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## Appendix C: An Advanced Example

This Appendix gives an example of a grammar using some of the advanced features discussed in Section 10. The desk calculator example in Appendix A is modified to provide a desk calculator that does floating point interval arithmetic. The calculator understands floating point constants, the arithmetic operations +, -, *, /, unary -, and = (assignment), and has 26 floating point variables, ``a'' through ``z''. Moreover, it also understands intervals , written

```	( x , y )
```
where x is less than or equal to y . There are 26 interval valued variables ``A'' through ``Z'' that may also be used. The usage is similar to that in Appendix A; assignments return no value, and print nothing, while expressions print the (floating or interval) value.

This example explores a number of interesting features of Yacc and C. Intervals are represented by a structure, consisting of the left and right endpoint values, stored as double 's. This structure is given a type name, INTERVAL, by using typedef . The Yacc value stack can also contain floating point scalars, and integers (used to index into the arrays holding the variable values). Notice that this entire strategy depends strongly on being able to assign structures and unions in C. In fact, many of the actions call functions that return structures as well.

It is also worth noting the use of YYERROR to handle error conditions: division by an interval containing 0, and an interval presented in the wrong order. In effect, the error recovery mechanism of Yacc is used to throw away the rest of the offending line.

In addition to the mixing of types on the value stack, this grammar also demonstrates an interesting use of syntax to keep track of the type (e.g. scalar or interval) of intermediate expressions. Note that a scalar can be automatically promoted to an interval if the context demands an interval value. This causes a large number of conflicts when the grammar is run through Yacc: 18 Shift/Reduce and 26 Reduce/Reduce. The problem can be seen by looking at the two input lines:

```	2.5 + ( 3.5 - 4. )
```
and
```	2.5 + ( 3.5 , 4. )
```
Notice that the 2.5 is to be used in an interval valued expression in the second example, but this fact is not known until the ``,'' is read; by this time, 2.5 is finished, and the parser cannot go back and change its mind. More generally, it might be necessary to look ahead an arbitrary number of tokens to decide whether to convert a scalar to an interval. This problem is evaded by having two rules for each binary interval valued operator: one when the left operand is a scalar, and one when the left operand is an interval. In the second case, the right operand must be an interval, so the conversion will be applied automatically. Despite this evasion, there are still many cases where the conversion may be applied or not, leading to the above conflicts. They are resolved by listing the rules that yield scalars first in the specification file; in this way, the conflicts will be resolved in the direction of keeping scalar valued expressions scalar valued until they are forced to become intervals.

This way of handling multiple types is very instructive, but not very general. If there were many kinds of expression types, instead of just two, the number of rules needed would increase dramatically, and the conflicts even more dramatically. Thus, while this example is instructive, it is better practice in a more normal programming language environment to keep the type information as part of the value, and not as part of the grammar.

Finally, a word about the lexical analysis. The only unusual feature is the treatment of floating point constants. The C library routine atof is used to do the actual conversion from a character string to a double precision value. If the lexical analyzer detects an error, it responds by returning a token that is illegal in the grammar, provoking a syntax error in the parser, and thence error recovery.

```
%{

#  include  <stdio.h>
#  include  <ctype.h>

typedef  struct  interval  {
double  lo,  hi;
}  INTERVAL;

INTERVAL  vmul(),  vdiv();

double  atof();

double  dreg[ 26 ];
INTERVAL  vreg[ 26 ];

%}

%start    lines

%union    {
int  ival;
double  dval;
INTERVAL  vval;
}

%token  <ival>  DREG  VREG	/*  indices  into  dreg,  vreg  arrays  */

%token  <dval>  CONST		/*  floating  point  constant  */

%type  <dval>  dexp		/*  expression  */

%type  <vval>  vexp		/*  interval  expression  */

/*  precedence  information  about  the  operators  */

%left	'+'  '-'
%left	'*'  '/'
%left	UMINUS        /*  precedence  for  unary  minus  */

%%

lines	:	/*  empty  */
|	lines  line
;

line	:	dexp  '\n'
{	printf(  "%15.8f\n",  \$1  );  }
|	vexp  '\n'
{	printf(  "(%15.8f  ,  %15.8f  )\n",  \$1.lo,  \$1.hi  );  }
|	DREG  '='  dexp  '\n'
{	dreg[\$1]  =  \$3;  }
|	VREG  '='  vexp  '\n'
{	vreg[\$1]  =  \$3;  }
|	error  '\n'
{	yyerrok;  }
;

dexp	:	CONST
|	DREG
{	\$\$  =  dreg[\$1];  }
|	dexp  '+'  dexp
{	\$\$  =  \$1  +  \$3;  }
|	dexp  '-'  dexp
{	\$\$  =  \$1  -  \$3;  }
|	dexp  '*'  dexp
{	\$\$  =  \$1  *  \$3;  }
|	dexp  '/'  dexp
{	\$\$  =  \$1  /  \$3;  }
|	'-'  dexp	%prec  UMINUS
{	\$\$  =  - \$2;  }
|	'('  dexp  ')'
{	\$\$  =  \$2;  }
;

vexp	:	dexp
{	\$\$.hi  =  \$\$.lo  =  \$1;  }
|	'('  dexp  ','  dexp  ')'
{
\$\$.lo  =  \$2;
\$\$.hi  =  \$4;
if(  \$\$.lo  >  \$\$.hi  ){
printf(  "interval  out  of  order\n"  );
YYERROR;
}
}
|	VREG
{	\$\$  =  vreg[\$1];    }
|	vexp  '+'  vexp
{	\$\$.hi  =  \$1.hi  +  \$3.hi;
\$\$.lo  =  \$1.lo  +  \$3.lo;    }
|	dexp  '+'  vexp
{	\$\$.hi  =  \$1  +  \$3.hi;
\$\$.lo  =  \$1  +  \$3.lo;    }
|	vexp  '-'  vexp
{	\$\$.hi  =  \$1.hi  -  \$3.lo;
\$\$.lo  =  \$1.lo  -  \$3.hi;    }
|	dexp  '-'  vexp
{	\$\$.hi  =  \$1  -  \$3.lo;
\$\$.lo  =  \$1  -  \$3.hi;    }
|	vexp  '*'  vexp
{	\$\$  =  vmul(  \$1.lo,  \$1.hi,  \$3  );  }
|	dexp  '*'  vexp
{	\$\$  =  vmul(  \$1,  \$1,  \$3  );  }
|	vexp  '/'  vexp
{	if(  dcheck(  \$3  )  )  YYERROR;
\$\$  =  vdiv(  \$1.lo,  \$1.hi,  \$3  );  }
|	dexp  '/'  vexp
{	if(  dcheck(  \$3  )  )  YYERROR;
\$\$  =  vdiv(  \$1,  \$1,  \$3  );  }
|	'-'  vexp	%prec  UMINUS
{	\$\$.hi  =  -\$2.lo;    \$\$.lo  =  -\$2.hi;    }
|	'('  vexp  ')'
{	\$\$  =  \$2;  }
;

%%

#  define  BSZ  50        /*  buffer  size  for  floating  point  numbers  */

/*  lexical  analysis  */

yylex(){
register  c;

while(  (c=getchar())  ==  ' '  ){  /*  skip  over  blanks  */  }

if(  isupper(  c  )  ){
yylval.ival  =  c  -  'A';
return(  VREG  );
}
if(  islower(  c  )  ){
yylval.ival  =  c  -  'a';
return(  DREG  );
}

if(  isdigit(  c  )  ||  c=='.'  ){
/*  gobble  up  digits,  points,  exponents  */

char  buf[BSZ+1],  *cp  =  buf;
int  dot  =  0,  exp  =  0;

for(  ;  (cp-buf)<BSZ  ;  ++cp,c=getchar()  ){

*cp  =  c;
if(  isdigit(  c  )  )  continue;
if(  c  ==  '.'  ){
if(  dot++  ||  exp  )  return(  '.'  );    /*  will  cause  syntax  error  */
continue;
}

if(  c  ==  'e'  ){
if(  exp++  )  return(  'e'  );    /*  will  cause  syntax  error  */
continue;
}

/*  end  of  number  */
break;
}
*cp  =  '\0';
if(  (cp-buf)  >=  BSZ  )  printf(  "constant  too  long:  truncated\n"  );
else  ungetc(  c,  stdin  );    /*  push  back  last  char  read  */
yylval.dval  =  atof(  buf  );
return(  CONST  );
}
return(  c  );
}

INTERVAL  hilo(  a,  b,  c,  d  )  double  a,  b,  c,  d;  {
/*  returns  the  smallest  interval  containing  a,  b,  c,  and  d  */
/*  used  by  *,  /  routines  */
INTERVAL  v;

if(  a>b  )  {  v.hi  =  a;    v.lo  =  b;  }
else  {  v.hi  =  b;    v.lo  =  a;  }

if(  c>d  )  {
if(  c>v.hi  )  v.hi  =  c;
if(  d<v.lo  )  v.lo  =  d;
}
else  {
if(  d>v.hi  )  v.hi  =  d;
if(  c<v.lo  )  v.lo  =  c;
}
return(  v  );
}

INTERVAL  vmul(  a,  b,  v  )  double  a,  b;    INTERVAL  v;  {
return(  hilo(  a*v.hi,  a*v.lo,  b*v.hi,  b*v.lo  )  );
}

dcheck(  v  )  INTERVAL  v;  {
if(  v.hi  >=  0.  &&  v.lo  <=  0.  ){
printf(  "divisor  interval  contains  0.\n"  );
return(  1  );
}
return(  0  );
}

INTERVAL  vdiv(  a,  b,  v  )  double  a,  b;    INTERVAL  v;  {
return(  hilo(  a/v.hi,  a/v.lo,  b/v.hi,  b/v.lo  )  );
}
```

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