simple_v_extension/vector_ops.mdwn

   1 [[!tag standards]]
   2
   3 # Vector Operations Extension to SV
   4
   5 This extension defines vector operations that would otherwise take several cycles to complete in software. With 3D priorities being to compute as many pixels per clock as possible, the normal RISC rules (reduce opcode count and make heavy use of macro op fusion) do not necessarily apply.
   6
   7 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.
   8
   9 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.
  10
  11 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.
  12
  13 However given that some of the parameters are vectors (with and without SUBVL set), and some are scalars (where SUBVL will not apply), some clear rules need to be defined as to how the operations work.
  14
  15 Examples which can require SUBVL include cross product and may in future involve complex numbers.
  16
  17 ## CORDIC
  18
  19 6 opcode options (fmt3):
  20
  21 * CORDIC.lin.rot vd, vs, beta
  22 * CORDIC.cir.rot vd, vs, beta
  23 * CORDIC.hyp.rot vd, vs, beta
  24 * CORDIC.lin.vec vd, vs, beta
  25 * CORDIC.cir.vec vd, vs, beta
  26 * CORDIC.hyp.vec vd, vs, beta
  27
  28
  29 | Instr | result | src1 | src2 | SUBVL | VL | Notes |
  30 | ------------------ | ------ | ---- | ---- | ----- | -- | ------ |
  31 | CORDIC.x.t vd, vs1, rs2 | vec2 | vec2 | scal | 2 | any | src2 ignores SUBVL |
  32
  33 SUBVL must be set to 2 and applies to vd and vs. SUBVL is *ignored* on beta.  vd and vs must be marked as vectors.
  34
  35 VL may be applied.  beta as a scalar is ok (applies across all vectors vd and vs). Predication is also ok (single predication) sourced from vd. Use of swizzle is also ok.
  36
  37 Non vector args vd, vs are reserved encodings.
  38
  39 CORDIC is an extremely general-purpose algorithm useful for a huge number
  40 of diverse purposes.  In its full form it does however require quite a
  41 few parameters, one of which is a vector, making it awkward to include in
  42 a standard "scalar" ISA.  Additionally the coordinates can be set to circular,
  43 linear or hyperbolic, producing three different modes, and the algorithm
  44 may also be run in either "vector" mode or "rotation" mode.  See [[discussion]]
  45
  46 CORDIC can also be used for performing DCT.  See
  47 <https://arxiv.org/abs/1606.02424>
  48
  49 CORDIC has several RADIX-4 papers for efficient pipelining.  Each stage requires its own ROM tables which can get costly.  Two combinatorial blocks may be chained together to double the RADIX and halve the pipeline depth, at the cost of doubling the latency.
  50
  51 Also, to get good accuracy, particularly at the limits of CORDIC input range, requires double the bitwidth of the output in internal computations. This similar to how MUL requires double the bitwidth to compute.
  52
  53 Links:
  54
  55 * <http://www.myhdl.org/docs/examples/sinecomp/>
  56 * <https://www.atlantis-press.com/proceedings/jcis2006/232>
  57
  58 ## Vector cross product
  59
  60 * VCROSS vd, vs1, vs1
  61
  62 Result is the cross product of x and y.
  63
  64 SUBVL must be set to 3, and all regs must be vectors. VL nonzero produces multiple results in vd.
  65
  66 | Instr | result | src1 | src2 | SUBVL | VL |
  67 | ------------------ | ------ | ---- | ---- | ----- | -- |
  68 | VCROSS vd, vs1, vs2 | vec3 | vec3 | vec3 | 3 | any |
  69
  70 The resulting components are, in order:
  71
  72     x[1] * y[2] - y[1] * x[2]
  73     x[2] * y[0] - y[2] * x[0]
  74     x[0] * y[1] - y[0] * x[1]
  75
  76 All the operands must be vectors of 3 components of a floating-point type.
  77
  78 Pseudocode:
  79
  80     vec3 a, b; // elements in order a.x, a.y, a.z
  81     // compute a cross b:
  82     vec3 t1 = a.yzx; // produce vector [a.y, a.z, a.x]
  83     vec3 t2 = b.zxy;
  84     vec3 t3 = a.zxy;
  85     vec3 t4 = b.yzx;
  86     vec3 p = t3 * t4;
  87     vec3 cross = t1 * t2 - p;
  88
  89 Assembler:
  90
  91     fswizzlei,2130 F4, F1
  92     fswizzlei,1320 F5, F1
  93     fswizzlei,2130 F6, F2
  94     fswizzlei,1320 F7, F2
  95     fmul F8, F5, F6
  96     fmulsub F3, F4, F7, F8
  97
  98 ## Vector dot product
  99
 100 * VDOT rd, vs1, vs2
 101
 102 Computes the dot product of two vectors. Internal accuracy must be
 103 greater than the input vectors and the result.
 104
 105 There are two possible argument options:
 106
 107 * SUBVL=2,3,4 vs1 and vs2 set as vectors,  multiple results are generated. When VL is set, only the first (unpredicated) SUBVector is used to create a result, if rd is scalar (standard behaviour for single predication). Otherwise, if rd is a vector, multiple scalar results are calculated (i.e. SUBVL is always ignored for rd). Swizzling may be applied.
 108 * When rd=scalar, SUBVL=1 and vs1=vec, vs2=vec, one scalar result is generated from the entire src vectors.  Predication is allowed on the src vectors.
 109
 110
 111 | Instr | result | src1 | src2 | SUBVL | VL |
 112 | ------------------ | ------ | ---- | ---- | ----- | -- |
 113 | VDOT rd, vs1, vs2 | scal | vec  | vec | 2-4 | any |
 114 | VDOT rd, vs1, vs2 | scal | vec  | vec | 1 | any |
 115
 116 Pseudocode in python:
 117
 118     from operator import mul
 119     sum(map(mul, A, B))
 120
 121 Pseudocode in c:
 122
 123     double dot_product(float v[], float u[], int n)
 124     {
 125         double result = 0.0;
 126         for (int i = 0; i < n; i++)
 127             result += v[i] * u[i];
 128         return result;
 129     }
 130
 131 ## Vector Normalisation (not included)
 132
 133 Vector normalisation may be performed through dot product, recip square root and multiplication:
 134
 135     fdot F3, F1, F1 # vector dot with self
 136     rcpsqrta F3, F3
 137     fscale,0 F2, F3, F1
 138
 139 Or it may be performed through VLEN (Vector length) and division.
 140
 141 ## Vector length
 142
 143 * rd=scalar, vs1=vec (SUBVL=1)
 144 * rd=scalar, vs1=vec (SUBVL=2,3,4) only 1 (predication rules apply)
 145 * rd=vec, SUBVL ignored; vs1=vec, SUBVL=2,3,4
 146 * rd=vec, SUBVL ignored; vs1=vec, SUBVL=1: reserved encoding.
 147
 148 * VLEN rd, vs1
 149
 150 The scalar length of a vector:
 151
 152     sqrt(x[0]^2 + x[1]^2 + ...).
 153
 154 One option is for this to be a macro op fusion sequence, with inverse-sqrt also being a second macro op sequence suitable for normalisation.
 155
 156 ## Vector distance
 157
 158 * VDIST rd, vs1, vs2
 159
 160 The scalar distance between two vectors. Subtracts one vector from the
 161 other and returns length:
 162
 163     length(v0 - v1)
 164
 165 ## Vector LERP
 166
 167 * VLERP vd, vs1, rs2 # SUBVL=2: vs1.v0 vs1.v1
 168
 169 | Instr | result | src1 | src2 | SUBVL | VL |
 170 | ------------------ | ------ | ---- | ---- | ----- | -- |
 171 | VLERP vd, vs1, rs2 | vec2 | vec2 | scal | 2 | any |
 172
 173 Known as **fmix** in GLSL.
 174
 175 <https://en.m.wikipedia.org/wiki/Linear_interpolation>
 176
 177 Pseudocode:
 178
 179     // Imprecise method, which does not guarantee v = v1 when t = 1,
 180     // due to floating-point arithmetic error.
 181     // This form may be used when the hardware has a native fused
 182     // multiply-add instruction.
 183     float lerp(float v0, float v1, float t) {
 184       return v0 + t * (v1 - v0);
 185     }
 186
 187     // Precise method, which guarantees v = v1 when t = 1.
 188     float lerp(float v0, float v1, float t) {
 189       return (1 - t) * v0 + t * v1;
 190     }
 191
 192 ## Vector SLERP
 193
 194 * VSLERP vd, vs1, vs2, rs3
 195
 196 Not recommended as it is not commonly used and has several trigonometric
 197 functions, although CORDIC in vector rotate circular mode is designed for this purpose. Also a costly 4 arg operation.
 198
 199 <https://en.m.wikipedia.org/wiki/Slerp>
 200
 201 Pseudocode:
 202
 203     Quaternion slerp(Quaternion v0, Quaternion v1, double t) {
 204         // Only unit quaternions are valid rotations.
 205         // Normalize to avoid undefined behavior.
 206         v0.normalize();
 207         v1.normalize();
 208
 209         // Compute the cosine of the angle between the two vectors.
 210         double dot = dot_product(v0, v1);
 211
 212         // If the dot product is negative, slerp won't take
 213         // the shorter path. Note that v1 and -v1 are equivalent when
 214         // the negation is applied to all four components. Fix by
 215         // reversing one quaternion.
 216         if (dot < 0.0f) {
 217             v1 = -v1;
 218             dot = -dot;
 219         }
 220
 221         const double DOT_THRESHOLD = 0.9995;
 222         if (dot > DOT_THRESHOLD) {
 223             // If the inputs are too close for comfort, linearly interpolate
 224             // and normalize the result.
 225
 226             Quaternion result = v0 + t*(v1 - v0);
 227             result.normalize();
 228             return result;
 229         }
 230
 231         // Since dot is in range [0, DOT_THRESHOLD], acos is safe
 232         double theta_0 = acos(dot);        // theta_0 = angle between input vectors
 233         double theta = theta_0*t;          // theta = angle between v0 and result
 234         double sin_theta = sin(theta);     // compute this value only once
 235         double sin_theta_0 = sin(theta_0); // compute this value only once
 236
 237         double s0 = cos(theta) - dot * sin_theta / sin_theta_0;  // == sin(theta_0 - theta) / sin(theta_0)
 238         double s1 = sin_theta / sin_theta_0;
 239
 240         return (s0 * v0) + (s1 * v1);
 241     }
 242
 243 However this algorithm does not involve transcendentals except in
 244 the computation of the tables: <https://en.wikipedia.org/wiki/CORDIC#Rotation_mode>
 245
 246     function v = cordic(beta,n)
 247         % This function computes v = [cos(beta), sin(beta)] (beta in radians)
 248         % using n iterations. Increasing n will increase the precision.
 249
 250         if beta < -pi/2 || beta > pi/2
 251             if beta < 0
 252                 v = cordic(beta + pi, n);
 253             else
 254                 v = cordic(beta - pi, n);
 255             end
 256             v = -v; % flip the sign for second or third quadrant
 257             return
 258         end
 259
 260         % Initialization of tables of constants used by CORDIC
 261         % need a table of arctangents of negative powers of two, in radians:
 262         % angles = atan(2.^-(0:27));
 263         angles =  [  ...
 264             0.78539816339745   0.46364760900081
 265             0.24497866312686   0.12435499454676 ...
 266             0.06241880999596   0.03123983343027
 267             0.01562372862048   0.00781234106010 ...
 268             0.00390623013197   0.00195312251648
 269             0.00097656218956   0.00048828121119 ...
 270             0.00024414062015   0.00012207031189
 271             0.00006103515617   0.00003051757812 ...
 272             0.00001525878906   0.00000762939453
 273             0.00000381469727   0.00000190734863 ...
 274             0.00000095367432   0.00000047683716
 275             0.00000023841858   0.00000011920929 ...
 276             0.00000005960464   0.00000002980232
 277             0.00000001490116   0.00000000745058 ];
 278         % and a table of products of reciprocal lengths of vectors [1, 2^-2j]:
 279         % Kvalues = cumprod(1./abs(1 + 1j*2.^(-(0:23))))
 280         Kvalues = [ ...
 281             0.70710678118655   0.63245553203368
 282             0.61357199107790   0.60883391251775 ...
 283             0.60764825625617   0.60735177014130
 284             0.60727764409353   0.60725911229889 ...
 285             0.60725447933256   0.60725332108988
 286             0.60725303152913   0.60725295913894 ...
 287             0.60725294104140   0.60725293651701
 288             0.60725293538591   0.60725293510314 ...
 289             0.60725293503245   0.60725293501477
 290             0.60725293501035   0.60725293500925 ...
 291             0.60725293500897   0.60725293500890
 292             0.60725293500889   0.60725293500888 ];
 293         Kn = Kvalues(min(n, length(Kvalues)));
 294
 295         % Initialize loop variables:
 296         v = [1;0]; % start with 2-vector cosine and sine of zero
 297         poweroftwo = 1;
 298         angle = angles(1);
 299
 300         % Iterations
 301         for j = 0:n-1;
 302             if beta < 0
 303                 sigma = -1;
 304             else
 305                 sigma = 1;
 306             end
 307             factor = sigma * poweroftwo;
 308             % Note the matrix multiplication can be done using scaling by
 309             % powers of two and addition subtraction
 310             R = [1, -factor; factor, 1];
 311             v = R * v; % 2-by-2 matrix multiply
 312             beta = beta - sigma * angle; % update the remaining angle
 313             poweroftwo = poweroftwo / 2;
 314             % update the angle from table, or eventually by just dividing by two
 315             if j+2 > length(angles)
 316                 angle = angle / 2;
 317             else
 318                 angle = angles(j+2);
 319             end
 320         end
 321
 322         % Adjust length of output vector to be [cos(beta), sin(beta)]:
 323         v = v * Kn;
 324         return
 325
 326     endfunction
 327
 328 2x2 matrix multiply can be done with shifts and adds:
 329
 330     x = v[0] - sigma * (v[1] * 2^(-j));
 331     y = sigma * (v[0] * 2^(-j)) + v[1];
 332     v = [x; y];
 333
 334 The technique is outlined in a paper as being applicable to 3D:
 335 <https://www.atlantis-press.com/proceedings/jcis2006/232>
 336
 337 # Expensive 3-operand OP32 operations
 338
 339 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
 340
 341 Another is to overwrite one of the src registers.
 342
 343 # Opcode Table
 344
 345 TODO
 346
 347 # Links
 348
 349 * <http://lists.libre-riscv.org/pipermail/libre-riscv-dev/2019-September/002736.html>
 350 * <http://lists.libre-riscv.org/pipermail/libre-riscv-dev/2019-September/002733.html>
 351 * <http://bugs.libre-riscv.org/show_bug.cgi?id=142>
 352
 353 Research Papers
 354
 355 * <https://www.researchgate.net/publication/2938554_PLX_FP_An_Efficient_Floating-Point_Instruction_Set_for_3D_Graphics>