[[!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. 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. 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 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: * ## 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] x[2] * y[0] - y[2] * x[0] x[0] * y[1] - y[0] * x[1] 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 * 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 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 + ...). ## Vector distance * 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. Pseudocode: // Imprecise method, which does not guarantee v = v1 when t = 1, // due to floating-point arithmetic error. // This form may be used when the hardware has a native fused // multiply-add instruction. float lerp(float v0, float v1, float t) { return v0 + t * (v1 - v0); } // Precise method, which guarantees v = v1 when t = 1. float lerp(float v0, float v1, float t) { return (1 - t) * v0 + t * v1; } ## 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. Pseudocode: Quaternion slerp(Quaternion v0, Quaternion v1, double t) { // Only unit quaternions are valid rotations. // Normalize to avoid undefined behavior. v0.normalize(); v1.normalize(); // Compute the cosine of the angle between the two vectors. double dot = dot_product(v0, v1); // If the dot product is negative, slerp won't take // the shorter path. Note that v1 and -v1 are equivalent when // the negation is applied to all four components. Fix by // reversing one quaternion. if (dot < 0.0f) { v1 = -v1; dot = -dot; } const double DOT_THRESHOLD = 0.9995; if (dot > DOT_THRESHOLD) { // If the inputs are too close for comfort, linearly interpolate // and normalize the result. Quaternion result = v0 + t*(v1 - v0); result.normalize(); return result; } // Since dot is in range [0, DOT_THRESHOLD], acos is safe double theta_0 = acos(dot); // theta_0 = angle between input vectors double theta = theta_0*t; // theta = angle between v0 and result double sin_theta = sin(theta); // compute this value only once double sin_theta_0 = sin(theta_0); // compute this value only once double s0 = cos(theta) - dot * sin_theta / sin_theta_0; // == sin(theta_0 - theta) / sin(theta_0) double s1 = sin_theta / sin_theta_0; return (s0 * v0) + (s1 * v1); } However this algorithm does not involve transcendentals except in the computation of the tables: 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: # 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 * * *