int odd_bits = (exponent % BIGNUM_DIGIT_LENGTH);
    if (odd_bits > 0)
      DTB_WRITE_DIGIT (1L << odd_bits);
    while (start < scan)
      {
	if (significand == 0)
	  {
	    while (start < scan)
	      (*--scan) = 0;
	    break;
	  }
	DTB_WRITE_DIGIT (BIGNUM_RADIX);
      }
    return (result);
  }
}

#undef DTB_WRITE_DIGIT

double
s48_bignum_to_double(bignum_type bignum)
{
  if (BIGNUM_ZERO_P (bignum))
    return (0);
  {
    double accumulator = 0;
    bignum_digit_type * start = (BIGNUM_START_PTR (bignum));
    bignum_digit_type * scan = (start + (BIGNUM_LENGTH (bignum)));
    while (start < scan)
      accumulator = ((accumulator * BIGNUM_RADIX) + (*--scan));
    return ((BIGNUM_NEGATIVE_P (bignum)) ? (-accumulator) : accumulator);
  }
}

int
s48_bignum_fits_in_word_p(bignum_type bignum, long word_length,
			  int twos_complement_p)
{
  unsigned int n_bits = (twos_complement_p ? (word_length - 1) : word_length);
  BIGNUM_ASSERT (n_bits > 0);
  {
    bignum_length_type length = (BIGNUM_LENGTH (bignum));
    bignum_length_type max_digits = (BIGNUM_BITS_TO_DIGITS (n_bits));
    bignum_digit_type msd, max;
    return
      ((length < max_digits) ||
       ((length == max_digits) &&
	((((msd = (BIGNUM_REF (bignum, (length - 1)))) <
	   (max = (1L << (n_bits - ((length - 1) * BIGNUM_DIGIT_LENGTH))))) ||
	  (twos_complement_p &&
	   (msd == max) &&
	   (BIGNUM_NEGATIVE_P (bignum)))))));
  }
}

bignum_type
s48_bignum_length_in_bits(bignum_type bignum)
{
  if (BIGNUM_ZERO_P (bignum))
    return (BIGNUM_ZERO ());
  {
    bignum_length_type index = ((BIGNUM_LENGTH (bignum)) - 1);
    bignum_digit_type digit = (BIGNUM_REF (bignum, index));
    bignum_type result = (bignum_allocate (2, 0));
    (BIGNUM_REF (result, 0)) = index;
    (BIGNUM_REF (result, 1)) = 0;
    bignum_destructive_scale_up (result, BIGNUM_DIGIT_LENGTH);
    while (digit > 0)
      {
	bignum_destructive_add (result, ((bignum_digit_type) 1));
	digit >>= 1;
      }
    return (bignum_trim (result));
  }
}

bignum_type
s48_bignum_length_upper_limit(void)
{
  bignum_type result = (bignum_allocate (2, 0));
  (BIGNUM_REF (result, 0)) = 0;
  (BIGNUM_REF (result, 1)) = BIGNUM_DIGIT_LENGTH;
  return (result);
}

bignum_type
s48_digit_stream_to_bignum(unsigned int n_digits,
			   unsigned int *producer(bignum_procedure_context),
			   bignum_procedure_context context,
			   unsigned int radix,
			   int negative_p)
{
  BIGNUM_ASSERT ((radix > 1) && (radix <= BIGNUM_RADIX_ROOT));
  if (n_digits == 0)
    return (BIGNUM_ZERO ());
  if (n_digits == 1)
    {
      long digit = ((long) ((*producer) (context)));
      return (s48_long_to_bignum (negative_p ? (- digit) : digit));
    }
  {
    bignum_length_type length;
    {
      unsigned int radix_copy = radix;
      unsigned int log_radix = 0;
      while (radix_copy > 0)
	{
	  radix_copy >>= 1;
	  log_radix += 1;
	}
      /* This length will be at least as large as needed. */
      length = (BIGNUM_BITS_TO_DIGITS (n_digits * log_radix));
    }
    {
      bignum_type result = (bignum_allocate_zeroed (length, negative_p));
      while ((n_digits--) > 0)
	{
	  bignum_destructive_scale_up (result, ((bignum_digit_type) radix));
	  bignum_destructive_add
	    (result, ((bignum_digit_type) ((*producer) (context))));
	}
      return (bignum_trim (result));
    }
  }
}

void
s48_bignum_to_digit_stream(bignum_type bignum,
			   unsigned int radix,
			   void *consumer(bignum_procedure_context,
					  bignum_digit_type),
			   bignum_procedure_context context)
{
  BIGNUM_ASSERT ((radix > 1) && (radix <= BIGNUM_RADIX_ROOT));
  if (! (BIGNUM_ZERO_P (bignum)))
    {
      bignum_type working_copy = (bignum_copy (bignum));
      bignum_digit_type * start = (BIGNUM_START_PTR (working_copy));
      bignum_digit_type * scan = (start + (BIGNUM_LENGTH (working_copy)));
      while (start < scan)
	{
	  if ((scan[-1]) == 0)
	    scan -= 1;
	  else
	    (*consumer)
	      (context, (bignum_destructive_scale_down (working_copy, radix)));
	}
      BIGNUM_DEALLOCATE (working_copy);
    }
  return;
}

long
s48_bignum_max_digit_stream_radix(void)
{
  return (BIGNUM_RADIX_ROOT);
}

/* Comparisons */

static int
bignum_equal_p_unsigned(bignum_type x, bignum_type y)
{
  bignum_length_type length = (BIGNUM_LENGTH (x));
  if (length != (BIGNUM_LENGTH (y)))
    return (0);
  else
    {
      bignum_digit_type * scan_x = (BIGNUM_START_PTR (x));
      bignum_digit_type * scan_y = (BIGNUM_START_PTR (y));
      bignum_digit_type * end_x = (scan_x + length);
      while (scan_x < end_x)
	if ((*scan_x++) != (*scan_y++))
	  return (0);
      return (1);
    }
}

static enum bignum_comparison
bignum_compare_unsigned(bignum_type x, bignum_type y)
{
  bignum_length_type x_length = (BIGNUM_LENGTH (x));
  bignum_length_type y_length = (BIGNUM_LENGTH (y));
  if (x_length < y_length)
    return (bignum_comparison_less);
  if (x_length > y_length)
    return (bignum_comparison_greater);
  {
    bignum_digit_type * start_x = (BIGNUM_START_PTR (x));
    bignum_digit_type * scan_x = (start_x + x_length);
    bignum_digit_type * scan_y = ((BIGNUM_START_PTR (y)) + y_length);
    while (start_x < scan_x)
      {
	bignum_digit_type digit_x = (*--scan_x);
	bignum_digit_type digit_y = (*--scan_y);
	if (digit_x < digit_y)
	  return (bignum_comparison_less);
	if (digit_x > digit_y)
	  return (bignum_comparison_greater);
      }
  }
  return (bignum_comparison_equal);
}

/* Addition */

static bignum_type
bignum_add_unsigned(bignum_type x, bignum_type y, int negative_p)
{
  if ((BIGNUM_LENGTH (y)) > (BIGNUM_LENGTH (x)))
    {
      bignum_type z = x;
      x = y;
      y = z;
    }
  {
    bignum_length_type x_length = (BIGNUM_LENGTH (x));
    bignum_type r = (bignum_allocate ((x_length + 1), negative_p));
    bignum_digit_type sum;
    bignum_digit_type carry = 0;
    bignum_digit_type * scan_x = (BIGNUM_START_PTR (x));
    bignum_digit_type * scan_r = (BIGNUM_START_PTR (r));
    {
      bignum_digit_type * scan_y = (BIGNUM_START_PTR (y));
      bignum_digit_type * end_y = (scan_y + (BIGNUM_LENGTH (y)));
      while (scan_y < end_y)
	{
	  sum = ((*scan_x++) + (*scan_y++) + carry);
	  if (sum < BIGNUM_RADIX)
	    {
	      (*scan_r++) = sum;
	      carry = 0;
	    }
	  else
	    {
	      (*scan_r++) = (sum - BIGNUM_RADIX);
	      carry = 1;
	    }
	}
    }
    {
      bignum_digit_type * end_x = ((BIGNUM_START_PTR (x)) + x_length);
      if (carry != 0)
	while (scan_x < end_x)
	  {
	    sum = ((*scan_x++) + 1);
	    if (sum < BIGNUM_RADIX)
	      {
		(*scan_r++) = sum;
		carry = 0;
		break;
	      }
	    else
	      (*scan_r++) = (sum - BIGNUM_RADIX);
	  }
      while (scan_x < end_x)
	(*scan_r++) = (*scan_x++);
    }
    if (carry != 0)
      {
	(*scan_r) = 1;
	return (r);
      }
    return (bignum_shorten_length (r, x_length));
  }
}

/* Subtraction */

static bignum_type
bignum_subtract_unsigned(bignum_type x, bignum_type y)
{
  int negative_p;
  switch (bignum_compare_unsigned (x, y))
    {
    case bignum_comparison_equal:
      return (BIGNUM_ZERO ());
    case bignum_comparison_less:
      {
	bignum_type z = x;
	x = y;
	y = z;
      }
      negative_p = 1;
      break;
    case bignum_comparison_greater:
      negative_p = 0;
      break;
    }
  {
    bignum_length_type x_length = (BIGNUM_LENGTH (x));
    bignum_type r = (bignum_allocate (x_length, negative_p));
    bignum_digit_type difference;
    bignum_digit_type borrow = 0;
    bignum_digit_type * scan_x = (BIGNUM_START_PTR (x));
    bignum_digit_type * scan_r = (BIGNUM_START_PTR (r));
    {
      bignum_digit_type * scan_y = (BIGNUM_START_PTR (y));
      bignum_digit_type * end_y = (scan_y + (BIGNUM_LENGTH (y)));
      while (scan_y < end_y)
	{
	  difference = (((*scan_x++) - (*scan_y++)) - borrow);
	  if (difference < 0)
	    {
	      (*scan_r++) = (difference + BIGNUM_RADIX);
	      borrow = 1;
	    }
	  else
	    {
	      (*scan_r++) = difference;
	      borrow = 0;
	    }
	}
    }
    {
      bignum_digit_type * end_x = ((BIGNUM_START_PTR (x)) + x_length);
      if (borrow != 0)
	while (scan_x < end_x)
	  {
	    difference = ((*scan_x++) - borrow);
	    if (difference < 0)
	      (*scan_r++) = (difference + BIGNUM_RADIX);
	    else
	      {
		(*scan_r++) = difference;
		borrow = 0;
		break;
	      }
	  }
      BIGNUM_ASSERT (borrow == 0);
      while (scan_x < end_x)
	(*scan_r++) = (*scan_x++);
    }
    return (bignum_trim (r));
  }
}

/* Multiplication
   Maximum value for product_low or product_high:
	((R * R) + (R * (R - 2)) + (R - 1))
   Maximum value for carry: ((R * (R - 1)) + (R - 1))
	where R == BIGNUM_RADIX_ROOT */

static bignum_type
bignum_multiply_unsigned(bignum_type x, bignum_type y, int negative_p)
{
  if ((BIGNUM_LENGTH (y)) > (BIGNUM_LENGTH (x)))
    {
      bignum_type z = x;
      x = y;
      y = z;
    }
  {
    bignum_digit_type carry;
    bignum_digit_type y_digit_low;
    bignum_digit_type y_digit_high;
    bignum_digit_type x_digit_low;
    bignum_digit_type x_digit_high;
    bignum_digit_type product_low;
    bignum_digit_type * scan_r;
    bignum_digit_type * scan_y;
    bignum_length_type x_length = (BIGNUM_LENGTH (x));
    bignum_length_type y_length = (BIGNUM_LENGTH (y));
    bignum_type r =
      (bignum_allocate_zeroed ((x_length + y_length), negative_p));
    bignum_digit_type * scan_x = (BIGNUM_START_PTR (x));
    bignum_digit_type * end_x = (scan_x + x_length);
    bignum_digit_type * start_y = (BIGNUM_START_PTR (y));
    bignum_digit_type * end_y = (start_y + y_length);
    bignum_digit_type * start_r = (BIGNUM_START_PTR (r));
#define x_digit x_digit_high
#define y_digit y_digit_high
#define product_high carry
    while (scan_x < end_x)
      {
	x_digit = (*scan_x++);
	x_digit_low = (HD_LOW (x_digit));
	x_digit_high = (HD_HIGH (x_digit));
	carry = 0;
	scan_y = start_y;
	scan_r = (start_r++);
	while (scan_y < end_y)
	  {
	    y_digit = (*scan_y++);
	    y_digit_low = (HD_LOW (y_digit));
	    y_digit_high = (HD_HIGH (y_digit));
	    product_low =
	      ((*scan_r) +
	       (x_digit_low * y_digit_low) +
	       (HD_LOW (carry)));
	    product_high =
	      ((x_digit_high * y_digit_low) +
	       (x_digit_low * y_digit_high) +
	       (HD_HIGH (product_low)) +
	       (HD_HIGH (carry)));
	    (*scan_r++) =
	      (HD_CONS ((HD_LOW (product_high)), (HD_LOW (product_low))));
	    carry =
	      ((x_digit_high * y_digit_high) +
	       (HD_HIGH (product_high)));
	  }
	(*scan_r) += carry;
      }
    return (bignum_trim (r));
#undef x_digit
#undef y_digit
#undef product_high
  }
}

static bignum_type
bignum_multiply_unsigned_small_factor(bignum_type x, bignum_digit_type y,
				      int negative_p)
{
  bignum_length_type length_x = (BIGNUM_LENGTH (x));
  bignum_type p = (bignum_allocate ((length_x + 1), negative_p));
  bignum_destructive_copy (x, p);
  (BIGNUM_REF (p, length_x)) = 0;
  bignum_destructive_scale_up (p, y);
  return (bignum_trim (p));
}

static void
bignum_destructive_scale_up(bignum_type bignum, bignum_digit_type factor)
{
  bignum_digit_type carry = 0;
  bignum_digit_type * scan = (BIGNUM_START_PTR (bignum));
  bignum_digit_type two_digits;
  bignum_digit_type product_low;
#define product_high carry
  bignum_digit_type * end = (scan + (BIGNUM_LENGTH (bignum)));
  BIGNUM_ASSERT ((factor > 1) && (factor < BIGNUM_RADIX_ROOT));
  while (scan < end)
    {
      two_digits = (*scan);
      product_low = ((factor * (HD_LOW (two_digits))) + (HD_LOW (carry)));
      product_high =
	((factor * (HD_HIGH (two_digits))) +
	 (HD_HIGH (product_low)) +
	 (HD_HIGH (carry)));
      (*scan++) = (HD_CONS ((HD_LOW (product_high)), (HD_LOW (product_low))));
      carry = (HD_HIGH (product_high));
    }
  /* A carry here would be an overflow, i.e. it would not fit.
     Hopefully the callers allocate enough space that this will
     never happen.
   */
  BIGNUM_ASSERT (carry == 0);
  return;
#undef product_high
}

static void
bignum_destructive_add(bignum_type bignum, bignum_digit_type n)
{
  bignum_digit_type * scan = (BIGNUM_START_PTR (bignum));
  bignum_digit_type digit;
  digit = ((*scan) + n);
  if (digit < BIGNUM_RADIX)
    {
      (*scan) = digit;
      return;
    }
  (*scan++) = (digit - BIGNUM_RADIX);
  while (1)
    {
      digit = ((*scan) + 1);
      if (digit < BIGNUM_RADIX)
	{
	  (*scan) = digit;
	  return;
	}
      (*scan++) = (digit - BIGNUM_RADIX);
    }
}

/* Division */

/* For help understanding this algorithm, see:
   Knuth, Donald E., "The Art of Computer Programming",
   volume 2, "Seminumerical Algorithms"
   section 4.3.1, "Multiple-Precision Arithmetic". */

static void
bignum_divide_unsigned_large_denominator(bignum_type numerator,
					 bignum_type denominator,
					 bignum_type * quotient,
					 bignum_type * remainder,
					 int q_negative_p,
					 int r_negative_p)
{
  bignum_length_type length_n = ((BIGNUM_LENGTH (numerator)) + 1);
  bignum_length_type length_d = (BIGNUM_LENGTH (denominator));
  bignum_type q =
    ((quotient != ((bignum_type *) 0))
     ? (bignum_allocate ((length_n - length_d), q_negative_p))
     : BIGNUM_OUT_OF_BAND);
  bignum_type u = (bignum_allocate (length_n, r_negative_p));
  int shift = 0;
  BIGNUM_ASSERT (length_d > 1);
  {
    bignum_digit_type v1 = (BIGNUM_REF ((denominator), (length_d - 1)));
    while (v1 < (BIGNUM_RADIX / 2))
      {
	v1 <<= 1;
	shift += 1;
      }
  }
  if (shift == 0)
    {
      bignum_destructive_copy (numerator, u);
      (BIGNUM_REF (u, (length_n - 1))) = 0;
      bignum_divide_unsigned_normalized (u, denominator, q);
    }
  else
    {
      bignum_type v = (bignum_allocate (length_d, 0));
      bignum_destructive_normalization (numerator, u, shift);
      bignum_destructive_normalization (denominator, v, shift);
      bignum_divide_unsigned_normalized (u, v, q);
      BIGNUM_DEALLOCATE (v);
      if (remainder != ((bignum_type *) 0))
	bignum_destructive_unnormalization (u, shift);
    }
  if (quotient != ((bignum_type *) 0))
    (*quotient) = (bignum_trim (q));
  if (remainder != ((bignum_type *) 0))
    (*remainder) = (bignum_trim (u));
  else
    BIGNUM_DEALLOCATE (u);
  return;
}

static void
bignum_divide_unsigned_normalized(bignum_type u, bignum_type v, bignum_type q)
{
  bignum_length_type u_length = (BIGNUM_LENGTH (u));
  bignum_length_type v_length = (BIGNUM_LENGTH (v));
  bignum_digit_type * u_start = (BIGNUM_START_PTR (u));
  bignum_digit_type * u_scan = (u_start + u_length);
  bignum_digit_type * u_scan_limit = (u_start + v_length);
  bignum_digit_type * u_scan_start = (u_scan - v_length);
  bignum_digit_type * v_start = (BIGNUM_START_PTR (v));
  bignum_digit_type * v_end = (v_start + v_length);
  bignum_digit_type * q_scan;
  bignum_digit_type v1 = (v_end[-1]);
  bignum_digit_type v2 = (v_end[-2]);
  bignum_digit_type ph;	/* high half of double-digit product */
  bignum_digit_type pl;	/* low half of double-digit product */
  bignum_digit_type guess;
  bignum_digit_type gh;	/* high half-digit of guess */
  bignum_digit_type ch;	/* high half of double-digit comparand */
  bignum_digit_type v2l = (HD_LOW (v2));
  bignum_digit_type v2h = (HD_HIGH (v2));
  bignum_digit_type cl;	/* low half of double-digit comparand */
#define gl ph			/* low half-digit of guess */
#define uj pl
#define qj ph
  bignum_digit_type gm;		/* memory loc for reference parameter */
  if (q != BIGNUM_OUT_OF_BAND)
    q_scan = ((BIGNUM_START_PTR (q)) + (BIGNUM_LENGTH (q)));
  while (u_scan_limit < u_scan)
    {
      uj = (*--u_scan);
      if (uj != v1)
	{
	  /* comparand =
	     (((((uj * BIGNUM_RADIX) + uj1) % v1) * BIGNUM_RADIX) + uj2);
	     guess = (((uj * BIGNUM_RADIX) + uj1) / v1); */
	  cl = (u_scan[-2]);
	  ch = (bignum_digit_divide (uj, (u_scan[-1]), v1, (&gm)));
	  guess = gm;
	}
      else
	{
	  cl = (u_scan[-2]);
	  ch = ((u_scan[-1]) + v1);
	  guess = (BIGNUM_RADIX - 1);
	}
      while (1)
	{
	  /* product = (guess * v2); */
	  gl = (HD_LOW (guess));
	  gh = (HD_HIGH (guess));
	  pl = (v2l * gl);
	  ph = ((v2l * gh) + (v2h * gl) + (HD_HIGH (pl)));
	  pl = (HD_CONS ((HD_LOW (ph)), (HD_LOW (pl))));
	  ph = ((v2h * gh) + (HD_HIGH (ph)));
	  /* if (comparand >= product) */
	  if ((ch > ph) || ((ch == ph) && (cl >= pl)))
	    break;
	  guess -= 1;
	  /* comparand += (v1 << BIGNUM_DIGIT_LENGTH) */
	  ch += v1;
	  /* if (comparand >= (BIGNUM_RADIX * BIGNUM_RADIX)) */
	  if (ch >= BIGNUM_RADIX)
	    break;
	}
      qj = (bignum_divide_subtract (v_start, v_end, guess, (--u_scan_start)));
      if (q != BIGNUM_OUT_OF_BAND)
	(*--q_scan) = qj;
    }
  return;
#undef gl
#undef uj
#undef qj
}

static bignum_digit_type
bignum_divide_subtract(bignum_digit_type * v_start,
		       bignum_digit_type * v_end,
		       bignum_digit_type guess,
		       bignum_digit_type * u_start)
{
  bignum_digit_type * v_scan = v_start;
  bignum_digit_type * u_scan = u_start;
  bignum_digit_type carry = 0;
  if (guess == 0) return (0);
  {
    bignum_digit_type gl = (HD_LOW (guess));
    bignum_digit_type gh = (HD_HIGH (guess));
    bignum_digit_type v;
    bignum_digit_type pl;
    bignum_digit_type vl;
#define vh v
#define ph carry
#define diff pl
    while (v_scan < v_end)
      {
	v = (*v_scan++);
	vl = (HD_LOW (v));
	vh = (HD_HIGH (v));
	pl = ((vl * gl) + (HD_LOW (carry)));
	ph = ((vl * gh) + (vh * gl) + (HD_HIGH (pl)) + (HD_HIGH (carry)));
	diff = ((*u_scan) - (HD_CONS ((HD_LOW (ph)), (HD_LOW (pl)))));
	if (diff < 0)
	  {
	    (*u_scan++) = (diff + BIGNUM_RADIX);
	    carry = ((vh * gh) + (HD_HIGH (ph)) + 1);
	  }
	else
	  {
	    (*u_scan++) = diff;
	    carry = ((vh * gh) + (HD_HIGH (ph)));
	  }
      }
    if (carry == 0)
      return (guess);
    diff = ((*u_scan) - carry);
    if (diff < 0)
      (*u_scan) = (diff + BIGNUM_RADIX);
    else
      {
	(*u_scan) = diff;
	return (guess);
      }
#undef vh
#undef ph
#undef diff
  }
  /* Subtraction generated carry, implying guess is one too large.
     Add v back in to bring it back down. */
  v_scan = v_start;
  u_scan = u_start;
  carry = 0;
  while (v_scan < v_end)
    {
      bignum_digit_type sum = ((*v_scan++) + (*u_scan) + carry);
      if (sum < BIGNUM_RADIX)
	{
	  (*u_scan++) = sum;
	  carry = 0;
	}
      else
	{
	  (*u_scan++) = (sum - BIGNUM_RADIX);
	  carry = 1;
	}
    }
  if (carry == 1)
    {
      bignum_digit_type sum = ((*u_scan) + carry);
      (*u_scan) = ((sum < BIGNUM_RADIX) ? sum : (sum - BIGNUM_RADIX));
    }
  return (guess - 1);
}

static void
bignum_divide_unsigned_medium_denominator(bignum_type numerator,
					  bignum_digit_type denominator,
					  bignum_type * quotient,
					  bignum_type * remainder,
					  int q_negative_p,
					  int r_negative_p)
{
  bignum_length_type length_n = (BIGNUM_LENGTH (numerator));
  bignum_length_type length_q;
  bignum_type q;
  int shift = 0;
  /* Because `bignum_digit_divide' requires a normalized denominator. */
  while (denominator < (BIGNUM_RADIX / 2))
    {
      denominator <<= 1;
      shift += 1;
    }
  if (shift == 0)
    {
      length_q = length_n;
      q = (bignum_allocate (length_q, q_negative_p));
      bignum_destructive_copy (numerator, q);
    }
  else
    {
      length_q = (length_n + 1);
      q = (bignum_allocate (length_q, q_negative_p));
      bignum_destructive_normalization (numerator, q, shift);
    }
  {
    bignum_digit_type r = 0;
    bignum_digit_type * start = (BIGNUM_START_PTR (q));
    bignum_digit_type * scan = (start + length_q);
    bignum_digit_type qj;
    if (quotient != ((bignum_type *) 0))
      {
	while (start < scan)
	  {
	    r = (bignum_digit_divide (r, (*--scan), denominator, (&qj)));
	    (*scan) = qj;
	  }
	(*quotient) = (bignum_trim (q));
      }
    else
      {
	while (start < scan)
	  r = (bignum_digit_divide (r, (*--scan), denominator, (&qj)));
	BIGNUM_DEALLOCATE (q);
      }
    if (remainder != ((bignum_type *) 0))
      {
	if (shift != 0)
	  r >>= shift;
	(*remainder) = (bignum_digit_to_bignum (r, r_negative_p));
      }
  }
  return;
}

static void
bignum_destructive_normalization(bignum_type source, bignum_type target,
				 int shift_left)
{
  bignum_digit_type digit;
  bignum_digit_type * scan_source = (BIGNUM_START_PTR (source));
  bignum_digit_type carry = 0;
  bignum_digit_type * scan_target = (BIGNUM_START_PTR (target));
  bignum_digit_type * end_source = (scan_source + (BIGNUM_LENGTH (source)));
  bignum_digit_type * end_target = (scan_target + (BIGNUM_LENGTH (target)));
  int shift_right = (BIGNUM_DIGIT_LENGTH - shift_left);
  bignum_digit_type mask = ((1L << shift_right) - 1);
  while (scan_source < end_source)
    {
      digit = (*scan_source++);
      (*scan_target++) = (((digit & mask) << shift_left) | carry);
      carry = (digit >> shift_right);
    }
  if (scan_target < end_target)
    (*scan_target) = carry;
  else
    BIGNUM_ASSERT (carry == 0);
  return;
}

static void
bignum_destructive_unnormalization(bignum_type bignum, int shift_right)
{
  bignum_digit_type * start = (BIGNUM_START_PTR (bignum));
  bignum_digit_type * scan = (start + (BIGNUM_LENGTH (bignum)));
  bignum_digit_type digit;
  bignum_digit_type carry = 0;
  int shift_left = (BIGNUM_DIGIT_LENGTH - shift_right);
  bignum_digit_type mask = ((1L << shift_right) - 1);
  while (start < scan)
    {
      digit = (*--scan);
      (*scan) = ((digit >> shift_right) | carry);
      carry = ((digit & mask) << shift_left);
    }
  BIGNUM_ASSERT (carry == 0);
  return;
}

/* This is a reduced version of the division algorithm, applied to the
   case of dividing two bignum digits by one bignum digit.  It is
   assumed that the numerator, denominator are normalized. */

#define BDD_STEP(qn, j)							\
{									\
  uj = (u[j]);								\
  if (uj != v1)								\
    {									\
      uj_uj1 = (HD_CONS (uj, (u[j + 1])));				\
      guess = (uj_uj1 / v1);						\
      comparand = (HD_CONS ((uj_uj1 % v1), (u[j + 2])));		\
    }									\
  else									\
    {									\
      guess = (BIGNUM_RADIX_ROOT - 1);					\
      comparand = (HD_CONS (((u[j + 1]) + v1), (u[j + 2])));		\
    }									\
  while ((guess * v2) > comparand)					\
    {									\
      guess -= 1;							\
      comparand += (v1 << BIGNUM_HALF_DIGIT_LENGTH);			\
      if (comparand >= BIGNUM_RADIX)					\
	break;								\
    }									\
  qn = (bignum_digit_divide_subtract (v1, v2, guess, (&u[j])));		\
}

static bignum_digit_type
bignum_digit_divide(bignum_digit_type uh, bignum_digit_type ul,
		    bignum_digit_type v,
		    bignum_digit_type * q) /* return value */
{
  bignum_digit_type guess;
  bignum_digit_type comparand;
  bignum_digit_type v1 = (HD_HIGH (v));
  bignum_digit_type v2 = (HD_LOW (v));
  bignum_digit_type uj;
  bignum_digit_type uj_uj1;
  bignum_digit_type q1;
  bignum_digit_type q2;
  bignum_digit_type u [4];
  if (uh == 0)
    {
      if (ul < v)
	{
	  (*q) = 0;
	  return (ul);
	}
      else if (ul == v)
	{
	  (*q) = 1;
	  return (0);
	}
    }
  (u[0]) = (HD_HIGH (uh));
  (u[1]) = (HD_LOW (uh));
  (u[2]) = (HD_HIGH (ul));
  (u[3]) = (HD_LOW (ul));
  v1 = (HD_HIGH (v));
  v2 = (HD_LOW (v));
  BDD_STEP (q1, 0);
  BDD_STEP (q2, 1);
  (*q) = (HD_CONS (q1, q2));
  return (HD_CONS ((u[2]), (u[3])));
}

#undef BDD_STEP

#define BDDS_MULSUB(vn, un, carry_in)					\
{									\
  product = ((vn * guess) + carry_in);					\
  diff = (un - (HD_LOW (product)));					\
  if (diff < 0)								\
    {									\
      un = (diff + BIGNUM_RADIX_ROOT);					\
      carry = ((HD_HIGH (product)) + 1);				\
    }									\
  else									\
    {									\
      un = diff;							\
      carry = (HD_HIGH (product));					\
    }									\
}

#define BDDS_ADD(vn, un, carry_in)					\
{									\
  sum = (vn + un + carry_in);						\
  if (sum < BIGNUM_RADIX_ROOT)						\
    {									\
      un = sum;								\
      carry = 0;							\
    }									\
  else									\
    {									\
      un = (sum - BIGNUM_RADIX_ROOT);					\
      carry = 1;							\
    }									\
}

static bignum_digit_type
bignum_digit_divide_subtract(bignum_digit_type v1, bignum_digit_type v2,
			     bignum_digit_type guess, bignum_digit_type * u)
{
  {
    bignum_digit_type product;
    bignum_digit_type diff;
    bignum_digit_type carry;
    BDDS_MULSUB (v2, (u[2]), 0);
    BDDS_MULSUB (v1, (u[1]), carry);
    if (carry == 0)
      return (guess);
    diff = ((u[0]) - carry);
    if (diff < 0)
      (u[0]) = (diff + BIGNUM_RADIX);
    else
      {
	(u[0]) = diff;
	return (guess);
      }
  }
  {
    bignum_digit_type sum;
    bignum_digit_type carry;
    BDDS_ADD(v2, (u[2]), 0);
    BDDS_ADD(v1, (u[1]), carry);
    if (carry == 1)
      (u[0]) += 1;
  }
  return (guess - 1);
}

#undef BDDS_MULSUB
#undef BDDS_ADD

static void
bignum_divide_unsigned_small_denominator(bignum_type numerator,
					 bignum_digit_type denominator,
					 bignum_type * quotient,
					 bignum_type * remainder,
					 int q_negative_p,
					 int r_negative_p)
{
  bignum_type q = (bignum_new_sign (numerator, q_negative_p));
  bignum_digit_type r = (bignum_destructive_scale_down (q, denominator));
  (*quotient) = (bignum_trim (q));
  if (remainder != ((bignum_type *) 0))
    (*remainder) = (bignum_digit_to_bignum (r, r_negative_p));
  return;
}

/* Given (denominator > 1), it is fairly easy to show that
   (quotient_high < BIGNUM_RADIX_ROOT), after which it is easy to see
   that all digits are < BIGNUM_RADIX. */

static bignum_digit_type
bignum_destructive_scale_down(bignum_type bignum, bignum_digit_type denominator)
{
  bignum_digit_type numerator;
  bignum_digit_type remainder = 0;
  bignum_digit_type two_digits;
#define quotient_high remainder
  bignum_digit_type * start = (BIGNUM_START_PTR (bignum));
  bignum_digit_type * scan = (start + (BIGNUM_LENGTH (bignum)));
  BIGNUM_ASSERT ((denominator > 1) && (denominator < BIGNUM_RADIX_ROOT));
  while (start < scan)
    {
      two_digits = (*--scan);
      numerator = (HD_CONS (remainder, (HD_HIGH (two_digits))));
      quotient_high = (numerator / denominator);
      numerator = (HD_CONS ((numerator % denominator), (HD_LOW (two_digits))));
      (*scan) = (HD_CONS (quotient_high, (numerator / denominator)));
      remainder = (numerator % denominator);
    }
  return (remainder);
#undef quotient_high
}

static bignum_type
bignum_remainder_unsigned_small_denominator(
       bignum_type n, bignum_digit_type d, int negative_p)
{
  bignum_digit_type two_digits;
  bignum_digit_type * start = (BIGNUM_START_PTR (n));
  bignum_digit_type * scan = (start + (BIGNUM_LENGTH (n)));
  bignum_digit_type r = 0;
  BIGNUM_ASSERT ((d > 1) && (d < BIGNUM_RADIX_ROOT));
  while (start < scan)
    {
      two_digits = (*--scan);
      r =
	((HD_CONS (((HD_CONS (r, (HD_HIGH (two_digits)))) % d),
		   (HD_LOW (two_digits))))
	 % d);
    }
  return (bignum_digit_to_bignum (r, negative_p));
}

static bignum_type
bignum_digit_to_bignum(bignum_digit_type digit, int negative_p)
{
  if (digit == 0)
    return (BIGNUM_ZERO ());
  else
    {
      bignum_type result = (bignum_allocate (1, negative_p));
      (BIGNUM_REF (result, 0)) = digit;
      return (result);
    }
}

/* Allocation */

static bignum_type
bignum_allocate(bignum_length_type length, int negative_p)
{
  BIGNUM_ASSERT ((length >= 0) || (length < BIGNUM_RADIX));
  {
    bignum_type result = (BIGNUM_ALLOCATE (length));
    BIGNUM_SET_HEADER (result, length, negative_p);
    return (result);
  }
}

static bignum_type
bignum_allocate_zeroed(bignum_length_type length, int negative_p)
{
  BIGNUM_ASSERT ((length >= 0) || (length < BIGNUM_RADIX));
  {
    bignum_type result = (BIGNUM_ALLOCATE (length));
    bignum_digit_type * scan = (BIGNUM_START_PTR (result));
    bignum_digit_type * end = (scan + length);
    BIGNUM_SET_HEADER (result, length, negative_p);
    while (scan < end)
      (*scan++) = 0;
    return (result);
  }
}

static bignum_type
bignum_shorten_length(bignum_type bignum, bignum_length_type length)
{
  bignum_length_type current_length = (BIGNUM_LENGTH (bignum));
  BIGNUM_ASSERT ((length >= 0) || (length <= current_length));
  if (length < current_length)
    {
      BIGNUM_SET_HEADER
	(bignum, length, ((length != 0) && (BIGNUM_NEGATIVE_P (bignum))));
      BIGNUM_REDUCE_LENGTH (bignum, bignum, length);
    }
  return (bignum);
}

static bignum_type
bignum_trim(bignum_type bignum)
{
  bignum_digit_type * start = (BIGNUM_START_PTR (bignum));
  bignum_digit_type * end = (start + (BIGNUM_LENGTH (bignum)));
  bignum_digit_type * scan = end;
  while ((start <= scan) && ((*--scan) == 0))
    ;
  scan += 1;
  if (scan < end)
    {
      bignum_length_type length = (scan - start);
      BIGNUM_SET_HEADER
	(bignum, length, ((length != 0) && (BIGNUM_NEGATIVE_P (bignum))));
      BIGNUM_REDUCE_LENGTH (bignum, bignum, length);
    }
  return (bignum);
}

/* Copying */

static bignum_type
bignum_copy(bignum_type source)
{
  bignum_type target =
    (bignum_allocate ((BIGNUM_LENGTH (source)), (BIGNUM_NEGATIVE_P (source))));
  bignum_destructive_copy (source, target);
  return (target);
}

static bignum_type
bignum_new_sign(bignum_type bignum, int negative_p)
{
  bignum_type result =
    (bignum_allocate ((BIGNUM_LENGTH (bignum)), negative_p));
  bignum_destructive_copy (bignum, result);
  return (result);
}

static bignum_type
bignum_maybe_new_sign(bignum_type bignum, int negative_p)
{
#ifndef BIGNUM_FORCE_NEW_RESULTS
  if ((BIGNUM_NEGATIVE_P (bignum)) ? negative_p : (! negative_p))
    return (bignum);
  else
#endif /* not BIGNUM_FORCE_NEW_RESULTS */
    {
      bignum_type result =
	(bignum_allocate ((BIGNUM_LENGTH (bignum)), negative_p));
      bignum_destructive_copy (bignum, result);
      return (result);
    }
}

static void
bignum_destructive_copy(bignum_type source, bignum_type target)
{
  bignum_digit_type * scan_source = (BIGNUM_START_PTR (source));
  bignum_digit_type * end_source =
    (scan_source + (BIGNUM_LENGTH (source)));
  bignum_digit_type * scan_target = (BIGNUM_START_PTR (target));
  while (scan_source < end_source)
    (*scan_target++) = (*scan_source++);
  return;
}

/* Unused
static void
bignum_destructive_zero(bignum_type bignum)
{
  bignum_digit_type * scan = (BIGNUM_START_PTR (bignum));
  bignum_digit_type * end = (scan + (BIGNUM_LENGTH (bignum)));
  while (scan < end)
    (*scan++) = 0;
  return;
}
*/

/*
 * Added bitwise operations (and oddp).
 */

int
s48_bignum_oddp (bignum_type bignum)
{
  return (BIGNUM_LENGTH (bignum) > 0) && (BIGNUM_REF (bignum, 0) & 1);
}

bignum_type
s48_bignum_bitwise_not(bignum_type x)
{
  return s48_bignum_subtract(BIGNUM_ONE(1), x);
}

bignum_type
s48_bignum_arithmetic_shift(bignum_type arg1, long n)
{
  if (BIGNUM_NEGATIVE_P(arg1) && n < 0)
    return
      s48_bignum_bitwise_not(bignum_magnitude_ash(s48_bignum_bitwise_not(arg1),
						  n));
  else
    return bignum_magnitude_ash(arg1, n);
}

/*
 * This uses a `long'-returning bignum_length_in_bits() which we don't have.
long
s48_bignum_integer_length(bignum_type arg1)
{
 return((BIGNUM_NEGATIVE_P (arg1)) 
        ? bignum_length_in_bits (s48_bignum_bitwise_not (arg1))
	: bignum_length_in_bits (arg1));
}
*/

long
s48_bignum_bit_count(bignum_type arg1)
{
 return((BIGNUM_NEGATIVE_P (arg1)) 
        ? bignum_unsigned_logcount (s48_bignum_bitwise_not (arg1))
	: bignum_unsigned_logcount (arg1));
}

#define AND_OP 0
#define IOR_OP 1
#define XOR_OP 2

bignum_type
s48_bignum_bitwise_and(bignum_type arg1, bignum_type arg2)
{
  return(
	 (BIGNUM_NEGATIVE_P (arg1))
	 ? (BIGNUM_NEGATIVE_P (arg2))
	   ? bignum_negneg_bitwise_op(AND_OP, arg1, arg2)
	   : bignum_posneg_bitwise_op(AND_OP, arg2, arg1)
	 : (BIGNUM_NEGATIVE_P (arg2))
	   ? bignum_posneg_bitwise_op(AND_OP, arg1, arg2)
	   : bignum_pospos_bitwise_op(AND_OP, arg1, arg2)
	 );
}

bignum_type
s48_bignum_bitwise_ior(bignum_type arg1, bignum_type arg2)
{
  return(
	 (BIGNUM_NEGATIVE_P (arg1))
	 ? (BIGNUM_NEGATIVE_P (arg2))
	   ? bignum_negneg_bitwise_op(IOR_OP, arg1, arg2)
	   : bignum_posneg_bitwise_op(IOR_OP, arg2, arg1)
	 : (BIGNUM_NEGATIVE_P (arg2))
	   ? bignum_posneg_bitwise_op(IOR_OP, arg1, arg2)
	   : bignum_pospos_bitwise_op(IOR_OP, arg1, arg2)
	 );
}

bignum_type
s48_bignum_bitwise_xor(bignum_type arg1, bignum_type arg2)
{
  return(
	 (BIGNUM_NEGATIVE_P (arg1))
	 ? (BIGNUM_NEGATIVE_P (arg2))
	   ? bignum_negneg_bitwise_op(XOR_OP, arg1, arg2)
	   : bignum_posneg_bitwise_op(XOR_OP, arg2, arg1)
	 : (BIGNUM_NEGATIVE_P (arg2))
	   ? bignum_posneg_bitwise_op(XOR_OP, arg1, arg2)
	   : bignum_pospos_bitwise_op(XOR_OP, arg1, arg2)
	 );
}

/* ash for the magnitude */
/* assume arg1 is a big number, n is a long */
static bignum_type
bignum_magnitude_ash(bignum_type arg1, long n)
{
  bignum_type result;
  bignum_digit_type *scan1;
  bignum_digit_type *scanr;
  bignum_digit_type *end;

  long digit_offset,bit_offset;

  if (BIGNUM_ZERO_P (arg1)) return (arg1);

  if (n > 0) {
    digit_offset = n / BIGNUM_DIGIT_LENGTH;
    bit_offset =   n % BIGNUM_DIGIT_LENGTH;
    
    result = bignum_allocate_zeroed (BIGNUM_LENGTH (arg1) + digit_offset + 1,
				     BIGNUM_NEGATIVE_P(arg1));

    scanr = BIGNUM_START_PTR (result) + digit_offset;
    scan1 = BIGNUM_START_PTR (arg1);
    end = scan1 + BIGNUM_LENGTH (arg1);
    
    while (scan1 < end) {
      *scanr = *scanr | (*scan1 & BIGNUM_DIGIT_MASK) << bit_offset;
      *scanr = *scanr & BIGNUM_DIGIT_MASK;
      scanr++;
      *scanr = *scan1++ >> (BIGNUM_DIGIT_LENGTH - bit_offset);
      *scanr = *scanr & BIGNUM_DIGIT_MASK;
    }
  }
  else if (n < 0
	   && (-n >= (BIGNUM_LENGTH (arg1) * (bignum_length_type) BIGNUM_DIGIT_LENGTH)))
    result = BIGNUM_ZERO ();

  else if (n < 0) {
    digit_offset = -n / BIGNUM_DIGIT_LENGTH;
    bit_offset =   -n % BIGNUM_DIGIT_LENGTH;
    
    result = bignum_allocate_zeroed (BIGNUM_LENGTH (arg1) - digit_offset,
				     BIGNUM_NEGATIVE_P(arg1));
    
    scanr = BIGNUM_START_PTR (result);
    scan1 = BIGNUM_START_PTR (arg1) + digit_offset;
    end = scanr + BIGNUM_LENGTH (result) - 1;
    
    while (scanr < end) {
      *scanr =  (*scan1++ & BIGNUM_DIGIT_MASK) >> bit_offset ;
      *scanr = (*scanr | 
	*scan1 << (BIGNUM_DIGIT_LENGTH - bit_offset)) & BIGNUM_DIGIT_MASK;
      scanr++;
    }
    *scanr =  (*scan1++ & BIGNUM_DIGIT_MASK) >> bit_offset ;
  }
  else if (n == 0) result = arg1;
  
  return (bignum_trim (result));
}

static bignum_type
bignum_pospos_bitwise_op(int op, bignum_type arg1, bignum_type arg2)
{
  bignum_type result;
  bignum_length_type max_length;

  bignum_digit_type *scan1, *end1, digit1;
  bignum_digit_type *scan2, *end2, digit2;
  bignum_digit_type *scanr, *endr;

  max_length =  (BIGNUM_LENGTH(arg1) > BIGNUM_LENGTH(arg2))
               ? BIGNUM_LENGTH(arg1) : BIGNUM_LENGTH(arg2);

  result = bignum_allocate(max_length, 0);

  scanr = BIGNUM_START_PTR(result);
  scan1 = BIGNUM_START_PTR(arg1);
  scan2 = BIGNUM_START_PTR(arg2);
  endr = scanr + max_length;
  end1 = scan1 + BIGNUM_LENGTH(arg1);
  end2 = scan2 + BIGNUM_LENGTH(arg2);

  while (scanr < endr) {
    digit1 = (scan1 < end1) ? *scan1++ : 0;
    digit2 = (scan2 < end2) ? *scan2++ : 0;
    /*
    fprintf(stderr, "[pospos op = %d, i = %ld, d1 = %lx, d2 = %lx]\n",
	    op, endr - scanr, digit1, digit2);
	    */
    *scanr++ = (op == 0) ? digit1 & digit2 :
               (op == 1) ? digit1 | digit2 :
                           digit1 ^ digit2;
  }
  return bignum_trim(result);
}

static bignum_type
bignum_posneg_bitwise_op(int op, bignum_type arg1, bignum_type arg2)
{
  bignum_type result;
  bignum_length_type max_length;

  bignum_digit_type *scan1, *end1, digit1;
  bignum_digit_type *scan2, *end2, digit2, carry2;
  bignum_digit_type *scanr, *endr;

  char neg_p = op == IOR_OP || op == XOR_OP;

  max_length =  (BIGNUM_LENGTH(arg1) > BIGNUM_LENGTH(arg2) + 1)
               ? BIGNUM_LENGTH(arg1) : BIGNUM_LENGTH(arg2) + 1;

  result = bignum_allocate(max_length, neg_p);

  scanr = BIGNUM_START_PTR(result);
  scan1 = BIGNUM_START_PTR(arg1);
  scan2 = BIGNUM_START_PTR(arg2);
  endr = scanr + max_length;
  end1 = scan1 + BIGNUM_LENGTH(arg1);
  end2 = scan2 + BIGNUM_LENGTH(arg2);

  carry2 = 1;

  while (scanr < endr) {
    digit1 = (scan1 < end1) ? *scan1++ : 0;
    digit2 = (~((scan2 < end2) ? *scan2++ : 0) & BIGNUM_DIGIT_MASK)
             + carry2;

    if (digit2 < BIGNUM_RADIX)
      carry2 = 0;
    else
      {
	digit2 = (digit2 - BIGNUM_RADIX);
	carry2 = 1;
      }
    
    *scanr++ = (op == AND_OP) ? digit1 & digit2 :
               (op == IOR_OP) ? digit1 | digit2 :
                                digit1 ^ digit2;
  }
  
  if (neg_p)
    bignum_negate_magnitude(result);

  return bignum_trim(result);
}

static bignum_type
bignum_negneg_bitwise_op(int op, bignum_type arg1, bignum_type arg2)
{
  bignum_type result;
  bignum_length_type max_length;

  bignum_digit_type *scan1, *end1, digit1, carry1;
  bignum_digit_type *scan2, *end2, digit2, carry2;
  bignum_digit_type *scanr, *endr;

  char neg_p = op == AND_OP || op == IOR_OP;

  max_length =  (BIGNUM_LENGTH(arg1) > BIGNUM_LENGTH(arg2))
               ? BIGNUM_LENGTH(arg1) + 1 : BIGNUM_LENGTH(arg2) + 1;

  result = bignum_allocate(max_length, neg_p);

  scanr = BIGNUM_START_PTR(result);
  scan1 = BIGNUM_START_PTR(arg1);
  scan2 = BIGNUM_START_PTR(arg2);
  endr = scanr + max_length;
  end1 = scan1 + BIGNUM_LENGTH(arg1);
  end2 = scan2 + BIGNUM_LENGTH(arg2);

  carry1 = 1;
  carry2 = 1;

  while (scanr < endr) {
    digit1 = (~((scan1 < end1) ? *scan1++ : 0) & BIGNUM_DIGIT_MASK) + carry1;
    digit2 = (~((scan2 < end2) ? *scan2++ : 0) & BIGNUM_DIGIT_MASK) + carry2;

    if (digit1 < BIGNUM_RADIX)
      carry1 = 0;
    else
      {
	digit1 = (digit1 - BIGNUM_RADIX);
	carry1 = 1;
      }
    
    if (digit2 < BIGNUM_RADIX)
      carry2 = 0;
    else
      {
	digit2 = (digit2 - BIGNUM_RADIX);
	carry2 = 1;
      }
    
    *scanr++ = (op == 0) ? digit1 & digit2 :
               (op == 1) ? digit1 | digit2 :
                           digit1 ^ digit2;
  }

  if (neg_p)
    bignum_negate_magnitude(result);

  return bignum_trim(result);
}

static void
bignum_negate_magnitude(bignum_type arg)
{
  bignum_digit_type *scan;
  bignum_digit_type *end;
  bignum_digit_type digit;
  bignum_digit_type carry;

  scan = BIGNUM_START_PTR(arg);
  end = scan + BIGNUM_LENGTH(arg);

  carry = 1;

  while (scan < end) {
    digit = (~*scan & BIGNUM_DIGIT_MASK) + carry;

    if (digit < BIGNUM_RADIX)
      carry = 0;
    else
      {
	digit = (digit - BIGNUM_RADIX);
	carry = 1;
      }
    
    *scan++ = digit;
  }
}

static long
bignum_unsigned_logcount(bignum_type arg)
{

  bignum_digit_type *scan;
  bignum_digit_type *end;
  bignum_digit_type digit;

  /* sufficient for any reasonable big number */
  long result;
  int i;

  if (BIGNUM_ZERO_P (arg)) return (0L);

  scan = BIGNUM_START_PTR (arg);
  end = scan + BIGNUM_LENGTH (arg);
  result = 0L;
    
  while (scan < end) {
      digit = *scan++ & BIGNUM_DIGIT_MASK;
      for (i = 0; i++ < BIGNUM_DIGIT_LENGTH; digit = digit >> 1L)
	  result += digit & 1L;
  }

  return (result);
}

int
bignum_logbitp(int shift, bignum_type arg)
{
  return((BIGNUM_NEGATIVE_P (arg)) 
	 ? !bignum_unsigned_logbitp (shift, s48_bignum_bitwise_not (arg))
	 : bignum_unsigned_logbitp (shift,arg));
}

static int
bignum_unsigned_logbitp(int shift, bignum_type bignum)
{
  bignum_length_type len = (BIGNUM_LENGTH (bignum));
  bignum_digit_type digit;
  int index = shift / BIGNUM_DIGIT_LENGTH;
  int p;
  if (index >= len)
    return 0;
  digit = (BIGNUM_REF (bignum, index));
  p = shift % BIGNUM_DIGIT_LENGTH;
  return digit & (1 << p);
}