denominator = value.denominator
else:
value = FixedPoint.cast(value)
- # 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)
numerator = value.bits
denominator = 1 << value.frac_wid
if denominator < 0:
iter_count: int
"""the total number of iterations of the division algorithm's loop"""
- # tuple to be immutable
- table: "tuple[FixedPoint, ...]" = field(init=False)
+ # 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)
+ 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."""
assert isinstance(addr, int)
assert 0 <= addr < self.table_addr_count
_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)
- max_numerator = min_numerator + values_per_table_entry
+ 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_value, max_value = self.table_input_exact_range(addr)
+ min_input, max_input = self.table_input_exact_range(addr)
# division swaps min/max
- return 1 / max_value, 1 / min_value
+ 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)
RoundDir.DOWN))
# we have to use object.__setattr__ since frozen=True
object.__setattr__(self, "table", tuple(table))
- object.__setattr__(self, "ops", tuple(_goldschmidt_div_ops(self)))
-
- @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
- 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:
- pass
- raise ValueError(f"can't find working parameters for a goldschmidt "
- f"division algorithm with io_width={io_width}")
+ object.__setattr__(self, "ops", tuple(self.__make_ops()))
@property
def expanded_width(self):
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
"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)
+ return self._shrink_max(retval)
@cache_on_self
def max_d(self, i):
"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)
+ return self._shrink_max(retval)
@cache_on_self
def max_f(self, i):
# `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)
+ return self._shrink_max(retval)
@cache_on_self
def max_delta(self, i):
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)
+ # `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):
# `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)
+ min_prod = Fraction(1)
for j in range(i):
max_n_j = self.max_n(j)
max_d_j = self.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
+ retval = 1 - (1 - max_n_i) * min_prod
+ return self._shrink_max(retval)
@cached_property
def max_n_shift(self):
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):
assert False, f"unimplemented GoldschmidtDivOp: {self}"
-def _goldschmidt_div_ops(params):
- """ 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
-
- arguments:
- params: GoldschmidtDivParams
- the parameters for the algorithm
-
- yields: GoldschmidtDivOp
- the operations needed to perform the division.
- """
- assert isinstance(params, GoldschmidtDivParams)
-
- # 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(params.expanded_width > 4)
-
- # 3. require `abs(e[0]) + 3 * d[0] / 2 + f[0] < 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)
-
- for addr in range(params.table_addr_count):
- f_prime_m1 = params.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(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
- 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
-
-
def goldschmidt_div(n, d, params):
""" Goldschmidt division algorithm.
projects / soc.git / blobdiff