def OLDgetPartitionNumber( n ):
if n < 0:
return 0
if n < 2:
return 1
result = mpmathify( 0 )
for k in arange( 1, n + 1 ):
#n1 = n - k * ( 3 * k - 1 ) / 2
n1 = fsub( n, fdiv( fmul( k, fsub( fmul( 3, k ), 1 ) ), 2 ) )
#n2 = n - k * ( 3 * k + 1 ) / 2
n2 = fsub( n, fdiv( fmul( k, fadd( fmul( 3, k ), 1 ) ), 2 ) )
result = fadd( result, fmul( power( -1, fadd( k, 1 ) ), fadd( getPartitionNumber( n1 ), getPartitionNumber( n2 ) ) ) )
if n1 <= 0:
break
#old = NOT_QUITE_AS_OLDgetPartitionNumber( n )
#
#if ( old != result ):
# raise ValueError( "It's broke." )
return result
def generateRealArgument( range = [ 0, 10 ], allowNegative = True ):
factor = 1
if allowNegative and getRandomInteger( 2 ) == 1:
factor = -1
return nstr( fmul( exp( fmul( getRandomNumber( ), random.uniform( *range ) ) ), factor ) )
def getRobbinsConstant( ):
robbins = fsub( fsub( fadd( 4, fmul( 17, sqrt( 2 ) ) ), fmul( 6, sqrt( 3 ) ) ), fmul( 7, pi ) )
robbins = fdiv( robbins, 105 )
robbins = fadd( robbins, fdiv( log( fadd( 1, sqrt( 2 ) ) ), 5 ) )
robbins = fadd( robbins, fdiv( fmul( 2, log( fadd( 2, sqrt( 3 ) ) ) ), 5 ) )
return robbins
def getNthNonagonalOctagonalNumber( n ):
sqrt6 = sqrt( 6 )
sqrt7 = sqrt( 7 )
return nint( floor( fdiv( fmul( fsub( fmul( 11, sqrt7 ), fmul( 9, sqrt6 ) ),
power( fadd( sqrt6, sqrt7 ), fsub( fmul( 8, real_int( n ) ), 5 ) ) ),
672 ) ) )
def calculateWindChill( measurement1, measurement2 ):
validUnitTypes = [
[ 'velocity', 'temperature' ],
]
arguments = matchUnitTypes( [ measurement1, measurement2 ], validUnitTypes )
if not arguments:
raise ValueError( '\'wind_chill\' requires velocity and temperature measurements' )
wind_speed = arguments[ 'velocity' ].convert( 'miles/hour' ).value
temperature = arguments[ 'temperature' ].convert( 'degrees_F' ).value
if wind_speed < 3:
raise ValueError( '\'wind_chill\' is not defined for wind speeds less than 3 mph' )
if temperature > 50:
raise ValueError( '\'wind_chill\' is not defined for temperatures over 50 degrees fahrenheit' )
result = fsum( [ 35.74, fmul( temperature, 0.6215 ), fneg( fmul( 35.75, power( wind_speed, 0.16 ) ) ),
fprod( [ 0.4275, temperature, power( wind_speed, 0.16 ) ] ) ] )
# in case someone puts in a silly velocity
if result < -459.67:
result = -459.67
return RPNMeasurement( result, 'degrees_F' ).convert( arguments[ 'temperature' ].units )
def getNSphereRadius( n, k ):
if real_int( k ) < 3:
raise ValueError( 'the number of dimensions must be at least 3' )
if not isinstance( n, RPNMeasurement ):
return RPNMeasurement( n, 'meter' )
dimensions = n.getDimensions( )
if dimensions == { 'length' : 1 }:
return n
elif dimensions == { 'length' : int( k - 1 ) }:
m2 = n.convertValue( RPNMeasurement( 1, [ { 'meter' : int( k - 1 ) } ] ) )
result = root( fdiv( fmul( m2, gamma( fdiv( k, 2 ) ) ),
fmul( 2, power( pi, fdiv( k, 2 ) ) ) ), fsub( k, 1 ) )
return RPNMeasurement( result, [ { 'meter' : 1 } ] )
elif dimensions == { 'length' : int( k ) }:
m3 = n.convertValue( RPNMeasurement( 1, [ { 'meter' : int( k ) } ] ) )
result = root( fmul( fdiv( gamma( fadd( fdiv( k, 2 ), 1 ) ),
power( pi, fdiv( k, 2 ) ) ),
m3 ), k )
return RPNMeasurement( result, [ { 'meter' : 1 } ] )
else:
raise ValueError( 'incompatible measurement type for computing the radius: ' +
str( dimensions ) )
def getNthOctagonalTriangularNumber( n ):
sign = power( -1, real( n ) )
return nint( floor( fdiv( fmul( fsub( 7, fprod( [ 2, sqrt( 6 ), sign ] ) ),
power( fadd( sqrt( 3 ), sqrt( 2 ) ),
fsub( fmul( 4, real_int( n ) ), 2 ) ) ),
96 ) ) )
def getInvertedBits( n ):
value = real_int( n )
# determine how many groups of bits we will be looking at
if value == 0:
groupings = 1
else:
groupings = int( fadd( floor( fdiv( ( log( value, 2 ) ), g.bitwiseGroupSize ) ), 1 ) )
placeValue = mpmathify( 1 << g.bitwiseGroupSize )
multiplier = mpmathify( 1 )
remaining = value
result = mpmathify( 0 )
for i in range( 0, groupings ):
# Let's let Python do the actual inverting
group = fmod( ~int( fmod( remaining, placeValue ) ), placeValue )
result += fmul( group, multiplier )
remaining = floor( fdiv( remaining, placeValue ) )
multiplier = fmul( multiplier, placeValue )
return result
def getSkyLocation( n, k ):
if not isinstance( n, ephem.Body ) or not isinstance( k, RPNDateTime ):
raise ValueError( '\'sky_location\' expects an astronomical object and a date-time' )
n.compute( k.to( 'utc' ).format( ) )
return [ fdiv( fmul( mpmathify( n.ra ), 180 ), pi ), fdiv( fmul( mpmathify( n.dec ), 180 ), pi ) ]
def findNthPolygonalNumber( n, k ):
if real_int( k ) < 3:
raise ValueError( 'the number of sides of the polygon cannot be less than 3,' )
return nint( fdiv( fsum( [ sqrt( fsum( [ power( k, 2 ), fprod( [ 8, k, real( n ) ] ),
fneg( fmul( 8, k ) ), fneg( fmul( 16, n ) ), 16 ] ) ),
k, -4 ] ), fmul( 2, fsub( k, 2 ) ) ) )
def getEclipseTotality( body1, body2, location, date ):
'''Returns the angular size of an astronomical object in radians.'''
if isinstance( location, str ):
location = getLocation( location )
if not isinstance( body1, RPNAstronomicalObject ) or not isinstance( body2, RPNAstronomicalObject ) and \
not isinstance( location, RPNLocation ) or not isinstance( date, RPNDateTime ):
raise ValueError( 'expected two astronomical objects, a location and a date-time' )
separation = body1.getAngularSeparation( body2, location, date ).value
radius1 = body1.getAngularSize( ).value
radius2 = body2.getAngularSize( ).value
if separation > fadd( radius1, radius2 ):
return 0
distance1 = body1.getDistanceFromEarth( date )
distance2 = body2.getDistanceFromEarth( date )
area1 = fmul( pi, power( radius1, 2 ) )
area2 = fmul( pi, power( radius2, 2 ) )
area_of_intersection = fadd( getCircleIntersectionTerm( radius1, radius2, separation ),
getCircleIntersectionTerm( radius2, radius1, separation ) )
if distance1 > distance2:
result = fdiv( area_of_intersection, area1 )
else:
result = fdiv( area_of_intersection, area2 )
if result > 1:
return 1
else:
return result
def getSkyLocation( n, k ):
'''Returns the location of an astronomical object in the sky in terms of right ascension and declination.'''
if not isinstance( n, ephem.Body ) or not isinstance( k, RPNDateTime ):
raise ValueError( '\'sky_location\' expects an astronomical object and a date-time' )
n.compute( k.to( 'utc' ).format( ) )
return [ fdiv( fmul( mpmathify( n.ra ), 180 ), pi ), fdiv( fmul( mpmathify( n.dec ), 180 ), pi ) ]
def getEulerPhi( n ):
if real( n ) < 2:
return n
if g.ecm:
return reduce( fmul, ( fmul( fsub( i[ 0 ], 1 ), power( i[ 0 ], fsub( i[ 1 ], 1 ) ) ) for i in getECMFactors( n ) ) )
else:
return reduce( fmul, ( fmul( fsub( i[ 0 ], 1 ), power( i[ 0 ], fsub( i[ 1 ], 1 ) ) ) for i in getFactors( n ) ) )
def getLocationInfo( location ):
if isinstance( location, str ):
location = getLocation( location )
elif not isinstance( location. RPNLocation ):
raise ValueError( 'location name or location object expected' )
return [ fdiv( fmul( mpmathify( location.observer.lat ), 180 ), pi ),
fdiv( fmul( mpmathify( location.observer.long ), 180 ), pi ) ]
def getNthMotzkinNumber( n ):
result = 0
for j in arange( 0, floor( fdiv( real( n ), 3 ) ) + 1 ):
result = fadd( result, fprod( [ power( -1, j ), binomial( fadd( n, 1 ), j ),
binomial( fsub( fmul( 2, n ), fmul( 3, j ) ), n ) ] ) )
return fdiv( result, fadd( n, 1 ) )
def get_prob_poisson(events, length, rate):
""" P(k, lambda = t * rate) = """
avg_events = mpmath.fmul(rate, length) # lambda
prob = mpmath.fmul((-1), avg_events)
for i in range(1, events + 1):
prob = mpmath.fadd(prob, mpmath.log(mpmath.fdiv(avg_events, i)))
prob = mpmath.exp(prob)
return prob
def findCenteredPolygonalNumber( n, k ):
if real_int( k ) < 3:
raise ValueError( 'the number of sides of the polygon cannot be less than 3,' )
s = fdiv( k, 2 )
return nint( fdiv( fadd( sqrt( s ),
sqrt( fsum( [ fmul( 4, real( n ) ), s, -4 ] ) ) ), fmul( 2, sqrt( s ) ) ) )
def getNthDecagonalNonagonalNumber( n ):
dps = 8 * int( real_int( n ) )
if mp.dps < dps:
mp.dps = dps
return nint( floor( fdiv( fmul( fadd( 15, fmul( 2, sqrt( 14 ) ) ),
power( fadd( fmul( 2, sqrt( 2 ) ), sqrt( 7 ) ),
fsub( fmul( 8, n ), 6 ) ) ), 448 ) ) )
def calculateOrbitalRadius( measurement1, measurement2 ):
'''
To solve the radius of a circular orbit, we need Newton's gravitational
constant and two of the following three items:
G = Newton's gravitational constant
m = planetary mass (i.e., mass of the thing being orbited)
T = orbital period
v = orbital velocity
---- radius in terms of period and mass
r = cbrt( T^2*G*m/4*pi^2 )
---- radius in terms of velocity and mass
r = G*m/v^2
---- radius in terms of velocity and period
r = v*T/2*pi
'''
validUnitTypes = [
[ 'mass', 'time' ],
[ 'velocity', 'time' ],
[ 'mass', 'velocity' ],
]
arguments = matchUnitTypes( [ measurement1, measurement2 ], validUnitTypes )
if not arguments:
raise ValueError( '\'orbital_radius\' requires specific measurement types (see help)' )
if 'mass' in arguments:
mass = arguments[ 'mass' ]
if 'time' in arguments:
bPeriod = True
period = arguments[ 'time' ]
else:
bPeriod = False
velocity = arguments[ 'velocity' ]
else:
# period and velocity
period = arguments[ 'time' ]
velocity = arguments[ 'velocity' ]
radius = divide( multiply( velocity, period ), fmul( 2, pi ) )
return radius.convert( 'meter' )
if bPeriod:
# period and mass
term = divide( getProduct( [ getPower( period, 2 ), getNewtonsConstant( ), mass ] ),
fmul( 4, power( pi, 2 ) ) )
radius = getRoot( term, 3 )
else:
# velocity and mass
radius = divide( multiply( getNewtonsConstant( ), mass ), getPower( velocity, 2 ) )
return radius.convert( 'meter' )
def getNthNonagonalPentagonalNumber( n ):
sqrt21 = sqrt( 21 )
sign = power( -1, real_int( n ) )
return nint( floor( fdiv( fprod( [ fadd( 25, fmul( 4, sqrt21 ) ),
fsub( 5, fmul( sqrt21, sign ) ),
power( fadd( fmul( 2, sqrt( 7 ) ), fmul( 3, sqrt( 3 ) ) ),
fsub( fmul( 4, n ), 4 ) ) ] ),
336 ) ) )
def getNthDecagonalHeptagonalNumber( n ):
sqrt10 = sqrt( 10 )
return nint( floor( fdiv( fprod( [ fsub( 11,
fmul( fmul( 2, sqrt10 ),
power( -1, real_int( n ) ) ) ),
fadd( 1, sqrt10 ),
power( fadd( 3, sqrt10 ),
fsub( fmul( 4, n ), 3 ) ) ] ), 320 ) ) )
def joinNumber( digits, base ):
place = 1
result = 0
for digit in digits:
result = fadd( result, fmul( digit, place ) )
place = fmul( place, base )
return result
def getNthMenageNumber( n ):
if n < 0:
raise ValueError( '\'menage\' requires a non-negative argument' )
elif n in [ 1, 2 ]:
return 0
elif n in [ 0, 3 ]:
return 1
else:
return nsum( lambda k: fdiv( fprod( [ power( -1, k ), fmul( 2, n ), binomial( fsub( fmul( 2, n ), k ), k ),
fac( fsub( n, k ) ) ] ), fsub( fmul( 2, n ), k ) ), [ 0, n ] )
def getNthNonagonalTriangularNumber( n ):
a = fmul( 3, sqrt( 7 ) )
b = fadd( 8, a )
c = fsub( 8, a )
return nint( fsum( [ fdiv( 5, 14 ),
fmul( fdiv( 9, 28 ), fadd( power( b, real_int( n ) ), power( c, n ) ) ),
fprod( [ fdiv( 3, 28 ),
sqrt( 7 ),
fsub( power( b, n ), power( c, n ) ) ] ) ] ) )
def getNthSquareTriangularNumber( n ):
neededPrecision = int( real_int( n ) * 3.5 ) # determined by experimentation
if mp.dps < neededPrecision:
setAccuracy( neededPrecision )
sqrt2 = sqrt( 2 )
return nint( power( fdiv( fsub( power( fadd( 1, sqrt2 ), fmul( 2, n ) ),
power( fsub( 1, sqrt2 ), fmul( 2, n ) ) ),
fmul( 4, sqrt2 ) ), 2 ) )
def getNthDecagonalCenteredSquareNumber( n ):
sqrt10 = sqrt( 10 )
dps = 7 * int( real_int( n ) )
if mp.dps < dps:
mp.dps = dps
return nint( floor( fsum( [ fdiv( 1, 8 ),
fmul( fdiv( 7, 16 ), power( fsub( 721, fmul( 228, sqrt10 ) ), fsub( n, 1 ) ) ),
fmul( fmul( fdiv( 1, 8 ), power( fsub( 721, fmul( 228, sqrt10 ) ), fsub( n, 1 ) ) ), sqrt10 ),
fmul( fmul( fdiv( 1, 8 ), power( fadd( 721, fmul( 228, sqrt10 ) ), fsub( n, 1 ) ) ), sqrt10 ),
fmul( fdiv( 7, 16 ), power( fadd( 721, fmul( 228, sqrt10 ) ), fsub( n, 1 ) ) ) ] ) ) )
def getNthMotzkinNumber( n ):
'''
http://oeis.org/A001006
a(n) = sum((-1)^j*binomial(n+1, j)*binomial(2n-3j, n), j=0..floor(n/3))/(n+1)
'''
result = 0
for j in arange( 0, floor( fdiv( real( n ), 3 ) ) + 1 ):
result = fadd( result, fprod( [ power( -1, j ), binomial( fadd( n, 1 ), j ),
binomial( fsub( fmul( 2, n ), fmul( 3, j ) ), n ) ] ) )
return fdiv( result, fadd( n, 1 ) )
def multiply( self, other ):
if isinstance( other, RPNMeasurement ):
newValue = fmul( self.value, other.value )
factor, newUnits = combineUnits( self.getUnits( ), other.getUnits( ) )
self = RPNMeasurement( fmul( newValue, factor ), newUnits )
else:
newValue = fmul( self.value, other )
self = RPNMeasurement( newValue, self.getUnits( ),
self.getUnitName( ), self.getPluralUnitName( ) )
return self.normalizeUnits( )
def makePythagoreanTriple( n, k ):
if real( n ) < 0 or real( k ) < 0:
raise ValueError( "'make_pyth_3' requires positive arguments" )
if n == k:
raise ValueError( "'make_pyth_3' requires unequal arguments" )
result = [ ]
result.append( fprod( [ 2, n, k ] ) )
result.append( fabs( fsub( fmul( n, n ), fmul( k, k ) ) ) )
result.append( fadd( fmul( n, n ), fmul( k, k ) ) )
return sorted( result )
def solveQuadraticPolynomial( a, b, c ):
if a == 0:
if b == 0:
raise ValueError( 'invalid expression, no variable coefficients' )
else:
# linear equation, one root
return [ fdiv( fneg( c ), b ) ]
else:
d = sqrt( fsub( power( b, 2 ), fmul( 4, fmul( a, c ) ) ) )
x1 = fdiv( fadd( fneg( b ), d ), fmul( 2, a ) )
x2 = fdiv( fsub( fneg( b ), d ), fmul( 2, a ) )
return [ x1, x2 ]
def single_term_func(ti, ia, c):
if c < sys.float_info.max:
return ti**(-ia - 1) * c
else:
return fmul(ti**(-ia - 1), c)
