1 # Vector Operations Extension to SV
3 This extension is usually dependent on SV SUBVL being implemented. When SUBVL is set to define the length of a subvector the operations in this extension interpret the elements as a single vector.
5 Normally in SV all operations are scalar and independent, and the operations on them may inherently be independently parallelised, with the result being a vector of length exactly equal to the input vectors.
7 In this extension, the subvector itself is typically the unit, although some operations will work on scalars or standard vectors as well, or the result is a scalar that is dependent on all elements within the vector arguments.
9 Examples which can require SUBVL include cross product and may in future involve complex numbers.
13 CORDIC is an extremely general-purpose algorithm useful for a huge number
14 of diverse purposes. In its full form it does however require quite a
15 few parameters, one of which is a vector, making it awkward to include in
16 a standard "scalar" ISA. Additionally the coordinates can be set to circular,
17 linear or hyperbolic, producing three different modes, and the algorithm
18 may also be run in either "vector" mode or "rotation" mode. See [[discussion]]
20 CORDIC can also be used for performing DCT. See
21 <https://arxiv.org/abs/1606.02424>
23 vx, vy = CORDIC(vx, vy, coordinate\_mode, beta)
26 int iterations = 0; // Number of times to run the algorithm
27 float arctanTable[iterations]; // in Radians
28 float K = 0.6073; // K
29 float v_x,v_y; // Vector v; x and y components
31 for(i=0; i < iterations; i++) {
32 arctanTable[i] = atan(pow(2,-i));
35 float vnew_x; // To store the new value of x;
36 for(i = 0; i < iterations; i++) {
37 // If beta is negative, we need to do a counter-clockwise rotation:
39 vnew_x = v_x + (v_y*pow(2,-i));
40 v_y -= (v_x*pow(2,-i));
41 beta += arctanTable[i];
43 // If beta is positive, we need to do a clockwise rotation:
45 vnew_x = v_x - (v_y*pow(2,-i));
46 v_y += (v_x*pow(2,-i));
47 beta -= arctanTable[i];
56 * <http://www.myhdl.org/docs/examples/sinecomp/>
58 ## Vector cross product
60 Result is the cross product of x and y, i.e., the resulting components are, in order:
62 x[1] * y[2] - y[1] * x[2]
63 x[2] * y[0] - y[2] * x[0]
64 x[0] * y[1] - y[0] * x[1]
66 All the operands must be vectors of 3 components of a floating-point type.
70 vec3 a, b; // elements in order a.x, a.y, a.z
72 vec3 t1 = a.yzx; // produce vector [a.y, a.z, a.x]
77 vec3 cross = t1 * t2 - p;
81 Computes the dot product of two vectors. Internal accuracy must be
82 greater than the input vectors and the result.
86 from operator import mul
91 double dot_product(float v[], float u[], int n)
94 for (int i = 0; i < n; i++)
95 result += v[i] * u[i];
101 The scalar length of a vector:
103 sqrt(x[0]^2 + x[1]^2 + ...).
107 The scalar distance between two vectors. Subtracts one vector from the
108 other and returns length:
114 Known as **fmix** in GLSL.
116 <https://en.m.wikipedia.org/wiki/Linear_interpolation>
120 // Imprecise method, which does not guarantee v = v1 when t = 1,
121 // due to floating-point arithmetic error.
122 // This form may be used when the hardware has a native fused
123 // multiply-add instruction.
124 float lerp(float v0, float v1, float t) {
125 return v0 + t * (v1 - v0);
128 // Precise method, which guarantees v = v1 when t = 1.
129 float lerp(float v0, float v1, float t) {
130 return (1 - t) * v0 + t * v1;
135 Not recommended as it is not commonly used and has several trigonometric
138 <https://en.m.wikipedia.org/wiki/Slerp>
142 Quaternion slerp(Quaternion v0, Quaternion v1, double t) {
143 // Only unit quaternions are valid rotations.
144 // Normalize to avoid undefined behavior.
148 // Compute the cosine of the angle between the two vectors.
149 double dot = dot_product(v0, v1);
151 // If the dot product is negative, slerp won't take
152 // the shorter path. Note that v1 and -v1 are equivalent when
153 // the negation is applied to all four components. Fix by
154 // reversing one quaternion.
160 const double DOT_THRESHOLD = 0.9995;
161 if (dot > DOT_THRESHOLD) {
162 // If the inputs are too close for comfort, linearly interpolate
163 // and normalize the result.
165 Quaternion result = v0 + t*(v1 - v0);
170 // Since dot is in range [0, DOT_THRESHOLD], acos is safe
171 double theta_0 = acos(dot); // theta_0 = angle between input vectors
172 double theta = theta_0*t; // theta = angle between v0 and result
173 double sin_theta = sin(theta); // compute this value only once
174 double sin_theta_0 = sin(theta_0); // compute this value only once
176 double s0 = cos(theta) - dot * sin_theta / sin_theta_0; // == sin(theta_0 - theta) / sin(theta_0)
177 double s1 = sin_theta / sin_theta_0;
179 return (s0 * v0) + (s1 * v1);
182 However this algorithm does not involve transcendentals except in
183 the computation of the tables: <https://en.wikipedia.org/wiki/CORDIC#Rotation_mode>
185 function v = cordic(beta,n)
186 % This function computes v = [cos(beta), sin(beta)] (beta in radians)
187 % using n iterations. Increasing n will increase the precision.
189 if beta < -pi/2 || beta > pi/2
191 v = cordic(beta + pi, n);
193 v = cordic(beta - pi, n);
195 v = -v; % flip the sign for second or third quadrant
199 % Initialization of tables of constants used by CORDIC
200 % need a table of arctangents of negative powers of two, in radians:
201 % angles = atan(2.^-(0:27));
203 0.78539816339745 0.46364760900081
204 0.24497866312686 0.12435499454676 ...
205 0.06241880999596 0.03123983343027
206 0.01562372862048 0.00781234106010 ...
207 0.00390623013197 0.00195312251648
208 0.00097656218956 0.00048828121119 ...
209 0.00024414062015 0.00012207031189
210 0.00006103515617 0.00003051757812 ...
211 0.00001525878906 0.00000762939453
212 0.00000381469727 0.00000190734863 ...
213 0.00000095367432 0.00000047683716
214 0.00000023841858 0.00000011920929 ...
215 0.00000005960464 0.00000002980232
216 0.00000001490116 0.00000000745058 ];
217 % and a table of products of reciprocal lengths of vectors [1, 2^-2j]:
218 % Kvalues = cumprod(1./abs(1 + 1j*2.^(-(0:23))))
220 0.70710678118655 0.63245553203368
221 0.61357199107790 0.60883391251775 ...
222 0.60764825625617 0.60735177014130
223 0.60727764409353 0.60725911229889 ...
224 0.60725447933256 0.60725332108988
225 0.60725303152913 0.60725295913894 ...
226 0.60725294104140 0.60725293651701
227 0.60725293538591 0.60725293510314 ...
228 0.60725293503245 0.60725293501477
229 0.60725293501035 0.60725293500925 ...
230 0.60725293500897 0.60725293500890
231 0.60725293500889 0.60725293500888 ];
232 Kn = Kvalues(min(n, length(Kvalues)));
234 % Initialize loop variables:
235 v = [1;0]; % start with 2-vector cosine and sine of zero
246 factor = sigma * poweroftwo;
247 % Note the matrix multiplication can be done using scaling by
248 % powers of two and addition subtraction
249 R = [1, -factor; factor, 1];
250 v = R * v; % 2-by-2 matrix multiply
251 beta = beta - sigma * angle; % update the remaining angle
252 poweroftwo = poweroftwo / 2;
253 % update the angle from table, or eventually by just dividing by two
254 if j+2 > length(angles)
261 % Adjust length of output vector to be [cos(beta), sin(beta)]:
267 2x2 matrix multiply can be done with shifts and adds:
269 x = v[0] - sigma * (v[1] * 2^(-j));
270 y = sigma * (v[0] * 2^(-j)) + v[1];
273 The technique is outlined in a paper as being applicable to 3D:
274 <https://www.atlantis-press.com/proceedings/jcis2006/232>
276 # Expensive 3-operand OP32 operations
278 3-operand operations are extremely expensive in terms of OP32 encoding space. A potential idea is to embed 3 RVC register formats across two out of three 5-bit fields rs1/rs2/rd