# Part of Odoo. See LICENSE file for full copyright and licensing details. import builtins import math __all__ = [ "float_compare", "float_is_zero", "float_repr", "float_round", "float_split", "float_split_str", ] def round(f): # P3's builtin round differs from P2 in the following manner: # * it rounds half to even rather than up (away from 0) # * round(-0.) loses the sign (it returns -0 rather than 0) # * round(x) returns an int rather than a float # # this compatibility shim implements Python 2's round in terms of # Python 3's so that important rounding error under P3 can be # trivially fixed, assuming the P2 behaviour to be debugged and # correct. roundf = builtins.round(f) if builtins.round(f + 1) - roundf != 1: return f + math.copysign(0.5, f) # copysign ensures round(-0.) -> -0 *and* result is a float return math.copysign(roundf, f) def _float_check_precision(precision_digits=None, precision_rounding=None): if precision_rounding is not None and precision_digits is None: assert precision_rounding > 0,\ f"precision_rounding must be positive, got {precision_rounding}" elif precision_digits is not None and precision_rounding is None: # TODO: `int`s will also get the `is_integer` method starting from python 3.12 assert float(precision_digits).is_integer() and precision_digits >= 0,\ f"precision_digits must be a non-negative integer, got {precision_digits}" precision_rounding = 10 ** -precision_digits else: msg = "exactly one of precision_digits and precision_rounding must be specified" raise AssertionError(msg) return precision_rounding def float_round(value, precision_digits=None, precision_rounding=None, rounding_method='HALF-UP'): """Return ``value`` rounded to ``precision_digits`` decimal digits, minimizing IEEE-754 floating point representation errors, and applying the tie-breaking rule selected with ``rounding_method``, by default HALF-UP (away from zero). Precision must be given by ``precision_digits`` or ``precision_rounding``, not both! :param float value: the value to round :param int precision_digits: number of fractional digits to round to. :param float precision_rounding: decimal number representing the minimum non-zero value at the desired precision (for example, 0.01 for a 2-digit precision). :param rounding_method: the rounding method used: - 'HALF-UP' will round to the closest number with ties going away from zero. - 'HALF-DOWN' will round to the closest number with ties going towards zero. - 'HALF_EVEN' will round to the closest number with ties going to the closest even number. - 'UP' will always round away from 0. - 'DOWN' will always round towards 0. :return: rounded float """ rounding_factor = _float_check_precision(precision_digits=precision_digits, precision_rounding=precision_rounding) if rounding_factor == 0 or value == 0: return 0.0 # NORMALIZE - ROUND - DENORMALIZE # In order to easily support rounding to arbitrary 'steps' (e.g. coin values), # we normalize the value before rounding it as an integer, and de-normalize # after rounding: e.g. float_round(1.3, precision_rounding=.5) == 1.5 def normalize(val): return val / rounding_factor def denormalize(val): return val * rounding_factor # inverting small rounding factors reduces rounding errors if rounding_factor < 1: rounding_factor = float_invert(rounding_factor) normalize, denormalize = denormalize, normalize normalized_value = normalize(value) # Due to IEEE-754 float/double representation limits, the approximation of the # real value may be slightly below the tie limit, resulting in an error of # 1 unit in the last place (ulp) after rounding. # For example 2.675 == 2.6749999999999998. # To correct this, we add a very small epsilon value, scaled to the # the order of magnitude of the value, to tip the tie-break in the right # direction. # Credit: discussion with OpenERP community members on bug 882036 epsilon_magnitude = math.log2(abs(normalized_value)) # `2**(epsilon_magnitude - 52)` would be the minimal size, but we increase it to be # more tolerant of inaccuracies accumulated after multiple floating point operations epsilon = 2**(epsilon_magnitude - 50) match rounding_method: case 'HALF-UP': # 0.5 rounds away from 0 result = round(normalized_value + math.copysign(epsilon, normalized_value)) case 'HALF-EVEN': # 0.5 rounds towards closest even number integral = math.floor(normalized_value) remainder = abs(normalized_value - integral) is_half = abs(0.5 - remainder) < epsilon # if is_half & integral is odd, add odd bit to make it even result = integral + (integral & 1) if is_half else round(normalized_value) case 'HALF-DOWN': # 0.5 rounds towards 0 result = round(normalized_value - math.copysign(epsilon, normalized_value)) case 'UP': # round to number furthest from zero result = math.trunc(normalized_value + math.copysign(1 - epsilon, normalized_value)) case 'DOWN': # round to number closest to zero result = math.trunc(normalized_value + math.copysign(epsilon, normalized_value)) case _: msg = f"unknown rounding method: {rounding_method}" raise ValueError(msg) return denormalize(result) def float_is_zero(value, precision_digits=None, precision_rounding=None): """Returns true if ``value`` is small enough to be treated as zero at the given precision (smaller than the corresponding *epsilon*). The precision (``10**-precision_digits`` or ``precision_rounding``) is used as the zero *epsilon*: values less than that are considered to be zero. Precision must be given by ``precision_digits`` or ``precision_rounding``, not both! Warning: ``float_is_zero(value1-value2)`` is not equivalent to ``float_compare(value1,value2) == 0``, as the former will round after computing the difference, while the latter will round before, giving different results for e.g. 0.006 and 0.002 at 2 digits precision. :param int precision_digits: number of fractional digits to round to. :param float precision_rounding: decimal number representing the minimum non-zero value at the desired precision (for example, 0.01 for a 2-digit precision). :param float value: value to compare with the precision's zero :return: True if ``value`` is considered zero """ epsilon = _float_check_precision(precision_digits=precision_digits, precision_rounding=precision_rounding) return value == 0.0 or abs(float_round(value, precision_rounding=epsilon)) < epsilon def float_compare(value1, value2, precision_digits=None, precision_rounding=None): """Compare ``value1`` and ``value2`` after rounding them according to the given precision. A value is considered lower/greater than another value if their rounded value is different. This is not the same as having a non-zero difference! Precision must be given by ``precision_digits`` or ``precision_rounding``, not both! Example: 1.432 and 1.431 are equal at 2 digits precision, so this method would return 0 However 0.006 and 0.002 are considered different (this method returns 1) because they respectively round to 0.01 and 0.0, even though 0.006-0.002 = 0.004 which would be considered zero at 2 digits precision. Warning: ``float_is_zero(value1-value2)`` is not equivalent to ``float_compare(value1,value2) == 0``, as the former will round after computing the difference, while the latter will round before, giving different results for e.g. 0.006 and 0.002 at 2 digits precision. :param float value1: first value to compare :param float value2: second value to compare :param int precision_digits: number of fractional digits to round to. :param float precision_rounding: decimal number representing the minimum non-zero value at the desired precision (for example, 0.01 for a 2-digit precision). :return: (resp.) -1, 0 or 1, if ``value1`` is (resp.) lower than, equal to, or greater than ``value2``, at the given precision. """ rounding_factor = _float_check_precision(precision_digits=precision_digits, precision_rounding=precision_rounding) # equal numbers round equally, so we can skip that step # doing this after _float_check_precision to validate parameters first if value1 == value2: return 0 value1 = float_round(value1, precision_rounding=rounding_factor) value2 = float_round(value2, precision_rounding=rounding_factor) delta = value1 - value2 if float_is_zero(delta, precision_rounding=rounding_factor): return 0 return -1 if delta < 0.0 else 1 def float_repr(value, precision_digits): """Returns a string representation of a float with the given number of fractional digits. This should not be used to perform a rounding operation (this is done via :func:`~.float_round`), but only to produce a suitable string representation for a float. :param float value: :param int precision_digits: number of fractional digits to include in the output """ # Can't use str() here because it seems to have an intrinsic # rounding to 12 significant digits, which causes a loss of # precision. e.g. str(123456789.1234) == str(123456789.123)!! return "%.*f" % (precision_digits, value) def float_split_str(value, precision_digits): """Splits the given float 'value' in its unitary and decimal parts, returning each of them as a string, rounding the value using the provided ``precision_digits`` argument. The length of the string returned for decimal places will always be equal to ``precision_digits``, adding zeros at the end if needed. In case ``precision_digits`` is zero, an empty string is returned for the decimal places. Examples: 1.432 with precision 2 => ('1', '43') 1.49 with precision 1 => ('1', '5') 1.1 with precision 3 => ('1', '100') 1.12 with precision 0 => ('1', '') :param float value: value to split. :param int precision_digits: number of fractional digits to round to. :return: returns the tuple(, ) of the given value :rtype: tuple(str, str) """ value = float_round(value, precision_digits=precision_digits) value_repr = float_repr(value, precision_digits) return tuple(value_repr.split('.')) if precision_digits else (value_repr, '') def float_split(value, precision_digits): """ same as float_split_str() except that it returns the unitary and decimal parts as integers instead of strings. In case ``precision_digits`` is zero, 0 is always returned as decimal part. :rtype: tuple(int, int) """ units, cents = float_split_str(value, precision_digits) if not cents: return int(units), 0 return int(units), int(cents) def json_float_round(value, precision_digits, rounding_method='HALF-UP'): """Not suitable for float calculations! Similar to float_repr except that it returns a float suitable for json dump This may be necessary to produce "exact" representations of rounded float values during serialization, such as what is done by `json.dumps()`. Unfortunately `json.dumps` does not allow any form of custom float representation, nor any custom types, everything is serialized from the basic JSON types. :param int precision_digits: number of fractional digits to round to. :param rounding_method: the rounding method used: 'HALF-UP', 'UP' or 'DOWN', the first one rounding up to the closest number with the rule that number>=0.5 is rounded up to 1, the second always rounding up and the latest one always rounding down. :return: a rounded float value that must not be used for calculations, but is ready to be serialized in JSON with minimal chances of representation errors. """ rounded_value = float_round(value, precision_digits=precision_digits, rounding_method=rounding_method) rounded_repr = float_repr(rounded_value, precision_digits=precision_digits) # As of Python 3.1, rounded_repr should be the shortest representation for our # rounded float, so we create a new float whose repr is expected # to be the same value, or a value that is semantically identical # and will be used in the json serialization. # e.g. if rounded_repr is '3.1750', the new float repr could be 3.175 # but not 3.174999999999322452. # Cfr. bpo-1580: https://bugs.python.org/issue1580 return float(rounded_repr) _INVERTDICT = { 1e-1: 1e+1, 1e-2: 1e+2, 1e-3: 1e+3, 1e-4: 1e+4, 1e-5: 1e+5, 1e-6: 1e+6, 1e-7: 1e+7, 1e-8: 1e+8, 1e-9: 1e+9, 1e-10: 1e+10, 2e-1: 5e+0, 2e-2: 5e+1, 2e-3: 5e+2, 2e-4: 5e+3, 2e-5: 5e+4, 2e-6: 5e+5, 2e-7: 5e+6, 2e-8: 5e+7, 2e-9: 5e+8, 2e-10: 5e+9, 5e-1: 2e+0, 5e-2: 2e+1, 5e-3: 2e+2, 5e-4: 2e+3, 5e-5: 2e+4, 5e-6: 2e+5, 5e-7: 2e+6, 5e-8: 2e+7, 5e-9: 2e+8, 5e-10: 2e+9, } def float_invert(value): """Inverts a floating point number with increased accuracy. :param float value: value to invert. :param bool store: whether store the result in memory for future calls. :return: rounded float. """ result = _INVERTDICT.get(value) if result is None: coefficient, exponent = f'{value:.15e}'.split('e') # invert exponent by changing sign, and coefficient by dividing by its square result = float(f'{coefficient}e{-int(exponent)}') / float(coefficient)**2 return result if __name__ == "__main__": import time start = time.time() count = 0 def try_round(amount, expected, precision_digits=3): result = float_repr(float_round(amount, precision_digits=precision_digits), precision_digits=precision_digits) if result != expected: print('###!!! Rounding error: got %s , expected %s' % (result, expected)) return complex(1, 1) return 1 # Extended float range test, inspired by Cloves Almeida's test on bug #882036. fractions = [.0, .015, .01499, .675, .67499, .4555, .4555, .45555] expecteds = ['.00', '.02', '.01', '.68', '.67', '.46', '.456', '.4556'] precisions = [2, 2, 2, 2, 2, 2, 3, 4] for magnitude in range(7): for frac, exp, prec in zip(fractions, expecteds, precisions): for sign in [-1, 1]: for x in range(0, 10000, 97): n = x * 10**magnitude f = sign * (n + frac) f_exp = ('-' if f != 0 and sign == -1 else '') + str(n) + exp count += try_round(f, f_exp, precision_digits=prec) stop = time.time() count, errors = int(count.real), int(count.imag) # Micro-bench results: # 47130 round calls in 0.422306060791 secs, with Python 2.6.7 on Core i3 x64 # with decimal: # 47130 round calls in 6.612248100021 secs, with Python 2.6.7 on Core i3 x64 print(count, " round calls, ", errors, "errors, done in ", (stop-start), 'secs')