================================ WEEK 5: ADDENDA =============================


In these supplementary notes for the Week 5 lecture, we have the 
two algorithms for adding and multiplying floating-point numbers seen in class.

Floating-point addition
-----------------------

Given: base b,
       floating-point numbers: u = f_u * b^e_u, that is u = (f_u, e_u)
 			       v = f_v * b^e_v, that is v = (f_v, e_v)
       p-digit mantissa

Find the sum w = u (+) v 

Algorithm:

   A1. [Unpack] Separate exponent and fraction parts of the 
           representation of u and v:
                 
                   u		v
                 ------------------
       exponent  e_u            e_v
       mantissa  f_u            f_v

   A2. [Assume e_u >= e_v] If e_u < e_v, then interchange roles of u and v.

   A3. [Set e_w]  e_w = e_u. 

   A4. [Test e_u - e_v] If e_u - e_v >= p+2  (that is,  e_u >> e_v), then 
            set f_w = f_u.
            Go to Step A7.

   A5. [Scale right] Shift f_v to the right e_u - e_v places; this in reality 
            means that f_v is being divided by b^(e_u-e_v).

   A6. [Add] Set f_w = f_u + f_v.

   A7. [Normalize] Here round the last (that is, the p-th digit) of the 
            mantissa with respect to the value of the p+1-th digit, and 
            truncate the mantissa to keep only its first p digits.  



Example:

Given:   base 10
         u = 0.3472e+2, v = 0.11202-1
         4-digit mantissa

Find the sum w = u (+) v

   A1. [Unpack]                  
                   u		v
                 ------------------
       exponent  +2              -1
       mantissa  0.3472      0.1120

   A2. [Assume e_u >= e_v] We in fact have e_u >= e_v; therefore no 
            interchange of the roles of u and v.

   A3. [Set e_w]  e_w = +2. 

   A4. [Test e_u - e_v] We don't have e_u >> e_v); thus we don't  
            set f_w = 0.3472 and we go to the next step which is A5.

   A5. [Scale right] Shift 0.1120 to the right 2 - (-1) places; this in 
            reality means that 0.1120 is being divided by 10^(2-(-1)) = 10^3.
            So we get f_v = 0.1120/(10^3) = 0.0001120.

   A6. [Add] Set f_w = 0.3472 + 0.0001120 = 0.3473120.

   A7. [Normalize] Here round the 4-th digit of the 
            mantissa with respect to the value of the 5-th digit, and 
            truncate the mantissa to keep only its first 4 digits. So 
            the final result is: 

                  f_w =  0.3473
                  e_w =  +2
                  
                  w = 0.3473e+2 
  
  


Floating-point multiplication
-----------------------------

Given: base b,
       floating-point numbers: u = f_u * b^e_u, that is u = (f_u, e_u)
 			       v = f_v * b^e_v, that is v = (f_v, e_v)
       p-digit mantissa

Find the product w = u (x) v  (or u (*) v)

Algorithm:

   M1. [Unpack] Separate exponent and fraction parts of the 
           representation of u and v:
                 
                   u		v
                 ------------------
       exponent  e_u            e_v
       mantissa  f_u            f_v

   M2. [Operate] Set e_w = e_u + e_v
                     f_w = f_u * f_v
        
                 Truncate f_w to p+2 digit at this point if necessary.

   M3. [Normalize] Same as for the operation (+).


Example:

Given:   base 10
         u = 0.1344e+2, v = 0.71881e+1
         4-digit mantissa

Find the product w = u (*) v

   M1. [Unpack]                  
                   u		v
                 ------------------
       exponent  +2              +1
       mantissa  0.1344      0.71881

   M2. [Operate] Set e_w = 2+1
                     f_w = 0.1344 * 0.71881 = 0.09660672

   M3. [Normalize]  We get w = 0.9661e+2