Method in randomness

where 9 signifies any significant hit and the leftmost g in the left cell could be 1. Note that the inequalities (12) show axb~ + 2-'*V(alba I? aebe) < 2 *. Now we divide this result by b> Since we must have a double-lentO* quotient and since each fixed-point division which we are to use ~ow gives a single-length quotient and a single-length remainder, we first divide the double-length dividend by the single-length divisor b b and obtain the single-length quotient ql and a single-length remainder. We then augment this remainder with a single-length zero (thus obtaining the double.-length number) and divide this double-length number by bl to obtain the single-length quotient 92 • The q: :rod qz constitute the desired double-length quotient (alb2 .q-2-47(alba q-aebc))/bt. Thus we now have V in the form qi (l 2 ? T T Step 3, Compute U-V. To compute U-V by adding the complemented V to U, namely, U-V U ÷ (-V), we proceed as follows. First, suit-tract 1 from the rightmost bit of the leftmost word of U (borrow) arid add 1 to the rightmost bit of the rightmost word of U (end-a.round carry). Call the result U'. Next complement each hit g of V in (1)16). Call the result V'. where ~ means the complemented 9. Add U' ((P14)) and V' bit-by-bit using fixed-point operations, where arty carry bit that might be t~rodueed froin the addition of the lower twn words must be properly added to the next upper wet(1. WE again arrange the result (=U-V) in the form ¢,-t t 'r Step 4. Compute U-V/hi = Aiils. %Ve mtist now divide the triple-length nunlber UP by the single-length number bt to obtain the triple-length quotient. We accomplish this by three successive fixed-point divisions, each time obt~dning a single-length quotient and a siligle-length remainder. Hence it is vital to obtain the correct remainder at least from the first two divisions. To obtain this we regard the dividend U-V and the divisor b~ as integers and use the integer divide operation. (The fractional divide operation may not retain the last big of the remainder. It is then necessary to rearrange U-V of (P18) in the form T ? ? T an extra zero (We do not have to rearrange bt+) Now the proposed division tU-V)/bt can be easily performed by three successive single precision fixed point divisions, thus obtaining the triple-length quotient. …