add WIP HDL version of goldschmidt division -- it's currently broken

[soc.git] / src / soc / fu / div / experiment / goldschmidt_div_sqrt.py

# SPDX-License-Identifier: LGPL-3-or-later
# Copyright 2022 Jacob Lifshay programmerjake@gmail.com

# Funded by NLnet Assure Programme 2021-02-052, https://nlnet.nl/assure part
# of Horizon 2020 EU Programme 957073.

from collections import defaultdict
from dataclasses import dataclass, field, fields, replace
import logging
import math
import enum
from fractions import Fraction
from types import FunctionType
from functools import lru_cache
from nmigen.hdl.ast import Signal, unsigned, Mux, signed
from nmigen.hdl.dsl import Module, Elaboratable
from nmigen.hdl.mem import Memory
from nmutil.clz import CLZ

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, Any
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
class RoundDir(enum.Enum):
    DOWN = enum.auto()
    UP = enum.auto()
    NEAREST_TIES_UP = enum.auto()
    ERROR_IF_INEXACT = enum.auto()


@dataclass(frozen=True)
class FixedPoint:
    bits: int
    frac_wid: int

    def __post_init__(self):
        # called by the autogenerated __init__
        assert isinstance(self.bits, int)
        assert isinstance(self.frac_wid, int) and self.frac_wid >= 0

    @staticmethod
    def cast(value):
        """convert `value` to a fixed-point number with enough fractional
        bits to preserve its value."""
        if isinstance(value, FixedPoint):
            return value
        if isinstance(value, int):
            return FixedPoint(value, 0)
        if isinstance(value, str):
            value = value.strip()
            neg = value.startswith("-")
            if neg or value.startswith("+"):
                value = value[1:]
            if value.startswith(("0x", "0X")) and "." in value:
                value = value[2:]
                got_dot = False
                bits = 0
                frac_wid = 0
                for digit in value:
                    if digit == "_":
                        continue
                    if got_dot:
                        if digit == ".":
                            raise ValueError("too many `.` in string")
                        frac_wid += 4
                    if digit == ".":
                        got_dot = True
                        continue
                    if not digit.isalnum():
                        raise ValueError("invalid hexadecimal digit")
                    bits <<= 4
                    bits |= int("0x" + digit, base=16)
            else:
                bits = int(value, base=0)
                frac_wid = 0
            if neg:
                bits = -bits
            return FixedPoint(bits, frac_wid)

        if isinstance(value, float):
            n, d = value.as_integer_ratio()
            log2_d = d.bit_length() - 1
            assert d == 1 << log2_d, ("d isn't a power of 2 -- won't ever "
                                      "fail with float being IEEE 754")
            return FixedPoint(n, log2_d)
        raise TypeError("can't convert type to FixedPoint")

    @staticmethod
    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`."""
        assert isinstance(frac_wid, int) and frac_wid >= 0
        assert isinstance(round_dir, RoundDir)
        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:
            pass
        elif round_dir == RoundDir.UP:
            if remainder != 0:
                bits += 1
        elif round_dir == RoundDir.NEAREST_TIES_UP:
            if remainder * 2 >= denominator:
                bits += 1
        elif round_dir == RoundDir.ERROR_IF_INEXACT:
            if remainder != 0:
                raise ValueError("inexact conversion")
        else:
            assert False, "unimplemented round_dir"
        return FixedPoint(bits, frac_wid)

    def to_frac_wid(self, frac_wid, round_dir=RoundDir.ERROR_IF_INEXACT):
        """convert to the nearest fixed-point number with `frac_wid`
        fractional bits, rounding according to `round_dir`."""
        return FixedPoint.with_frac_wid(self, frac_wid, round_dir)

    def __float__(self):
        # 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
        greater than rhs, zero if self is equal to rhs, and a negative integer
        if self is less than rhs."""
        rhs = FixedPoint.cast(rhs)
        common_frac_wid = max(self.frac_wid, rhs.frac_wid)
        lhs = self.to_frac_wid(common_frac_wid)
        rhs = rhs.to_frac_wid(common_frac_wid)
        return lhs.bits - rhs.bits

    def __eq__(self, rhs):
        return self.cmp(rhs) == 0

    def __ne__(self, rhs):
        return self.cmp(rhs) != 0

    def __gt__(self, rhs):
        return self.cmp(rhs) > 0

    def __lt__(self, rhs):
        return self.cmp(rhs) < 0

    def __ge__(self, rhs):
        return self.cmp(rhs) >= 0

    def __le__(self, rhs):
        return self.cmp(rhs) <= 0

    def fract(self):
        """return the fractional part of `self`.
        that is `self - math.floor(self)`.
        """
        fract_mask = (1 << self.frac_wid) - 1
        return FixedPoint(self.bits & fract_mask, self.frac_wid)

    def __str__(self):
        if self < 0:
            return "-" + str(-self)
        digit_bits = 4
        frac_digit_count = (self.frac_wid + digit_bits - 1) // digit_bits
        fract = self.fract().to_frac_wid(frac_digit_count * digit_bits)
        frac_str = hex(fract.bits)[2:].zfill(frac_digit_count)
        return hex(math.floor(self)) + "." + frac_str

    def __repr__(self):
        return f"FixedPoint.with_frac_wid({str(self)!r}, {self.frac_wid})"

    def __add__(self, rhs):
        rhs = FixedPoint.cast(rhs)
        common_frac_wid = max(self.frac_wid, rhs.frac_wid)
        lhs = self.to_frac_wid(common_frac_wid)
        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)

    def __sub__(self, rhs):
        rhs = FixedPoint.cast(rhs)
        common_frac_wid = max(self.frac_wid, rhs.frac_wid)
        lhs = self.to_frac_wid(common_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 div(self, rhs, frac_wid, round_dir=RoundDir.ERROR_IF_INEXACT):
        assert isinstance(frac_wid, int) and frac_wid >= 0
        assert isinstance(round_dir, RoundDir)
        rhs = FixedPoint.cast(rhs)
        return FixedPoint.with_frac_wid(self.as_fraction()
                                        / rhs.as_fraction(),
                                        frac_wid, round_dir)

    def sqrt(self, round_dir=RoundDir.ERROR_IF_INEXACT):
        assert isinstance(round_dir, RoundDir)
        if self < 0:
            raise ValueError("can't compute sqrt of negative number")
        if self == 0:
            return self
        retval = FixedPoint(0, self.frac_wid)
        int_part_wid = self.bits.bit_length() - self.frac_wid
        first_bit_index = -(-int_part_wid // 2)  # division rounds up
        last_bit_index = -self.frac_wid
        for bit_index in range(first_bit_index, last_bit_index - 1, -1):
            trial = retval + FixedPoint(1 << (bit_index + self.frac_wid),
                                        self.frac_wid)
            if trial * trial <= self:
                retval = trial
        if round_dir == RoundDir.DOWN:
            pass
        elif round_dir == RoundDir.UP:
            if retval * retval < self:
                retval += FixedPoint(1, self.frac_wid)
        elif round_dir == RoundDir.NEAREST_TIES_UP:
            half_way = retval + FixedPoint(1, self.frac_wid + 1)
            if half_way * half_way <= self:
                retval += FixedPoint(1, self.frac_wid)
        elif round_dir == RoundDir.ERROR_IF_INEXACT:
            if retval * retval != self:
                raise ValueError("inexact sqrt")
        else:
            assert False, "unimplemented round_dir"
        return retval

    def rsqrt(self, round_dir=RoundDir.ERROR_IF_INEXACT):
        """compute the reciprocal-sqrt of `self`"""
        assert isinstance(round_dir, RoundDir)
        if self < 0:
            raise ValueError("can't compute rsqrt of negative number")
        if self == 0:
            raise ZeroDivisionError("can't compute rsqrt of zero")
        retval = FixedPoint(0, self.frac_wid)
        first_bit_index = -(-self.frac_wid // 2)  # division rounds up
        last_bit_index = -self.frac_wid
        for bit_index in range(first_bit_index, last_bit_index - 1, -1):
            trial = retval + FixedPoint(1 << (bit_index + self.frac_wid),
                                        self.frac_wid)
            if trial * trial * self <= 1:
                retval = trial
        if round_dir == RoundDir.DOWN:
            pass
        elif round_dir == RoundDir.UP:
            if retval * retval * self < 1:
                retval += FixedPoint(1, self.frac_wid)
        elif round_dir == RoundDir.NEAREST_TIES_UP:
            half_way = retval + FixedPoint(1, self.frac_wid + 1)
            if half_way * half_way * self <= 1:
                retval += FixedPoint(1, self.frac_wid)
        elif round_dir == RoundDir.ERROR_IF_INEXACT:
            if retval * retval * self != 1:
                raise ValueError("inexact rsqrt")
        else:
            assert False, "unimplemented round_dir"
        return retval


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 GoldschmidtDivParamsBase:
    """parameters for a Goldschmidt division algorithm, excluding derived
    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"""


@dataclass(frozen=True, unsafe_hash=True)
class GoldschmidtDivParams(GoldschmidtDivParamsBase):
    """parameters for a Goldschmidt division algorithm.
    Use `GoldschmidtDivParams.get` to find a efficient set of parameters.
    """

    # tuple to be immutable, repr=False so repr() works for debugging even when
    # __post_init__ hasn't finished running yet
    table: "tuple[FixedPoint, ...]" = field(init=False, repr=False)
    """the lookup-table"""

    ops: "tuple[GoldschmidtDivOp, ...]" = field(init=False, repr=False)
    """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_accuracy(self.io_width >= 1, "io_width out of range")
        _assert_accuracy(self.extra_precision >= 0,
                         "extra_precision out of range")
        _assert_accuracy(self.table_addr_bits >= 1,
                         "table_addr_bits out of range")
        _assert_accuracy(self.table_data_bits >= 1,
                         "table_data_bits out of range")
        _assert_accuracy(self.iter_count >= 1, "iter_count out of range")
        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

    @property
    def n_d_f_int_wid(self):
        """the number of bits in the integer part of `state.n`, `state.d`, and
        `state.f` during the main iteration loop.
        """
        return 2

    @property
    def n_d_f_total_wid(self):
        """the total number of bits (both integer and fraction bits) in
        `state.n`, `state.d`, and `state.f` during the main iteration loop.
        """
        return self.n_d_f_int_wid + self.expanded_width

    @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

    @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.

            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`:
        # the conditions on n_hat are satisfied by construction.
        for i in range(self.iter_count):
            _assert_accuracy(self.max_f(i) == 0)
            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 * self.n_hat / 2)
        for _ in range(i):
            power = self._shrink_max(power * power)

        max_rel_error = (2 * i) * self.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

    @cache_on_self
    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 += 5e7 * self.iter_count
        return cost

    @staticmethod
    @lru_cache(maxsize=1 << 16)
    def __cached_new(base_params):
        assert isinstance(base_params, GoldschmidtDivParamsBase)
        # can't use dataclasses.asdict, since it's recursive and will also give
        # child class fields too, which we don't want.
        kwargs = {}
        for field in fields(GoldschmidtDivParamsBase):
            kwargs[field.name] = getattr(base_params, field.name)
        try:
            return GoldschmidtDivParams(**kwargs), None
        except ParamsNotAccurateEnough as e:
            return None, e

    @staticmethod
    def __raise(e):  # type: (ParamsNotAccurateEnough) -> Any
        raise e

    @staticmethod
    def cached_new(base_params, handle_error=__raise):
        assert isinstance(base_params, GoldschmidtDivParamsBase)
        params, error = GoldschmidtDivParams.__cached_new(base_params)
        if error is None:
            return params
        else:
            return handle_error(error)

    @staticmethod
    def get(io_width, cost_fn=default_cost_fn, max_table_addr_bits=12):
        """ find efficient parameters for a goldschmidt division algorithm
        with `params.io_width == io_width`.

        arguments:
        io_width: int
            bit-width of the input divisor and the result.
            the input numerator is `2 * io_width`-bits wide.
        cost_fn: Callable[[GoldschmidtDivParams], float]
            return the estimated cost on an arbitrary scale of implementing
            goldschmidt division with the specified parameters. larger cost
            values mean worse parameters.
        max_table_addr_bits: int
            maximum allowable value of `table_addr_bits`
        """
        assert isinstance(io_width, int) and io_width >= 1
        assert callable(cost_fn)

        last_error = None
        last_error_params = None

        def cached_new(base_params):
            def handle_error(e):
                nonlocal last_error, last_error_params
                last_error = e
                last_error_params = base_params
                return None

            retval = GoldschmidtDivParams.cached_new(base_params, handle_error)
            if retval is None:
                logging.debug(f"GoldschmidtDivParams.get: err: {base_params}")
            else:
                logging.debug(f"GoldschmidtDivParams.get: ok: {base_params}")
            return retval

        @lru_cache(maxsize=None)
        def get_cost(base_params):
            params = cached_new(base_params)
            if params is None:
                return math.inf
            retval = cost_fn(params)
            logging.debug(f"GoldschmidtDivParams.get: cost={retval}: {params}")
            return retval

        # start with parameters big enough to always work.
        initial_extra_precision = io_width * 2 + 4
        initial_params = GoldschmidtDivParamsBase(
            io_width=io_width,
            extra_precision=initial_extra_precision,
            table_addr_bits=min(max_table_addr_bits, io_width),
            table_data_bits=io_width + initial_extra_precision,
            iter_count=1 + io_width.bit_length())

        if cached_new(initial_params) is None:
            raise ValueError(f"initial goldschmidt division algorithm "
                             f"parameters are invalid: {initial_params}"
                             ) from last_error

        # find good initial `iter_count`
        params = initial_params
        for iter_count in range(1, initial_params.iter_count):
            trial_params = replace(params, iter_count=iter_count)
            if cached_new(trial_params) is not None:
                params = trial_params
                break

        # now find `table_addr_bits`
        cost = get_cost(params)
        for table_addr_bits in range(1, max_table_addr_bits):
            trial_params = replace(params, table_addr_bits=table_addr_bits)
            trial_cost = get_cost(trial_params)
            if trial_cost < cost:
                params = trial_params
                cost = trial_cost
                break

        # check one higher `iter_count` to see if it has lower cost
        for table_addr_bits in range(1, max_table_addr_bits + 1):
            trial_params = replace(params,
                                   table_addr_bits=table_addr_bits,
                                   iter_count=params.iter_count + 1)
            trial_cost = get_cost(trial_params)
            if trial_cost < cost:
                params = trial_params
                cost = trial_cost
                break

        # now shrink `table_data_bits`
        while True:
            trial_params = replace(params,
                                   table_data_bits=params.table_data_bits - 1)
            trial_cost = get_cost(trial_params)
            if trial_cost < cost:
                params = trial_params
                cost = trial_cost
            else:
                break

        # and shrink `extra_precision`
        while True:
            trial_params = replace(params,
                                   extra_precision=params.extra_precision - 1)
            trial_cost = get_cost(trial_params)
            if trial_cost < cost:
                params = trial_params
                cost = trial_cost
            else:
                break

        return cached_new(params)


def clz(v, wid):
    assert isinstance(wid, int)
    assert isinstance(v, int) and 0 <= v < (1 << wid)
    return (1 << wid).bit_length() - v.bit_length()


@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 gen_hdl(self, params, state, sync_rom):
        # FIXME: finish getting hdl/simulation to work
        """generate the hdl for this operation.

        arguments:
        params: GoldschmidtDivParams
            the goldschmidt division parameters.
        state: GoldschmidtDivHDLState
            the input/output state
        sync_rom: bool
            true if the rom should be read synchronously rather than
            combinatorially, incurring an extra clock cycle of latency.
        """
        assert isinstance(params, GoldschmidtDivParams)
        assert isinstance(state, GoldschmidtDivHDLState)
        m = state.m
        expanded_width = params.expanded_width
        table_addr_bits = params.table_addr_bits
        if self == GoldschmidtDivOp.Normalize:
            # normalize so 1 <= d < 2
            assert state.d.width == params.io_width
            assert state.n.width == 2 * params.io_width
            d_leading_zeros = CLZ(params.io_width)
            m.submodules.d_leading_zeros = d_leading_zeros
            m.d.comb += d_leading_zeros.sig_in.eq(state.d)
            d_shift_out = Signal.like(state.d)
            m.d.comb += d_shift_out.eq(state.d << d_leading_zeros.lz)
            state.d = Signal(params.n_d_f_total_wid)
            m.d.comb += state.d.eq(d_shift_out << (params.extra_precision
                                                   + params.n_d_f_int_wid))

            # normalize so 1 <= n < 2
            n_leading_zeros = CLZ(2 * params.io_width)
            m.submodules.n_leading_zeros = n_leading_zeros
            m.d.comb += n_leading_zeros.sig_in.eq(state.n)
            n_shift_s_v = (params.io_width + d_leading_zeros.lz
                           - n_leading_zeros.lz)
            n_shift_s = Signal.like(n_shift_s_v)
            state.n_shift = Signal(d_leading_zeros.lz.width)
            m.d.comb += [
                n_shift_s.eq(n_shift_s_v),
                state.n_shift.eq(Mux(n_shift_s < 0, 0, n_shift_s)),
            ]
            n = Signal(params.n_d_f_total_wid)
            shifted_n = state.n << state.n_shift
            fixed_shift = params.expanded_width - state.n.width
            m.d.comb += n.eq(shifted_n << fixed_shift)
            state.n = n
        elif self == GoldschmidtDivOp.FEqTableLookup:
            assert state.d.width == params.n_d_f_total_wid, "invalid d width"
            # compute initial f by table lookup

            # extra bit for table entries == 1.0
            table_width = 1 + params.table_data_bits
            table = Memory(width=table_width, depth=len(params.table),
                           init=[i.bits for i in params.table])
            addr = state.d[:-params.n_d_f_int_wid][-table_addr_bits:]
            if sync_rom:
                table_read = table.read_port()
                m.d.comb += table_read.addr.eq(addr)
                state.insert_pipeline_register()
            else:
                table_read = table.read_port(domain="comb")
                m.d.comb += table_read.addr.eq(addr)
            m.submodules.table_read = table_read
            state.f = Signal(params.n_d_f_int_wid + params.expanded_width)
            data_shift = (table_width - params.table_data_bits
                          + params.expanded_width)
            m.d.comb += state.f.eq(table_read.data << data_shift)
        elif self == GoldschmidtDivOp.MulNByF:
            assert state.n.width == params.n_d_f_total_wid, "invalid n width"
            assert state.f is not None
            assert state.f.width == params.n_d_f_total_wid, "invalid f width"
            n = Signal.like(state.n)
            m.d.comb += n.eq((state.n * state.f) >> params.expanded_width)
            state.n = n
        elif self == GoldschmidtDivOp.MulDByF:
            assert state.d.width == params.n_d_f_total_wid, "invalid d width"
            assert state.f is not None
            assert state.f.width == params.n_d_f_total_wid, "invalid f width"
            d = Signal.like(state.d)
            m.d.comb += d.eq((state.d * state.f) >> params.expanded_width)
            state.d = d
        elif self == GoldschmidtDivOp.FEq2MinusD:
            assert state.d.width == params.n_d_f_total_wid, "invalid d width"
            f = Signal.like(state.d)
            m.d.comb += f.eq((2 << params.expanded_width) - state.d)
            state.f = f
        elif self == GoldschmidtDivOp.CalcResult:
            assert state.n.width == params.n_d_f_total_wid, "invalid n width"
            assert state.n_shift is not None
            # scale to correct value
            n = state.n * (1 << state.n_shift)
            q_approx = Signal(params.io_width)
            # extra bit for if it's bigger than orig_d
            r_approx = Signal(params.io_width + 1)
            adjusted_r = Signal(signed(1 + params.io_width))
            m.d.comb += [
                q_approx.eq((state.n << state.n_shift)
                            >> params.expanded_width),
                r_approx.eq(state.orig_n - q_approx * state.orig_d),
                adjusted_r.eq(r_approx - state.orig_d),
            ]
            state.quotient = Signal(params.io_width)
            state.remainder = Signal(params.io_width)

            with m.If(adjusted_r >= 0):
                m.d.comb += [
                    state.quotient.eq(q_approx + 1),
                    state.remainder.eq(adjusted_r),
                ]
            with m.Else():
                m.d.comb += [
                    state.quotient.eq(q_approx),
                    state.remainder.eq(r_approx),
                ]
        else:
            assert False, f"unimplemented GoldschmidtDivOp: {self}"


@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.
    """


def goldschmidt_div(n, d, 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:
        n: int
            numerator. a `2*width`-bit unsigned integer.
            must be less than `d << width`, otherwise the quotient wouldn't
            fit in `width` bits.
        d: int
            denominator. a `width`-bit unsigned integer. must not be zero.
        width: int
            the bit-width of the inputs/outputs. must be a positive 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)
    assert isinstance(n, int) and 0 <= n < (d << params.io_width)

    # this whole algorithm is done with fixed-point arithmetic where values
    # 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.quotient is not None
    assert state.remainder is not None

    return state.quotient, state.remainder


@dataclass(eq=False)
class GoldschmidtDivHDLState:
    m: Module
    """The HDL Module"""

    orig_n: Signal
    """original numerator"""

    orig_d: Signal
    """original denominator"""

    n: Signal
    """numerator -- N_prime[i] in the paper's algorithm 2"""

    d: Signal
    """denominator -- D_prime[i] in the paper's algorithm 2"""

    f: "Signal | None" = None
    """current factor -- F_prime[i] in the paper's algorithm 2"""

    quotient: "Signal | None" = None
    """final quotient"""

    remainder: "Signal | None" = None
    """final remainder"""

    n_shift: "Signal | None" = None
    """amount the numerator needs to be left-shifted at the end of the
    algorithm.
    """

    old_signals: "defaultdict[str, list[Signal]]" = field(repr=False,
                                                          init=False)

    __signal_name_prefix: "str" = field(default="state_", repr=False,
                                        init=False)

    def __post_init__(self):
        # called by the autogenerated __init__
        self.old_signals = defaultdict(list)

    def __setattr__(self, name, value):
        assert isinstance(name, str)
        if name.startswith("_"):
            return super().__setattr__(name, value)
        try:
            old_signals = self.old_signals[name]
        except AttributeError:
            # haven't yet finished __post_init__
            return super().__setattr__(name, value)
        assert name != "m" and name != "old_signals", f"can't write to {name}"
        assert isinstance(value, Signal)
        value.name = f"{self.__signal_name_prefix}{name}_{len(old_signals)}"
        old_signal = getattr(self, name, None)
        if old_signal is not None:
            assert isinstance(old_signal, Signal)
            old_signals.append(old_signal)
        return super().__setattr__(name, value)

    def insert_pipeline_register(self):
        old_prefix = self.__signal_name_prefix
        try:
            for field in fields(GoldschmidtDivHDLState):
                if field.name.startswith("_") or field.name == "m":
                    continue
                old_sig = getattr(self, field.name, None)
                if old_sig is None:
                    continue
                assert isinstance(old_sig, Signal)
                new_sig = Signal.like(old_sig)
                setattr(self, field.name, new_sig)
                self.m.d.sync += new_sig.eq(old_sig)
        finally:
            self.__signal_name_prefix = old_prefix


class GoldschmidtDivHDL(Elaboratable):
    # FIXME: finish getting hdl/simulation to work
    """ 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

        attributes:
        params: GoldschmidtDivParams
            the goldschmidt division algorithm parameters.
        pipe_reg_indexes: list[int]
            the operation indexes where pipeline registers should be inserted.
            duplicate values mean multiple registers should be inserted for
            that operation index -- this is useful to allow yosys to spread a
            multiplication across those multiple pipeline stages.
        sync_rom: bool
            true if the rom should be read synchronously rather than
            combinatorially, incurring an extra clock cycle of latency.
        n: Signal(unsigned(2 * params.io_width))
            input numerator. a `2 * params.io_width`-bit unsigned integer.
            must be less than `d << params.io_width`, otherwise the quotient
            wouldn't fit in `params.io_width` bits.
        d: Signal(unsigned(params.io_width))
            input denominator. a `params.io_width`-bit unsigned integer.
            must not be zero.
        q: Signal(unsigned(params.io_width))
            output quotient. only valid when `n < (d << params.io_width)`.
        r: Signal(unsigned(params.io_width))
            output remainder. only valid when `n < (d << params.io_width)`.
    """

    @property
    def total_pipeline_registers(self):
        """the total number of pipeline registers"""
        return len(self.pipe_reg_indexes) + self.sync_rom

    def __init__(self, params, pipe_reg_indexes=(), sync_rom=False):
        assert isinstance(params, GoldschmidtDivParams)
        assert isinstance(sync_rom, bool)
        self.params = params
        self.pipe_reg_indexes = sorted(int(i) for i in pipe_reg_indexes)
        self.sync_rom = sync_rom
        self.n = Signal(unsigned(2 * params.io_width))
        self.d = Signal(unsigned(params.io_width))
        self.q = Signal(unsigned(params.io_width))
        self.r = Signal(unsigned(params.io_width))

    def elaborate(self, platform):
        m = Module()
        state = GoldschmidtDivHDLState(
            m=m,
            orig_n=self.n,
            orig_d=self.d,
            n=self.n,
            d=self.d)

        # copy and reverse
        pipe_reg_indexes = list(reversed(self.pipe_reg_indexes))

        for op_index, op in enumerate(self.params.ops):
            while len(pipe_reg_indexes) > 0 \
                    and pipe_reg_indexes[-1] <= op_index:
                pipe_reg_indexes.pop()
                state.insert_pipeline_register()
            op.gen_hdl(self.params, state, self.sync_rom)

        while len(pipe_reg_indexes) > 0:
            pipe_reg_indexes.pop()
            state.insert_pipeline_register()

        m.d.comb += self.q.eq(state.quotient)
        m.d.comb += self.r.eq(state.remainder)
        return m


GOLDSCHMIDT_SQRT_RSQRT_TABLE_ADDR_INT_WID = 2


@lru_cache()
def goldschmidt_sqrt_rsqrt_table(table_addr_bits, table_data_bits):
    """Generate the look-up table needed for Goldschmidt's square-root and
    reciprocal-square-root algorithm.

    arguments:
    table_addr_bits: int
        the number of address bits for the look-up table.
    table_data_bits: int
        the number of data bits for the look-up table.
    """
    assert isinstance(table_addr_bits, int) and \
        table_addr_bits >= GOLDSCHMIDT_SQRT_RSQRT_TABLE_ADDR_INT_WID
    assert isinstance(table_data_bits, int) and table_data_bits >= 1
    table = []
    table_len = 1 << table_addr_bits
    for addr in range(table_len):
        if addr == 0:
            value = FixedPoint(0, table_data_bits)
        elif (addr << 2) < table_len:
            value = None  # table entries should be unused
        else:
            table_addr_frac_wid = table_addr_bits
            table_addr_frac_wid -= GOLDSCHMIDT_SQRT_RSQRT_TABLE_ADDR_INT_WID
            max_input_value = FixedPoint(addr + 1, table_addr_bits - 2)
            max_frac_wid = max(max_input_value.frac_wid, table_data_bits)
            value = max_input_value.to_frac_wid(max_frac_wid)
            value = value.rsqrt(RoundDir.DOWN)
            value = value.to_frac_wid(table_data_bits, RoundDir.DOWN)
        table.append(value)

    # tuple for immutability
    return tuple(table)

# FIXME: add code to calculate error bounds and check that the algorithm will
# actually work (like in the goldschmidt division algorithm).
# FIXME: add code to calculate a good set of parameters based on the error
# bounds checking.


def goldschmidt_sqrt_rsqrt(radicand, io_width, frac_wid, extra_precision,
                           table_addr_bits, table_data_bits, iter_count):
    """Goldschmidt's square-root and reciprocal-square-root algorithm.

    uses algorithm based on second method at:
    https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Goldschmidt%E2%80%99s_algorithm

    arguments:
    radicand: FixedPoint(frac_wid=frac_wid)
        the input value to take the square-root and reciprocal-square-root of.
    io_width: int
        the number of bits in the input (`radicand`) and output values.
    frac_wid: int
        the number of fraction bits in the input (`radicand`) and output
        values.
    extra_precision: int
        the number of bits of internal extra precision.
    table_addr_bits: int
        the number of address bits for the look-up table.
    table_data_bits: int
        the number of data bits for the look-up table.

    returns: tuple[FixedPoint, FixedPoint]
        the square-root and reciprocal-square-root, rounded down to the
        nearest representable value. If `radicand == 0`, then the
        reciprocal-square-root value returned is zero.
    """
    assert (isinstance(radicand, FixedPoint)
            and radicand.frac_wid == frac_wid
            and 0 <= radicand.bits < (1 << io_width))
    assert isinstance(io_width, int) and io_width >= 1
    assert isinstance(frac_wid, int) and 0 <= frac_wid < io_width
    assert isinstance(extra_precision, int) and extra_precision >= io_width
    assert isinstance(table_addr_bits, int) and table_addr_bits >= 1
    assert isinstance(table_data_bits, int) and table_data_bits >= 1
    assert isinstance(iter_count, int) and iter_count >= 0
    expanded_frac_wid = frac_wid + extra_precision
    s = radicand.to_frac_wid(expanded_frac_wid)
    sqrt_rshift = extra_precision
    rsqrt_rshift = extra_precision
    while s != 0 and s < 1:
        s = (s * 4).to_frac_wid(expanded_frac_wid)
        sqrt_rshift += 1
        rsqrt_rshift -= 1
    while s >= 4:
        s = s.div(4, expanded_frac_wid)
        sqrt_rshift -= 1
        rsqrt_rshift += 1
    table = goldschmidt_sqrt_rsqrt_table(table_addr_bits=table_addr_bits,
                                         table_data_bits=table_data_bits)
    # core goldschmidt sqrt/rsqrt algorithm:
    # initial setup:
    table_addr_frac_wid = table_addr_bits
    table_addr_frac_wid -= GOLDSCHMIDT_SQRT_RSQRT_TABLE_ADDR_INT_WID
    addr = s.to_frac_wid(table_addr_frac_wid, RoundDir.DOWN)
    assert 0 <= addr.bits < (1 << table_addr_bits), "table addr out of range"
    f = table[addr.bits]
    assert f is not None, "accessed invalid table entry"
    # use with_frac_wid to fix IDE type deduction
    f = FixedPoint.with_frac_wid(f, expanded_frac_wid, RoundDir.DOWN)
    x = (s * f).to_frac_wid(expanded_frac_wid, RoundDir.DOWN)
    h = (f * 0.5).to_frac_wid(expanded_frac_wid, RoundDir.DOWN)
    for _ in range(iter_count):
        # iteration step:
        f = (1.5 - x * h).to_frac_wid(expanded_frac_wid, RoundDir.DOWN)
        x = (x * f).to_frac_wid(expanded_frac_wid, RoundDir.DOWN)
        h = (h * f).to_frac_wid(expanded_frac_wid, RoundDir.DOWN)
    r = 2 * h
    # now `x` is approximately `sqrt(s)` and `r` is approximately `rsqrt(s)`

    sqrt = FixedPoint(x.bits >> sqrt_rshift, frac_wid)
    rsqrt = FixedPoint(r.bits >> rsqrt_rshift, frac_wid)

    next_sqrt = FixedPoint(sqrt.bits + 1, frac_wid)
    if next_sqrt * next_sqrt <= radicand:
        sqrt = next_sqrt

    next_rsqrt = FixedPoint(rsqrt.bits + 1, frac_wid)
    if next_rsqrt * next_rsqrt * radicand <= 1 and radicand != 0:
        rsqrt = next_rsqrt
    return sqrt, rsqrt