def _add(a, b):
assert isinstance(a, FixedPoint)
assert isinstance(b, FixedPoint)
promoted_qformat = QFormat.from_qformats(a.qformat, b.qformat)
result_qformat = QFormat(promoted_qformat.integer_bits + 1,
promoted_qformat.fraction_bits)
lhs_op = FixedPoint(a, result_qformat)
rhs_op = FixedPoint(b, result_qformat)
result_numerator = lhs_op._numerator + rhs_op._numerator
return FixedPoint._from_numerator(result_numerator, result_qformat)
def _add(a, b):
assert isinstance(a, FixedPoint)
assert isinstance(b, FixedPoint)
promoted_qformat = QFormat.from_qformats(a.qformat, b.qformat)
result_qformat = QFormat(promoted_qformat.integer_bits + 1, promoted_qformat.fraction_bits)
lhs_op = FixedPoint(a, result_qformat)
rhs_op = FixedPoint(b, result_qformat)
result_numerator = lhs_op._numerator + rhs_op._numerator
return FixedPoint._from_numerator(result_numerator, result_qformat)
def _truediv(dividend, divisor):
assert isinstance(dividend, FixedPoint)
assert isinstance(divisor, FixedPoint)
result_qformat = QFormat(
dividend.qformat.integer_bits + divisor.qformat.fraction_bits + 1,
divisor.qformat.integer_bits + dividend.qformat.fraction_bits)
working_qformat = QFormat.from_qformats(dividend.qformat, divisor.qformat,
result_qformat)
lhs_op = FixedPoint(dividend, working_qformat)
rhs_op = FixedPoint(divisor, working_qformat)
# We use Fraction's round() here rather than floor division to get correct rounding
working_numerator = round(
Fraction(lhs_op._numerator * working_qformat.denominator,
rhs_op._numerator))
working_result = FixedPoint._from_numerator(working_numerator,
working_qformat)
return FixedPoint(working_result, result_qformat)
def _mul(a, b):
assert isinstance(a, FixedPoint)
assert isinstance(b, FixedPoint)
result_qformat = QFormat(
a.qformat.integer_bits + b.qformat.integer_bits + 1,
a.qformat.fraction_bits + b.qformat.fraction_bits)
lhs_op = FixedPoint(a, result_qformat)
rhs_op = FixedPoint(b, result_qformat)
result_numerator = (lhs_op._numerator *
rhs_op._numerator) // result_qformat.denominator
return FixedPoint._from_numerator(result_numerator, result_qformat)
def _from_integer(cls, i):
"""Create a FixedPoint using a QFormat with sufficient precision to represent the integer.
Args:
i: The integer to be represented in FixedPoint.
Returns:
A FixedPoint representation with sufficient precision to represent i.
"""
assert isinstance(i, int)
num_bits = i.bit_length() + 1 # Additional bit for sign information
qformat = QFormat(num_bits, 0)
return cls._from_numerator(i, qformat)
def _truediv(dividend, divisor):
assert isinstance(dividend, FixedPoint)
assert isinstance(divisor, FixedPoint)
result_qformat = QFormat(dividend.qformat.integer_bits + divisor.qformat.fraction_bits + 1,
divisor.qformat.integer_bits + dividend.qformat.fraction_bits)
working_qformat = QFormat.from_qformats(dividend.qformat, divisor.qformat, result_qformat)
lhs_op = FixedPoint(dividend, working_qformat)
rhs_op = FixedPoint(divisor, working_qformat)
# We use Fraction's round() here rather than floor division to get correct rounding
working_numerator = round(Fraction(lhs_op._numerator * working_qformat.denominator, rhs_op._numerator))
working_result = FixedPoint._from_numerator(working_numerator, working_qformat)
return FixedPoint(working_result, result_qformat)
def _pow(base, exponent):
assert isinstance(base, FixedPoint)
assert isinstance(base, FixedPoint)
if exponent.is_integer():
integer_exponent = abs(floor(exponent))
result_qformat = QFormat(
max(base.qformat.integer_bits - 1, 0) * integer_exponent + 1,
base.qformat.fraction_bits * integer_exponent)
result_numerator = base._numerator**integer_exponent
positive_result = FixedPoint._from_numerator(result_numerator,
result_qformat)
if exponent >= 0:
return positive_result
else:
reciprocal_result = 1 / positive_result
return reciprocal_result
return float(base)**float(exponent)
def _from_float(cls, f):
"""Create a FixedPoint using a QFormat without loss of precision.
Args:
f (float): A float of which to create an equivalent FixedPoint representation.
Returns:
A FixedPoint object the QFormat for which will have sufficient precision to
exactly represent the float.
Raises:
OverflowError: If f is NaN or infinite.
"""
assert isinstance(f, Real)
if isnan(f) or isinf(f):
raise OverflowError("{} cannot be represented by {}".format(
f, cls.__name__))
# 1. Work out where the binary point is
fr, exp = frexp(f)
numerator = int(fr * (2**53)) # 53 binary places in the fraction
binary_point_index = 53 - exp # Is one based
# 2. Work out how many significant figures there are to the left of the binary point; call this m
most_significant_set_index = numerator.bit_length() - 1
num_leading_bits = 1 + most_significant_set_index - binary_point_index
m = max(num_leading_bits, 0) + 1 # One to accommodate sign
# 3. Work out how many significant figures there are to the right of the binary point; call this n
lowest_bit = lowest_set_bit(numerator)
least_significant_set_index = lowest_bit.bit_length() - 1
num_trailing_bits = binary_point_index - least_significant_set_index
n = max(num_trailing_bits, 0)
# 4. Make QFormat(m, n)
qformat = QFormat(m, n)
# 5. Shift the numerator to fit Qm.n
# We want the binary point to be at index n
shift = binary_point_index - n
shifted_numerator = signed_left_shift(numerator, shift)
return cls._from_numerator(shifted_numerator, qformat)
def _from_rational_exact(cls, r):
"""Create a FixedPoint using a QFormat with sufficient precision to represent the integer.
Args:
r: A rational number to be represented exactly.
Returns:
An exact FixedPoint representation of r.
Raises:
ValueError: If r cannot be represented exactly.
"""
assert isinstance(r, Rational)
integer_part = trunc(r)
fraction_part = abs(r - integer_part)
binary_numerator, binary_denominator = fraction_with_base(
fraction_part, base=2)
qformat = QFormat(integer_part.bit_length() + 1,
int(log2(binary_denominator)))
return cls._from_numerator(binary_numerator, qformat)
def __neg__(self):
# This can overflow for the most negative value of the current QFormat - the positive value can't be
# represented - so the result must have one additional bit of precision.
result_qformat = QFormat(self._qformat.integer_bits + 1,
self._qformat.fraction_bits)
return FixedPoint._from_numerator(-self._numerator, result_qformat)