[perl5.git] / lib / bigint.pl

package bigint;
#
# This library is no longer being maintained, and is included for backward
# compatibility with Perl 4 programs which may require it.
#
# In particular, this should not be used as an example of modern Perl
# programming techniques.
# This legacy library is deprecated and will be removed in a future
# release of perl.
#
# Suggested alternative:  Math::BigInt

# arbitrary size integer math package
#
# by Mark Biggar
#
# Canonical Big integer value are strings of the form
#       /^[+-]\d+$/ with leading zeros suppressed
# Input values to these routines may be strings of the form
#       /^\s*[+-]?[\d\s]+$/.
# Examples:
#   '+0'                            canonical zero value
#   '   -123 123 123'               canonical value '-123123123'
#   '1 23 456 7890'                 canonical value '+1234567890'
# Output values always in canonical form
#
# Actual math is done in an internal format consisting of an array
#   whose first element is the sign (/^[+-]$/) and whose remaining 
#   elements are base 100000 digits with the least significant digit first.
# The string 'NaN' is used to represent the result when input arguments 
#   are not numbers, as well as the result of dividing by zero
#
# routines provided are:
#
#   bneg(BINT) return BINT              negation
#   babs(BINT) return BINT              absolute value
#   bcmp(BINT,BINT) return CODE         compare numbers (undef,<0,=0,>0)
#   badd(BINT,BINT) return BINT         addition
#   bsub(BINT,BINT) return BINT         subtraction
#   bmul(BINT,BINT) return BINT         multiplication
#   bdiv(BINT,BINT) return (BINT,BINT)  division (quo,rem) just quo if scalar
#   bmod(BINT,BINT) return BINT         modulus
#   bgcd(BINT,BINT) return BINT         greatest common divisor
#   bnorm(BINT) return BINT             normalization
#

# overcome a floating point problem on certain osnames (posix-bc, os390)
BEGIN {
    my $x = 100000.0;
    my $use_mult = int($x*1e-5)*1e5 == $x ? 1 : 0;
}

$zero = 0;

\f
# normalize string form of number.   Strip leading zeros.  Strip any
#   white space and add a sign, if missing.
# Strings that are not numbers result the value 'NaN'.

sub main'bnorm { #(num_str) return num_str
    local($_) = @_;
    s/\s+//g;                           # strip white space
    if (s/^([+-]?)0*(\d+)$/$1$2/) {     # test if number
	substr($_,0,0) = '+' unless $1; # Add missing sign
	s/^-0/+0/;
	$_;
    } else {
	'NaN';
    }
}

# Convert a number from string format to internal base 100000 format.
#   Assumes normalized value as input.
sub internal { #(num_str) return int_num_array
    local($d) = @_;
    ($is,$il) = (substr($d,0,1),length($d)-2);
    substr($d,0,1) = '';
    ($is, reverse(unpack("a" . ($il%5+1) . ("a5" x ($il/5)), $d)));
}

# Convert a number from internal base 100000 format to string format.
#   This routine scribbles all over input array.
sub external { #(int_num_array) return num_str
    $es = shift;
    grep($_ > 9999 || ($_ = substr('0000'.$_,-5)), @_);   # zero pad
    &'bnorm(join('', $es, reverse(@_)));    # reverse concat and normalize
}

# Negate input value.
sub main'bneg { #(num_str) return num_str
    local($_) = &'bnorm(@_);
    vec($_,0,8) ^= ord('+') ^ ord('-') unless $_ eq '+0';
    s/^./N/ unless /^[-+]/; # works both in ASCII and EBCDIC
    $_;
}

# Returns the absolute value of the input.
sub main'babs { #(num_str) return num_str
    &abs(&'bnorm(@_));
}

sub abs { # post-normalized abs for internal use
    local($_) = @_;
    s/^-/+/;
    $_;
}
\f
# Compares 2 values.  Returns one of undef, <0, =0, >0. (suitable for sort)
sub main'bcmp { #(num_str, num_str) return cond_code
    local($x,$y) = (&'bnorm($_[0]),&'bnorm($_[1]));
    if ($x eq 'NaN') {
	undef;
    } elsif ($y eq 'NaN') {
	undef;
    } else {
	&cmp($x,$y);
    }
}

sub cmp { # post-normalized compare for internal use
    local($cx, $cy) = @_;
    return 0 if ($cx eq $cy);

    local($sx, $sy) = (substr($cx, 0, 1), substr($cy, 0, 1));
    local($ld);

    if ($sx eq '+') {
      return  1 if ($sy eq '-' || $cy eq '+0');
      $ld = length($cx) - length($cy);
      return $ld if ($ld);
      return $cx cmp $cy;
    } else { # $sx eq '-'
      return -1 if ($sy eq '+');
      $ld = length($cy) - length($cx);
      return $ld if ($ld);
      return $cy cmp $cx;
    }

}

sub main'badd { #(num_str, num_str) return num_str
    local(*x, *y); ($x, $y) = (&'bnorm($_[0]),&'bnorm($_[1]));
    if ($x eq 'NaN') {
	'NaN';
    } elsif ($y eq 'NaN') {
	'NaN';
    } else {
	@x = &internal($x);             # convert to internal form
	@y = &internal($y);
	local($sx, $sy) = (shift @x, shift @y); # get signs
	if ($sx eq $sy) {
	    &external($sx, &add(*x, *y)); # if same sign add
	} else {
	    ($x, $y) = (&abs($x),&abs($y)); # make abs
	    if (&cmp($y,$x) > 0) {
		&external($sy, &sub(*y, *x));
	    } else {
		&external($sx, &sub(*x, *y));
	    }
	}
    }
}

sub main'bsub { #(num_str, num_str) return num_str
    &'badd($_[0],&'bneg($_[1]));    
}

# GCD -- Euclids algorithm Knuth Vol 2 pg 296
sub main'bgcd { #(num_str, num_str) return num_str
    local($x,$y) = (&'bnorm($_[0]),&'bnorm($_[1]));
    if ($x eq 'NaN' || $y eq 'NaN') {
	'NaN';
    } else {
	($x, $y) = ($y,&'bmod($x,$y)) while $y ne '+0';
	$x;
    }
}
\f
# routine to add two base 1e5 numbers
#   stolen from Knuth Vol 2 Algorithm A pg 231
#   there are separate routines to add and sub as per Kunth pg 233
sub add { #(int_num_array, int_num_array) return int_num_array
    local(*x, *y) = @_;
    $car = 0;
    for $x (@x) {
	last unless @y || $car;
	$x -= 1e5 if $car = (($x += shift(@y) + $car) >= 1e5) ? 1 : 0;
    }
    for $y (@y) {
	last unless $car;
	$y -= 1e5 if $car = (($y += $car) >= 1e5) ? 1 : 0;
    }
    (@x, @y, $car);
}

# subtract base 1e5 numbers -- stolen from Knuth Vol 2 pg 232, $x > $y
sub sub { #(int_num_array, int_num_array) return int_num_array
    local(*sx, *sy) = @_;
    $bar = 0;
    for $sx (@sx) {
	last unless @y || $bar;
	$sx += 1e5 if $bar = (($sx -= shift(@sy) + $bar) < 0);
    }
    @sx;
}

# multiply two numbers -- stolen from Knuth Vol 2 pg 233
sub main'bmul { #(num_str, num_str) return num_str
    local(*x, *y); ($x, $y) = (&'bnorm($_[0]), &'bnorm($_[1]));
    if ($x eq 'NaN') {
	'NaN';
    } elsif ($y eq 'NaN') {
	'NaN';
    } else {
	@x = &internal($x);
	@y = &internal($y);
	local($signr) = (shift @x ne shift @y) ? '-' : '+';
	@prod = ();
	for $x (@x) {
	    ($car, $cty) = (0, 0);
	    for $y (@y) {
		$prod = $x * $y + $prod[$cty] + $car;
                if ($use_mult) {
		    $prod[$cty++] =
		        $prod - ($car = int($prod * 1e-5)) * 1e5;
                }
                else {
		    $prod[$cty++] =
		        $prod - ($car = int($prod / 1e5)) * 1e5;
                }
	    }
	    $prod[$cty] += $car if $car;
	    $x = shift @prod;
	}
	&external($signr, @x, @prod);
    }
}

# modulus
sub main'bmod { #(num_str, num_str) return num_str
    (&'bdiv(@_))[1];
}
\f
sub main'bdiv { #(dividend: num_str, divisor: num_str) return num_str
    local (*x, *y); ($x, $y) = (&'bnorm($_[0]), &'bnorm($_[1]));
    return wantarray ? ('NaN','NaN') : 'NaN'
	if ($x eq 'NaN' || $y eq 'NaN' || $y eq '+0');
    return wantarray ? ('+0',$x) : '+0' if (&cmp(&abs($x),&abs($y)) < 0);
    @x = &internal($x); @y = &internal($y);
    $srem = $y[0];
    $sr = (shift @x ne shift @y) ? '-' : '+';
    $car = $bar = $prd = 0;
    if (($dd = int(1e5/($y[$#y]+1))) != 1) {
	for $x (@x) {
	    $x = $x * $dd + $car;
            if ($use_mult) {
	    $x -= ($car = int($x * 1e-5)) * 1e5;
            }
            else {
	    $x -= ($car = int($x / 1e5)) * 1e5;
            }
	}
	push(@x, $car); $car = 0;
	for $y (@y) {
	    $y = $y * $dd + $car;
            if ($use_mult) {
	    $y -= ($car = int($y * 1e-5)) * 1e5;
            }
            else {
	    $y -= ($car = int($y / 1e5)) * 1e5;
            }
	}
    }
    else {
	push(@x, 0);
    }
    @q = (); ($v2,$v1) = @y[-2,-1];
    while ($#x > $#y) {
	($u2,$u1,$u0) = @x[-3..-1];
	$q = (($u0 == $v1) ? 99999 : int(($u0*1e5+$u1)/$v1));
	--$q while ($v2*$q > ($u0*1e5+$u1-$q*$v1)*1e5+$u2);
	if ($q) {
	    ($car, $bar) = (0,0);
	    for ($y = 0, $x = $#x-$#y-1; $y <= $#y; ++$y,++$x) {
		$prd = $q * $y[$y] + $car;
                if ($use_mult) {
		$prd -= ($car = int($prd * 1e-5)) * 1e5;
                }
                else {
		$prd -= ($car = int($prd / 1e5)) * 1e5;
                }
		$x[$x] += 1e5 if ($bar = (($x[$x] -= $prd + $bar) < 0));
	    }
	    if ($x[$#x] < $car + $bar) {
		$car = 0; --$q;
		for ($y = 0, $x = $#x-$#y-1; $y <= $#y; ++$y,++$x) {
		    $x[$x] -= 1e5
			if ($car = (($x[$x] += $y[$y] + $car) > 1e5));
		}
	    }   
	}
	pop(@x); unshift(@q, $q);
    }
    if (wantarray) {
	@d = ();
	if ($dd != 1) {
	    $car = 0;
	    for $x (reverse @x) {
		$prd = $car * 1e5 + $x;
		$car = $prd - ($tmp = int($prd / $dd)) * $dd;
		unshift(@d, $tmp);
	    }
	}
	else {
	    @d = @x;
	}
	(&external($sr, @q), &external($srem, @d, $zero));
    } else {
	&external($sr, @q);
    }
}
1;
Commit	Line	Data
5303340c	1	package bigint;
a6d71656 GS	2	#
	3	# This library is no longer being maintained, and is included for backward
	4	# compatibility with Perl 4 programs which may require it.
	5	#
	6	# In particular, this should not be used as an example of modern Perl
	7	# programming techniques.
b6dbc3e0 S	8	# This legacy library is deprecated and will be removed in a future
b6dbc3e0 S	9	# release of perl.
a6d71656 GS	10	#
a6d71656 GS	11	# Suggested alternative: Math::BigInt
b6dbc3e0	12
5303340c LW	13	# arbitrary size integer math package
	14	#
	15	# by Mark Biggar
	16	#
	17	# Canonical Big integer value are strings of the form
	18	# /^[+-]\d+$/ with leading zeros suppressed
	19	# Input values to these routines may be strings of the form
	20	# /^\s*[+-]?[\d\s]+$/.
	21	# Examples:
	22	# '+0' canonical zero value
	23	# ' -123 123 123' canonical value '-123123123'
	24	# '1 23 456 7890' canonical value '+1234567890'
d1be9408	25	# Output values always in canonical form
5303340c LW	26	#
	27	# Actual math is done in an internal format consisting of an array
	28	# whose first element is the sign (/^[+-]$/) and whose remaining
	29	# elements are base 100000 digits with the least significant digit first.
	30	# The string 'NaN' is used to represent the result when input arguments
	31	# are not numbers, as well as the result of dividing by zero
	32	#
	33	# routines provided are:
	34	#
	35	# bneg(BINT) return BINT negation
	36	# babs(BINT) return BINT absolute value
	37	# bcmp(BINT,BINT) return CODE compare numbers (undef,<0,=0,>0)
	38	# badd(BINT,BINT) return BINT addition
	39	# bsub(BINT,BINT) return BINT subtraction
	40	# bmul(BINT,BINT) return BINT multiplication
	41	# bdiv(BINT,BINT) return (BINT,BINT) division (quo,rem) just quo if scalar
	42	# bmod(BINT,BINT) return BINT modulus
	43	# bgcd(BINT,BINT) return BINT greatest common divisor
	44	# bnorm(BINT) return BINT normalization
	45	#
8990e307	46
1f45ae4a PP	47	# overcome a floating point problem on certain osnames (posix-bc, os390)
	48	BEGIN {
	49	my $x = 100000.0;
	50	my $use_mult = int($x1e-5)1e5 == $x ? 1 : 0;
	51	}
	52
8990e307 LW	53	$zero = 0;
8990e307 LW	54
5303340c LW	55	\f
	56	# normalize string form of number. Strip leading zeros. Strip any
	57	# white space and add a sign, if missing.
	58	# Strings that are not numbers result the value 'NaN'.
8990e307	59
5303340c LW	60	sub main'bnorm { #(num_str) return num_str
	61	local($_) = @_;
	62	s/\s+//g; # strip white space
	63	if (s/^([+-]?)0*(\d+)$/$1$2/) { # test if number
859172fe	64	substr($_,0,0) = '+' unless $1; # Add missing sign
5303340c LW	65	s/^-0/+0/;
	66	$_;
	67	} else {
	68	'NaN';
	69	}
	70	}
	71
	72	# Convert a number from string format to internal base 100000 format.
	73	# Assumes normalized value as input.
	74	sub internal { #(num_str) return int_num_array
	75	local($d) = @_;
859172fe Z	76	($is,$il) = (substr($d,0,1),length($d)-2);
859172fe Z	77	substr($d,0,1) = '';
5303340c LW	78	($is, reverse(unpack("a" . ($il%5+1) . ("a5" x ($il/5)), $d)));
	79	}
	80
	81	# Convert a number from internal base 100000 format to string format.
	82	# This routine scribbles all over input array.
	83	sub external { #(int_num_array) return num_str
	84	$es = shift;
	85	grep($_ > 9999 \|\| ($_ = substr('0000'.$_,-5)), @_); # zero pad
	86	&'bnorm(join('', $es, reverse(@_))); # reverse concat and normalize
	87	}
	88
	89	# Negate input value.
	90	sub main'bneg { #(num_str) return num_str
	91	local($_) = &'bnorm(@_);
	92	vec($_,0,8) ^= ord('+') ^ ord('-') unless $_ eq '+0';
9d116dd7	93	s/^./N/ unless /^[-+]/; # works both in ASCII and EBCDIC
5303340c LW	94	$_;
	95	}
	96
	97	# Returns the absolute value of the input.
	98	sub main'babs { #(num_str) return num_str
	99	&abs(&'bnorm(@_));
	100	}
	101
	102	sub abs { # post-normalized abs for internal use
	103	local($_) = @_;
	104	s/^-/+/;
	105	$_;
	106	}
	107	\f
	108	# Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
	109	sub main'bcmp { #(num_str, num_str) return cond_code
859172fe	110	local($x,$y) = (&'bnorm($_[0]),&'bnorm($_[1]));
5303340c LW	111	if ($x eq 'NaN') {
	112	undef;
	113	} elsif ($y eq 'NaN') {
	114	undef;
	115	} else {
	116	&cmp($x,$y);
	117	}
	118	}
	119
	120	sub cmp { # post-normalized compare for internal use
	121	local($cx, $cy) = @_;
1e2e1ae8 JH	122	return 0 if ($cx eq $cy);
	123
	124	local($sx, $sy) = (substr($cx, 0, 1), substr($cy, 0, 1));
	125	local($ld);
	126
	127	if ($sx eq '+') {
	128	return 1 if ($sy eq '-' \|\| $cy eq '+0');
	129	$ld = length($cx) - length($cy);
	130	return $ld if ($ld);
	131	return $cx cmp $cy;
	132	} else { # $sx eq '-'
	133	return -1 if ($sy eq '+');
	134	$ld = length($cy) - length($cx);
	135	return $ld if ($ld);
	136	return $cy cmp $cx;
	137	}
	138
5303340c LW	139	}
	140
	141	sub main'badd { #(num_str, num_str) return num_str
859172fe	142	local(x, y); ($x, $y) = (&'bnorm($_[0]),&'bnorm($_[1]));
5303340c LW	143	if ($x eq 'NaN') {
	144	'NaN';
	145	} elsif ($y eq 'NaN') {
	146	'NaN';
	147	} else {
	148	@x = &internal($x); # convert to internal form
	149	@y = &internal($y);
	150	local($sx, $sy) = (shift @x, shift @y); # get signs
	151	if ($sx eq $sy) {
	152	&external($sx, &add(x, y)); # if same sign add
	153	} else {
	154	($x, $y) = (&abs($x),&abs($y)); # make abs
	155	if (&cmp($y,$x) > 0) {
	156	&external($sy, &sub(y, x));
	157	} else {
	158	&external($sx, &sub(x, y));
	159	}
	160	}
	161	}
	162	}
	163
	164	sub main'bsub { #(num_str, num_str) return num_str
859172fe	165	&'badd($_[0],&'bneg($_[1]));
5303340c LW	166	}
	167
	168	# GCD -- Euclids algorithm Knuth Vol 2 pg 296
	169	sub main'bgcd { #(num_str, num_str) return num_str
859172fe	170	local($x,$y) = (&'bnorm($_[0]),&'bnorm($_[1]));
68decaef	171	if ($x eq 'NaN' \|\| $y eq 'NaN') {
5303340c	172	'NaN';
68decaef	173	} else {
5303340c LW	174	($x, $y) = ($y,&'bmod($x,$y)) while $y ne '+0';
	175	$x;
	176	}
	177	}
	178	\f
68decaef	179	# routine to add two base 1e5 numbers
5303340c LW	180	# stolen from Knuth Vol 2 Algorithm A pg 231
	181	# there are separate routines to add and sub as per Kunth pg 233
	182	sub add { #(int_num_array, int_num_array) return int_num_array
	183	local(x, y) = @_;
	184	$car = 0;
	185	for $x (@x) {
	186	last unless @y \|\| $car;
55497cff	187	$x -= 1e5 if $car = (($x += shift(@y) + $car) >= 1e5) ? 1 : 0;
5303340c LW	188	}
	189	for $y (@y) {
	190	last unless $car;
55497cff	191	$y -= 1e5 if $car = (($y += $car) >= 1e5) ? 1 : 0;
5303340c LW	192	}
	193	(@x, @y, $car);
	194	}
	195
68decaef	196	# subtract base 1e5 numbers -- stolen from Knuth Vol 2 pg 232, $x > $y
5303340c LW	197	sub sub { #(int_num_array, int_num_array) return int_num_array
	198	local(sx, sy) = @_;
	199	$bar = 0;
	200	for $sx (@sx) {
	201	last unless @y \|\| $bar;
e334a159	202	$sx += 1e5 if $bar = (($sx -= shift(@sy) + $bar) < 0);
5303340c LW	203	}
	204	@sx;
	205	}
	206
	207	# multiply two numbers -- stolen from Knuth Vol 2 pg 233
	208	sub main'bmul { #(num_str, num_str) return num_str
859172fe	209	local(x, y); ($x, $y) = (&'bnorm($_[0]), &'bnorm($_[1]));
5303340c LW	210	if ($x eq 'NaN') {
	211	'NaN';
	212	} elsif ($y eq 'NaN') {
	213	'NaN';
	214	} else {
	215	@x = &internal($x);
	216	@y = &internal($y);
	217	local($signr) = (shift @x ne shift @y) ? '-' : '+';
	218	@prod = ();
	219	for $x (@x) {
859172fe	220	($car, $cty) = (0, 0);
5303340c LW	221	for $y (@y) {
5303340c LW	222	$prod = $x * $y + $prod[$cty] + $car;
1f45ae4a PP	223	if ($use_mult) {
	224	$prod[$cty++] =
	225	$prod - ($car = int($prod * 1e-5)) * 1e5;
	226	}
	227	else {
	228	$prod[$cty++] =
	229	$prod - ($car = int($prod / 1e5)) * 1e5;
	230	}
5303340c LW	231	}
	232	$prod[$cty] += $car if $car;
	233	$x = shift @prod;
	234	}
	235	&external($signr, @x, @prod);
	236	}
	237	}
	238
	239	# modulus
	240	sub main'bmod { #(num_str, num_str) return num_str
859172fe	241	(&'bdiv(@_))[1];
5303340c LW	242	}
	243	\f
	244	sub main'bdiv { #(dividend: num_str, divisor: num_str) return num_str
859172fe	245	local (x, y); ($x, $y) = (&'bnorm($_[0]), &'bnorm($_[1]));
5303340c LW	246	return wantarray ? ('NaN','NaN') : 'NaN'
	247	if ($x eq 'NaN' \|\| $y eq 'NaN' \|\| $y eq '+0');
	248	return wantarray ? ('+0',$x) : '+0' if (&cmp(&abs($x),&abs($y)) < 0);
	249	@x = &internal($x); @y = &internal($y);
859172fe	250	$srem = $y[0];
5303340c LW	251	$sr = (shift @x ne shift @y) ? '-' : '+';
5303340c LW	252	$car = $bar = $prd = 0;
68decaef	253	if (($dd = int(1e5/($y[$#y]+1))) != 1) {
5303340c LW	254	for $x (@x) {
5303340c LW	255	$x = $x * $dd + $car;
1f45ae4a	256	if ($use_mult) {
68decaef	257	$x -= ($car = int($x * 1e-5)) * 1e5;
1f45ae4a PP	258	}
	259	else {
	260	$x -= ($car = int($x / 1e5)) * 1e5;
	261	}
5303340c LW	262	}
	263	push(@x, $car); $car = 0;
	264	for $y (@y) {
	265	$y = $y * $dd + $car;
1f45ae4a	266	if ($use_mult) {
68decaef	267	$y -= ($car = int($y * 1e-5)) * 1e5;
1f45ae4a PP	268	}
	269	else {
	270	$y -= ($car = int($y / 1e5)) * 1e5;
	271	}
5303340c LW	272	}
	273	}
	274	else {
	275	push(@x, 0);
	276	}
ed6116ce	277	@q = (); ($v2,$v1) = @y[-2,-1];
5303340c	278	while ($#x > $#y) {
ed6116ce	279	($u2,$u1,$u0) = @x[-3..-1];
68decaef LW	280	$q = (($u0 == $v1) ? 99999 : int(($u0*1e5+$u1)/$v1));
68decaef LW	281	--$q while ($v2$q > ($u01e5+$u1-$q$v1)1e5+$u2);
5303340c LW	282	if ($q) {
5303340c LW	283	($car, $bar) = (0,0);
859172fe	284	for ($y = 0, $x = $#x-$#y-1; $y <= $#y; ++$y,++$x) {
5303340c	285	$prd = $q * $y[$y] + $car;
1f45ae4a	286	if ($use_mult) {
68decaef	287	$prd -= ($car = int($prd * 1e-5)) * 1e5;
1f45ae4a PP	288	}
	289	else {
	290	$prd -= ($car = int($prd / 1e5)) * 1e5;
	291	}
68decaef	292	$x[$x] += 1e5 if ($bar = (($x[$x] -= $prd + $bar) < 0));
5303340c LW	293	}
	294	if ($x[$#x] < $car + $bar) {
	295	$car = 0; --$q;
859172fe	296	for ($y = 0, $x = $#x-$#y-1; $y <= $#y; ++$y,++$x) {
68decaef LW	297	$x[$x] -= 1e5
68decaef LW	298	if ($car = (($x[$x] += $y[$y] + $car) > 1e5));
5303340c LW	299	}
	300	}
	301	}
	302	pop(@x); unshift(@q, $q);
	303	}
	304	if (wantarray) {
	305	@d = ();
	306	if ($dd != 1) {
	307	$car = 0;
	308	for $x (reverse @x) {
68decaef	309	$prd = $car * 1e5 + $x;
5303340c LW	310	$car = $prd - ($tmp = int($prd / $dd)) * $dd;
	311	unshift(@d, $tmp);
	312	}
	313	}
	314	else {
	315	@d = @x;
	316	}
8990e307	317	(&external($sr, @q), &external($srem, @d, $zero));
5303340c LW	318	} else {
	319	&external($sr, @q);
	320	}
	321	}
	322	1;