[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.
#
# 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) = '+' 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,$[,1),length($d)-2);
    substr($d,$[,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,$y) = (&'bnorm($_[$[]),&'bnorm($_[$[+1]));
    if ($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, $y) = (&'bnorm($_[$[]),&'bnorm($_[$[+1]));
    if ($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($_[$[],&'bneg($_[$[+1]));    
}

# GCD -- Euclids algorithm Knuth Vol 2 pg 296
sub main'bgcd { #(num_str, num_str) return num_str
    local($x,$y) = (&'bnorm($_[$[]),&'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, $y) = (&'bnorm($_[$[]), &'bnorm($_[$[+1]));
    if ($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, $[);
	    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, $y) = (&'bnorm($_[$[]), &'bnorm($_[$[+1]));
    return wantarray ? ('NaN','NaN') : 'NaN'
	if ($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 = $y[$[];
    $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 = $[, $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 = $[, $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.
	8	#
	9	# Suggested alternative: Math::BigInt
	10	#
5303340c LW	11	# arbitrary size integer math package
	12	#
	13	# by Mark Biggar
	14	#
	15	# Canonical Big integer value are strings of the form
	16	# /^[+-]\d+$/ with leading zeros suppressed
	17	# Input values to these routines may be strings of the form
	18	# /^\s*[+-]?[\d\s]+$/.
	19	# Examples:
	20	# '+0' canonical zero value
	21	# ' -123 123 123' canonical value '-123123123'
	22	# '1 23 456 7890' canonical value '+1234567890'
d1be9408	23	# Output values always in canonical form
5303340c LW	24	#
	25	# Actual math is done in an internal format consisting of an array
	26	# whose first element is the sign (/^[+-]$/) and whose remaining
	27	# elements are base 100000 digits with the least significant digit first.
	28	# The string 'NaN' is used to represent the result when input arguments
	29	# are not numbers, as well as the result of dividing by zero
	30	#
	31	# routines provided are:
	32	#
	33	# bneg(BINT) return BINT negation
	34	# babs(BINT) return BINT absolute value
	35	# bcmp(BINT,BINT) return CODE compare numbers (undef,<0,=0,>0)
	36	# badd(BINT,BINT) return BINT addition
	37	# bsub(BINT,BINT) return BINT subtraction
	38	# bmul(BINT,BINT) return BINT multiplication
	39	# bdiv(BINT,BINT) return (BINT,BINT) division (quo,rem) just quo if scalar
	40	# bmod(BINT,BINT) return BINT modulus
	41	# bgcd(BINT,BINT) return BINT greatest common divisor
	42	# bnorm(BINT) return BINT normalization
	43	#
8990e307	44
1f45ae4a PP	45	# overcome a floating point problem on certain osnames (posix-bc, os390)
	46	BEGIN {
	47	my $x = 100000.0;
	48	my $use_mult = int($x1e-5)1e5 == $x ? 1 : 0;
	49	}
	50
8990e307 LW	51	$zero = 0;
8990e307 LW	52
5303340c LW	53	\f
	54	# normalize string form of number. Strip leading zeros. Strip any
	55	# white space and add a sign, if missing.
	56	# Strings that are not numbers result the value 'NaN'.
8990e307	57
5303340c LW	58	sub main'bnorm { #(num_str) return num_str
	59	local($_) = @_;
	60	s/\s+//g; # strip white space
	61	if (s/^([+-]?)0*(\d+)$/$1$2/) { # test if number
79072805	62	substr($_,$[,0) = '+' unless $1; # Add missing sign
5303340c LW	63	s/^-0/+0/;
	64	$_;
	65	} else {
	66	'NaN';
	67	}
	68	}
	69
	70	# Convert a number from string format to internal base 100000 format.
	71	# Assumes normalized value as input.
	72	sub internal { #(num_str) return int_num_array
	73	local($d) = @_;
79072805 LW	74	($is,$il) = (substr($d,$[,1),length($d)-2);
79072805 LW	75	substr($d,$[,1) = '';
5303340c LW	76	($is, reverse(unpack("a" . ($il%5+1) . ("a5" x ($il/5)), $d)));
	77	}
	78
	79	# Convert a number from internal base 100000 format to string format.
	80	# This routine scribbles all over input array.
	81	sub external { #(int_num_array) return num_str
	82	$es = shift;
	83	grep($_ > 9999 \|\| ($_ = substr('0000'.$_,-5)), @_); # zero pad
	84	&'bnorm(join('', $es, reverse(@_))); # reverse concat and normalize
	85	}
	86
	87	# Negate input value.
	88	sub main'bneg { #(num_str) return num_str
	89	local($_) = &'bnorm(@_);
	90	vec($_,0,8) ^= ord('+') ^ ord('-') unless $_ eq '+0';
9d116dd7	91	s/^./N/ unless /^[-+]/; # works both in ASCII and EBCDIC
5303340c LW	92	$_;
	93	}
	94
	95	# Returns the absolute value of the input.
	96	sub main'babs { #(num_str) return num_str
	97	&abs(&'bnorm(@_));
	98	}
	99
	100	sub abs { # post-normalized abs for internal use
	101	local($_) = @_;
	102	s/^-/+/;
	103	$_;
	104	}
	105	\f
	106	# Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
	107	sub main'bcmp { #(num_str, num_str) return cond_code
79072805	108	local($x,$y) = (&'bnorm($_[$[]),&'bnorm($_[$[+1]));
5303340c LW	109	if ($x eq 'NaN') {
	110	undef;
	111	} elsif ($y eq 'NaN') {
	112	undef;
	113	} else {
	114	&cmp($x,$y);
	115	}
	116	}
	117
	118	sub cmp { # post-normalized compare for internal use
	119	local($cx, $cy) = @_;
1e2e1ae8 JH	120	return 0 if ($cx eq $cy);
	121
	122	local($sx, $sy) = (substr($cx, 0, 1), substr($cy, 0, 1));
	123	local($ld);
	124
	125	if ($sx eq '+') {
	126	return 1 if ($sy eq '-' \|\| $cy eq '+0');
	127	$ld = length($cx) - length($cy);
	128	return $ld if ($ld);
	129	return $cx cmp $cy;
	130	} else { # $sx eq '-'
	131	return -1 if ($sy eq '+');
	132	$ld = length($cy) - length($cx);
	133	return $ld if ($ld);
	134	return $cy cmp $cx;
	135	}
	136
5303340c LW	137	}
	138
	139	sub main'badd { #(num_str, num_str) return num_str
79072805	140	local(x, y); ($x, $y) = (&'bnorm($_[$[]),&'bnorm($_[$[+1]));
5303340c LW	141	if ($x eq 'NaN') {
	142	'NaN';
	143	} elsif ($y eq 'NaN') {
	144	'NaN';
	145	} else {
	146	@x = &internal($x); # convert to internal form
	147	@y = &internal($y);
	148	local($sx, $sy) = (shift @x, shift @y); # get signs
	149	if ($sx eq $sy) {
	150	&external($sx, &add(x, y)); # if same sign add
	151	} else {
	152	($x, $y) = (&abs($x),&abs($y)); # make abs
	153	if (&cmp($y,$x) > 0) {
	154	&external($sy, &sub(y, x));
	155	} else {
	156	&external($sx, &sub(x, y));
	157	}
	158	}
	159	}
	160	}
	161
	162	sub main'bsub { #(num_str, num_str) return num_str
79072805	163	&'badd($_[$[],&'bneg($_[$[+1]));
5303340c LW	164	}
	165
	166	# GCD -- Euclids algorithm Knuth Vol 2 pg 296
	167	sub main'bgcd { #(num_str, num_str) return num_str
79072805	168	local($x,$y) = (&'bnorm($_[$[]),&'bnorm($_[$[+1]));
68decaef	169	if ($x eq 'NaN' \|\| $y eq 'NaN') {
5303340c	170	'NaN';
68decaef	171	} else {
5303340c LW	172	($x, $y) = ($y,&'bmod($x,$y)) while $y ne '+0';
	173	$x;
	174	}
	175	}
	176	\f
68decaef	177	# routine to add two base 1e5 numbers
5303340c LW	178	# stolen from Knuth Vol 2 Algorithm A pg 231
	179	# there are separate routines to add and sub as per Kunth pg 233
	180	sub add { #(int_num_array, int_num_array) return int_num_array
	181	local(x, y) = @_;
	182	$car = 0;
	183	for $x (@x) {
	184	last unless @y \|\| $car;
55497cff	185	$x -= 1e5 if $car = (($x += shift(@y) + $car) >= 1e5) ? 1 : 0;
5303340c LW	186	}
	187	for $y (@y) {
	188	last unless $car;
55497cff	189	$y -= 1e5 if $car = (($y += $car) >= 1e5) ? 1 : 0;
5303340c LW	190	}
	191	(@x, @y, $car);
	192	}
	193
68decaef	194	# subtract base 1e5 numbers -- stolen from Knuth Vol 2 pg 232, $x > $y
5303340c LW	195	sub sub { #(int_num_array, int_num_array) return int_num_array
	196	local(sx, sy) = @_;
	197	$bar = 0;
	198	for $sx (@sx) {
	199	last unless @y \|\| $bar;
e334a159	200	$sx += 1e5 if $bar = (($sx -= shift(@sy) + $bar) < 0);
5303340c LW	201	}
	202	@sx;
	203	}
	204
	205	# multiply two numbers -- stolen from Knuth Vol 2 pg 233
	206	sub main'bmul { #(num_str, num_str) return num_str
79072805	207	local(x, y); ($x, $y) = (&'bnorm($_[$[]), &'bnorm($_[$[+1]));
5303340c LW	208	if ($x eq 'NaN') {
	209	'NaN';
	210	} elsif ($y eq 'NaN') {
	211	'NaN';
	212	} else {
	213	@x = &internal($x);
	214	@y = &internal($y);
	215	local($signr) = (shift @x ne shift @y) ? '-' : '+';
	216	@prod = ();
	217	for $x (@x) {
79072805	218	($car, $cty) = (0, $[);
5303340c LW	219	for $y (@y) {
5303340c LW	220	$prod = $x * $y + $prod[$cty] + $car;
1f45ae4a PP	221	if ($use_mult) {
	222	$prod[$cty++] =
	223	$prod - ($car = int($prod * 1e-5)) * 1e5;
	224	}
	225	else {
	226	$prod[$cty++] =
	227	$prod - ($car = int($prod / 1e5)) * 1e5;
	228	}
5303340c LW	229	}
	230	$prod[$cty] += $car if $car;
	231	$x = shift @prod;
	232	}
	233	&external($signr, @x, @prod);
	234	}
	235	}
	236
	237	# modulus
	238	sub main'bmod { #(num_str, num_str) return num_str
79072805	239	(&'bdiv(@_))[$[+1];
5303340c LW	240	}
	241	\f
	242	sub main'bdiv { #(dividend: num_str, divisor: num_str) return num_str
79072805	243	local (x, y); ($x, $y) = (&'bnorm($_[$[]), &'bnorm($_[$[+1]));
5303340c LW	244	return wantarray ? ('NaN','NaN') : 'NaN'
	245	if ($x eq 'NaN' \|\| $y eq 'NaN' \|\| $y eq '+0');
	246	return wantarray ? ('+0',$x) : '+0' if (&cmp(&abs($x),&abs($y)) < 0);
	247	@x = &internal($x); @y = &internal($y);
79072805	248	$srem = $y[$[];
5303340c LW	249	$sr = (shift @x ne shift @y) ? '-' : '+';
5303340c LW	250	$car = $bar = $prd = 0;
68decaef	251	if (($dd = int(1e5/($y[$#y]+1))) != 1) {
5303340c LW	252	for $x (@x) {
5303340c LW	253	$x = $x * $dd + $car;
1f45ae4a	254	if ($use_mult) {
68decaef	255	$x -= ($car = int($x * 1e-5)) * 1e5;
1f45ae4a PP	256	}
	257	else {
	258	$x -= ($car = int($x / 1e5)) * 1e5;
	259	}
5303340c LW	260	}
	261	push(@x, $car); $car = 0;
	262	for $y (@y) {
	263	$y = $y * $dd + $car;
1f45ae4a	264	if ($use_mult) {
68decaef	265	$y -= ($car = int($y * 1e-5)) * 1e5;
1f45ae4a PP	266	}
	267	else {
	268	$y -= ($car = int($y / 1e5)) * 1e5;
	269	}
5303340c LW	270	}
	271	}
	272	else {
	273	push(@x, 0);
	274	}
ed6116ce	275	@q = (); ($v2,$v1) = @y[-2,-1];
5303340c	276	while ($#x > $#y) {
ed6116ce	277	($u2,$u1,$u0) = @x[-3..-1];
68decaef LW	278	$q = (($u0 == $v1) ? 99999 : int(($u0*1e5+$u1)/$v1));
68decaef LW	279	--$q while ($v2$q > ($u01e5+$u1-$q$v1)1e5+$u2);
5303340c LW	280	if ($q) {
5303340c LW	281	($car, $bar) = (0,0);
79072805	282	for ($y = $[, $x = $#x-$#y+$[-1; $y <= $#y; ++$y,++$x) {
5303340c	283	$prd = $q * $y[$y] + $car;
1f45ae4a	284	if ($use_mult) {
68decaef	285	$prd -= ($car = int($prd * 1e-5)) * 1e5;
1f45ae4a PP	286	}
	287	else {
	288	$prd -= ($car = int($prd / 1e5)) * 1e5;
	289	}
68decaef	290	$x[$x] += 1e5 if ($bar = (($x[$x] -= $prd + $bar) < 0));
5303340c LW	291	}
	292	if ($x[$#x] < $car + $bar) {
	293	$car = 0; --$q;
79072805	294	for ($y = $[, $x = $#x-$#y+$[-1; $y <= $#y; ++$y,++$x) {
68decaef LW	295	$x[$x] -= 1e5
68decaef LW	296	if ($car = (($x[$x] += $y[$y] + $car) > 1e5));
5303340c LW	297	}
	298	}
	299	}
	300	pop(@x); unshift(@q, $q);
	301	}
	302	if (wantarray) {
	303	@d = ();
	304	if ($dd != 1) {
	305	$car = 0;
	306	for $x (reverse @x) {
68decaef	307	$prd = $car * 1e5 + $x;
5303340c LW	308	$car = $prd - ($tmp = int($prd / $dd)) * $dd;
	309	unshift(@d, $tmp);
	310	}
	311	}
	312	else {
	313	@d = @x;
	314	}
8990e307	315	(&external($sr, @q), &external($srem, @d, $zero));
5303340c LW	316	} else {
	317	&external($sr, @q);
	318	}
	319	}
	320	1;