src/ieee754/part_mul_add/multiply.py

   1 # SPDX-License-Identifier: LGPL-2.1-or-later
   2 # See Notices.txt for copyright information
   3 """Integer Multiplication."""
   4
   5 from nmigen import Signal, Module, Value, Elaboratable, Cat, C, Mux, Repl
   6 from nmigen.hdl.ast import Assign
   7 from abc import ABCMeta, abstractmethod
   8 from nmigen.cli import main
   9
  10
  11 class PartitionPoints(dict):
  12     """Partition points and corresponding ``Value``s.
  13
  14     The points at where an ALU is partitioned along with ``Value``s that
  15     specify if the corresponding partition points are enabled.
  16
  17     For example: ``{1: True, 5: True, 10: True}`` with
  18     ``width == 16`` specifies that the ALU is split into 4 sections:
  19     * bits 0 <= ``i`` < 1
  20     * bits 1 <= ``i`` < 5
  21     * bits 5 <= ``i`` < 10
  22     * bits 10 <= ``i`` < 16
  23
  24     If the partition_points were instead ``{1: True, 5: a, 10: True}``
  25     where ``a`` is a 1-bit ``Signal``:
  26     * If ``a`` is asserted:
  27         * bits 0 <= ``i`` < 1
  28         * bits 1 <= ``i`` < 5
  29         * bits 5 <= ``i`` < 10
  30         * bits 10 <= ``i`` < 16
  31     * Otherwise
  32         * bits 0 <= ``i`` < 1
  33         * bits 1 <= ``i`` < 10
  34         * bits 10 <= ``i`` < 16
  35     """
  36
  37     def __init__(self, partition_points=None):
  38         """Create a new ``PartitionPoints``.
  39
  40         :param partition_points: the input partition points to values mapping.
  41         """
  42         super().__init__()
  43         if partition_points is not None:
  44             for point, enabled in partition_points.items():
  45                 if not isinstance(point, int):
  46                     raise TypeError("point must be a non-negative integer")
  47                 if point < 0:
  48                     raise ValueError("point must be a non-negative integer")
  49                 self[point] = Value.wrap(enabled)
  50
  51     def like(self, name=None, src_loc_at=0):
  52         """Create a new ``PartitionPoints`` with ``Signal``s for all values.
  53
  54         :param name: the base name for the new ``Signal``s.
  55         """
  56         if name is None:
  57             name = Signal(src_loc_at=1+src_loc_at).name  # get variable name
  58         retval = PartitionPoints()
  59         for point, enabled in self.items():
  60             retval[point] = Signal(enabled.shape(), name=f"{name}_{point}")
  61         return retval
  62
  63     def eq(self, rhs):
  64         """Assign ``PartitionPoints`` using ``Signal.eq``."""
  65         if set(self.keys()) != set(rhs.keys()):
  66             raise ValueError("incompatible point set")
  67         for point, enabled in self.items():
  68             yield enabled.eq(rhs[point])
  69
  70     def as_mask(self, width):
  71         """Create a bit-mask from `self`.
  72
  73         Each bit in the returned mask is clear only if the partition point at
  74         the same bit-index is enabled.
  75
  76         :param width: the bit width of the resulting mask
  77         """
  78         bits = []
  79         for i in range(width):
  80             if i in self:
  81                 bits.append(~self[i])
  82             else:
  83                 bits.append(True)
  84         return Cat(*bits)
  85
  86     def get_max_partition_count(self, width):
  87         """Get the maximum number of partitions.
  88
  89         Gets the number of partitions when all partition points are enabled.
  90         """
  91         retval = 1
  92         for point in self.keys():
  93             if point < width:
  94                 retval += 1
  95         return retval
  96
  97     def fits_in_width(self, width):
  98         """Check if all partition points are smaller than `width`."""
  99         for point in self.keys():
 100             if point >= width:
 101                 return False
 102         return True
 103
 104
 105 class FullAdder(Elaboratable):
 106     """Full Adder.
 107
 108     :attribute in0: the first input
 109     :attribute in1: the second input
 110     :attribute in2: the third input
 111     :attribute sum: the sum output
 112     :attribute carry: the carry output
 113     """
 114
 115     def __init__(self, width):
 116         """Create a ``FullAdder``.
 117
 118         :param width: the bit width of the input and output
 119         """
 120         self.in0 = Signal(width)
 121         self.in1 = Signal(width)
 122         self.in2 = Signal(width)
 123         self.sum = Signal(width)
 124         self.carry = Signal(width)
 125
 126     def elaborate(self, platform):
 127         """Elaborate this module."""
 128         m = Module()
 129         m.d.comb += self.sum.eq(self.in0 ^ self.in1 ^ self.in2)
 130         m.d.comb += self.carry.eq((self.in0 & self.in1)
 131                                   | (self.in1 & self.in2)
 132                                   | (self.in2 & self.in0))
 133         return m
 134
 135
 136 class PartitionedAdder(Elaboratable):
 137     """Partitioned Adder.
 138
 139     :attribute width: the bit width of the input and output. Read-only.
 140     :attribute a: the first input to the adder
 141     :attribute b: the second input to the adder
 142     :attribute output: the sum output
 143     :attribute partition_points: the input partition points. Modification not
 144         supported, except for by ``Signal.eq``.
 145     """
 146
 147     def __init__(self, width, partition_points):
 148         """Create a ``PartitionedAdder``.
 149
 150         :param width: the bit width of the input and output
 151         :param partition_points: the input partition points
 152         """
 153         self.width = width
 154         self.a = Signal(width)
 155         self.b = Signal(width)
 156         self.output = Signal(width)
 157         self.partition_points = PartitionPoints(partition_points)
 158         if not self.partition_points.fits_in_width(width):
 159             raise ValueError("partition_points doesn't fit in width")
 160         expanded_width = 0
 161         for i in range(self.width):
 162             if i in self.partition_points:
 163                 expanded_width += 1
 164             expanded_width += 1
 165         self._expanded_width = expanded_width
 166         self._expanded_a = Signal(expanded_width)
 167         self._expanded_b = Signal(expanded_width)
 168         self._expanded_output = Signal(expanded_width)
 169
 170     def elaborate(self, platform):
 171         """Elaborate this module."""
 172         m = Module()
 173         expanded_index = 0
 174         # store bits in a list, use Cat later.  graphviz is much cleaner
 175         al = []
 176         bl = []
 177         ol = []
 178         ea = []
 179         eb = []
 180         eo = []
 181         # partition points are "breaks" (extra zeros) in what would otherwise
 182         # be a massive long add.
 183         for i in range(self.width):
 184             if i in self.partition_points:
 185                 # add extra bit set to 0 + 0 for enabled partition points
 186                 # and 1 + 0 for disabled partition points
 187                 ea.append(self._expanded_a[expanded_index])
 188                 al.append(~self.partition_points[i])
 189                 eb.append(self._expanded_b[expanded_index])
 190                 bl.append(C(0))
 191                 expanded_index += 1
 192             ea.append(self._expanded_a[expanded_index])
 193             al.append(self.a[i])
 194             eb.append(self._expanded_b[expanded_index])
 195             bl.append(self.b[i])
 196             eo.append(self._expanded_output[expanded_index])
 197             ol.append(self.output[i])
 198             expanded_index += 1
 199         # combine above using Cat
 200         m.d.comb += Cat(*ea).eq(Cat(*al))
 201         m.d.comb += Cat(*eb).eq(Cat(*bl))
 202         m.d.comb += Cat(*ol).eq(Cat(*eo))
 203         # use only one addition to take advantage of look-ahead carry and
 204         # special hardware on FPGAs
 205         m.d.comb += self._expanded_output.eq(
 206             self._expanded_a + self._expanded_b)
 207         return m
 208
 209
 210 FULL_ADDER_INPUT_COUNT = 3
 211
 212
 213 class AddReduce(Elaboratable):
 214     """Add list of numbers together.
 215
 216     :attribute inputs: input ``Signal``s to be summed. Modification not
 217         supported, except for by ``Signal.eq``.
 218     :attribute register_levels: List of nesting levels that should have
 219         pipeline registers.
 220     :attribute output: output sum.
 221     :attribute partition_points: the input partition points. Modification not
 222         supported, except for by ``Signal.eq``.
 223     """
 224
 225     def __init__(self, inputs, output_width, register_levels, partition_points):
 226         """Create an ``AddReduce``.
 227
 228         :param inputs: input ``Signal``s to be summed.
 229         :param output_width: bit-width of ``output``.
 230         :param register_levels: List of nesting levels that should have
 231             pipeline registers.
 232         :param partition_points: the input partition points.
 233         """
 234         self.inputs = list(inputs)
 235         self._resized_inputs = [
 236             Signal(output_width, name=f"resized_inputs[{i}]")
 237             for i in range(len(self.inputs))]
 238         self.register_levels = list(register_levels)
 239         self.output = Signal(output_width)
 240         self.partition_points = PartitionPoints(partition_points)
 241         if not self.partition_points.fits_in_width(output_width):
 242             raise ValueError("partition_points doesn't fit in output_width")
 243         self._reg_partition_points = self.partition_points.like()
 244         max_level = AddReduce.get_max_level(len(self.inputs))
 245         for level in self.register_levels:
 246             if level > max_level:
 247                 raise ValueError(
 248                     "not enough adder levels for specified register levels")
 249
 250     @staticmethod
 251     def get_max_level(input_count):
 252         """Get the maximum level.
 253
 254         All ``register_levels`` must be less than or equal to the maximum
 255         level.
 256         """
 257         retval = 0
 258         while True:
 259             groups = AddReduce.full_adder_groups(input_count)
 260             if len(groups) == 0:
 261                 return retval
 262             input_count %= FULL_ADDER_INPUT_COUNT
 263             input_count += 2 * len(groups)
 264             retval += 1
 265
 266     def next_register_levels(self):
 267         """``Iterable`` of ``register_levels`` for next recursive level."""
 268         for level in self.register_levels:
 269             if level > 0:
 270                 yield level - 1
 271
 272     @staticmethod
 273     def full_adder_groups(input_count):
 274         """Get ``inputs`` indices for which a full adder should be built."""
 275         return range(0,
 276                      input_count - FULL_ADDER_INPUT_COUNT + 1,
 277                      FULL_ADDER_INPUT_COUNT)
 278
 279     def elaborate(self, platform):
 280         """Elaborate this module."""
 281         m = Module()
 282
 283         # resize inputs to correct bit-width and optionally add in
 284         # pipeline registers
 285         resized_input_assignments = [self._resized_inputs[i].eq(self.inputs[i])
 286                                      for i in range(len(self.inputs))]
 287         if 0 in self.register_levels:
 288             m.d.sync += resized_input_assignments
 289             m.d.sync += self._reg_partition_points.eq(self.partition_points)
 290         else:
 291             m.d.comb += resized_input_assignments
 292             m.d.comb += self._reg_partition_points.eq(self.partition_points)
 293
 294         groups = AddReduce.full_adder_groups(len(self.inputs))
 295         # if there are no full adders to create, then we handle the base cases
 296         # and return, otherwise we go on to the recursive case
 297         if len(groups) == 0:
 298             if len(self.inputs) == 0:
 299                 # use 0 as the default output value
 300                 m.d.comb += self.output.eq(0)
 301             elif len(self.inputs) == 1:
 302                 # handle single input
 303                 m.d.comb += self.output.eq(self._resized_inputs[0])
 304             else:
 305                 # base case for adding 2 or more inputs, which get recursively
 306                 # reduced to 2 inputs
 307                 assert len(self.inputs) == 2
 308                 adder = PartitionedAdder(len(self.output),
 309                                          self._reg_partition_points)
 310                 m.submodules.final_adder = adder
 311                 m.d.comb += adder.a.eq(self._resized_inputs[0])
 312                 m.d.comb += adder.b.eq(self._resized_inputs[1])
 313                 m.d.comb += self.output.eq(adder.output)
 314             return m
 315         # go on to handle recursive case
 316         intermediate_terms = []
 317
 318         def add_intermediate_term(value):
 319             intermediate_term = Signal(
 320                 len(self.output),
 321                 name=f"intermediate_terms[{len(intermediate_terms)}]")
 322             intermediate_terms.append(intermediate_term)
 323             m.d.comb += intermediate_term.eq(value)
 324
 325         # store mask in intermediary (simplifies graph)
 326         part_mask = Signal(len(self.output), reset_less=True)
 327         mask = self._reg_partition_points.as_mask(len(self.output))
 328         m.d.comb += part_mask.eq(mask)
 329
 330         # create full adders for this recursive level.
 331         # this shrinks N terms to 2 * (N // 3) plus the remainder
 332         for i in groups:
 333             adder_i = FullAdder(len(self.output))
 334             setattr(m.submodules, f"adder_{i}", adder_i)
 335             m.d.comb += adder_i.in0.eq(self._resized_inputs[i])
 336             m.d.comb += adder_i.in1.eq(self._resized_inputs[i + 1])
 337             m.d.comb += adder_i.in2.eq(self._resized_inputs[i + 2])
 338             add_intermediate_term(adder_i.sum)
 339             shifted_carry = adder_i.carry << 1
 340             # mask out carry bits to prevent carries between partitions
 341             add_intermediate_term((adder_i.carry << 1) & part_mask)
 342         # handle the remaining inputs.
 343         if len(self.inputs) % FULL_ADDER_INPUT_COUNT == 1:
 344             add_intermediate_term(self._resized_inputs[-1])
 345         elif len(self.inputs) % FULL_ADDER_INPUT_COUNT == 2:
 346             # Just pass the terms to the next layer, since we wouldn't gain
 347             # anything by using a half adder since there would still be 2 terms
 348             # and just passing the terms to the next layer saves gates.
 349             add_intermediate_term(self._resized_inputs[-2])
 350             add_intermediate_term(self._resized_inputs[-1])
 351         else:
 352             assert len(self.inputs) % FULL_ADDER_INPUT_COUNT == 0
 353         # recursive invocation of ``AddReduce``
 354         next_level = AddReduce(intermediate_terms,
 355                                len(self.output),
 356                                self.next_register_levels(),
 357                                self._reg_partition_points)
 358         m.submodules.next_level = next_level
 359         m.d.comb += self.output.eq(next_level.output)
 360         return m
 361
 362
 363 OP_MUL_LOW = 0
 364 OP_MUL_SIGNED_HIGH = 1
 365 OP_MUL_SIGNED_UNSIGNED_HIGH = 2  # a is signed, b is unsigned
 366 OP_MUL_UNSIGNED_HIGH = 3
 367
 368
 369 def get_term(value, shift=0, enabled=None):
 370     if enabled is not None:
 371         value = Mux(enabled, value, 0)
 372     if shift > 0:
 373         value = Cat(Repl(C(0, 1), shift), value)
 374     else:
 375         assert shift == 0
 376     return value
 377
 378
 379 def g_add_term(m, terms, value, shift=0, enabled=None):
 380     term = Signal(128, reset_less=True)
 381     terms.append(term)
 382     m.d.comb += term.eq(get_term(value, shift, enabled))
 383
 384
 385 class Term(Elaboratable):
 386     def __init__(self, width, twidth, pbwid, a_index, b_index):
 387         self.a_index = a_index
 388         self.b_index = b_index
 389         self.width = width
 390         self.a = Signal(width, reset_less=True)
 391         self.b = Signal(width, reset_less=True)
 392         self.term = Signal(twidth, reset_less=True)
 393         self.pb_en = Signal(pbwid, reset_less=True)
 394
 395     def elaborate(self, platform):
 396
 397         m = Module()
 398         product = Signal(self.width*2, reset_less=True)
 399         m.d.comb += product.eq(self.a * self.b)
 400
 401         tl = []
 402         min_index = min(self.a_index, self.b_index)
 403         max_index = max(self.a_index, self.b_index)
 404         for i in range(min_index, max_index):
 405             tl.append(self.pb_en[i])
 406         name = "te_%d_%d" % (self.a_index, self.b_index)
 407         if len(tl) > 0:
 408             term_enabled = Signal(name=name, reset_less=True)
 409             m.d.comb += term_enabled.eq(~(Cat(*tl).bool()))
 410         else:
 411             term_enabled = None
 412         shift = 8 * (self.a_index + self.b_index)
 413         m.d.comb += self.term.eq(get_term(product, shift, term_enabled))
 414
 415         return m
 416
 417
 418 class Mul8_16_32_64(Elaboratable):
 419     """Signed/Unsigned 8/16/32/64-bit partitioned integer multiplier.
 420
 421     Supports partitioning into any combination of 8, 16, 32, and 64-bit
 422     partitions on naturally-aligned boundaries. Supports the operation being
 423     set for each partition independently.
 424
 425     :attribute part_pts: the input partition points. Has a partition point at
 426         multiples of 8 in 0 < i < 64. Each partition point's associated
 427         ``Value`` is a ``Signal``. Modification not supported, except for by
 428         ``Signal.eq``.
 429     :attribute part_ops: the operation for each byte. The operation for a
 430         particular partition is selected by assigning the selected operation
 431         code to each byte in the partition. The allowed operation codes are:
 432
 433         :attribute OP_MUL_LOW: the LSB half of the product. Equivalent to
 434             RISC-V's `mul` instruction.
 435         :attribute OP_MUL_SIGNED_HIGH: the MSB half of the product where both
 436             ``a`` and ``b`` are signed. Equivalent to RISC-V's `mulh`
 437             instruction.
 438         :attribute OP_MUL_SIGNED_UNSIGNED_HIGH: the MSB half of the product
 439             where ``a`` is signed and ``b`` is unsigned. Equivalent to RISC-V's
 440             `mulhsu` instruction.
 441         :attribute OP_MUL_UNSIGNED_HIGH: the MSB half of the product where both
 442             ``a`` and ``b`` are unsigned. Equivalent to RISC-V's `mulhu`
 443             instruction.
 444     """
 445
 446     def __init__(self, register_levels= ()):
 447         self.part_pts = PartitionPoints()
 448         for i in range(8, 64, 8):
 449             self.part_pts[i] = Signal(name=f"part_pts_{i}")
 450         self.part_ops = [Signal(2, name=f"part_ops_{i}") for i in range(8)]
 451         self.a = Signal(64)
 452         self.b = Signal(64)
 453         self.output = Signal(64)
 454         self.register_levels = list(register_levels)
 455         self._intermediate_output = Signal(128)
 456         self._delayed_part_ops = [
 457             [Signal(2, name=f"_delayed_part_ops_{delay}_{i}")
 458              for i in range(8)]
 459             for delay in range(1 + len(self.register_levels))]
 460         self._part_8 = [Signal(name=f"_part_8_{i}") for i in range(8)]
 461         self._part_16 = [Signal(name=f"_part_16_{i}") for i in range(4)]
 462         self._part_32 = [Signal(name=f"_part_32_{i}") for i in range(2)]
 463         self._part_64 = [Signal(name=f"_part_64")]
 464         self._delayed_part_8 = [
 465             [Signal(name=f"_delayed_part_8_{delay}_{i}")
 466              for i in range(8)]
 467             for delay in range(1 + len(self.register_levels))]
 468         self._delayed_part_16 = [
 469             [Signal(name=f"_delayed_part_16_{delay}_{i}")
 470              for i in range(4)]
 471             for delay in range(1 + len(self.register_levels))]
 472         self._delayed_part_32 = [
 473             [Signal(name=f"_delayed_part_32_{delay}_{i}")
 474              for i in range(2)]
 475             for delay in range(1 + len(self.register_levels))]
 476         self._delayed_part_64 = [
 477             [Signal(name=f"_delayed_part_64_{delay}")]
 478             for delay in range(1 + len(self.register_levels))]
 479         self._output_64 = Signal(64)
 480         self._output_32 = Signal(64)
 481         self._output_16 = Signal(64)
 482         self._output_8 = Signal(64)
 483         self._a_signed = [Signal(name=f"_a_signed_{i}") for i in range(8)]
 484         self._b_signed = [Signal(name=f"_b_signed_{i}") for i in range(8)]
 485         self._not_a_term_8 = Signal(128)
 486         self._neg_lsb_a_term_8 = Signal(128)
 487         self._not_b_term_8 = Signal(128)
 488         self._neg_lsb_b_term_8 = Signal(128)
 489         self._not_a_term_16 = Signal(128)
 490         self._neg_lsb_a_term_16 = Signal(128)
 491         self._not_b_term_16 = Signal(128)
 492         self._neg_lsb_b_term_16 = Signal(128)
 493         self._not_a_term_32 = Signal(128)
 494         self._neg_lsb_a_term_32 = Signal(128)
 495         self._not_b_term_32 = Signal(128)
 496         self._neg_lsb_b_term_32 = Signal(128)
 497         self._not_a_term_64 = Signal(128)
 498         self._neg_lsb_a_term_64 = Signal(128)
 499         self._not_b_term_64 = Signal(128)
 500         self._neg_lsb_b_term_64 = Signal(128)
 501
 502     def _part_byte(self, index):
 503         if index == -1 or index == 7:
 504             return C(True, 1)
 505         assert index >= 0 and index < 8
 506         return self.part_pts[index * 8 + 8]
 507
 508     def elaborate(self, platform):
 509         m = Module()
 510
 511         # collect part-bytes
 512         pbs = Signal(8, reset_less=True)
 513         tl = []
 514         for i in range(8):
 515             pb = Signal(name="pb%d" % i, reset_less=True)
 516             m.d.comb += pb.eq(self._part_byte(i))
 517             tl.append(pb)
 518         m.d.comb += pbs.eq(Cat(*tl))
 519
 520         for i in range(len(self.part_ops)):
 521             m.d.comb += self._delayed_part_ops[0][i].eq(self.part_ops[i])
 522             m.d.sync += [self._delayed_part_ops[j + 1][i]
 523                          .eq(self._delayed_part_ops[j][i])
 524                          for j in range(len(self.register_levels))]
 525
 526         def add_intermediate_value(value):
 527             intermediate_value = Signal(len(value), reset_less=True)
 528             m.d.comb += intermediate_value.eq(value)
 529             return intermediate_value
 530
 531         for parts, delayed_parts in [(self._part_64, self._delayed_part_64),
 532                                      (self._part_32, self._delayed_part_32),
 533                                      (self._part_16, self._delayed_part_16),
 534                                      (self._part_8, self._delayed_part_8)]:
 535             byte_count = 8 // len(parts)
 536             for i in range(len(parts)):
 537                 pb = pbs[i * byte_count - 1]
 538                 value = add_intermediate_value(pb)
 539                 for j in range(i * byte_count, (i + 1) * byte_count - 1):
 540                     pb = add_intermediate_value(~pbs[j])
 541                     value = add_intermediate_value(value & pb)
 542                 pb = pbs[(i + 1) * byte_count - 1]
 543                 value = add_intermediate_value(value & pb)
 544                 m.d.comb += parts[i].eq(value)
 545                 m.d.comb += delayed_parts[0][i].eq(parts[i])
 546                 m.d.sync += [delayed_parts[j + 1][i].eq(delayed_parts[j][i])
 547                              for j in range(len(self.register_levels))]
 548
 549         terms = []
 550
 551         for a_index in range(8):
 552             for b_index in range(8):
 553                 t = Term(8, 128, 8, a_index, b_index)
 554                 setattr(m.submodules, "term_%d_%d" % (a_index, b_index), t)
 555
 556                 m.d.comb += t.a.eq(self.a.bit_select(a_index * 8, 8))
 557                 m.d.comb += t.b.eq(self.b.bit_select(b_index * 8, 8))
 558                 m.d.comb += t.pb_en.eq(pbs)
 559
 560                 terms.append(t.term)
 561
 562         def add_term(value, shift=0, enabled=None):
 563             g_add_term(m, terms, value, shift, enabled)
 564
 565         for i in range(8):
 566             a_signed = self.part_ops[i] != OP_MUL_UNSIGNED_HIGH
 567             b_signed = (self.part_ops[i] == OP_MUL_LOW) \
 568                         | (self.part_ops[i] == OP_MUL_SIGNED_HIGH)
 569             m.d.comb += self._a_signed[i].eq(a_signed)
 570             m.d.comb += self._b_signed[i].eq(b_signed)
 571
 572         # it's fine to bitwise-or these together since they are never enabled
 573         # at the same time
 574         add_term(self._not_a_term_8 | self._not_a_term_16
 575                  | self._not_a_term_32 | self._not_a_term_64)
 576         add_term(self._neg_lsb_a_term_8 | self._neg_lsb_a_term_16
 577                  | self._neg_lsb_a_term_32 | self._neg_lsb_a_term_64)
 578         add_term(self._not_b_term_8 | self._not_b_term_16
 579                  | self._not_b_term_32 | self._not_b_term_64)
 580         add_term(self._neg_lsb_b_term_8 | self._neg_lsb_b_term_16
 581                  | self._neg_lsb_b_term_32 | self._neg_lsb_b_term_64)
 582
 583         for not_a_term, \
 584             neg_lsb_a_term, \
 585             not_b_term, \
 586             neg_lsb_b_term, \
 587             parts in [
 588                 (self._not_a_term_8,
 589                  self._neg_lsb_a_term_8,
 590                  self._not_b_term_8,
 591                  self._neg_lsb_b_term_8,
 592                  self._part_8),
 593                 (self._not_a_term_16,
 594                  self._neg_lsb_a_term_16,
 595                  self._not_b_term_16,
 596                  self._neg_lsb_b_term_16,
 597                  self._part_16),
 598                 (self._not_a_term_32,
 599                  self._neg_lsb_a_term_32,
 600                  self._not_b_term_32,
 601                  self._neg_lsb_b_term_32,
 602                  self._part_32),
 603                 (self._not_a_term_64,
 604                  self._neg_lsb_a_term_64,
 605                  self._not_b_term_64,
 606                  self._neg_lsb_b_term_64,
 607                  self._part_64),
 608                 ]:
 609             byte_width = 8 // len(parts)
 610             bit_width = 8 * byte_width
 611             for i in range(len(parts)):
 612                 be = parts[i] & self.a[(i + 1) * bit_width - 1] \
 613                     & self._a_signed[i * byte_width]
 614                 ae = parts[i] & self.b[(i + 1) * bit_width - 1] \
 615                     & self._b_signed[i * byte_width]
 616                 a_enabled = Signal(name="a_en_%d" % i, reset_less=True)
 617                 b_enabled = Signal(name="b_en_%d" % i, reset_less=True)
 618                 m.d.comb += a_enabled.eq(ae)
 619                 m.d.comb += b_enabled.eq(be)
 620
 621                 # for 8-bit values: form a * 0xFF00 by using -a * 0x100, the
 622                 # negation operation is split into a bitwise not and a +1.
 623                 # likewise for 16, 32, and 64-bit values.
 624                 m.d.comb += [
 625                     not_a_term.bit_select(bit_width * 2 * i, bit_width * 2)
 626                     .eq(Mux(a_enabled,
 627                             Cat(Repl(0, bit_width),
 628                                 ~self.a.bit_select(bit_width * i, bit_width)),
 629                             0)),
 630
 631                     neg_lsb_a_term.bit_select(bit_width * 2 * i, bit_width * 2)
 632                     .eq(Cat(Repl(0, bit_width), a_enabled)),
 633
 634                     not_b_term.bit_select(bit_width * 2 * i, bit_width * 2)
 635                     .eq(Mux(b_enabled,
 636                             Cat(Repl(0, bit_width),
 637                                 ~self.b.bit_select(bit_width * i, bit_width)),
 638                             0)),
 639
 640                     neg_lsb_b_term.bit_select(bit_width * 2 * i, bit_width * 2)
 641                     .eq(Cat(Repl(0, bit_width), b_enabled))]
 642
 643         expanded_part_pts = PartitionPoints()
 644         for i, v in self.part_pts.items():
 645             signal = Signal(name=f"expanded_part_pts_{i*2}", reset_less=True)
 646             expanded_part_pts[i * 2] = signal
 647             m.d.comb += signal.eq(v)
 648
 649         add_reduce = AddReduce(terms,
 650                                128,
 651                                self.register_levels,
 652                                expanded_part_pts)
 653         m.submodules.add_reduce = add_reduce
 654         m.d.comb += self._intermediate_output.eq(add_reduce.output)
 655         m.d.comb += self._output_64.eq(
 656             Mux(self._delayed_part_ops[-1][0] == OP_MUL_LOW,
 657                 self._intermediate_output.bit_select(0, 64),
 658                 self._intermediate_output.bit_select(64, 64)))
 659
 660         # create _output_32
 661         ol = []
 662         for i in range(2):
 663             op = Signal(32, reset_less=True, name="op32_%d" % i)
 664             m.d.comb += op.eq(
 665                 Mux(self._delayed_part_ops[-1][4 * i] == OP_MUL_LOW,
 666                     self._intermediate_output.bit_select(i * 64, 32),
 667                     self._intermediate_output.bit_select(i * 64 + 32, 32)))
 668             ol.append(op)
 669         m.d.comb += self._output_32.eq(Cat(*ol))
 670
 671         # create _output_16
 672         ol = []
 673         for i in range(4):
 674             op = Signal(16, reset_less=True, name="op16_%d" % i)
 675             m.d.comb += op.eq(
 676                 Mux(self._delayed_part_ops[-1][2 * i] == OP_MUL_LOW,
 677                     self._intermediate_output.bit_select(i * 32, 16),
 678                     self._intermediate_output.bit_select(i * 32 + 16, 16)))
 679             ol.append(op)
 680         m.d.comb += self._output_16.eq(Cat(*ol))
 681
 682         # create _output_8
 683         ol = []
 684         for i in range(8):
 685             op = Signal(8, reset_less=True, name="op8_%d" % i)
 686             m.d.comb += op.eq(
 687                 Mux(self._delayed_part_ops[-1][i] == OP_MUL_LOW,
 688                     self._intermediate_output.bit_select(i * 16, 8),
 689                     self._intermediate_output.bit_select(i * 16 + 8, 8)))
 690             ol.append(op)
 691         m.d.comb += self._output_8.eq(Cat(*ol))
 692
 693         # final output
 694         ol = []
 695         for i in range(8):
 696             op = Signal(8, reset_less=True, name="op%d" % i)
 697             m.d.comb += op.eq(
 698                 Mux(self._delayed_part_8[-1][i]
 699                     | self._delayed_part_16[-1][i // 2],
 700                     Mux(self._delayed_part_8[-1][i],
 701                         self._output_8.bit_select(i * 8, 8),
 702                         self._output_16.bit_select(i * 8, 8)),
 703                     Mux(self._delayed_part_32[-1][i // 4],
 704                         self._output_32.bit_select(i * 8, 8),
 705                         self._output_64.bit_select(i * 8, 8))))
 706             ol.append(op)
 707         m.d.comb += self.output.eq(Cat(*ol))
 708         return m
 709
 710
 711 if __name__ == "__main__":
 712     m = Mul8_16_32_64()
 713     main(m, ports=[m.a,
 714                    m.b,
 715                    m._intermediate_output,
 716                    m.output,
 717                    *m.part_ops,
 718                    *m.part_pts.values()])