def multiply(self, x, y):
# Use 2 * wordlength long multipliers
z = Representation(mantissa=x.mantissa * y.mantissa,
format_=Format(msb=self.format_.msb * 2 + 1,
lsb=self.format_.lsb * 2,
signed=self.format_.signed))
mantissa, (underflow, overflow) = self.format_.represent(
z, rounding_method=self.rounding_method)
if underflow and not self.multiply.allows_underflow:
raise UnderflowError(f'{x} * {y} underflows in {self.format_}',
mpfloat(z), self.format_.value_epsilon)
if overflow:
if not self.multiply.allows_overflow:
raise OverflowError(f'{x} * {y} overflows in {self.format_}',
mpfloat(z), self.format_.value_interval)
mantissa, _ = self.overflow_behavior(
mantissa, range_=self.format_.mantissa_interval)
return type(x)(mantissa, self.format_)
def __le__(self, other):
if not other.is_Number:
return super().__le__(other)
if not other:
return self._args[0] <= 0
if isinstance(other, type(self)):
return self._args[0] <= other._args[0]
if isinstance(self._args[0], Basic):
return self._args[0] <= other
return self._args[0] <= mpfloat(other)
def __eq__(self, other):
if not other.is_Number:
return False
if not other:
return not self._args[0]
if isinstance(other, type(self)):
return self._args[0] == other._args[0]
if isinstance(self._args[0], Basic):
return self._args[0] == other
return self._args[0] == mpfloat(other)
def computation_error_bounds(self, input_range, state_ranges=None):
if state_ranges is None and self.states:
state_ranges = []
for model in self.state_observer_models:
state_range = output_range(model, mpfloat(input_range))
state_range = interval(
lower_bound=asvector_if_possible(state_range.lower_bound),
upper_bound=asvector_if_possible(state_range.upper_bound),
)
state_ranges.append(state_range)
return super().computation_error_bounds([input_range], state_ranges)
def nearest(self, x):
if x.is_integer:
return x
value = nearest_integer(mpfloat(x))
mantissa, (_, overflow) = self.format_.represent(value)
if overflow:
if not self.nearest.allows_overflow:
raise OverflowError(f'round({x}) overflows in {self.format_}',
mantissa * self.format_.value_epsilon,
self.format_.value_interval)
mantissa, _ = self.overflow_behavior(
mantissa, range_=self.format_.mantissa_interval)
return type(x)(mantissa, format_)
def __mul__(self, other):
if not isinstance(other, Number):
other = Number(other)
result = self.number * other.number
snumber = mpfloat(self.number)
onumber = mpfloat(other.number)
result_error_bounds = ((snumber + self.error_bounds) *
(onumber + other.error_bounds)).difference(
snumber * onumber)
if isinstance(result, (FixedPointNumber, FixedPointSymbol)):
exact = (self.number == -1 or self.number == 1 or self.number == 0
or other.number == -1 or other.number == 1
or other.number == 0)
if not exact:
rounding = ProcessingUnit.active().rounding_method
if isinstance(self.number, (FixedPointNumber, FixedPointSymbol)) and \
isinstance(other.number, (FixedPointNumber, FixedPointSymbol)):
result_error_bounds += rounding.error_bounds(
result.format_.lsb,
self.number.format_.lsb + other.number.format_.lsb)
else:
result_error_bounds += rounding.error_bounds(
result.format_.lsb)
return Number(result, result_error_bounds)
def represent(self, rvalue, rounding_method=round):
from ltitop.arithmetic.fixed_point.representation import Representation
if isinstance(rvalue, Representation) and hasattr(
rounding_method, 'shift'):
lvalue = rvalue.mantissa
quantize = functools.partial(rounding_method.shift,
n=rvalue.format_.lsb - self.lsb)
else:
lvalue = mpfloat(rvalue)
quantize = functools.partial(mpquantize,
nbits=-self.lsb,
rounding_method=rounding_method)
if not self.signed and not np.all(lvalue >= 0):
raise ValueError(f'Unsigned format cannot represent {rvalue}')
mantissa = quantize(lvalue)
underflow = bool(np.any(np.logical_and(mantissa == 0, lvalue != 0)))
overflow = bool(np.any(mantissa not in self.mantissa_interval))
return mantissa, (underflow, overflow)
def best(cls,
value,
wordlength,
rounding_method=nearest_integer,
signed=True):
"""
See "Reliable Implementation of Linear Filters with Fixed-Point Arithmetic",
Hilaire and Lopez, 2013
"""
# TODO(hidmic): optimize when value is a Representation
with mpmath.workprec(20 * wordlength): # enough bits
mpvalue = mpfloat(value)
if not np.all(mpvalue >= 0) and not signed:
raise ValueError(f'Unsigned format cannot represent {value}')
# estimate MSB
msb = np.max(mpmsb(mpvalue, signed=signed))
if np.isneginf(msb):
msb = 0 # arbitrary
msb = int(msb)
# estimate LSB
lsb = msb - wordlength + int(signed)
# compute mantissa
mantissa = mpquantize(mpvalue,
nbits=-lsb,
rounding_method=rounding_method)
# adjust MSB if limits were overpassed
adjusted_msb = msb # no adjustment
upper_limit = 2**(wordlength - int(signed))
if not np.all(mantissa < upper_limit):
adjusted_msb = msb + 1
if np.all(mantissa < 0):
lower_limit = -2**(wordlength - 2)
if np.all(mantissa > lower_limit):
adjusted_msb = msb - 1
if adjusted_msb != msb:
# adjust LSB and mantissa
msb = adjusted_msb
lsb = msb - wordlength + int(signed)
mantissa = mpquantize(mpvalue,
nbits=-lsb,
rounding_method=mpmath.nint)
return mantissa, cls(msb=msb, lsb=lsb, signed=signed)
def _as_mpf_val(self, prec):
with mpmath.workprec(prec):
return mpfloat(self._args[0])
class Number:
number: Any
error_bounds: Interval = Interval(mpfloat(0))
def __add__(self, other):
if not isinstance(other, Number):
other = Number(other)
result = self.number + other.number
result_error_bounds = self.error_bounds + other.error_bounds
if isinstance(result, (FixedPointNumber, FixedPointSymbol)):
exact = self.number == 0 or other.number == 0
if not exact:
rounding = ProcessingUnit.active().rounding_method
if isinstance(self.number,
(FixedPointNumber, FixedPointSymbol)):
if result.format_.lsb > self.number.format_.lsb:
result_error_bounds += rounding.error_bounds(
result.format_.lsb, self.number.format_.lsb)
elif self.number != 0:
result_error_bounds += rounding.error_bounds(
result.format_.lsb)
if isinstance(other.number,
(FixedPointNumber, FixedPointSymbol)):
if result.format_.lsb > other.number.format_.lsb:
result_error_bounds += rounding.error_bounds(
result.format_.lsb, other.number.format_.lsb)
elif other.number != 0:
result_error_bounds += rounding.error_bounds(
result.format_.lsb)
return Number(result, result_error_bounds)
__radd__ = __add__
def __sub__(self, other):
if not isinstance(other, Number):
other = Number(other)
result = self.number - other.number
result_error_bounds = self.error_bounds + other.error_bounds
if isinstance(result, (FixedPointNumber, FixedPointSymbol)):
exact = self.number == 0 or other.number == 0
if not exact:
rounding = ProcessingUnit.active().rounding_method
if isinstance(self.number,
(FixedPointNumber, FixedPointSymbol)):
if result.format_.lsb > self.number.format_.lsb:
result_error_bounds += rounding.error_bounds(
result.format_.lsb, self.number.format_.lsb)
elif self.number != 0:
result_error_bounds += rounding.error_bounds(
result.format_.lsb)
if isinstance(other.number,
(FixedPointNumber, FixedPointSymbol)):
if result.format_.lsb > other.number.format_.lsb:
result_error_bounds += rounding.error_bounds(
result.format_.lsb, other.number.format_.lsb)
elif other.number != 0:
result_error_bounds += rounding.error_bounds(
result.format_.lsb)
return Number(result, result_error_bounds)
def __rsub__(self, other):
return Number(other) - self
def __mul__(self, other):
if not isinstance(other, Number):
other = Number(other)
result = self.number * other.number
snumber = mpfloat(self.number)
onumber = mpfloat(other.number)
result_error_bounds = ((snumber + self.error_bounds) *
(onumber + other.error_bounds)).difference(
snumber * onumber)
if isinstance(result, (FixedPointNumber, FixedPointSymbol)):
exact = (self.number == -1 or self.number == 1 or self.number == 0
or other.number == -1 or other.number == 1
or other.number == 0)
if not exact:
rounding = ProcessingUnit.active().rounding_method
if isinstance(self.number, (FixedPointNumber, FixedPointSymbol)) and \
isinstance(other.number, (FixedPointNumber, FixedPointSymbol)):
result_error_bounds += rounding.error_bounds(
result.format_.lsb,
self.number.format_.lsb + other.number.format_.lsb)
else:
result_error_bounds += rounding.error_bounds(
result.format_.lsb)
return Number(result, result_error_bounds)
__rmul__ = __mul__
def __div__(self, other):
# TODO(hidmic): implement
return NotImplemented
__truediv__ = __div__
def __rdiv__(self, other):
return Number(other) / self
__rtruediv__ = __rdiv__
def __mod__(self, other):
# TODO(hidmic): implement
return NotImplemented
def __rmod__(self, other):
return Number(other) % self
def __apply_rounding_method__(self, method):
number = method(self.number)
error_bounds = self.error_bounds
lsb = None
if isinstance(self.number, (FixedPointNumber, FixedPointSymbol)):
lsb = self.number.format_.lsb
error_bounds += method.error_bounds(0, lsb)
return Number(number, error_bounds)
def __trunc__(self):
return self.__apply_rounding_method__(truncate)
def __ceil__(self):
return self.__apply_rounding_method__(ceil)
def __floor__(self):
return self.__apply_rounding_method__(floor)
def __round__(self):
return self.__apply_rounding_method__(nearest_integer)
def __neg__(self):
return Number(-self.number, -self.error_bounds)
def __float__(self):
return float(self.number)
def __mpfloat__(self):
return mpfloat(self.number)
def __int__(self):
return int(self.number)
__long__ = __int__
def __nonzero__(self):
return bool(self.number)
__bool__ = __nonzero__
_iterable = False # tell sympy to not iterate this
def __getitem__(self, key):
try:
error_bounds = self.error_bounds[key]
except TypeError:
error_bounds = self.error_bounds
return Number(self.number[key], error_bounds)
def __eq__(self, other):
if not isinstance(other, Number):
other = Number(other)
return self.number == other.number and \
self.error_bounds == other.error_bounds == 0
def __ne__(self, other):
return not (self == other)
def __lt__(self, other):
if not isinstance(other, Number):
other = Number(other)
return mpfloat(self.number) + self.error_bounds < \
mpfloat(other.number) + other.error_bounds
def __le__(self, other):
return self < other or self == other
def __gt__(self, other):
return not (self <= other)
def __ge__(self, other):
return not (self < other)
def __lshift__(self, n):
return Number(self.number << n, self.error_bounds * 2**n)
def __rshift__(self, n):
# TODO(hidmic): implement
return NotImplemented
def __lt__(self, other):
if not isinstance(other, Number):
other = Number(other)
return mpfloat(self.number) + self.error_bounds < \
mpfloat(other.number) + other.error_bounds
def __mpfloat__(self):
return mpfloat(self.number)