# Funded by NLnet Assure Programme 2021-02-052, https://nlnet.nl/assure part
# of Horizon 2020 EU Programme 957073.
-from dataclasses import dataclass
+from dataclasses import dataclass, field
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
def with_frac_wid(value, frac_wid, round_dir=RoundDir.ERROR_IF_INEXACT):
"""convert `value` to the nearest fixed-point number with `frac_wid`
fractional bits, rounding according to `round_dir`."""
- value = FixedPoint.cast(value)
assert isinstance(frac_wid, int) and frac_wid >= 0
assert isinstance(round_dir, RoundDir)
- # compute number of bits that should be removed from value
- del_bits = value.frac_wid - frac_wid
- if del_bits == 0:
- return value
- if del_bits < 0: # add bits
- return FixedPoint(value.bits << -del_bits,
- frac_wid)
+ if isinstance(value, Fraction):
+ numerator = value.numerator
+ denominator = value.denominator
+ else:
+ value = FixedPoint.cast(value)
+ numerator = value.bits
+ denominator = 1 << value.frac_wid
+ if denominator < 0:
+ numerator = -numerator
+ denominator = -denominator
+ bits, remainder = divmod(numerator << frac_wid, denominator)
if round_dir == RoundDir.DOWN:
- bits = value.bits >> del_bits
+ pass
elif round_dir == RoundDir.UP:
- bits = -((-value.bits) >> del_bits)
+ if remainder != 0:
+ bits += 1
elif round_dir == RoundDir.NEAREST_TIES_UP:
- bits = value.bits >> (del_bits - 1)
- bits += 1
- bits >>= 1
+ if remainder * 2 >= denominator:
+ bits += 1
elif round_dir == RoundDir.ERROR_IF_INEXACT:
- bits = value.bits >> del_bits
- if bits << del_bits != value.bits:
+ if remainder != 0:
raise ValueError("inexact conversion")
else:
assert False, "unimplemented round_dir"
return FixedPoint.with_frac_wid(self, frac_wid, round_dir)
def __float__(self):
- return self.bits * 2.0 ** -self.frac_wid
+ # use truediv to get correct result even when bits
+ # and frac_wid are huge
+ return float(self.bits / (1 << self.frac_wid))
+
+ def as_fraction(self):
+ return Fraction(self.bits, 1 << self.frac_wid)
def cmp(self, rhs):
"""compare self with rhs, returning a positive integer if self is
rhs = rhs.to_frac_wid(common_frac_wid)
return FixedPoint(lhs.bits + rhs.bits, common_frac_wid)
+ def __radd__(self, lhs):
+ # symmetric
+ return self.__add__(lhs)
+
def __neg__(self):
return FixedPoint(-self.bits, self.frac_wid)
rhs = rhs.to_frac_wid(common_frac_wid)
return FixedPoint(lhs.bits - rhs.bits, common_frac_wid)
+ def __rsub__(self, lhs):
+ # a - b == -(b - a)
+ return -self.__sub__(lhs)
+
def __mul__(self, rhs):
rhs = FixedPoint.cast(rhs)
return FixedPoint(self.bits * rhs.bits, self.frac_wid + rhs.frac_wid)
+ def __rmul__(self, lhs):
+ # symmetric
+ return self.__mul__(lhs)
+
def __floor__(self):
return self.bits >> self.frac_wid
-def goldschmidt_div(n, d, width):
+@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"""
+
+ 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.
+ """
+
+
+class ParamsNotAccurateEnough(Exception):
+ """raised when the parameters aren't accurate enough to have goldschmidt
+ division work."""
+
+
+def _assert_accuracy(condition, msg="not accurate enough"):
+ if condition:
+ return
+ raise ParamsNotAccurateEnough(msg)
+
+
+@dataclass(frozen=True, unsafe_hash=True)
+class GoldschmidtDivParams:
+ """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, default so repr() works for debugging even when
+ # __post_init__ hasn't finished running yet
+ table: "tuple[FixedPoint, ...]" = field(init=False, default=NotImplemented)
+ """the lookup-table"""
+
+ ops: "tuple[GoldschmidtDivOp, ...]" = field(init=False,
+ default=NotImplemented)
+ """the operations needed to perform the goldschmidt division algorithm."""
+
+ def _shrink_bound(self, bound, round_dir):
+ """prevent fractions from having huge numerators/denominators by
+ rounding to a `FixedPoint` and converting back to a `Fraction`.
+
+ This is intended only for values used to compute bounds, and not for
+ values that end up in the hardware.
+ """
+ assert isinstance(bound, (Fraction, int))
+ assert round_dir is RoundDir.DOWN or round_dir is RoundDir.UP, \
+ "you shouldn't use that round_dir on bounds"
+ frac_wid = self.io_width * 4 + 100 # should be enough precision
+ fixed = FixedPoint.with_frac_wid(bound, frac_wid, round_dir)
+ return fixed.as_fraction()
+
+ def _shrink_min(self, min_bound):
+ """prevent fractions used as minimum bounds from having huge
+ numerators/denominators by rounding down to a `FixedPoint` and
+ converting back to a `Fraction`.
+
+ This is intended only for values used to compute bounds, and not for
+ values that end up in the hardware.
+ """
+ return self._shrink_bound(min_bound, RoundDir.DOWN)
+
+ def _shrink_max(self, max_bound):
+ """prevent fractions used as maximum bounds from having huge
+ numerators/denominators by rounding up to a `FixedPoint` and
+ converting back to a `Fraction`.
+
+ This is intended only for values used to compute bounds, and not for
+ values that end up in the hardware.
+ """
+ return self._shrink_bound(max_bound, RoundDir.UP)
+
+ @property
+ def table_addr_count(self):
+ """number of distinct addresses in the lookup-table."""
+ # used while computing self.table, so can't just do len(self.table)
+ return 1 << self.table_addr_bits
+
+ def table_input_exact_range(self, addr):
+ """return the range of inputs as `Fraction`s used for the table entry
+ with address `addr`."""
+ assert isinstance(addr, int)
+ assert 0 <= addr < self.table_addr_count
+ _assert_accuracy(self.io_width >= self.table_addr_bits)
+ addr_shift = self.io_width - self.table_addr_bits
+ min_numerator = (1 << self.io_width) + (addr << addr_shift)
+ denominator = 1 << self.io_width
+ values_per_table_entry = 1 << addr_shift
+ max_numerator = min_numerator + values_per_table_entry - 1
+ min_input = Fraction(min_numerator, denominator)
+ max_input = Fraction(max_numerator, denominator)
+ min_input = self._shrink_min(min_input)
+ max_input = self._shrink_max(max_input)
+ assert 1 <= min_input <= max_input < 2
+ return min_input, max_input
+
+ def table_value_exact_range(self, addr):
+ """return the range of values as `Fraction`s used for the table entry
+ with address `addr`."""
+ min_input, max_input = self.table_input_exact_range(addr)
+ # division swaps min/max
+ min_value = 1 / max_input
+ max_value = 1 / min_input
+ min_value = self._shrink_min(min_value)
+ max_value = self._shrink_max(max_value)
+ assert 0.5 < min_value <= max_value <= 1
+ return min_value, max_value
+
+ def table_exact_value(self, index):
+ min_value, max_value = self.table_value_exact_range(index)
+ # we round down
+ return min_value
+
+ def __post_init__(self):
+ # called by the autogenerated __init__
+ assert self.io_width >= 1
+ 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),
+ self.table_data_bits,
+ RoundDir.DOWN))
+ # we have to use object.__setattr__ since frozen=True
+ object.__setattr__(self, "table", tuple(table))
+ object.__setattr__(self, "ops", tuple(self.__make_ops()))
+
+ @property
+ def expanded_width(self):
+ """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)
+ min_e0 = self._shrink_min(min_e0)
+ max_e0 = self._shrink_max(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
+
+ return self._shrink_max(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))
+
+ return self._shrink_max(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)
+
+ return self._shrink_max(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)
+
+ # `delta[i]` has to be smaller than one otherwise errors would go off
+ # to infinity
+ _assert_accuracy(retval < 1)
+
+ return self._shrink_max(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(1)
+ 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"
+ retval = 1 - (1 - max_n_i) * min_prod
+ return self._shrink_max(retval)
+
+ @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
+
+ def __make_ops(self):
+ """ Goldschmidt division algorithm.
+
+ based on:
+ Even, G., Seidel, P. M., & Ferguson, W. E. (2003).
+ A Parametric Error Analysis of Goldschmidt's Division Algorithm.
+ https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.90.1238&rep=rep1&type=pdf
+
+ yields: GoldschmidtDivOp
+ the operations needed to perform the division.
+ """
+ # establish assumptions of the paper's error analysis (section 3.1):
+
+ # 1. normalize so A (numerator) and B (denominator) are in [1, 2)
+ yield GoldschmidtDivOp.Normalize
+
+ # 2. ensure all relative errors from directed rounding are <= 1 / 4.
+ # the assumption is met by multipliers with > 4-bits precision
+ _assert_accuracy(self.expanded_width > 4)
+
+ # 3. require `abs(e[0]) + 3 * d[0] / 2 + f[0] < 1 / 2`.
+ _assert_accuracy(self.max_abs_e0 + 3 * self.max_d(0) / 2
+ + self.max_f(0) < Fraction(1, 2))
+
+ # 4. the initial approximation F'[-1] of 1/B is in [1/2, 1].
+ # (B is the denominator)
+
+ for addr in range(self.table_addr_count):
+ f_prime_m1 = self.table[addr]
+ _assert_accuracy(0.5 <= f_prime_m1 <= 1)
+
+ yield GoldschmidtDivOp.FEqTableLookup
+
+ # 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(self.iter_count):
+ _assert_accuracy(self.max_f(i) == 0)
+ n_hat = max(n_hat, self.max_n(i), self.max_d(i))
+ yield GoldschmidtDivOp.MulNByF
+ if i != self.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 = self.iter_count - 1 # last used `i`
+ # compute power manually to prevent huge intermediate values
+ power = self._shrink_max(self.max_abs_e0 + 3 * n_hat / 2)
+ for _ in range(i):
+ power = self._shrink_max(power * power)
+
+ max_rel_error = (2 * i) * n_hat + power
+
+ min_a_over_b = Fraction(1, 2)
+ max_a_over_b = Fraction(2)
+ max_allowed_abs_error = max_a_over_b / (1 << self.max_n_shift)
+ max_allowed_rel_error = max_allowed_abs_error / min_a_over_b
+
+ _assert_accuracy(max_rel_error < max_allowed_rel_error,
+ f"not accurate enough: max_rel_error={max_rel_error}"
+ f" max_allowed_rel_error={max_allowed_rel_error}")
+
+ yield GoldschmidtDivOp.CalcResult
+
+ @staticmethod
+ def get(io_width):
+ """ find efficient parameters for a goldschmidt division algorithm
+ with `params.io_width == io_width`.
+ """
+ assert isinstance(io_width, int) and io_width >= 1
+ last_params = None
+ last_error = None
+ 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
+ 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 as e:
+ last_params = (f"GoldschmidtDivParams("
+ f"io_width={io_width!r}, "
+ f"extra_precision={extra_precision!r}, "
+ f"table_addr_bits={table_addr_bits!r}, "
+ f"table_data_bits={table_data_bits!r}, "
+ f"iter_count={iter_count!r})")
+ last_error = e
+ raise ValueError(f"can't find working parameters for a goldschmidt "
+ f"division algorithm: last params: {last_params}"
+ ) from last_error
+
+
+@enum.unique
+class GoldschmidtDivOp(enum.Enum):
+ Normalize = "n, d, n_shift = normalize(n, d)"
+ FEqTableLookup = "f = table_lookup(d)"
+ MulNByF = "n *= f"
+ MulDByF = "d *= f"
+ FEq2MinusD = "f = 2 - d"
+ CalcResult = "result = unnormalize_and_round(n)"
+
+ def run(self, params, state):
+ assert isinstance(params, GoldschmidtDivParams)
+ assert isinstance(state, GoldschmidtDivState)
+ expanded_width = params.expanded_width
+ table_addr_bits = params.table_addr_bits
+ if self == GoldschmidtDivOp.Normalize:
+ # normalize so 1 <= d < 2
+ # can easily be done with count-leading-zeros and left shift
+ while state.d < 1:
+ state.n = (state.n * 2).to_frac_wid(expanded_width)
+ state.d = (state.d * 2).to_frac_wid(expanded_width)
+
+ state.n_shift = 0
+ # normalize so 1 <= n < 2
+ while state.n >= 2:
+ state.n = (state.n * 0.5).to_frac_wid(expanded_width)
+ state.n_shift += 1
+ elif self == GoldschmidtDivOp.FEqTableLookup:
+ # compute initial f by table lookup
+ d_m_1 = state.d - 1
+ d_m_1 = d_m_1.to_frac_wid(table_addr_bits, RoundDir.DOWN)
+ assert 0 <= d_m_1.bits < (1 << params.table_addr_bits)
+ state.f = params.table[d_m_1.bits]
+ elif self == GoldschmidtDivOp.MulNByF:
+ assert state.f is not None
+ n = state.n * state.f
+ state.n = n.to_frac_wid(expanded_width, round_dir=RoundDir.DOWN)
+ elif self == GoldschmidtDivOp.MulDByF:
+ assert state.f is not None
+ d = state.d * state.f
+ state.d = d.to_frac_wid(expanded_width, round_dir=RoundDir.UP)
+ elif self == GoldschmidtDivOp.FEq2MinusD:
+ state.f = (2 - state.d).to_frac_wid(expanded_width)
+ elif self == GoldschmidtDivOp.CalcResult:
+ assert state.n_shift is not None
+ # scale to correct value
+ n = state.n * (1 << state.n_shift)
+
+ 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}"
+
+
+def goldschmidt_div(n, d, params):
""" Goldschmidt division algorithm.
based on:
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(width, int) and width >= 1
- assert isinstance(d, int) and 0 < d < (1 << width)
- assert isinstance(n, int) and 0 <= n < (d << width)
+ assert isinstance(params, GoldschmidtDivParams)
+ assert isinstance(d, int) and 0 < d < (1 << params.io_width)
+ assert isinstance(n, int) and 0 <= n < (d << params.io_width)
- # FIXME: calculate best values for extra_precision, table_addr_bits, and
- # table_data_bits -- these are wrong
- extra_precision = width + 3
- table_addr_bits = 4
- table_data_bits = 8
+ # this whole algorithm is done with fixed-point arithmetic where values
+ # have `width` fractional bits
- width += extra_precision
+ state = GoldschmidtDivState(
+ orig_n=n,
+ orig_d=d,
+ n=FixedPoint(n, params.io_width),
+ d=FixedPoint(d, params.io_width),
+ )
- table = []
- for i in range(1 << table_addr_bits):
- value = 1 / (1 + i * 2 ** -table_addr_bits)
- table.append(FixedPoint.with_frac_wid(value, table_data_bits,
- RoundDir.DOWN))
+ for op in params.ops:
+ op.run(params, state)
- # this whole algorithm is done with fixed-point arithmetic where values
- # have `width` fractional bits
+ assert state.quotient is not None
+ assert state.remainder is not None
- n = FixedPoint(n, width)
- d = FixedPoint(d, width)
-
- # normalize so 1 <= d < 2
- # can easily be done with count-leading-zeros and left shift
- while d < 1:
- n = (n * 2).to_frac_wid(width)
- d = (d * 2).to_frac_wid(width)
-
- n_shift = 0
- # normalize so 1 <= n < 2
- while n >= 2:
- n = (n * 0.5).to_frac_wid(width)
- n_shift += 1
-
- # compute initial f by table lookup
- f = table[(d - 1).to_frac_wid(table_addr_bits, RoundDir.DOWN).bits]
-
- min_bits_of_precision = 1
- while min_bits_of_precision < width * 2:
- # multiply both n and d by f
- n *= f
- d *= f
- n = n.to_frac_wid(width, round_dir=RoundDir.DOWN)
- d = d.to_frac_wid(width, round_dir=RoundDir.UP)
-
- # slightly less than 2 to make the computation just a bitwise not
- nearly_two = FixedPoint.with_frac_wid(2, width)
- nearly_two = FixedPoint(nearly_two.bits - 1, width)
- f = (nearly_two - d).to_frac_wid(width)
-
- min_bits_of_precision *= 2
-
- # scale to correct value
- n *= 1 << n_shift
-
- # avoid incorrectly rounding down
- n = n.to_frac_wid(width - extra_precision, round_dir=RoundDir.UP)
- return math.floor(n)
+ return state.quotient, state.remainder
projects / soc.git / blobdiff