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

2. Cordic Algorithms
A convergence method for evaluating trigonometric (and other) functions
if a unit-length vector with end point at (X,Y) = (1,0) is rotated by an angle Z, its new end point will be at (X,Y) = (cos Z, sin Z)
simple hardware - shifters, adders, lookup table
COordinate Rotations DIgital Computer
Family of algorithms: rotation, vector mode
circular rotations
linear rotations
hyperbolic rotations

3. Real Cordic Rotations
If vector OEi is rotated about the origin by an angle i, the new vector OEi+1 will have the
coordinates
Ei+1
(Xi+1, Yi+1) Ei
Ri+1
Real rotation: Ei+1
(Xi, Yi) i
Xi+1 = Xi cos i - Yi sin i
Yi+1 = Yi cos i + Xi sin i
Zi+1 = Zi - i Ri O
The problem is that the iterations are not simple operations - have two multiply operations in each = 4 multipliers, plus have to store a lot of constants (cos’s alpha i’s and sin’s of alpha i’s)
The variable Z allows us to keep track of the total rotation over several steps. If Z0 is the initial rotation goal and if the i angles are selected at each step such that after n iterations Zn tends to 0, then En will be the end point after rotation by angle Z0

4. Pseudo Cordic Rotations
Pseudo rotations increase the
vector length to
Ri+1 = Ri(1 + tan2i)1/2
E’i+1
(Xi+1, Yi+1)
Ri+1
Pseudo rotation: E’i+1 Ei
(Xi, Yi)
Xi+1 = Xi - Yi tan i
Yi+1 = Yi + Xi tan i
Zi+1 = Zi - i i Ri O
Now have simpler operations - at first glance looks like two multipliers and a table with the tan alpha values. The R term simplifies since the expansion factor is a constant for a constant set of angles - that is the real key. So, if we have a constant set of angles, we know the tan alpha values to store (and they are a relatively small set). Additionally, if we choose the alpha’s correctly, we can simplify the multiplications to shift (and add) operations!
(a + tan2a(i))1/2 = 1/cos a(i)
Hi,
   I downloaded your CSE 757 CORDIC PPS-turned-PDF as guided by Google. It helped me understand CORDIC better. However, there is an error on slide 4, the squared term within the expansion faction K should be alpha[i]^2 without calculating any tangent.
   Ernesto
Expansion factor K = (1 + tan2 i)1/2 depends on the rotation angles 1 ,2 …, n. If we always rotate by the same angles, K is a constant.

5. Basic Cordic Rotations
To simplify pseudo rotations, pick i such that tan i = di 2-i where di  {-1,1}. Then
Xi+1 = Xi - di Yi 2-i
Yi+1 = Yi + di Xi 2-i
Zi+1 = Zi - di tan-12-i
Computation of Xi+1 and Yi+1 requires an i-bit right shift and an add/subtract; Zi+1 only requires an add/subtract and one table lookup
Precompute and store the function tan-12-i

6. Choosing the Angles i i = 2-i Ei=tan-12-i di Zi - diEi = Zi+1 1
30.00 – = -1
= 11.57 2
11.57 – = -2.47 3
= 4.66 4
4.66 – 3.58 = 1.08 5
= -0.71 6
= 0.19 7
= -0.26 8
= -0.04 9
= 0.07
Zi+1  zero

7. Rotating the Angles
Illustration of the first three rotations for a Z of 30
30 O
(X0, Y0)
for class handout

8. Rotating the Angles
Illustration of the first three rotations for a Z of 30
30 O
(X0, Y0)
-45
(X1, Y1)
(X2, Y2)
+26.6
(X3, Y3)
-14
for lecture

9. Circular Rotation Mode
Choosing di = sign(Zi)  {-1,1} to force Z to 0 gives the rotation mode Cordic iterations
Xi+1 = Xi - di Yi 2-i
Yi+1 = Yi + di Xi 2-i
Zi+1 = Zi - di Ei where Ei = tan-12-i
After n iterations, when Zn is sufficiently close to 0, then we have Z = i and
Xn = K(X cos Z - Y sin Z) where K = …
Yn = K(Y cos Z + X sin Z)
Zn = 0
Rule: Choose di  {-1,1} such that Z  0
Computes cos Z and sin Z by starting with X = 1/K = … and Y = 0
For k bits of precision, run it k iterations since for large i > k, tan-12-i  2-i

10. Convergence Domain
Convergence of Z to 0 happens because each angle is more than half the previous angle.
The domain of convergence is   Z  99.7
which is the sum of all the angles given in slide 6
Outside this range, we can use trig identities to convert to the range
cos (Z  2j) = cos Z sin (Z  2j) = sin Z
cos (Z - ) = - cos Z sin (Z - ) = - sin Z
transformations are particularly easy if angles are represented and manipulated in multiples of pi radians
E.g., z = 0.2pi radian or 36degrees gives a domain of convergences of -1/2 to +1/2

11. Rotation Example
Computing cos 30 (= ) and sin 30 (= ) i di
Xi+1  cos
Yi+1  sin
Zi+1  0
1/K = 1 -1 2 3 4 5 6 7 8
. . .

12. Circular Vector Mode
Choosing di = - sign(Xi Yi)  {-1,1} to force Y to 0 gives the vector mode Cordic iterations
Xi+1 = Xi - di Yi 2-i
Yi+1 = Yi + di Xi 2-i
Zi+1 = Zi - di Ei where Ei = tan-12-i
After n iterations, when Yn is sufficiently close to 0, then we have tan (i) = -Y/Z
Xn = K (X2 + Y2) where K = …
Yn = 0
Zn = Z + tan-1 (Y/X)
Rule: Choose di  {-1,1} such that Y  0
Computes tan-1 Y by starting with X = 1 and Z = 0
For k bits of precision, run it k iterations since for large i > k, tan-12-i  2-i

13. Vector Example Computing tan-1 0.414 210 (= 22.499 830) di
Xi+1  K(X2 + Y2)
Yi+1  0
Zi+1  tan-1 -1 1 2 3 4 5 6 7 8
. . .
Need to check answers for bugs !!!

14. Circular Cordic HardwareX 
shift
Xi+1 = Xi – di Yi 2-i
i counter 
shift
Yi+1 = Yi – di Xi 2-i Y
lookup table stores values of tan-1 2-i Z 
lookup
table
Zi+1 = Zi – di Ei
values of
Ei= tan-12-i
di control

15. Generalized Cordic Generalized Cordic defined as
Xi+1 = Xi -  di Yi 2-i
Yi+1 = Yi + di Xi 2-i
Zi+1 = Zi - di Ei
Defining  and Ei gives
 = 1 Circular rotations (basic) Ei = tan-12-i
 = 0 Linear rotations Ei = 2-i
 = -1 Hyperbolic rotations Ei = tanh-12-i

16. Generalized Rotations
2(area EOF)
angle EOF =
(OW)2
2(area AOB)
angle AOB =
(OU)2 U A B
 = 1 W E F
 = -1 O
 = 0 V
vector OV

17. Enhanced Cordic Hardware 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

18. Circular Mode ( = 1) Circular vector mode Circular rotation mode
Xn = K(X cos Z - Y sin Z)
Yn = K(Y cos Z + X sin Z)
Zn = 0
Rule: Choose di  {-1,1} such that Z  0
Circular vector mode
Xn = K(X2 + Y2)1/2
Yn = 0
Zn = Z + tan-1 (Y/X)
Rule: Choose di  {-1,1} such that Y  0
Computes functions
cos Z and sin Z
X = 1/K and Y = 0
Computes function
tan-1 Y
X = 1 and Z = 0

19. Linear Mode ( = 0) Linear vector mode Linear rotation mode Xn = X
Yn = Y + X * Z
Zn = 0
Rule: Choose di  {-1,1} such that Z  0
Linear vector mode
Yn = 0
Zn = Z + Y/X
Rule: Choose di  {-1,1} such that Y  0
Computes functions
multiply and multiply-add
Y = 0
End point of the vector is kept on the line x = x(0) and the vector length is defined by R(i) = x(i)
Hence the true length of OV of the scaling factor is 1 (pseudorotations are true linear rotations)
Converges trivially for suitably restricted values of z (rotation) or y (vector)
Computes functions
divide and divide-add
Z = 0

20. Hyperbolic Mode ( = -1) Hyperbolic vector mode
Hyperbolic rotation mode
Xn = K’(X cosh Z + Y sinh Z)
Yn = K’(Y cosh Z + X sinh Z)
Zn = 0
Rule: Choose di  {-1,1} such that Z  0
K’ = …
Hyperbolic vector mode
Xn = K’(X2 - Y2)1/2
Yn = 0
Zn = Z + tanh-1 (Y/X)
Rule: Choose di  {-1,1} such that Y  0
Computes functions
hyperbolic sin and cos
X = 1/K’ and Y = 0
Vector length actually shrinks because cosh a(i) > 1
More difficult convergence guarantee – for details see book section 22.4
By repeating iterations 4, 13, 40, 121, …, j, 3j+1 (K’ has been computed to take these extra iterations into account) then convergence is guaranteed for |z| < 1.13 (rotation) or |y| < 0.81 (vector)
Computes functions
hyperbolic tanh-1 y
X = 1 and Z = 0

21. 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

22. Cordic Summary
Can compute virtually all trig functions of common interest
When the number of iterations is fixed, K and K’ are constants
Thus, we can not stop computing if Z (or Y) becomes 0 (except when  = 0); we always need k iterations for k digits of precision
Cordic 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)

23. Key References
Duprat and Muller, The CORDIC algorithm: New results for fast VLSI implementation, IEEE Trans. on Computers, 42(2): , 1993.
Parhami, Computer Arithmetic, Oxford Univ. Press, 1999.
Phatak, Double step branching CORDIC, IEEE Trans. on Computers, 47(5) , May 1998.
Vachss, The CORDIC magnification function, IEEE Micro, 7(5):83-83, Oct
Volder, The CORDIC trigonometric computing technique, IRE Trans. Elecronic Computers, Vol 8, pp , Sept
Walther, A unified algorithm for elementary functions, Proc. Spring Joint Computer Conf, pp , 1971.