import math
import enum
from fractions import Fraction
+from types import FunctionType
+
+try:
+ from functools import cached_property
+except ImportError:
+ from cached_property import cached_property
+
+# fix broken IDE type detection for cached_property
+from typing import TYPE_CHECKING
+if TYPE_CHECKING:
+ from functools import cached_property
+
+
+_NOT_FOUND = object()
+
+
+def cache_on_self(func):
+ """like `functools.cached_property`, except for methods. unlike
+ `lru_cache` the cache is per-class instance rather than a global cache
+ per-method."""
+
+ assert isinstance(func, FunctionType), \
+ "non-plain methods are not supported"
+
+ cache_name = func.__name__ + "__cache"
+
+ def wrapper(self, *args, **kwargs):
+ # specifically access through `__dict__` to bypass frozen=True
+ cache = self.__dict__.get(cache_name, _NOT_FOUND)
+ if cache is _NOT_FOUND:
+ self.__dict__[cache_name] = cache = {}
+ key = (args, *kwargs.items())
+ retval = cache.get(key, _NOT_FOUND)
+ if retval is _NOT_FOUND:
+ retval = func(self, *args, **kwargs)
+ cache[key] = retval
+ return retval
+
+ wrapper.__doc__ = func.__doc__
+ return wrapper
@enum.unique
@dataclass
class GoldschmidtDivState:
+ orig_n: int
+ """original numerator"""
+
+ orig_d: int
+ """original denominator"""
+
n: FixedPoint
"""numerator -- N_prime[i] in the paper's algorithm 2"""
+
d: FixedPoint
"""denominator -- D_prime[i] in the paper's algorithm 2"""
+
f: "FixedPoint | None" = None
"""current factor -- F_prime[i] in the paper's algorithm 2"""
- result: "int | None" = None
- """final result"""
+
+ quotient: "int | None" = None
+ """final quotient"""
+
+ remainder: "int | None" = None
+ """final remainder"""
+
n_shift: "int | None" = None
"""amount the numerator needs to be left-shifted at the end of the
algorithm.
"""parameters for a Goldschmidt division algorithm.
Use `GoldschmidtDivParams.get` to find a efficient set of parameters.
"""
+
io_width: int
"""bit-width of the input divisor and the result.
the input numerator is `2 * io_width`-bits wide.
"""
+
extra_precision: int
"""number of bits of additional precision used inside the algorithm."""
+
table_addr_bits: int
"""the number of address bits used in the lookup-table."""
+
table_data_bits: int
"""the number of data bits used in the lookup-table."""
+
+ iter_count: int
+ """the total number of iterations of the division algorithm's loop"""
+
# tuple to be immutable
table: "tuple[FixedPoint, ...]" = field(init=False)
"""the lookup-table"""
+
ops: "tuple[GoldschmidtDivOp, ...]" = field(init=False)
"""the operations needed to perform the goldschmidt division algorithm."""
with address `addr`."""
assert isinstance(addr, int)
assert 0 <= addr < self.table_addr_count
- assert self.io_width >= self.table_addr_bits
+ _assert_accuracy(self.io_width >= self.table_addr_bits)
min_numerator = (1 << self.table_addr_bits) + addr
denominator = 1 << self.table_addr_bits
values_per_table_entry = 1 << (self.io_width - self.table_addr_bits)
assert self.extra_precision >= 0
assert self.table_addr_bits >= 1
assert self.table_data_bits >= 1
+ assert self.iter_count >= 1
table = []
for addr in range(1 << self.table_addr_bits):
table.append(FixedPoint.with_frac_wid(self.table_exact_value(addr),
with `params.io_width == io_width`.
"""
assert isinstance(io_width, int) and io_width >= 1
- for extra_precision in range(io_width * 2):
- for table_addr_bits in range(3, 7 + 1):
+ for extra_precision in range(io_width * 2 + 4):
+ for table_addr_bits in range(1, 7 + 1):
table_data_bits = io_width + extra_precision
- try:
- return GoldschmidtDivParams(
- io_width=io_width,
- extra_precision=extra_precision,
- table_addr_bits=table_addr_bits,
- table_data_bits=table_data_bits)
- except ParamsNotAccurateEnough:
- pass
+ for iter_count in range(1, 2 * io_width.bit_length()):
+ try:
+ return GoldschmidtDivParams(
+ io_width=io_width,
+ extra_precision=extra_precision,
+ table_addr_bits=table_addr_bits,
+ table_data_bits=table_data_bits,
+ iter_count=iter_count)
+ except ParamsNotAccurateEnough:
+ pass
raise ValueError(f"can't find working parameters for a goldschmidt "
f"division algorithm with io_width={io_width}")
"""the total number of bits of precision used inside the algorithm."""
return self.io_width + self.extra_precision
+ @cache_on_self
+ def max_neps(self, i):
+ """maximum value of `neps[i]`.
+ `neps[i]` is defined to be `n[i] * N_prime[i - 1] * F_prime[i - 1]`.
+ """
+ assert isinstance(i, int) and 0 <= i < self.iter_count
+ return Fraction(1, 1 << self.expanded_width)
+
+ @cache_on_self
+ def max_deps(self, i):
+ """maximum value of `deps[i]`.
+ `deps[i]` is defined to be `d[i] * D_prime[i - 1] * F_prime[i - 1]`.
+ """
+ assert isinstance(i, int) and 0 <= i < self.iter_count
+ return Fraction(1, 1 << self.expanded_width)
+
+ @cache_on_self
+ def max_feps(self, i):
+ """maximum value of `feps[i]`.
+ `feps[i]` is defined to be `f[i] * (2 - D_prime[i - 1])`.
+ """
+ assert isinstance(i, int) and 0 <= i < self.iter_count
+ # zero, because the computation of `F_prime[i]` in
+ # `GoldschmidtDivOp.MulDByF.run(...)` is exact.
+ return Fraction(0)
+
+ @cached_property
+ def e0_range(self):
+ """minimum and maximum values of `e[0]`
+ (the relative error in `F_prime[-1]`)
+ """
+ min_e0 = Fraction(0)
+ max_e0 = Fraction(0)
+ for addr in range(self.table_addr_count):
+ # `F_prime[-1] = (1 - e[0]) / B`
+ # => `e[0] = 1 - B * F_prime[-1]`
+ min_b, max_b = self.table_input_exact_range(addr)
+ f_prime_m1 = self.table[addr].as_fraction()
+ assert min_b >= 0 and f_prime_m1 >= 0, \
+ "only positive quadrant of interval multiplication implemented"
+ min_product = min_b * f_prime_m1
+ max_product = max_b * f_prime_m1
+ # negation swaps min/max
+ cur_min_e0 = 1 - max_product
+ cur_max_e0 = 1 - min_product
+ min_e0 = min(min_e0, cur_min_e0)
+ max_e0 = max(max_e0, cur_max_e0)
+ return min_e0, max_e0
+
+ @cached_property
+ def min_e0(self):
+ """minimum value of `e[0]` (the relative error in `F_prime[-1]`)
+ """
+ min_e0, max_e0 = self.e0_range
+ return min_e0
+
+ @cached_property
+ def max_e0(self):
+ """maximum value of `e[0]` (the relative error in `F_prime[-1]`)
+ """
+ min_e0, max_e0 = self.e0_range
+ return max_e0
+
+ @cached_property
+ def max_abs_e0(self):
+ """maximum value of `abs(e[0])`."""
+ return max(abs(self.min_e0), abs(self.max_e0))
+
+ @cached_property
+ def min_abs_e0(self):
+ """minimum value of `abs(e[0])`."""
+ return Fraction(0)
+
+ @cache_on_self
+ def max_n(self, i):
+ """maximum value of `n[i]` (the relative error in `N_prime[i]`
+ relative to the previous iteration)
+ """
+ assert isinstance(i, int) and 0 <= i < self.iter_count
+ if i == 0:
+ # from Claim 10
+ # `n[0] = neps[0] / ((1 - e[0]) * (A / B))`
+ # `n[0] <= 2 * neps[0] / (1 - e[0])`
+
+ assert self.max_e0 < 1 and self.max_neps(0) >= 0, \
+ "only one quadrant of interval division implemented"
+ retval = 2 * self.max_neps(0) / (1 - self.max_e0)
+ elif i == 1:
+ # from Claim 10
+ # `n[1] <= neps[1] / ((1 - f[0]) * (1 - pi[0] - delta[0]))`
+ min_mpd = 1 - self.max_pi(0) - self.max_delta(0)
+ assert self.max_f(0) <= 1 and min_mpd >= 0, \
+ "only one quadrant of interval multiplication implemented"
+ prod = (1 - self.max_f(0)) * min_mpd
+ assert self.max_neps(1) >= 0 and prod > 0, \
+ "only one quadrant of interval division implemented"
+ retval = self.max_neps(1) / prod
+ else:
+ # from Claim 6
+ # `0 <= n[i] <= 2 * max_neps[i] / (1 - pi[i - 1] - delta[i - 1])`
+ min_mpd = 1 - self.max_pi(i - 1) - self.max_delta(i - 1)
+ assert self.max_neps(i) >= 0 and min_mpd > 0, \
+ "only one quadrant of interval division implemented"
+ retval = self.max_neps(i) / min_mpd
+
+ # we need Fraction to avoid using float by accident
+ # -- it also hints to the IDE to give the correct type
+ return Fraction(retval)
+
+ @cache_on_self
+ def max_d(self, i):
+ """maximum value of `d[i]` (the relative error in `D_prime[i]`
+ relative to the previous iteration)
+ """
+ assert isinstance(i, int) and 0 <= i < self.iter_count
+ if i == 0:
+ # from Claim 10
+ # `d[0] = deps[0] / (1 - e[0])`
+
+ assert self.max_e0 < 1 and self.max_deps(0) >= 0, \
+ "only one quadrant of interval division implemented"
+ retval = self.max_deps(0) / (1 - self.max_e0)
+ elif i == 1:
+ # from Claim 10
+ # `d[1] <= deps[1] / ((1 - f[0]) * (1 - delta[0] ** 2))`
+ assert self.max_f(0) <= 1 and self.max_delta(0) <= 1, \
+ "only one quadrant of interval multiplication implemented"
+ divisor = (1 - self.max_f(0)) * (1 - self.max_delta(0) ** 2)
+ assert self.max_deps(1) >= 0 and divisor > 0, \
+ "only one quadrant of interval division implemented"
+ retval = self.max_deps(1) / divisor
+ else:
+ # from Claim 6
+ # `0 <= d[i] <= max_deps[i] / (1 - delta[i - 1])`
+ assert self.max_deps(i) >= 0 and self.max_delta(i - 1) < 1, \
+ "only one quadrant of interval division implemented"
+ retval = self.max_deps(i) / (1 - self.max_delta(i - 1))
+
+ # we need Fraction to avoid using float by accident
+ # -- it also hints to the IDE to give the correct type
+ return Fraction(retval)
+
+ @cache_on_self
+ def max_f(self, i):
+ """maximum value of `f[i]` (the relative error in `F_prime[i]`
+ relative to the previous iteration)
+ """
+ assert isinstance(i, int) and 0 <= i < self.iter_count
+ if i == 0:
+ # from Claim 10
+ # `f[0] = feps[0] / (1 - delta[0])`
+
+ assert self.max_delta(0) < 1 and self.max_feps(0) >= 0, \
+ "only one quadrant of interval division implemented"
+ retval = self.max_feps(0) / (1 - self.max_delta(0))
+ elif i == 1:
+ # from Claim 10
+ # `f[1] = feps[1]`
+ retval = self.max_feps(1)
+ else:
+ # from Claim 6
+ # `f[i] <= max_feps[i]`
+ retval = self.max_feps(i)
+
+ # we need Fraction to avoid using float by accident
+ # -- it also hints to the IDE to give the correct type
+ return Fraction(retval)
+
+ @cache_on_self
+ def max_delta(self, i):
+ """ maximum value of `delta[i]`.
+ `delta[i]` is defined in Definition 4 of paper.
+ """
+ assert isinstance(i, int) and 0 <= i < self.iter_count
+ if i == 0:
+ # `delta[0] = abs(e[0]) + 3 * d[0] / 2`
+ retval = self.max_abs_e0 + Fraction(3, 2) * self.max_d(0)
+ else:
+ # `delta[i] = delta[i - 1] ** 2 + f[i - 1]`
+ prev_max_delta = self.max_delta(i - 1)
+ assert prev_max_delta >= 0
+ retval = prev_max_delta ** 2 + self.max_f(i - 1)
+
+ # we need Fraction to avoid using float by accident
+ # -- it also hints to the IDE to give the correct type
+ return Fraction(retval)
+
+ @cache_on_self
+ def max_pi(self, i):
+ """ maximum value of `pi[i]`.
+ `pi[i]` is defined right below Theorem 5 of paper.
+ """
+ assert isinstance(i, int) and 0 <= i < self.iter_count
+ # `pi[i] = 1 - (1 - n[i]) * prod`
+ # where `prod` is the product of,
+ # for `j` in `0 <= j < i`, `(1 - n[j]) / (1 + d[j])`
+ min_prod = Fraction(0)
+ for j in range(i):
+ max_n_j = self.max_n(j)
+ max_d_j = self.max_d(j)
+ assert max_n_j <= 1 and max_d_j > -1, \
+ "only one quadrant of interval division implemented"
+ min_prod *= (1 - max_n_j) / (1 + max_d_j)
+ max_n_i = self.max_n(i)
+ assert max_n_i <= 1 and min_prod >= 0, \
+ "only one quadrant of interval multiplication implemented"
+ return 1 - (1 - max_n_i) * min_prod
+
+ @cached_property
+ def max_n_shift(self):
+ """ maximum value of `state.n_shift`.
+ """
+ # input numerator is `2*io_width`-bits
+ max_n = (1 << (self.io_width * 2)) - 1
+ max_n_shift = 0
+ # normalize so 1 <= n < 2
+ while max_n >= 2:
+ max_n >>= 1
+ max_n_shift += 1
+ return max_n_shift
+
@enum.unique
class GoldschmidtDivOp(enum.Enum):
# scale to correct value
n = state.n * (1 << state.n_shift)
- # avoid incorrectly rounding down
- n = n.to_frac_wid(params.io_width, round_dir=RoundDir.UP)
- state.result = math.floor(n)
+ state.quotient = math.floor(n)
+ state.remainder = state.orig_n - state.quotient * state.orig_d
+ if state.remainder >= state.orig_d:
+ state.quotient += 1
+ state.remainder -= state.orig_d
else:
assert False, f"unimplemented GoldschmidtDivOp: {self}"
_assert_accuracy(params.expanded_width > 4)
# 3. require `abs(e[0]) + 3 * d[0] / 2 + f[0] < 1 / 2`.
-
- # maximum `abs(e[0])`
- max_abs_e0 = 0
- # maximum `d[0]`
- max_d0 = 0
- # `f[i] = 0` for all `i`
- fi = 0
- for addr in range(params.table_addr_count):
- # `F_prime[-1] = (1 - e[0]) / B`
- # => `e[0] = 1 - B * F_prime[-1]`
- min_b, max_b = params.table_input_exact_range(addr)
- f_prime_m1 = params.table[addr].as_fraction()
- assert min_b >= 0 and f_prime_m1 >= 0, \
- "only positive quadrant of interval multiplication implemented"
- min_product = min_b * f_prime_m1
- max_product = max_b * f_prime_m1
- # negation swaps min/max
- min_e0 = 1 - max_product
- max_e0 = 1 - min_product
- max_abs_e0 = max(max_abs_e0, abs(min_e0), abs(max_e0))
-
- # `D_prime[0] = (1 + d[0]) * B * F_prime[-1]`
- # `D_prime[0] = abs_round_err + B * F_prime[-1]`
- # => `d[0] = abs_round_err / (B * F_prime[-1])`
- max_abs_round_err = Fraction(1, 1 << params.expanded_width)
- assert min_product > 0 and max_abs_round_err >= 0, \
- "only positive quadrant of interval division implemented"
- # division swaps divisor's min/max
- max_d0 = max(max_d0, max_abs_round_err / min_product)
-
- _assert_accuracy(max_abs_e0 + 3 * max_d0 / 2 + fi < Fraction(1, 2))
+ _assert_accuracy(params.max_abs_e0 + 3 * params.max_d(0) / 2
+ + params.max_f(0) < Fraction(1, 2))
# 4. the initial approximation F'[-1] of 1/B is in [1/2, 1].
# (B is the denominator)
yield GoldschmidtDivOp.FEqTableLookup
- # we use Setting I (section 4.1 of the paper)
-
- min_bits_of_precision = 1
- # FIXME: calculate error and check if it's small enough
- while min_bits_of_precision < params.io_width * 2:
+ # we use Setting I (section 4.1 of the paper):
+ # Require `n[i] <= n_hat` and `d[i] <= n_hat` and `f[i] = 0`
+ n_hat = Fraction(0)
+ for i in range(params.iter_count):
+ _assert_accuracy(params.max_f(i) == 0)
+ n_hat = max(n_hat, params.max_n(i), params.max_d(i))
yield GoldschmidtDivOp.MulNByF
- yield GoldschmidtDivOp.MulDByF
- yield GoldschmidtDivOp.FEq2MinusD
-
- min_bits_of_precision *= 2
+ if i != params.iter_count - 1:
+ yield GoldschmidtDivOp.MulDByF
+ yield GoldschmidtDivOp.FEq2MinusD
+
+ # relative approximation error `p(N_prime[i])`:
+ # `p(N_prime[i]) = (A / B - N_prime[i]) / (A / B)`
+ # `0 <= p(N_prime[i])`
+ # `p(N_prime[i]) <= (2 * i) * n_hat \`
+ # ` + (abs(e[0]) + 3 * n_hat / 2) ** (2 ** i)`
+ i = params.iter_count - 1 # last used `i`
+ max_rel_error = (2 * i) * n_hat + \
+ (params.max_abs_e0 + 3 * n_hat / 2) ** (2 ** i)
+
+ min_a_over_b = Fraction(1, 2)
+ max_a_over_b = Fraction(2)
+ max_allowed_abs_error = max_a_over_b / (1 << params.max_n_shift)
+ max_allowed_rel_error = max_allowed_abs_error / min_a_over_b
+
+ _assert_accuracy(max_rel_error < max_allowed_rel_error)
yield GoldschmidtDivOp.CalcResult
width: int
the bit-width of the inputs/outputs. must be a positive integer.
- returns: int
- the quotient. a `width`-bit unsigned integer.
+ returns: tuple[int, int]
+ the quotient and remainder. a tuple of two `width`-bit unsigned
+ integers.
"""
assert isinstance(params, GoldschmidtDivParams)
assert isinstance(d, int) and 0 < d < (1 << params.io_width)
# have `width` fractional bits
state = GoldschmidtDivState(
+ orig_n=n,
+ orig_d=d,
n=FixedPoint(n, params.io_width),
d=FixedPoint(d, params.io_width),
)
for op in params.ops:
op.run(params, state)
- assert state.result is not None
+ assert state.quotient is not None
+ assert state.remainder is not None
- return state.result
+ return state.quotient, state.remainder
projects / soc.git / commitdiff