나눗셈 회로 문서의 구현 코드다. if __name__ == "__main__": 아래가 자체 테스트다.
# 21.나눗셈 회로: 복원 나눗셈을 레지스터 값 수준에서 흉내 낸다.
# 실제 회로가 아니라 파이썬 흉내다. div·mflo·mfhi는 작은 MIPS 흉내기로 확인한다.
# 실행: python3 21_divider-circuit_impl.py
REGS = ["zero", "at", "v0", "v1", "a0", "a1", "a2", "a3",
"t0", "t1", "t2", "t3", "t4", "t5", "t6", "t7",
"s0", "s1", "s2", "s3", "s4", "s5", "s6", "s7",
"t8", "t9", "k0", "k1", "gp", "sp", "fp", "ra"]
# opcode·funct 값은 MIPS Green Sheet(Reference Data)의 표를 따른다.
R_FUNCT = {"sll": 0x00, "srl": 0x02, "sra": 0x03, "jr": 0x08,
"mfhi": 0x10, "mflo": 0x12, "mult": 0x18, "multu": 0x19,
"div": 0x1A, "divu": 0x1B,
"add": 0x20, "addu": 0x21, "sub": 0x22, "subu": 0x23,
"and": 0x24, "or": 0x25, "xor": 0x26, "nor": 0x27,
"slt": 0x2A, "sltu": 0x2B}
I_OP = {"beq": 0x04, "bne": 0x05, "addi": 0x08, "addiu": 0x09,
"slti": 0x0A, "sltiu": 0x0B, "andi": 0x0C, "ori": 0x0D,
"xori": 0x0E, "lui": 0x0F, "lb": 0x20, "lw": 0x23,
"lbu": 0x24, "sb": 0x28, "sw": 0x2B}
J_OP = {"j": 0x02, "jal": 0x03}
FUNCT_R = {v: k for k, v in R_FUNCT.items()}
OP_I = {v: k for k, v in I_OP.items()}
OP_J = {v: k for k, v in J_OP.items()}
SHIFTS = ("sll", "srl", "sra")
ZERO_EXT = ("andi", "ori", "xori", "lui")
MEMOPS = ("lw", "sw", "lb", "lbu", "sb")
MASK = 0xFFFFFFFF
def reg(name):
name = name.strip()
assert name.startswith("$"), name
name = name[1:]
return int(name) if name.isdigit() else REGS.index(name)
def rname(n):
return "$" + REGS[n]
def signed16(x):
x &= 0xFFFF
return x - 0x10000 if x & 0x8000 else x
def signed32(x):
x &= MASK
return x - (1 << 32) if x & 0x80000000 else x
def fields_r(op, rs, rt, rd, shamt, funct):
return (op << 26) | (rs << 21) | (rt << 16) | (rd << 11) | (shamt << 6) | funct
def fields_i(op, rs, rt, imm):
return (op << 26) | (rs << 21) | (rt << 16) | (imm & 0xFFFF)
def split_instr(text):
text = text.split("#")[0].strip()
parts = text.split(None, 1)
args = [a.strip() for a in parts[1].split(",")] if len(parts) > 1 else []
return parts[0], args
def target_of(tok, labels):
return labels[tok] if labels and tok in labels else int(tok, 0)
def encode(text, addr=0, labels=None):
"""어셈블리 한 줄을 32비트 기계어로 바꾼다. addr는 이 명령이 놓인 주소."""
m, a = split_instr(text)
if m in R_FUNCT:
f = R_FUNCT[m]
if m in SHIFTS: # sll rd, rt, shamt
sh = int(a[2], 0)
assert 0 <= sh < 32
return fields_r(0, 0, reg(a[1]), reg(a[0]), sh, f)
if m == "jr":
return fields_r(0, reg(a[0]), 0, 0, 0, f)
if m in ("mult", "multu", "div", "divu"):
return fields_r(0, reg(a[0]), reg(a[1]), 0, 0, f)
if m in ("mfhi", "mflo"):
return fields_r(0, 0, 0, reg(a[0]), 0, f)
return fields_r(0, reg(a[1]), reg(a[2]), reg(a[0]), 0, f) # add rd, rs, rt
if m in I_OP:
op = I_OP[m]
if m in MEMOPS: # lw rt, offset(rs)
off, base = a[1].rstrip(")").split("(")
off = int(off, 0)
assert -32768 <= off <= 32767
return fields_i(op, reg(base), reg(a[0]), off)
if m in ("beq", "bne"): # beq rs, rt, label
diff = target_of(a[2], labels) - (addr + 4)
assert diff % 4 == 0
off = diff // 4
assert -32768 <= off <= 32767, "분기 거리가 16비트를 넘는다"
return fields_i(op, reg(a[0]), reg(a[1]), off)
if m == "lui":
imm = int(a[1], 0)
assert 0 <= imm <= 0xFFFF
return fields_i(op, 0, reg(a[0]), imm)
imm = int(a[2], 0)
if m in ZERO_EXT:
assert 0 <= imm <= 0xFFFF
else:
assert -32768 <= imm <= 32767
return fields_i(op, reg(a[1]), reg(a[0]), imm) # addi rt, rs, imm
if m in J_OP:
t = target_of(a[0], labels)
assert t % 4 == 0
assert (t & 0xF0000000) == ((addr + 4) & 0xF0000000), "위 4비트가 다르다"
return (J_OP[m] << 26) | ((t >> 2) & 0x03FFFFFF)
raise ValueError("모르는 명령: " + text)
def split_fields(word):
return {"op": word >> 26, "rs": (word >> 21) & 31, "rt": (word >> 16) & 31,
"rd": (word >> 11) & 31, "shamt": (word >> 6) & 31, "funct": word & 63,
"imm": word & 0xFFFF, "addr": word & 0x03FFFFFF}
def decode(word, addr=0):
"""32비트 기계어를 어셈블리 한 줄로 되돌린다. 분기·점프의 목적지는 16진 주소로 적는다."""
f = split_fields(word)
if f["op"] == 0:
m = FUNCT_R[f["funct"]]
if m in SHIFTS:
return "%s %s, %s, %d" % (m, rname(f["rd"]), rname(f["rt"]), f["shamt"])
if m == "jr":
return "jr %s" % rname(f["rs"])
if m in ("mult", "multu", "div", "divu"):
return "%s %s, %s" % (m, rname(f["rs"]), rname(f["rt"]))
if m in ("mfhi", "mflo"):
return "%s %s" % (m, rname(f["rd"]))
return "%s %s, %s, %s" % (m, rname(f["rd"]), rname(f["rs"]), rname(f["rt"]))
if f["op"] in OP_J:
t = ((addr + 4) & 0xF0000000) | (f["addr"] << 2)
return "%s 0x%08x" % (OP_J[f["op"]], t)
m = OP_I[f["op"]]
if m in MEMOPS:
return "%s %s, %d(%s)" % (m, rname(f["rt"]), signed16(f["imm"]), rname(f["rs"]))
if m in ("beq", "bne"):
t = (addr + 4 + signed16(f["imm"]) * 4) & MASK
return "%s %s, %s, 0x%08x" % (m, rname(f["rs"]), rname(f["rt"]), t)
if m == "lui":
return "lui %s, 0x%x" % (rname(f["rt"]), f["imm"])
if m in ZERO_EXT:
return "%s %s, %s, 0x%x" % (m, rname(f["rt"]), rname(f["rs"]), f["imm"])
return "%s %s, %s, %d" % (m, rname(f["rt"]), rname(f["rs"]), signed16(f["imm"]))
def assemble(source, base=0x00400000):
"""여러 줄을 두 번 훑어 (주소, 기계어, 원문) 목록과 라벨 표를 돌려준다."""
labels, lines, addr = {}, [], base
for raw in source.strip().splitlines():
line = raw.split("#")[0].strip()
while ":" in line:
lab, line = line.split(":", 1)
labels[lab.strip()] = addr
line = line.strip()
if line:
lines.append((addr, line))
addr += 4
return [(a, encode(t, a, labels), t) for a, t in lines], labels
class Machine:
"""MIPS 일부를 흉내 내는 작은 기계. 기계어를 해독해 실행한다.
지연 분기는 흉내 내지 않는다(jal은 PC+4를 $ra에 넣는다)."""
def __init__(self, source, base=0x00400000, big_endian=True):
self.code, self.labels = assemble(source, base)
self.words = {a: w for a, w, _ in self.code}
self.text = {a: t for a, _, t in self.code}
self.r = [0] * 32
self.hi = self.lo = 0
self.mem = {}
self.pc = base
self.big = big_endian
self.steps = 0
self.log = []
def load_word(self, addr):
assert addr % 4 == 0, "워드 주소는 4의 배수여야 한다"
bs = [self.mem.get(addr + i, 0) for i in range(4)]
if not self.big:
bs.reverse()
return (bs[0] << 24) | (bs[1] << 16) | (bs[2] << 8) | bs[3]
def store_word(self, addr, val):
assert addr % 4 == 0, "워드 주소는 4의 배수여야 한다"
bs = [(val >> s) & 0xFF for s in (24, 16, 8, 0)]
if not self.big:
bs.reverse()
for i, b in enumerate(bs):
self.mem[addr + i] = b
def set(self, n, val):
if n != 0: # $zero는 써도 0이다
self.r[n] = val & MASK
def step(self):
pc = self.pc
f = split_fields(self.words[pc])
r, npc = self.r, pc + 4
rs, rt = r[f["rs"]], r[f["rt"]]
simm = signed16(f["imm"])
if f["op"] == 0:
m = FUNCT_R[f["funct"]]
if m in ("add", "sub"):
v = signed32(rs) + signed32(rt) if m == "add" else signed32(rs) - signed32(rt)
assert -2**31 <= v < 2**31, "오버플로 예외"
self.set(f["rd"], v)
elif m == "addu": self.set(f["rd"], rs + rt)
elif m == "subu": self.set(f["rd"], rs - rt)
elif m == "and": self.set(f["rd"], rs & rt)
elif m == "or": self.set(f["rd"], rs | rt)
elif m == "xor": self.set(f["rd"], rs ^ rt)
elif m == "nor": self.set(f["rd"], ~(rs | rt))
elif m == "slt": self.set(f["rd"], int(signed32(rs) < signed32(rt)))
elif m == "sltu": self.set(f["rd"], int(rs < rt))
elif m == "sll": self.set(f["rd"], rt << f["shamt"])
elif m == "srl": self.set(f["rd"], rt >> f["shamt"])
elif m == "sra": self.set(f["rd"], signed32(rt) >> f["shamt"])
elif m == "jr": npc = rs
elif m == "mult":
p = (signed32(rs) * signed32(rt)) & 0xFFFFFFFFFFFFFFFF
self.hi, self.lo = p >> 32, p & MASK
elif m == "multu":
p = rs * rt
self.hi, self.lo = p >> 32, p & MASK
elif m == "div":
a, b = signed32(rs), signed32(rt)
q = abs(a) // abs(b)
q = -q if (a < 0) != (b < 0) else q
self.lo, self.hi = q & MASK, (a - q * b) & MASK
elif m == "divu":
self.lo, self.hi = rs // rt, rs % rt
elif m == "mfhi": self.set(f["rd"], self.hi)
elif m == "mflo": self.set(f["rd"], self.lo)
elif f["op"] in OP_J:
if OP_J[f["op"]] == "jal":
self.set(31, pc + 4)
npc = ((pc + 4) & 0xF0000000) | (f["addr"] << 2)
else:
m = OP_I[f["op"]]
if m == "addi":
v = signed32(rs) + simm
assert -2**31 <= v < 2**31, "오버플로 예외"
self.set(f["rt"], v)
elif m == "addiu": self.set(f["rt"], rs + simm)
elif m == "slti": self.set(f["rt"], int(signed32(rs) < simm))
elif m == "sltiu": self.set(f["rt"], int(rs < (simm & MASK)))
elif m == "andi": self.set(f["rt"], rs & f["imm"])
elif m == "ori": self.set(f["rt"], rs | f["imm"])
elif m == "xori": self.set(f["rt"], rs ^ f["imm"])
elif m == "lui": self.set(f["rt"], f["imm"] << 16)
elif m == "lw": self.set(f["rt"], self.load_word((rs + simm) & MASK))
elif m == "sw": self.store_word((rs + simm) & MASK, rt)
elif m == "lb":
b = self.mem.get((rs + simm) & MASK, 0)
self.set(f["rt"], b - 256 if b & 0x80 else b)
elif m == "lbu": self.set(f["rt"], self.mem.get((rs + simm) & MASK, 0))
elif m == "sb": self.mem[(rs + simm) & MASK] = rt & 0xFF
elif m == "beq":
if rs == rt: npc = pc + 4 + simm * 4
elif m == "bne":
if rs != rt: npc = pc + 4 + simm * 4
self.pc = npc & MASK
self.steps += 1
return pc
def run(self, max_steps=100000, hook=None):
"""PC가 프로그램 밖으로 나가면 멈춘다."""
while self.pc in self.words:
pc = self.step()
if hook:
hook(self, pc)
assert self.steps < max_steps, "끝나지 않는다"
return self
def reg(self, name):
return self.r[reg(name)]
def divide_restoring(dividend, divisor, n, trace=None):
"""복원 나눗셈: 나누는 수를 2n비트 칸의 위쪽 절반에 놓고 오른쪽으로 밀며 n+1번 돈다."""
assert divisor != 0
rem, div, quo = dividend, divisor << n, 0
if trace is not None:
trace.append((0, "처음", quo, div, rem))
for i in range(1, n + 2):
rem -= div
if rem < 0:
rem += div # 되돌린다(복원)
quo = (quo << 1) & ((1 << n) - 1)
what = "뺐더니 음수: 되돌리고 몫에 0"
else:
quo = ((quo << 1) | 1) & ((1 << n) - 1)
what = "뺐더니 0 이상: 그대로 두고 몫에 1"
div >>= 1
if trace is not None:
trace.append((i, what, quo, div, rem))
return quo, rem
def signed_divide(a, b, n):
"""부호 있는 나눗셈: 크기끼리 나눈 뒤 부호를 붙인다. 나머지는 나누어지는 수의 부호를 따른다."""
q, r = divide_restoring(abs(a), abs(b), n)
if (a < 0) != (b < 0):
q = -q
if a < 0:
r = -r
return q, r
def bits(x, w):
return format(x, "0%db" % w)
if __name__ == "__main__":
# 추적 표: 13 ÷ 5
tr = []
assert divide_restoring(13, 5, 4, tr) == (2, 3) and 13 == 2 * 5 + 3
assert [(i, bits(q, 4), bits(d, 8), bits(r, 8)) for i, _, q, d, r in tr] == [
(0, "0000", "01010000", "00001101"),
(1, "0000", "00101000", "00001101"),
(2, "0000", "00010100", "00001101"),
(3, "0000", "00001010", "00001101"),
(4, "0001", "00000101", "00000011"),
(5, "0010", "00000010", "00000011")]
assert [d for _, _, _, d, _ in tr[:5]] == [80, 40, 20, 10, 5]
restored = ["음수" in w for _, w, _, _, _ in tr[1:]]
assert restored == [True, True, True, False, True] and sum(restored) == 4
# 전수 검사
count = {}
for n in (4, 8):
count[n] = 0
for x in range(1 << n):
for y in range(1, 1 << n):
assert divide_restoring(x, y, n) == (x // y, x % y)
count[n] += 1
assert count == {4: 240, 8: 65280}
# 불변식: N = Q_i · D · 2^(n+1-i) + R_i, 0 <= R_i < D · 2^(n+1-i), 나누는 수 칸 = D · 2^(n-i) (내림)
n = 4
for x in range(16):
for y in range(1, 16):
tr = []
divide_restoring(x, y, n, tr)
for i, _, q, d, r in tr:
scale = y * 2 ** (n + 1 - i)
assert x == q * scale + r and 0 <= r < scale
assert d == (y << n) >> i
assert tr[1][2] == 0 # 첫 걸음은 늘 몫에 0
# 부호 규칙과 MIPS div
assert encode("div $s2, $s3") == 0x0253001A
want = {(47, 5): (9, 2), (-47, 5): (-9, -2), (47, -5): (-9, 2), (-47, -5): (9, -2)}
for (a, b), (q, r) in want.items():
assert signed_divide(a, b, 8) == (q, r) and q * b + r == a
m = Machine("div $s2, $s3\nmflo $s0\nmfhi $s1")
m.r[reg("$s2")], m.r[reg("$s3")] = a & MASK, b & MASK
m.run()
assert (signed32(m.reg("$s0")), signed32(m.reg("$s1"))) == (q, r)
assert -7 == -3 * 2 + -1 and -7 == -4 * 2 + 1 and -47 == -10 * 5 + 3
for a in range(-40, 41):
for b in list(range(-9, 0)) + list(range(1, 10)):
q, r = signed_divide(a, b, 8)
assert q * b + r == a and abs(r) < abs(b) and (r == 0 or (r < 0) == (a < 0))
# sra와 div의 버림 방향
m = Machine("sra $t0, $s2, 1\naddi $t1, $zero, 2\ndiv $s2, $t1\nmflo $t2")
m.r[reg("$s2")] = (-3) & MASK
m.run()
assert signed32(m.reg("$t0")) == -2 and signed32(m.reg("$t2")) == -1
# D = 0이면 몫 칸이 1로 가득 찬다(뜻 없는 값). 알고리즘의 입력 조건 밖이다.
rem, div, quo = 13, 0, 0
for _ in range(5):
rem -= div
quo = ((quo << 1) | 1) & 15 if rem >= 0 else (quo << 1) & 15
div >>= 1
assert quo == 0b1111
# 카드 C1
tr = []
assert divide_restoring(11, 4, 4, tr) == (2, 3)
assert [bits(q, 4) for _, _, q, _, _ in tr[1:]] == ["0000", "0000", "0000", "0001", "0010"]
assert [d for _, _, _, d, _ in tr[:5]] == [64, 32, 16, 8, 4]
print("ALL CHECKS PASSED")