Sign of symbolic algebraic expression

Question

Is there any algorithm that can find the sign of an arbitrary symbolic algebraic expression specified in the "Tree-form"?

I know that a general algorithm does not exist, because the problem of recognizing zero is insoluble for an arbitrary expression, but how do I approach the problem of finding the sign of an expression? (how is this done in computer algebra?)

For example: sign(sqrt(2)-1) = ?

Answer 1

Rate function value

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  variables[](id,name,value)
  constants[](id,name,value)
  numbers  [](id,    ,value)
  

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  :

  sign(sqrt(2)-1)
  

  :

  id type
  0  function
  1  number
  2  constant
  3  variable
  

  :

  id name   pointer
  0  '('    ???
  1  ')'    ???
  2  '+'    ???
  3  '-'    ???
  4  '*'    ???
  5  '/'    ???
  6  'sqrt' ???
  7  'sign' ???
  

  . :

  id value
  0  2  
  1  1
  

  :

  type id
  0    7   // sign(1 operand)
  0    6   // sqrt(1 operand)
  1    0   // 2
  0    3   // - (2 operands)
  1    1   // 1
  

• .

  • init

    op1=0`,`op2=0, // set all operands to zero (number depends on supported functions usually 2)
    opn=0          // actual operands number
    fx=none        // actual function (for example none=-1)
    fxn=0          // actual function operands number
    

  • (, , ), op? opn++.

    fx,fxn

  • if opn == fxn

    , fx

    op1=fxtab[fx].pointer(op1,op2,...)
    fx=none,fxn=1
    opn=1  (some spec functions can return more operands, then set op1,op2,.. opn=...)
    

  • goto # 2,

  • op1

( ++):

double sign(double op1) 
 { 
 if (op1>0.0) return +1.0; 
 if (op1<0.0) return -1.0; 
 return 0.0; 
 }
double sqrt1(double op1) { return sqrt(op1); }
double plus1(double op1) { return op1; }
double minus1(double op1) { return -op1; }
double plus2(double op1,double op2) { return op1+op2; }
double minus2(double op1,double op2) { return op1-op2; }

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