/usr/share/lilypond/2.18.2/python/rational.py is in lilypond-data 2.18.2-4.1.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 | """Implementation of rational arithmetic."""
from __future__ import division
import math as _math
def _gcf(a, b):
"""Returns the greatest common factor of a and b."""
a = abs(a)
b = abs(b)
while b:
a, b = b, a % b
return a
class Rational(object):
"""
This class provides an exact representation of rational numbers.
All of the standard arithmetic operators are provided. In mixed-type
expressions, an int or a long can be converted to a Rational without
loss of precision, and will be done as such.
Rationals can be implicity (using binary operators) or explicity
(using float(x) or x.decimal()) converted to floats or Decimals;
this may cause a loss of precision. The reverse conversions can be
done without loss of precision, and are performed with the
from_exact_float and from_exact_decimal static methods. However,
because of rounding error in the original values, this tends to
produce "ugly" fractions. "Nicer" conversions to Rational can be made
with approx_smallest_denominator or approx_smallest_error.
"""
def __init__(self, numerator, denominator=1):
"""Contructs the Rational object for numerator/denominator."""
if not isinstance(numerator, (int, long)):
raise TypeError('numerator must have integer type')
if not isinstance(denominator, (int, long)):
raise TypeError('denominator must have integer type')
if not denominator:
raise ZeroDivisionError('rational construction')
self._d = denominator
self._n = numerator
self.normalize_self()
# Cancel the fraction to reduced form
def normalize_self(self):
factor = _gcf(self._n, self._d)
self._n = self._n // factor
self._d = self._d // factor
if self._d < 0:
self._n = -self._n
self._d = -self._d
def numerator(self):
return self._n
def denominator(self):
return self._d
def __repr__(self):
if self._d == 1:
return "Rational(%d)" % self._n
else:
return "Rational(%d, %d)" % (self._n, self._d)
def __str__(self):
if self._d == 1:
return str(self._n)
else:
return "%d/%d" % (self._n, self._d)
def __hash__(self):
try:
return hash(float(self))
except OverflowError:
return hash(long(self))
def __float__(self):
return self._n / self._d
def __int__(self):
if self._n < 0:
return -int(-self._n // self._d)
else:
return int(self._n // self._d)
def __long__(self):
return long(int(self))
def __nonzero__(self):
return bool(self._n)
def __pos__(self):
return self
def __neg__(self):
return Rational(-self._n, self._d)
def __abs__(self):
if self._n < 0:
return -self
else:
return self
def __add__(self, other):
if isinstance(other, Rational):
return Rational(self._n * other._d + self._d * other._n,
self._d * other._d)
elif isinstance(other, (int, long)):
return Rational(self._n + self._d * other, self._d)
elif isinstance(other, (float, complex)):
return float(self) + other
else:
return NotImplemented
__radd__ = __add__
def __sub__(self, other):
if isinstance(other, Rational):
return Rational(self._n * other._d - self._d * other._n,
self._d * other._d)
elif isinstance(other, (int, long)):
return Rational(self._n - self._d * other, self._d)
elif isinstance(other, (float, complex)):
return float(self) - other
else:
return NotImplemented
def __rsub__(self, other):
if isinstance(other, (int, long)):
return Rational(other * self._d - self._n, self._d)
elif isinstance(other, (float, complex)):
return other - float(self)
else:
return NotImplemented
def __mul__(self, other):
if isinstance(other, Rational):
return Rational(self._n * other._n, self._d * other._d)
elif isinstance(other, (int, long)):
return Rational(self._n * other, self._d)
elif isinstance(other, (float, complex)):
return float(self) * other
else:
return NotImplemented
__rmul__ = __mul__
def __truediv__(self, other):
if isinstance(other, Rational):
return Rational(self._n * other._d, self._d * other._n)
elif isinstance(other, (int, long)):
return Rational(self._n, self._d * other)
elif isinstance(other, (float, complex)):
return float(self) / other
else:
return NotImplemented
__div__ = __truediv__
def __rtruediv__(self, other):
if isinstance(other, (int, long)):
return Rational(other * self._d, self._n)
elif isinstance(other, (float, complex)):
return other / float(self)
else:
return NotImplemented
__rdiv__ = __rtruediv__
def __floordiv__(self, other):
truediv = self / other
if isinstance(truediv, Rational):
return truediv._n // truediv._d
else:
return truediv // 1
def __rfloordiv__(self, other):
return (other / self) // 1
def __mod__(self, other):
return self - self // other * other
def __rmod__(self, other):
return other - other // self * self
def __divmod__(self, other):
return self // other, self % other
def __cmp__(self, other):
if other == 0:
return cmp(self._n, 0)
else:
return cmp(self - other, 0)
def __pow__(self, other):
if isinstance(other, (int, long)):
if other < 0:
return Rational(self._d ** -other, self._n ** -other)
else:
return Rational(self._n ** other, self._d ** other)
else:
return float(self) ** other
def __rpow__(self, other):
return other ** float(self)
def round(self, denominator):
"""Return self rounded to nearest multiple of 1/denominator."""
int_part, frac_part = divmod(self * denominator, 1)
round_direction = cmp(frac_part * 2, 1)
if round_direction == 0:
numerator = int_part + (int_part & 1) # round to even
elif round_direction < 0:
numerator = int_part
else:
numerator = int_part + 1
return Rational(numerator, denominator)
def rational_from_exact_float(x):
"""Returns the exact Rational equivalent of x."""
mantissa, exponent = _math.frexp(x)
mantissa = int(mantissa * 2 ** 53)
exponent -= 53
if exponent < 0:
return Rational(mantissa, 2 ** (-exponent))
else:
return Rational(mantissa * 2 ** exponent)
def rational_approx_smallest_denominator(x, tolerance):
"""
Returns a Rational approximation of x.
Minimizes the denominator given a constraint on the error.
x = the float or Decimal value to convert
tolerance = maximum absolute error allowed,
must be of the same type as x
"""
tolerance = abs(tolerance)
n = 1
while True:
m = int(round(x * n))
result = Rational(m, n)
if abs(result - x) < tolerance:
return result
n += 1
def rational_approx_smallest_error(x, maxDenominator):
"""
Returns a Rational approximation of x.
Minimizes the error given a constraint on the denominator.
x = the float or Decimal value to convert
maxDenominator = maximum denominator allowed
"""
result = None
minError = x
for n in xrange(1, maxDenominator + 1):
m = int(round(x * n))
r = Rational(m, n)
error = abs(r - x)
if error == 0:
return r
elif error < minError:
result = r
minError = error
return result
def divide(x, y):
"""Same as x/y, but returns a Rational if both are ints."""
if isinstance(x, (int, long)) and isinstance(y, (int, long)):
return Rational(x, y)
else:
return x / y
|