max_n_shift += 1
return max_n_shift
+ @cached_property
+ def n_hat(self):
+ """ maximum value of, for all `i`, `max_n(i)` and `max_d(i)`
+ """
+ n_hat = Fraction(0)
+ for i in range(self.iter_count):
+ n_hat = max(n_hat, self.max_n(i), self.max_d(i))
+ return self._shrink_max(n_hat)
+
def __make_ops(self):
""" Goldschmidt division algorithm.
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)
+ # Require `n[i] <= n_hat` and `d[i] <= n_hat` and `f[i] = 0`:
+ # the conditions on n_hat are satisfied by construction.
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
# ` + (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)
+ power = self._shrink_max(self.max_abs_e0 + 3 * self.n_hat / 2)
for _ in range(i):
power = self._shrink_max(power * power)
- max_rel_error = (2 * i) * n_hat + power
+ max_rel_error = (2 * i) * self.n_hat + power
min_a_over_b = Fraction(1, 2)
max_a_over_b = Fraction(2)
yield GoldschmidtDivOp.CalcResult
+ def default_cost_fn(self):
+ """ calculate the estimated cost on an arbitrary scale of implementing
+ goldschmidt division with the specified parameters. larger cost
+ values mean worse parameters.
+
+ This is the default cost function for `GoldschmidtDivParams.get`.
+
+ returns: float
+ """
+ rom_cells = self.table_data_bits << self.table_addr_bits
+ cost = float(rom_cells)
+ for op in self.ops:
+ if op == GoldschmidtDivOp.MulNByF \
+ or op == GoldschmidtDivOp.MulDByF:
+ mul_cost = self.expanded_width ** 2
+ mul_cost *= self.expanded_width.bit_length()
+ cost += mul_cost
+ cost += 1e6 * self.iter_count
+ return cost
+
@staticmethod
def get(io_width):
""" find efficient parameters for a goldschmidt division algorithm