1. CSE 575 Computer Arithmetic Spring 2003 Mary Jane Irwin (www. cse. psu
CSE 575 Computer Arithmetic Spring Mary Jane Irwin (

2. Student Presentations - April 291
9:45
Fast adders
Matt & Ki-Yong 2
10:10
Array multipliers
Jahan & Jaehyun 3
10:35
Divider
Wei-Lun & Ing-Chao
11:00
BREAK 4
11:15
Log PE
Swapna, Greg, Eric, & Theo 5
12:05
Nanosenor PE
Aman

3. Cordic Algorithms Review
A family of convergence algorithms for trig (and other) functions
Generalized Cordic defined as
Xi+1 = Xi -  di Yi 2-i
Yi+1 = Yi + di Xi 2-i
Zi+1 = Zi - di Ei
Simple hardware - shifters, adders, lookup table
When the number of iterations is fixed, K and K’ are constants
Thus, we always need k iterations for k digits of precision
Can be extended to higher radices (e.g., for base 4, di{-2,-1,1,2} and the number of iterations will be cut in half with essentially the same hardware)
COordinate Rotations DIgital Computer

4. Cordic Hardware Review X X
Xi+1 = Xi – di Yi 2-i
shift
i counter 
shift
Yi+1 = Yi + di Xi 2-i Y Y
Zi+1 = Zi - di Ei Z  Z
Ei lookup table
(tan-12-i, 2-i, tanh-12-i) 
di control

5. Cordic Computation Unit
Rotation mode: di=sign(Zi); Zi0
Vector mode: di= -sign(Xi  Yi); Yi0
Ciclr
 = 1
Ei =
tan-12-i
X K(X cosZ – Y sinZ)
Y K(Y cosZ + X sinZ) Z
For cos & sin, set X=1/K, Y=0
tanZ=sinZ/cosZ
X K (X2 + Y2) Y
Z Z + tan-1(Y/X)
For tan-1, set X=1, Z=0
cos-1W=tan-1[(1-W2)/W];sin-1W=tan-1[W/(1-W2)]
Linear
 = 0
Ei = 2-i
X X
Y Y + X*Z
For multiply, set Y=0
X X Y
Z Z + Y/X
For divide, set Z=0
Hprblc
 = -1
tanh-1 2-i
X K’(X coshZ – Y sinhZ)
Y K’(Y coshZ + X sinhZ)
For cosh & sinh, set X=1/K’, Y=0
tanhZ=sinhZ/coshZ; Wt=Et lnW
Ez=sinhZ+coshZ
X K’ (X2 - Y2) Y
Z Z + tanh-1(Y/X)
For tanh-1, set X=1, Z=0
cosh-1W=ln[W+(1-W2)];sinh-1W=ln[W+(1+W2)]
lnW=2tanh-1|(W-1)/(W+1)|;W=((W+¼)2-(W-¼)2)
In executing the iterations for  = -1, steps 4, 13, 40, 121, …, j, 3j+1, … must be repeated.
CORDIC
CORDIC
CORDIC
CORDIC
CORDIC
CORDIC

6. Continued Sums/Products
Characterized by
P Pni  Q
Q = =
M Mni  1
Approximation Term – Normalizing Term
Generation Term - Result Term
Goals
Family of alg’s mapping onto the same hardware
Simple iterations (add/subt, shift)
Easy selection rules (resulting in simple ops)
Linear (or quadratic) convergence
Goals are similar to the ones we saw with Cordic

7. Approximation Terms Approximation Term – Normalizing Term
form the multiplicative inverse of one of the independent variables by a product of approximating terms, e.g., Mni
normalizing to 0 or 1
can only be computed approximately
or form the additive inverse of one of the independent variables by a sum of approximating terms
can be computed exactly

8. Generation Terms Generation Term – Result Term
generate the result (e.g., Pni) by either a product or sum of generating terms which are functionally dependent on the approximating terms

9. CS Multiplication Mult for unsigned, normalized operands
Z = X * Y = (X - mi + mi) * Y
= (X - mi) * Y + mi * Y Z
approx. term gener. term
where the mi’s are chosen appropriately
mi = si 2-i where si  {0,1}
Z = (X -  si 2-i) * Y +  si 2-i * Y
Linear convergence (exactly) – k iterations for k digits of precision
Simple operations – add/subtract, shift
Consider the si selection rules – can be viewed as just scanning X from left (msd) to right (lsd) and subtracting off the 1 digits (forming the additive inverse)
X = . x1 x2 x3 … xn if xi = 1 then
M = . s1 s2 s3 … sn si = 1
If xi = 1 then si = 1 so the sum of si 2-1 converges to X and we obviously get the product from the generating term
SAME AS REGULAR ADD AND SHIFT MULTIPLY (except working from ms end of multiplier)
What if si is chosen from the digit set –1, 0, 1 ?

10. CS Multiplication Approx. Term
X - mi  Xi = Xi-1 – si2-i
e.g., X1 = X0 – s12-1
X2 = (X0 – s12-1 ) – s22-2
And to guarantee convergence
X0 = X ½ for ½  X < ¾
X1 = X0 – m1 where m1 =
1 for ¾  X < 1
so if X  [½ , 1)  X1  [- ¼ , ¼)
Operations in Xi iteration are subtract/add and barrel shift
Note that X1 is forced to be symmetric about zero – REQUIRED since we are normalizing to zero

11. CS Multiplication Gener. Term
Gener. Term  Product Z
mi * Y   si 2-i *  yi 2-i
Z0 = 0
Z1 = Z0 + Ym (m1 = ½ or 1)
Zi = Zi-1 + Y si 2-i (barrel shift and add)
Selection Rules
Need |X1|  2/3 to ensure convergence
1 if 2/3 2-(i-1)  Xi-1  1/3 2-(i-1)
si = otherwise
-1 if - 2/3 2-(i-1)  Xi-1 < - 1/3 2-(i-1)
Get X1 bound via previous slide’s initialization step (when we force X1 to be symmetric about zero)
In general, just pick an “easy to implement” initialization step to get this condition
The choice of the section values 2/3 and 1/3 will become clear on the two next slides!

12. Selection Rules si picked to force Xi to 0
And (as in SRT division) can guarantee convergence since if | X1 |  2/3 * 20 then | Xi |  2/3 * 2-(i-1)
si = 0
2/3 2-(i-1)
1/3 2-(i-1)
-1/3 2-(i-1)
-2/3 2-(i-1)
si = -1 so add
si = 1 so subtract
For lecture
choosing si = 1 kicks the next selection into the zero range (can actually be either side of the zero breakpoint)

13. Convergence Proof PROOF (by induction)
For i = 1 by initialization
For i-1 if 1) Xi-1  1/3 2-(i-1) then choose si = 1 so
Xi = Xi-1 – 2-i  1/3 2-(i-1) - 2-i = 1/3 2-i i = -1/3 2-i
so Xi  -1/3 2-i so that si+1 = 0
and Xi+1 = Xi  -1/3 2-i = -2/3 2-(i+1) so that si+2 = -1 or 0
if 2) Xi-1 < - 1/3 2-(i-1) then choose si = -1 so etc QED
Note that no two 1’s (or –1’s) for si are ever selected in a row  no two adjacent 1’s of the same sign  canonical recoding of multiplier

14. CS Multiply Example Z = X * Y X = 0.0111 (< 2/3) Y = 0.1101
i si Xi = Xi-1 – si Zi = Zi-1 + Ysi2-i
X0 =  1/3* Z0 = 0
X1 = * Z1 = *1*2-1
= > -1/3* =
X2 = * Z2 = *0*2-2
= > -1/3* =
For lecture
Multiplier NOT normalized, so can skip special initialization step.
Notice that the multiplier, X, has been canonically recoded > 100-1
Have a msd multiplication algorithm with X recoded on-the-fly to canonical form - how can this be??? (since we know canonical recoding is left directed)
Also gives insight into why the Selection Rules use +-1/3
X0 = > 1/3 2**0 says choose s1 =1 (note that x0 would recode to )
X3 = * Z3 = *0*2-3
= < -1/3* =
X4 = * Z4 = *-1*2-4
= done =

15. ?Easy? Selection Rules
To choose si’s have to compare Xi-1 to the stored constants i 1/3*2-(i-1), -1/3*2-(i-1)
Constants vary depending on iteration
Full precision comparison – too expensive
Can reduce the number of selection constants by scaling Xi by 2i making the selection rules independent of the iteration
Can eliminate the full precision comparison by picking a selection value close to  1/3 that needs only a few bits to represent, like  3/8
Selection constant of 3/8 gives a 3 bit comparison and near CANONICAL recoding !!

16. Scaled CS Multiplication
Scaled Approx. Term  0
U0 = X = X  [½ , 1)
U1 = X = (X0 - m1)*2  [- ½ , ½ )
Ui = Xi2i = (Xi-1 - si2-i)2i = 2iXi-1 – si = 2*2i-1Xi-1 – si
Ui = 2Ui-1 – si
Generating Term  Product, Z
Z0 = 0 and Z1 = Z0 + Y m1
Z = mi * Y where mi = si2-i  Zi = Zi-1 + Y si 2-i
Selection Rules
1 if Ui-1  1/  3/8
si = otherwise
-1 if Ui-1 < - 1/  - 3/8
Now have only two comparison constants to store - + 1/3 and – 1/3
0.100
0.011 = +3/8
0.010
0.001
0.000
1.111
1.110
1.101 = -3/8
Linear convergence; simple operations; easy selection rules
Continued sums approach, so mi = si2**-i
Forcing Xi to zero, so scale by Ui = 2**i Xi

17. Scaled CS Multiply Example
Z = X * Y X = (< 2/3) Y =
i si Ui = 2Ui-1 – si Zi = Zi-1 + Ysi2-i
U0 =  3/ Z0 = 0
U1 = 2(0.0111) Z1 = *1*2-1
= > -3/ =
U2 = 2(-0.001) Z2 = *0*2-2
= > -3/ =
For lecture
Selection rules
S a1 a2 a3 a4
0.1xx x > 3/8 -> 1
> 3/8 -> !s a1 + !s !a1 a2 a3
= 3/8 -> 1
0.010 x < 3/8 -> 0
0.00x x < 3/8 -> 0
1.11x x > -3/8 -> 0
1.10x x > -3/8 -> 0
= -3/8 -> 0
1.100 x < -3/8 -> s a1 !a2 !a3 + s !a1
1.0xx x < -3/8 -> -1
U3 = 2(-0.01) Z3 = *0*2-3
= -0.1 < -3/ =
U4 = 2(-0.1) Z4 = *-1*2-4
= done =

18. CS Multiply Hardware Ui = 2Ui-1 – si Zi = Zi-1 + Ysi 2-i Zi si i Zi-1
n-b adder
barrel sft
complement
mini-adder
1-b shifter
n-b register
ROM
Ui-1 si
 3/8
ms 4 bits
Ui-1 Ui
For lecture
What is involved in a mini adder design?  what you are doing is adding 1 to a fraction, so, depending on the representation it can be as simple as complementing the high order (sign) bit
Complement box is selective and controlled by si (xor word gate)
What if the results are generated in carry-save form (to speed up addition)?

19. CP Division Division for unsigned, normalized operands
Z = Y/X = (Y  di) / (X  di) Z
gener term approx term
where the di’s are chosen appropriately
di = (1 + si 2-i) where si  {-1,0,1}
Z = (Y  (1 + si 2-i)) / (X  (1 + si 2-i))
Linear convergence (approximately) – k iterations for k digits of precision
Simple operations – add/subtract, shift
Consider the si selection rules (forming the multiplicative inverse)
Xi = Xi-1 + 2**-i Xi-1 for si = 1
Xi-1 = x1 x2 x3 … xn
si= -i - x1 x2 x3 …
so a simple rule would be if Xi-1 <1 pick si = 1 or if Xi-1 >=1 pick si = 0
We will see that this could give us problems with convergence (to 1) - (what to do if Xi-1>1 – would really like to pick si = -1)
What if si is chosen from the digit set –1, 0, 1 ? – easier to get convergence

20. CP Division Approximating Term
Approx. Term  1
X  (1 + si 2-i)  Xi = Xi-1(1 + si 2-i)
X0 = X ½  X < ¾
X1 = X0d1 where d1 =
1 ¾  X < 1
if X  [½ , 1)  X1  [¾ , 6/4)
remember trying to force Xi  1 so need to initialize it that way
Operations in Xi iteration are subtract/add and barrel shift
Note that X1 is forced to be symmetric about one – REQUIRED since we are normalizing to one
could have gotten it “more symmetic” around one by using d1=2 for X [1/2,2/3) and d1=1 for X [2/3,1) resulting in X1 [2/3,4/3) but the selection rule for d1 would require a full precision comparison, so we compromise on the simpler rule given in slide

21. CP Division Generating Term
Gener. Term  Quotient Z
Y  (1 + si 2-i)  Zi = Zi-1(1 + si 2-i)
Z0 = Y
Z1 = Z0d1
Selection Rules
-1 if Xi-1  1 + 1/3 2-(i-1)
si = otherwise
1 if Xi-1 < 1 - 1/3 2-(i-1)
 again unwieldy selection comparison constants
The choice of the section values +1/3 and -1/3 in once again to get canonical recoding on si

22. Scaled CP Division Scaled Approx. Term  0 Selection Rules
Scale factor when norm to 1 is Ui = 2i(Xi – 1)
forcing Ui to 0 will force to Xi to 1
U0 = 20(X0–1) = X0–1  [-½ , ½ )
U1 = 21(X1–1) = 2(X0d1–1)  [- ½ , 1 )
Ui = 2i(Xi-1(1+si2-i)–1) = 2 2i-1(Xi-1-1)+Xi-1si = 2Ui-1 + Xi-1si
= 2Ui-1+2 2i-1(Xi-1 –1) si2-i+si = 2Ui-1+2Ui-1si2-i+si
Ui = 2Ui-1(1 + si2-i) + si
Selection Rules
-1 if Ui-1  1/  3/8
si = otherwise
1 if Ui-1 < - 1/  - 3/8
For the d1 shown two slides ago, x1 [3/4,6/4) used to kick X1 symmetric about one gives u1 = 2([3/4,6/4)-1) = [-1/2,1) not quite symmetric about zero (maybe should have just chosen d1 = 1)
note that when si = 0 which is hopefully will be 2/3rds of the time, only have to do a simple shift

23. Scaled CP Division Example
Z = Y / X X = = Y = = 0.5
i si Ui = 2Ui-1(1 + si2-i) + si Zi = Zi-1(1 + si2-i)
U0 = Z0 = 0.5
U1 = 2(2*0.6-1) Z1 = 1.0
= 0.4  3/8
U2 = 2*0.4(1-1/4) Z2 = 1.0(1-1/4) = 0.75
= -0.4  -3/8
U3 = 2*-0.4(1+1/8) Z3 = 0.75(1+1/8)
= -0.1 > -3/ =
For lecture
U4 = 2*0.1=0.2 < 3/ Z4 =
U5 = 2*0.2=0.4  3/ Z5 =
U6 = 2*0.4(1-1/64) Z6 = (1-1/64)
= =

24. CP Divide Hardware Ui = 2Ui-1(1+si2-i)+si Zi = Zi-1(1+si2-i) n-b adder
barrel sft
complement
mini-adder
1-b shifter
n-b register
Ui-1 Ui si i Zi si i
Zi-1
n-b adder
barrel sft
complement
mini-adder
1-b shifter
n-b register
n-b adder
barrel sft
complement
mini-adder
1-b shifter
n-b register
n-b adder
barrel sft
complement
mini-adder
1-b shifter
n-b register
Ui-1
 3/8
ms 4 bits
ROM
 3/8
for lecture
look closer at design of mini-adder  what you are doing is adding 1 to a fraction, so, depending on the representation it can be as simple as complementing the high order (sign) bit
What if results generated in carry-save form? si

25. CS/P Mult/Divide Hardware
Ui = 2Ui-1 – si
Ui = 2Ui-1(1+si2-i)+si
Zi = Zi-1(1+si2-i)
Zi = Zi-1 + Ysi 2-i
CS/P Mult/Divide Hardware
n-b adder
barrel sft
complement
mini-adder
1-b shifter
n-b register
Ui-1 Ui si i Zi si i
Zi-1 i Y
n-b adder
barrel sft
complement
mini-adder
1-b shifter
n-b register
n-b adder
barrel sft
complement
mini-adder
1-b shifter
n-b register
n-b adder
barrel sft
complement
mini-adder
1-b shifter
n-b register
Ui-1
 3/8
ms 4 bits
ROM
 3/8
For lecture
merged mult/divide interconnections
Show were muxes have to go if * and / are combined si

26. Bit Slice Design Approach
n-b adder
barrel sft
complement
mini-adder
1-b shifter
n-b register
Ui-1 Ui si i Zi si i
Zi-1 i Y
n-b adder
barrel sft
complement
mini-adder
1-b shifter
n-b register
n-b adder
barrel sft
complement
mini-adder
1-b shifter
n-b register
n-b adder
barrel sft
complement
mini-adder
1-b shifter
n-b register
Ui-1
ROM
 3/8
leave routing channels over the bit slice to handle all necessary interconnects instead of “routing around” ! si

27. Continued Sums/Products Review
Approximation Term – Normalizing term
form the additive inverse of one of the independent variables by a sum of approximating terms (si2-i)
form the multiplicative inverse of one of the independent variables by a product of approximating terms (1 + si2-i)
Generation term
generates terms which are functionally dependent on the approximating the result by either a sum or a product of generating terms
Goals
Family of alg’s mapping onto the same hardware
Simple iterations (add/subt, shift)
Easy selection rules (resulting in simple operations)
Linear (or quadratic) convergence

28. CP Exponentiation Z = eX where ½  |X| < ln 2 so eX  [½, 2)
X = (X – ln( ei)) + ln( ei)
where ei = (1 + si 2-i) for si  {-1,0,1}
so that eX = eX – ln( ei) eln( ei) Z
1 gener term
approx term
eX = eX+-ln ei *  ei
Linear convergence (approximately) – k iterations for k digits of precision
Simple operations – add/subtract, shift
Have to precompute and store (along with the comparison constants) the values of –ln ei = - ln (1 + si 2**-1)
Tricks
eln value = value
– ln( ei) = +-ln ei

29. CP Exponentiation, con’t
Scaled Approx. Term  0
X + -ln ei  Xi = Xi-1 + (- ln(1 + si 2-i))
X0 = X and X1 = X0 - ln e1
and applying the norm. to 0 scale factor Ui=Xi2i
U0 = 20X0 = X0  [-½ , ln 2 )
U1 = 21X1 = 2(X0–ln e1)
Ui = 2iXi-1= 2i(Xi-1+(-ln ei)) = 2 2i-1Xi-1+ 2i(-ln ei)
= 2Ui-1+ 2i(-ln ei) = 2Ui-1+ 2i(-ln (1 + si 2-i))
Ui = 2Ui-1+ 2i(-ln (1 + si 2-i))
Operations in Xi iteration are subtract/add and barrel shift plus have to have access to the extra stored constants
stored constants for
i = 2, …,n (when si=0 constant is –ln 1=0)

30. CS/P Exponentiation, con’t
Gener. Term  Exponential Z
 ei  Zi = Zi-1(1 + si 2-i)
Z0 = 1
Z1 = Z0e1
Selection Rules
1 if Ui-1  3/8
si = otherwise
-1 if Ui-1 < - 3/8
The choice of the section values +3/8 and –3/8 in once again to get close to canonical recoding on si with easy comparison constants

31. Exp Initialization Rules
If we have |X|[½, ln2) and we want
X1[-½, ½) since we are normalizing to 0
X e ln e X1 = X0 - ln e1
[½, ln 2) e½ ½ [0, ln2–½)
[¼, ½) e¼ ¼ [0, ¼)
[-¼, ¼) [-¼, ¼)
[-½, ¼) e-¼ ¼ [-¼, 0)
[½, ln 2) e-½ ½ [-ln2+½, 0)

32. Stored Constants How many stored constants do we need?
 3/8 for si selection
½, ¼, -½, -¼ for e1 selection, e½, e¼, e-½, e-¼ to form Z1
-ln(1+si2-i) in support of Ui computation
when i is large, values converge for some n
ln(1+) =  - ½2 + 1/33 – ¼4 + … for –1< 1
if i > (n-3)/2 where n is the precision then
ln(1+) =  = si 2-i to machine accuracy
so only need to store -ln(12-i) for i = 2, …, (n-3)/2
e.g., for n=16, i=2,3,4,5,6  10 values
Remember –ln 1 = 0 !!

33. CS/P Exponentiation Hardware
Ui = 2Ui-1+2i(-ln(1+si2-i))
Zi = Zi-1(1+si2-i)
CS/P Exponentiation Hardware
n-b adder
barrel sft
complement
mini-adder
1-b shifter
n-b register
n-b adder
barrel sft
complement
mini-adder
1-b shifter
n-b register
n-b adder
barrel sft
complement
mini-adder
1-b shifter
n-b register si si i i i
Ui-1 si
3/8
-ln(12-i)
½,-½,¼,-¼ si
Ui-1
Zi-1 Ui Zi

34. CS/P Square Root Derivation
Z = X where X  [¼, 1)
Try Z = (Xri2 * ri-2) = (Xri2)* ri-1 Z
where ri = (1+si2-i) for si  {-1,0,1}
but forming ri-1 would be hard, so try again!
for lecture
Z = X/X = (Xri)/(Xri2) Z
gener term approx term

35. CS/P Square Root Scaled Approx. Term (Ui = 2i(Xi – 1))  0
(Xri2)  Ui = 2Ui-1 + 2si(1+Ui-12-i+1) + 2-isi2(1+Ui-12-i+1)
U0 = 20(X0–1) = X-1 and U1 = 21(X r1 -1)
Gener. Term  Square root Z
Xri  Zi = Zi-1(1+si2-i)
Z0 = X and Z1 = Z0r1
Selection Rules  Initialization Rule
-1 if Ui-1  3/ X[¼, 1) want X1[½, 2)
si = otherwise if ¼  X0 < ½
1 if Ui-1 < - 3/ r1 = 1 if ½  X0 < 1
approx term derivation: X0 = X; X1 = X0 r1
Xi = Xi-1(1 + si 2**-i)**2 -> 1 so Ui = 2**i (Xi - 1) -> 0
U0 = 2**0 (X0 - 1) = X-1 and U1 = 2**1 (X0 r1 - 1) = 2 X r1 - 2 = 2(X1 r1 - 1)
Ui = 2**i (Xi - 1) = 2**i (Xi-1 (1 + si 2**-i)**2 - 1) = 2**i (Xi Xi-1 (2 si 2**-i + si**2 2**-2i)
= 2 * 2**i-1 (Xi-1 - 1) + 2**i Xi-1 (2 si 2**-2 + si**2 2**-2i) = 2Ui-1 +2si Xi-1 + si**2 Xi-1 2**-1
= {2Ui-1} + {2*2*2**i-1 (Xi-1 -1)si 2**-i + si 2} + {2**-i 2 2**i-1 (Xi-1 -1) si**2 2**-i + si**2 2**-i}
1st term nd term rd term
= 2Ui-1 + 2**-i+2 si Ui-1 + si 2 + 2**-2i+1 si**2 Ui-1 + 2**-i si**2
= 2Ui si (1 + Ui-1 2**-i+1) + 2**-i si**2 (1 + Ui-1 2**-i+1)
Want X1 symmetric about 1

36. CS/P Square Root Hardware
Ui = 2Ui-1 + 2si(1+Ui-12-(i-1)) + 2-isi2(1+Ui-12-(i-1))
Zi = Zi-1(1+si2-i)
CS/P Square Root Hardware si
n-b adder
barrel sft
complement
adder
1-b shifter
n-b register
n-b adder
barrel sft
complement
mini-adder
1-b shifter
n-b register
n-b adder
barrel sft
complement
mini-adder
1-b shifter
n-b register si
si2
i-2 i
2i-1
Ui-1 si si
si2
3/8
Requires two cycles!! And a MUCH bigger mini-adder. Could use third bank, maybe. But note for large i (like i > n/2) don’t have to worry about the red term since it doesn’t affect result precision. si
Ui-1
Zi-1 Ui Zi

37. CS/P Sin&Cos Hardware Ui = 2Ui-1+si2i (-tan-12-i) Ri = Ri-1-si2-iIi-1
Ii = Ii-1+si2-iRi-1
CS/P Sin&Cos Hardware
n-b adder
barrel sft
complement
mini-adder
1-b shifter
n-b register
n-b adder
barrel sft
complement
mini-adder
1-b shifter
n-b register
n-b adder
barrel sft
complement
mini-adder
1-b shifter
n-b register si
!si si i i i i
Ui-1 si
!si si
3/8
-tan-12-i
tan /4
R = cos X and I = sin X si
Ui-1
Ri-1
Ii-1 Ui Ri Ii

38. Continued Sums/Products Review
Approximation Term – Normalizing term
form the additive inverse of one of the independent variables by a sum of approximating terms (si2-i)
form the multiplicative inverse of one of the independent variables by a product of approximating terms (1 + si2-i)
Generation term
generates the result by either a sum or a product of generating terms which are functionally dependent on the approximating terms
Goals
Family of alg’s mapping onto the same hardware
Simple iterations (add/subt, shift, table lookup)
Easy selection rules (requires scaling of approx. term)
Linear convergence

39. Key References
Baker, More efficient radix-2 algorithms for some elementary functions, IEEE Trans. on Computers, 24(11), 1975.
Chen, Automatic computation of exponentials, logarithms, ratios and square roots, IBM J. Research and Development, Vol 16, pp , 1972.
DeLugish, A class of algorithms for automatic evaluation of certain elementary functions, PhD Thesis, CS, UIUC, June 1970.
Ercegovac, Radix-16 evaluation of certain elementary functions, IEEE Trans. on Computers, 22(6): , 1973.
Koren, Zinaty, Evaluating elementary functions in a numerical coprocessor based on rational approximations, IEEE Trans. on Computers, 39(8): , 1990.
Parhami, Computer Arithmetic, Oxford Univ. Press, 1999.