/*
 * *****************************************************************************
 *
 * SPDX-License-Identifier: BSD-2-Clause
 *
 * Copyright (c) 2018-2020 Gavin D. Howard and contributors.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions are met:
 *
 * * Redistributions of source code must retain the above copyright notice, this
 *   list of conditions and the following disclaimer.
 *
 * * Redistributions in binary form must reproduce the above copyright notice,
 *   this list of conditions and the following disclaimer in the documentation
 *   and/or other materials provided with the distribution.
 *
 * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
 * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
 * ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE
 * LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
 * CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
 * SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
 * INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
 * CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
 * POSSIBILITY OF SUCH DAMAGE.
 *
 * *****************************************************************************
 *
 * The second bc math library.
 *
 */

define p(x,y){
	auto a
	a=y$
	if(y==a)return (x^a)@scale
	return e(y*l(x))
}
define r(x,p){
	auto t,n
	if(x==0)return x
	p=abs(p)$
	n=(x<0)
	x=abs(x)
	t=x@p
	if(p<scale(x)&&x-t>=5>>p+1)t+=1>>p
	if(n)t=-t
	return t
}
define ceil(x,p){
	auto t,n
	if(x==0)return x
	p=abs(p)$
	n=(x<0)
	x=abs(x)
	t=(x+((x@p<x)>>p))@p
	if(n)t=-t
	return t
}
define f(n){
	auto r
	n=abs(n)$
	for(r=1;n>1;--n)r*=n
	return r
}
define perm(n,k){
	auto f,g,s
	if(k>n)return 0
	n=abs(n)$
	k=abs(k)$
	f=f(n)
	g=f(n-k)
	s=scale
	scale=0
	f/=g
	scale=s
	return f
}
define comb(n,r){
	auto s,f,g,h
	if(r>n)return 0
	n=abs(n)$
	r=abs(r)$
	s=scale
	scale=0
	f=f(n)
	h=f(r)
	g=f(n-r)
	f/=h*g
	scale=s
	return f
}
define log(x,b){
	auto p,s
	s=scale
	if(scale<K)scale=K
	if(scale(x)>scale)scale=scale(x)
	scale*=2
	p=l(x)/l(b)
	scale=s
	return p@s
}
define l2(x){return log(x,2)}
define l10(x){return log(x,A)}
define root(x,n){
	auto s,m,r,q,p
	if(n<0)sqrt(n)
	n=n$
	if(n==0)x/n
	if(n==1)return x
	if(n==2)return sqrt(x)
	s=scale
	scale=0
	if(x<0&&n%2==0)sqrt(x)
	scale=s+2
	m=(x<0)
	x=abs(x)
	p=n-1
	q=10^ceil((length(x$)/n)$,0)
	while(r!=q){
		r=q
		q=(p*r+x/r^p)/n
	}
	if(m)r=-r
	scale=s
	return r@s
}
define cbrt(x){return root(x,3)}
define pi(s){
	auto t,v
	if(s==0)return 3
	s=abs(s)$
	t=scale
	scale=s+1
	v=4*a(1)
	scale=t
	return v@s
}
define t(x){
	auto s,c,l
	l=scale
	scale+=2
	s=s(x)
	c=c(x)
	scale=l
	return s/c
}
define a2(y,x){
	auto a,p
	if(!x&&!y)y/x
	if(x<=0){
		p=pi(scale+2)
		if(y<0)p=-p
	}
	if(x==0)a=p/2
	else{
		scale+=2
		a=a(y/x)+p
		scale-=2
	}
	return a@scale
}
define sin(x){return s(x)}
define cos(x){return c(x)}
define atan(x){return a(x)}
define tan(x){return t(x)}
define atan2(y,x){return a2(y,x)}
define r2d(x){
	auto r,i,s
	s=scale
	scale+=5
	i=ibase
	ibase=A
	r=x*180/pi(scale)
	ibase=i
	scale=s
	return r@s
}
define d2r(x){
	auto r,i,s
	s=scale
	scale+=5
	i=ibase
	ibase=A
	r=x*pi(scale)/180
	ibase=i
	scale=s
	return r@s
}
define frand(p){
	p=abs(p)$
	return irand(10^p)>>p
}
define ifrand(i,p){return irand(abs(i)$)+frand(p)}
define srand(x){
	if(irand(2))return -x
	return x
}
define brand(){return irand(2)}
define void output(x,b){
	auto c
	c=obase
	obase=b
	x
	obase=c
}
define void hex(x){output(x,G)}
define void binary(x){output(x,2)}
define ubytes(x){
	auto p,b,i
	b=ibase
	ibase=A
	x=abs(x)$
	i=2^8
	for(p=1;i-1<x;p*=2){i*=i}
	ibase=b
	return p
}
define sbytes(x){
	auto p,b,n,z
	z=(x<0)
	x=abs(x)
	x=x$
	n=ubytes(x)
	b=ibase
	ibase=A
	p=2^(n*8-1)
	if(x>p||(!z&&x==p))n*=2
	ibase=b
	return n
}
define void output_byte(x,i){
	auto j,p,y,b
	j=ibase
	ibase=A
	s=scale
	scale=0
	x=abs(x)$
	b=x/(2^(i*8))
	b%=2^8
	y=log(256,obase)
	if(b>1)p=log(b,obase)+1
	else p=b
	for(i=y-p;i>0;--i)print 0
	if(b)print b
	scale=s
	ibase=j
}
define void output_uint(x,n){
	auto i,b
	b=ibase
	ibase=A
	for(i=n-1;i>=0;--i){
		output_byte(x,i)
		if(i)print" "
		else print"\n"
	}
	ibase=b
}
define void hex_uint(x,n){
	auto o
	o=obase
	obase=G
	output_uint(x,n)
	obase=o
}
define void binary_uint(x,n){
	auto o
	o=obase
	obase=2
	output_uint(x,n)
	obase=o
}
define void uintn(x,n){
	if(scale(x)){
		print"Error: ",x," is not an integer.\n"
		return
	}
	if(x<0){
		print"Error: ",x," is negative.\n"
		return
	}
	if(x>=2^(n*8)){
		print"Error: ",x," cannot fit into ",n," unsigned byte(s).\n"
		return
	}
	binary_uint(x,n)
	hex_uint(x,n)
}
define void intn(x,n){
	auto t
	if(scale(x)){
		print"Error: ",x," is not an integer.\n"
		return
	}
	t=2^(n*8-1)
	if(abs(x)>=t&&(x>0||x!=-t)){
		print "Error: ",x," cannot fit into ",n," signed byte(s).\n"
		return
	}
	if(x<0)x=2^(n*8)-(-x)
	binary_uint(x,n)
	hex_uint(x,n)
}
define void uint8(x){uintn(x,1)}
define void int8(x){intn(x,1)}
define void uint16(x){uintn(x,2)}
define void int16(x){intn(x,2)}
define void uint32(x){uintn(x,4)}
define void int32(x){intn(x,4)}
define void uint64(x){uintn(x,8)}
define void int64(x){intn(x,8)}
define void uint(x){uintn(x,ubytes(x))}
define void int(x){intn(x,sbytes(x))}