sicp-all-tasks/sicp/2_002e13

Exercise 2.13: Show that under the assumption of
small percentage tolerances there is a simple formula for the approximate
percentage tolerance of the product of two intervals in terms of the tolerances
of the factors.  You may simplify the problem by assuming that all numbers are
positive.

After considerable work, Alyssa P. Hacker delivers her finished system.
Several years later, after she has forgotten all about it, she gets a frenzied
call from an irate user, Lem E. Tweakit.  It seems that Lem has noticed that
the formula for parallel resistors can be written in two algebraically
equivalent ways:

  
    
      
        
          R
          1
        
        
          R
          2
        
      
      
        
          R
          1
        
        +
        
          R
          2
        
      
    
  

and

  
    
      
        1
        
          1
          
            /
          
          
            R
            1
          
          +
          1
          
            /
          
          
            R
            2
          
        
      
    
    .
  

He has written the following two programs, each of which computes the
parallel-resistors formula differently:


(define (par1 r1 r2)
  (div-interval 
   (mul-interval r1 r2)
   (add-interval r1 r2)))

(define (par2 r1 r2)
  (let ((one (make-interval 1 1)))
    (div-interval 
     one
     (add-interval 
      (div-interval one r1) 
      (div-interval one r2)))))

Lem complains that Alyssa’s program gives different answers for the two ways of
computing. This is a serious complaint.