+[[!tag standards]]
+
# Vector Operations Extension to SV
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.
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.
+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.
+
Examples which can require SUBVL include cross product and may in future involve complex numbers.
+## CORDIC
+
+* SUBVL=2, vd, vs; SUBVL ignored on beta.
+* VL nonzero ok. beta as scalar ok (applies across all vectors)
+* non vector args vd, vs, or SUBVL!=2 reserved.
+
+6 opcode options (fmt3):
+
+* CORDIC.lin.rot vd, vs, beta
+* CORDIC.cir.rot vd, vs, beta
+* CORDIC.hyp.rot vd, vs, beta
+* CORDIC.lin.vec vd, vs, beta
+* CORDIC.cir.vec vd, vs, beta
+* CORDIC.hyp.vec vd, vs, beta
+
+CORDIC is an extremely general-purpose algorithm useful for a huge number
+of diverse purposes. In its full form it does however require quite a
+few parameters, one of which is a vector, making it awkward to include in
+a standard "scalar" ISA. Additionally the coordinates can be set to circular,
+linear or hyperbolic, producing three different modes, and the algorithm
+may also be run in either "vector" mode or "rotation" mode. See [[discussion]]
+
+CORDIC can also be used for performing DCT. See
+<https://arxiv.org/abs/1606.02424>
+
+vx, vy = CORDIC(vx, vy, coordinate\_mode, beta)
+
+ int i = 0;
+ int iterations = 0; // Number of times to run the algorithm
+ float arctanTable[iterations]; // in Radians
+ float K = 0.6073; // K
+ float v_x,v_y; // Vector v; x and y components
+
+ for(i=0; i < iterations; i++) {
+ arctanTable[i] = atan(pow(2,-i));
+ }
+
+ float vnew_x; // To store the new value of x;
+ for(i = 0; i < iterations; i++) {
+ // If beta is negative, we need to do a counter-clockwise rotation:
+ if( beta < 0) {
+ vnew_x = v_x + (v_y*pow(2,-i));
+ v_y -= (v_x*pow(2,-i));
+ beta += arctanTable[i];
+ }
+ // If beta is positive, we need to do a clockwise rotation:
+ else {
+ vnew_x = v_x - (v_y*pow(2,-i));
+ v_y += (v_x*pow(2,-i));
+ beta -= arctanTable[i];
+ }
+ v_x = vnew_x;
+ }
+ v_x *= K;
+ v_y *= K;
+
+Links:
+
+* <http://www.myhdl.org/docs/examples/sinecomp/>
+* <https://www.atlantis-press.com/proceedings/jcis2006/232>
+
## Vector cross product
+SUBVL=3, all regs. VL nonzero produces multiple vd results.
+
+* VCROSS vd, vs1, vs1
+
Result is the cross product of x and y, i.e., the resulting components are, in order:
x[1] * y[2] - y[1] * x[2]
All the operands must be vectors of 3 components of a floating-point type.
+Pseudocode:
+
+ vec3 a, b; // elements in order a.x, a.y, a.z
+ // compute a cross b:
+ vec3 t1 = a.yzx; // produce vector [a.y, a.z, a.x]
+ vec3 t2 = b.zxy;
+ vec3 t3 = a.zxy;
+ vec3 t4 = b.yzx;
+ vec3 p = t3 * t4;
+ vec3 cross = t1 * t2 - p;
+
+Assembler:
+
+ fpermute,2130 F4, F1
+ fpermute,1320 F5, F1
+ fpermute,2130 F6, F2
+ fpermute,1320 F7, F2
+ fmul F8, F5, F6
+ fmulsub F3, F4, F7, F8
+
## Vector dot product
-Computes the dot product of two vectors. Internal accuracy must be greater than the
-input vectors and the result.
+* SUBVL ignored on rd. SUBVL=2,3,4 vs1,vs2, if all vectors, multiple results generated. If rd scalar, only first (unpredicated) SUBVector is used.
+* rd=scalar, SUBVL=1 and vs1, vs2=vec will produce one scalar result. Predication allowed on src vectors.
+
+* VDOT rd, vs1, vs2
+
+Computes the dot product of two vectors. Internal accuracy must be
+greater than the input vectors and the result.
+
+Pseudocode in python:
+
+ from operator import mul
+ sum(map(mul, A, B))
+
+Pseudocode in c:
+
+ double dot_product(float v[], float u[], int n)
+ {
+ double result = 0.0;
+ for (int i = 0; i < n; i++)
+ result += v[i] * u[i];
+ return result;
+ }
+
+## Vector Normalisation (not included)
+
+Vector normalisation may be performed through dot product, recip square root and multiplication:
+
+ fdot F3, F1, F1 # vector dot with self
+ rcpsqrta F3, F3
+ fscale,0 F2, F3, F1
+
+Or it may be performed through VLEN (Vector length) and division.
## Vector length
+* rd=scalar, vs1=vec (SUBVL=1)
+* rd=scalar, vs1=vec (SUBVL=2,3,4) only 1 (predication rules apply)
+* rd=vec, SUBVL ignored; vs1=vec, SUBVL=2,3,4
+* rd=vec, SUBVL ignored; vs1=vec, SUBVL=1: reserved encoding.
+
+* VLEN rd, vs1
+
The scalar length of a vector:
sqrt(x[0]^2 + x[1]^2 + ...).
+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.
+
## Vector distance
-The scalar distance between two vectors. Subtracts one vector from the other and returns length
+* VDIST rd, vs1, vs2
+
+The scalar distance between two vectors. Subtracts one vector from the
+other and returns length:
+
+ length(v0 - v1)
## Vector LERP
+* VLERP rd, vs1, rs2 # SUBVL=2: vs1.v0 vs1.v1
+
Known as **fmix** in GLSL.
<https://en.m.wikipedia.org/wiki/Linear_interpolation>
## Vector SLERP
+* VSLERP vd, vs1, vs2, rs3
+
+Not recommended as it is not commonly used and has several trigonometric
+functions, although CORDIC in vector rotate circular mode is designed for this purpose. Also a costly 4 arg operation.
+
<https://en.m.wikipedia.org/wiki/Slerp>
Pseudocode:
return (s0 * v0) + (s1 * v1);
}
+However this algorithm does not involve transcendentals except in
+the computation of the tables: <https://en.wikipedia.org/wiki/CORDIC#Rotation_mode>
+
+ function v = cordic(beta,n)
+ % This function computes v = [cos(beta), sin(beta)] (beta in radians)
+ % using n iterations. Increasing n will increase the precision.
+
+ if beta < -pi/2 || beta > pi/2
+ if beta < 0
+ v = cordic(beta + pi, n);
+ else
+ v = cordic(beta - pi, n);
+ end
+ v = -v; % flip the sign for second or third quadrant
+ return
+ end
+
+ % Initialization of tables of constants used by CORDIC
+ % need a table of arctangents of negative powers of two, in radians:
+ % angles = atan(2.^-(0:27));
+ angles = [ ...
+ 0.78539816339745 0.46364760900081
+ 0.24497866312686 0.12435499454676 ...
+ 0.06241880999596 0.03123983343027
+ 0.01562372862048 0.00781234106010 ...
+ 0.00390623013197 0.00195312251648
+ 0.00097656218956 0.00048828121119 ...
+ 0.00024414062015 0.00012207031189
+ 0.00006103515617 0.00003051757812 ...
+ 0.00001525878906 0.00000762939453
+ 0.00000381469727 0.00000190734863 ...
+ 0.00000095367432 0.00000047683716
+ 0.00000023841858 0.00000011920929 ...
+ 0.00000005960464 0.00000002980232
+ 0.00000001490116 0.00000000745058 ];
+ % and a table of products of reciprocal lengths of vectors [1, 2^-2j]:
+ % Kvalues = cumprod(1./abs(1 + 1j*2.^(-(0:23))))
+ Kvalues = [ ...
+ 0.70710678118655 0.63245553203368
+ 0.61357199107790 0.60883391251775 ...
+ 0.60764825625617 0.60735177014130
+ 0.60727764409353 0.60725911229889 ...
+ 0.60725447933256 0.60725332108988
+ 0.60725303152913 0.60725295913894 ...
+ 0.60725294104140 0.60725293651701
+ 0.60725293538591 0.60725293510314 ...
+ 0.60725293503245 0.60725293501477
+ 0.60725293501035 0.60725293500925 ...
+ 0.60725293500897 0.60725293500890
+ 0.60725293500889 0.60725293500888 ];
+ Kn = Kvalues(min(n, length(Kvalues)));
+
+ % Initialize loop variables:
+ v = [1;0]; % start with 2-vector cosine and sine of zero
+ poweroftwo = 1;
+ angle = angles(1);
+
+ % Iterations
+ for j = 0:n-1;
+ if beta < 0
+ sigma = -1;
+ else
+ sigma = 1;
+ end
+ factor = sigma * poweroftwo;
+ % Note the matrix multiplication can be done using scaling by
+ % powers of two and addition subtraction
+ R = [1, -factor; factor, 1];
+ v = R * v; % 2-by-2 matrix multiply
+ beta = beta - sigma * angle; % update the remaining angle
+ poweroftwo = poweroftwo / 2;
+ % update the angle from table, or eventually by just dividing by two
+ if j+2 > length(angles)
+ angle = angle / 2;
+ else
+ angle = angles(j+2);
+ end
+ end
+
+ % Adjust length of output vector to be [cos(beta), sin(beta)]:
+ v = v * Kn;
+ return
+
+ endfunction
+
+2x2 matrix multiply can be done with shifts and adds:
+
+ x = v[0] - sigma * (v[1] * 2^(-j));
+ y = sigma * (v[0] * 2^(-j)) + v[1];
+ v = [x; y];
+
+The technique is outlined in a paper as being applicable to 3D:
+<https://www.atlantis-press.com/proceedings/jcis2006/232>
+
# Expensive 3-operand OP32 operations
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
+
+Another is to overwrite one of the src registers.
+
+# Opcode Table
+
+TODO
+
+# Links
+
+* <http://lists.libre-riscv.org/pipermail/libre-riscv-dev/2019-September/002736.html>
+* <http://lists.libre-riscv.org/pipermail/libre-riscv-dev/2019-September/002733.html>
+* <http://bugs.libre-riscv.org/show_bug.cgi?id=142>
+
+Research Papers
+
+* <https://www.researchgate.net/publication/2938554_PLX_FP_An_Efficient_Floating-Point_Instruction_Set_for_3D_Graphics>
projects / libreriscv.git / blobdiff