# Israel Koren

Yüklə 605 b.
 tarix 05.03.2018 ölçüsü 605 b.

• ## Also, multiplication by bi should be simple and fast

• Use the form bi =(1+si 2 ), where si  {-1,0,1}
• Multiplication reduced to shift and add
• ## The term ln bi=ln (1+si2 ) can be positive or negative

• Must ensure convergence of positive or negative xi+1 to 0

• ## If x0 positive fraction - simple scheme for selecting si - one-sided selection rule: si  {0,1}

• Similar to quotient-bit selection in restoring division
• ## In step i+1 form difference D=xi-ln (1+2 )

• If D positive or zero - set si=1 and xi+1=D
• If D negative - set si=0 and xi+1=xi
• ## Analyze subtract in xi+1 - examine Taylor series:

• At step i can cancel bit of weight 2 by subtracting ln (1+si 2 ) from xi
• Up to n steps needed to ensure convergence of n-bit fraction to 0 - convergence linear and m=n

• ## 2 is easily dealt with

• Incorporation into exponent part of floating-point number
• Through a shift operation for fixed-point arithmetic

• ## Example:

• k=1 : yi+1  y0 / x0 - divide operation
• k=1/2 : yi+1  y0 /  x0 - reciprocal of square root
• y0=x0 : yi+1   x0 - square root calculated

• ## tan (si 2 )= si tan (2 )

• n constants stored in ROM

• ## Selection rule becomes

• Similar to rule for nonrestoring division
• ## Examine convergence rate of xi+1 to 0 - Taylor series expansion of tan (si2 ) (in radians):

• Linear convergence

• ## Approximation error not larger than 2

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