def getNthNonagonalSquareNumberOperator(n):
p = fsum([fmul(8, sqrt(7)), fmul(9, sqrt(14)), fmul(-7, sqrt(2)), -28])
q = fsum([fmul(7, sqrt(2)), fmul(9, sqrt(14)), fmul(-8, sqrt(7)), -28])
sign = power(-1, n)
index = fdiv(
fsub(
fmul(fadd(p, fmul(q, sign)),
power(fadd(fmul(2, sqrt(2)), sqrt(7)), n)),
fmul(fsub(p, fmul(q, sign)),
power(fsub(fmul(2, sqrt(2)), sqrt(7)), fsub(n, 1)))), 112)
return nint(power(nint(index), 2))
def getNthNonagonalSquareNumber( n ):
if real( n ) < 0:
ValueError( '' )
p = fsum( [ fmul( 8, sqrt( 7 ) ), fmul( 9, sqrt( 14 ) ), fmul( -7, sqrt( 2 ) ), -28 ] )
q = fsum( [ fmul( 7, sqrt( 2 ) ), fmul( 9, sqrt( 14 ) ), fmul( -8, sqrt( 7 ) ), -28 ] )
sign = power( -1, real_int( n ) )
index = fdiv( fsub( fmul( fadd( p, fmul( q, sign ) ),
power( fadd( fmul( 2, sqrt( 2 ) ), sqrt( 7 ) ), n ) ),
fmul( fsub( p, fmul( q, sign ) ),
power( fsub( fmul( 2, sqrt( 2 ) ), sqrt( 7 ) ), fsub( n, 1 ) ) ) ), 112 )
return nint( power( nint( index ), 2 ) )
def getNthDecagonalCenteredSquareNumberOperator(n):
sqrt10 = sqrt(10)
dps = 7 * 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 _view_cone_calc(lat_geoc, lon_geoc, sat_pos, sat_vel, q_max, m):
"""Semi-private: Performs the viewing cone visibility calculation for the day defined by m.
Note: This function is based on a paper titled "rapid satellite-to-site visibility determination
based on self-adaptive interpolation technique" with some variation to account for interaction
of viewing cone with the satellite orbit.
Args:
lat_geoc (float): site location in degrees at the start of POI
lon_geoc (float): site location in degrees at the start of POI
sat_pos (Vector3D): position of satellite (at the same time as sat_vel)
sat_vel (Vector3D): velocity of satellite (at the same time as sat_pos)
q_max (float): maximum orbital radius
m (int): interval offsets (number of days after initial condition)
Returns:
Returns 4 numbers representing times at which the orbit is tangent to the viewing cone,
Raises:
ValueError: if any of the 4 formulas has a complex answer. This happens when the orbit and
viewing cone do not intersect or only intersect twice.
Note: With more analysis it should be possible to find a correct interval even in the case
where there are only two intersections but this is beyond the current scope of the project.
"""
lat_geoc = (lat_geoc * mp.pi) / 180
lon_geoc = (lon_geoc * mp.pi) / 180
# P vector (also referred to as orbital angular momentum in the paper) calculations
p_unit_x, p_unit_y, p_unit_z = cross(sat_pos, sat_vel) / (mp.norm(sat_pos) * mp.norm(sat_vel))
# Following are equations from Viewing cone section of referenced paper
r_site_magnitude = earth_radius_at_geocetric_lat(lat_geoc)
gamma1 = THETA_NAUGHT + mp.asin((r_site_magnitude * mp.sin((mp.pi / 2) + THETA_NAUGHT)) / q_max)
gamma2 = mp.pi - gamma1
# Note: atan2 instead of atan to get the correct quadrant.
arctan_term = mp.atan2(p_unit_x, p_unit_y)
arcsin_term_gamma, arcsin_term_gamma2 = [(mp.asin((mp.cos(gamma) - p_unit_z * mp.sin(lat_geoc))
/ (mp.sqrt((p_unit_x ** 2) + (p_unit_y ** 2)) *
mp.cos(lat_geoc)))) for gamma in [gamma1, gamma2]]
angle_1 = (arcsin_term_gamma - lon_geoc - arctan_term + 2 * mp.pi * m)
angle_2 = (mp.pi - arcsin_term_gamma - lon_geoc - arctan_term + 2 * mp.pi * m)
angle_3 = (arcsin_term_gamma2 - lon_geoc - arctan_term + 2 * mp.pi * m)
angle_4 = (mp.pi - arcsin_term_gamma2 - lon_geoc - arctan_term + 2 * mp.pi * m)
angles = [angle_1, angle_2, angle_3, angle_4]
# Check for complex answers
if any([not isinstance(angle, mp.mpf) for angle in angles]):
raise ValueError()
# Map all angles to 0 to 2*pi
for idx in range(len(angles)):
while angles[idx] < 0:
angles[idx] += 2 * mp.pi
while angles[idx] > 2 * mp.pi:
angles[idx] -= 2 * mp.pi
# Calculate the corresponding time for each angle and return
return [mp.nint((1 / ANGULAR_VELOCITY_EARTH) * angle) for angle in angles]
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 getMPFIntegerAsString(n):
'''Turns an mpmath mpf integer value into a string for use by lexicographic
operators.'''
if n == 0:
return '0'
return nstr(nint(n), int(floor(log10(n) + 10)))[:-2]
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 getMPFIntegerAsString( n ):
'''Turns an mpmath mpf integer value into a string for use by lexicographic
operators.'''
if n == 0:
return '0'
else:
return nstr( nint( n ), int( floor( log10( n ) + 10 ) ) )[ : -2 ]
def getNthOctagonalHeptagonalNumberOperator(n):
return nint(
floor(
fdiv(
fmul(fadd(17, fmul(sqrt(30), 2)),
power(fadd(sqrt(5), sqrt(6)), fsub(fmul(8, n), 6))),
480)))
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 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 getNthOctagonalTriangularNumberOperator(n):
sign = power(-1, n)
return nint(
floor(
fdiv(
fmul(fsub(7, fprod([2, sqrt(6), sign])),
power(fadd(sqrt(3), sqrt(2)), fsub(fmul(4, n), 2))), 96)))
def duplicateOperation( valueList ):
if g.duplicateOperations > 0:
raise ValueError( "'dupop' must be followed by another operation" )
if isinstance( valueList[ -1 ], list ):
raise ValueError( "'dupop' cannot accept a list argument" )
g.duplicateOperations = nint( floor( valueList.pop( ) ) )
def getNthNonagonalHeptagonalNumberOperator(n):
sqrt35 = sqrt(35)
return nint(
floor(
fdiv(
fmul(fadd(39, fmul(4, sqrt35)),
power(fadd(6, sqrt35), fsub(fmul(4, n), 3))), 560)))
def quantize(c, shr, dtype=numpy.int64):
assert numpy.all(shr >= 1)
c = mpmath.matrix(
[[mpmath.ldexp(c[i, j] + .5, int(shr[j]))
for j in range(c.cols - 1)] + [c[i, c.cols - 1]]
for i in range(c.rows)])
return numpy.array([[int(mpmath.nint(c[i, j])) for j in range(c.cols)]
for i in range(c.rows)], dtype)
def getNthDecagonalTriangularNumberOperator(n):
return nint(
floor(
fdiv(
fmul(fadd(9, fprod([4, sqrt(2),
power(-1, fadd(n, 1))])),
power(fadd(1, sqrt(2)), fsub(fmul(4, fadd(n, 1)), 6))),
64)))
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 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 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 __init__(self, width, depth, height, xpoz, ypoz, zpoz, wElements, dElements, hElements):
self.smallBlocksStructure = []
self.xpoz = mp.mpmathify(xpoz)
self.ypoz = mp.mpmathify(ypoz)
self.zpoz = mp.mpmathify(zpoz)
self.width = mp.mpmathify(width)
self.depth = mp.mpmathify(depth)
self.height = mp.mpmathify(height)
self.wElements = mp.mpmathify(mp.nint(wElements))
self.dElements = mp.mpmathify(mp.nint(dElements))
self.hElements = mp.mpmathify(mp.nint(hElements))
self.widthSmall = mp.mpmathify(self.width / self.wElements)
self.depthSmall = mp.mpmathify(self.depth / self.dElements)
self.heightSmall = mp.mpmathify(self.height / self.hElements)
self.nElements = mp.nint((self.width * self.depth * self.height) / (self.widthSmall * self.depthSmall * self.heightSmall))
def getNthNonagonalOctagonalNumberOperator(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, n), 5))), 672)))
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 getNthDecagonalHeptagonalNumberOperator(n):
sqrt10 = sqrt(10)
return nint(
floor(
fdiv(
fprod([
fsub(11, fmul(fmul(2, sqrt10), power(-1, n))),
fadd(1, sqrt10),
power(fadd(3, sqrt10), fsub(fmul(4, n), 3))
]), 320)))
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 drawbox():
global a, listindex, centerx, centery, pixdict, z, red, green, blue, canv, verbose
if verbose:
print('Outputting: ' + mpmath.nstr(mpmath.chop(a[listindex])))
if mpmath.isnan(a[listindex]) or mpmath.isinf(a[listindex]):
return
try:
x = int(mpmath.nint(mpmath.re(a[listindex]))) + centerx
y = int(mpmath.nint(mpmath.im(a[listindex]))) + centery
if (x, y) in pixdict:
canv.delete(pixdict[(x, y)])
del pixdict[(x, y)]
pixdict[(x, y)] = canv.create_rectangle(x * z - z + 1,
y * z - z + 1,
x * z + 1,
y * z + 1,
fill=rgb(red, green, blue),
width=0)
except OverflowError:
#the number is so huge it's not going to be displayed anyway, so just suppress the error and move on
pass
def duplicateOperation( valueList ):
if g.duplicateOperations > 0:
raise ValueError( "'duplicate_operator' must be followed by another operation" )
if isinstance( valueList[ -1 ], list ):
raise ValueError( "'duplicate_operator' cannot accept a list argument" )
argument = nint( floor( valueList.pop( ) ) )
if argument < 1:
raise ValueError( "'duplicate_operator' requires an argument of 1 or more (n.b., 1 has no effect)" )
g.duplicateOperations = argument
def getNthSquareTriangularNumberOperator(n):
neededPrecision = int(n * 3.5) # determined by experimentation
if mp.dps < neededPrecision:
mp.dps = 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 getNthNonagonalPentagonalNumberOperator(n):
sqrt21 = sqrt(21)
sign = power(-1, 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 __distance(self, a, b, type):
if a == b:
return 0
xd = a[0] - b[0]
yd = a[1] - b[1]
dij = None
if type == "EUC_2D":
"""
xd= x[i]-x[j];
yd= y[i]-y[j];
dij= nint( sqrt( xd*xd + yd*yd));
"""
dij = nint(math.sqrt(pow(xd, 2) + (pow(yd, 2))))
else:
if type == "ATT":
"""
xd= x[i]-x[j];
yd= y[i]-y[j];
rij= sqrt( (xd*xd + yd*yd)/10.0 );
tij= nint( rij );
if(tij<rij)
dij= tij+1;
else
dij= tij;
"""
rij = math.sqrt((pow(xd, 2) + (pow(yd, 2))) / 10.0)
tij = nint(rij)
if (tij < rij):
dij = tij + 1
else:
dij = tij
return dij
def getNthDecagonalNonagonalNumberOperator(n):
dps = 8 * 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 mpIntDigits(num):
if not mpmath.almosteq(num,0):
a = mpmath.log(abs(num), b=10)
b = mpmath.nint(a)
if mpmath.almosteq(a,b):
return int(b)+1
else:
c = mpmath.ceil(a)
try:
return int(c)
except:
pass
else:
return 0
def mpIntDigits(num):
if not mpmath.almosteq(num, 0):
a = mpmath.log(abs(num), b=10)
b = mpmath.nint(a)
if mpmath.almosteq(a, b):
return int(b) + 1
else:
c = mpmath.ceil(a)
try:
return int(c)
except:
pass
else:
return 0
def lst_to_nb(el, exact=False):
#Devolve o logaritmo do numero com os expoentes na lista el
if exact:
res = 1
for i in range(len(el)):
res *= pow(nint(exp(primes[i])), el[i])
else:
#print(el)
res = 0
for i in range(len(el)):
#print(primes[i], el[i], res)
res += primes[i] * el[i]
return res
def global_to_map(self, global_coordinates):
"""
Maps global coordinates to map coordinates. Mainly used to tell which
voxel the mouse pointer is on. Implementation based on article:
http://www.alcove-games.com/advanced-tutorials/isometric-tile-picking/
"""
(gx, gy) = global_coordinates
#depth info comes from the currently activated layer
mz = self._active_layer
#turn the global coordinates into a homogenous vector
g_coord = mp.matrix([[gx],
[gy + mz * self._voxel_dimensions[DEPTH]],
[1]])
#Apply the isometric unprojection matrix from common_util
g_coord = UNPROJECT * g_coord
#divide and round the coordinates to get integer indexes.
mx = int(mp.nint(g_coord[X]) / self._side_length)
my = int(mp.nint(g_coord[Y]) / self._side_length)
return (mx, my, mz)
def getNthPadovanNumber( arg ):
n = fadd( real( arg ), 4 )
a = root( fsub( fdiv( 27, 2 ), fdiv( fmul( 3, sqrt( 69 ) ), 2 ) ), 3 )
b = root( fdiv( fadd( 9, sqrt( 69 ) ), 2 ), 3 )
c = fadd( 1, fmul( mpc( 0, 1 ), sqrt( 3 ) ) )
d = fsub( 1, fmul( mpc( 0, 1 ), sqrt( 3 ) ) )
e = power( 3, fdiv( 2, 3 ) )
r = fadd( fdiv( a, 3 ), fdiv( b, e ) )
s = fsub( fmul( fdiv( d, -6 ), a ), fdiv( fmul( c, b ), fmul( 2, e ) ) )
t = fsub( fmul( fdiv( c, -6 ), a ), fdiv( fmul( d, b ), fmul( 2, e ) ) )
return nint( re( fsum( [ fdiv( power( r, n ), fadd( fmul( 2, r ), 3 ) ),
fdiv( power( s, n ), fadd( fmul( 2, s ), 3 ) ),
fdiv( power( t, n ), fadd( fmul( 2, t ), 3 ) ) ] ) ) )
def getCompositions( n, k ):
value = int( real_int( n ) )
count = int( real_int( k ) )
if count < 1:
raise ValueError( "'compositions' expects a size argument greater than 0'" )
if count == 1:
return [ [ value ] ]
else:
result = [ ]
for i in range( 1, int( ( value - count ) + 2 ) ):
result.extend( [ [ nint( i ) ] + comp for comp in getCompositions( n - i, count - 1 ) ] )
return result
def getCompositionsGenerator(n, k):
value = int(n)
count = int(k)
if count < 1:
raise ValueError(
"'compositions' expects a size argument greater than 0'")
if count == 1:
yield [value]
else:
#result = [ ]
for i in range(1, int((value - count) + 2)):
#result.extend( [ [ nint( i ) ] + comp for comp in getCompositions( n - i, count - 1 ) ] )
for comp in getCompositionsGenerator(n - i, count - 1):
yield [nint(i)] + comp
def convertToNonintegerBase(num, base):
epsilon = power(10, -(mp.dps - 3))
minPlace = -(floor(mp.dps / log(base)))
output = ''
integer = ''
remaining = num
# find starting place
place = int(floor(log(remaining, base)))
while remaining > epsilon:
if place < minPlace:
break
if place == -1:
integer = output
output = ''
placeValue = power(base, place)
value = fdiv(remaining, placeValue)
value = fmul(value, power(10, mp.dps - 3))
value = nint(value)
value = fdiv(value, power(10, mp.dps - 3))
value = floor(value)
remaining = chop(fsub(remaining, fmul(placeValue, value)))
output += str(value)[:-2]
place -= 1
if place >= 0:
integer = output + '0' * (place + 1)
output = ''
if integer == '':
return output, ''
return integer, output
def __init__(self, resource_handler, voxel_handler,
world_dimensions = (64, 64, 16),
voxel_dimensions = (72, 36, 36),
active_layer = 0,
visibility_flag = ONLY_SHOW_EXPOSED):
self._world_dimensions = world_dimensions
self._voxel_dimensions = voxel_dimensions
grid_length = world_dimensions[WIDTH] * world_dimensions[HEIGHT] * world_dimensions[DEPTH]
self._active_layer = active_layer
self.visibility_flag = visibility_flag
#The unprojected length of the horizontal side of each voxel
self._side_length = mp.nint(self._voxel_dimensions[WIDTH] * mp.sin(mp.pi / 4))
WorldBase.__init__(self, resource_handler, voxel_handler,
grid_length, (0, 0))
def convertToNonintegerBase( num, base ):
epsilon = power( 10, -( mp.dps - 3 ) )
minPlace = -( floor( mp.dps / log( base ) ) )
output = ''
integer = ''
remaining = num
# find starting place
place = int( floor( log( remaining, base ) ) )
while remaining > epsilon:
if place < minPlace:
break
if place == -1:
integer = output
output = ''
placeValue = power( base, place )
value = fdiv( remaining, placeValue )
value = fmul( value, power( 10, mp.dps - 3 ) )
value = nint( value )
value = fdiv( value, power( 10, mp.dps - 3 ) )
value = floor( value )
remaining = chop( fsub( remaining, fmul( placeValue, value ) ) )
output += str( value )[ : -2 ]
place -= 1
if place >= 0:
integer = output + '0' * ( place + 1 )
output = ''
if integer == '':
return output, ''
else:
return integer, output
def getNthHeptagonalPentagonalNumberOperator(n):
return nint(
floor(
fdiv(
fmul(power(fadd(2, sqrt(15)), 2),
power(fadd(4, sqrt(15)), fsub(fmul(4, n), 3))), 240)))
def _distance(num): return mp.fabs(mp.fsub(num, mp.nint(num)))
def getNearestInt( n ):
if isinstance( n, RPNMeasurement ):
return RPNMeasurement( nint( n.value ), n.units )
else:
return nint( n )
def _isInteger(num, tolerance=mp.power(10, -6)):
return mp.almosteq(num, mp.nint(num), tolerance)
def getNthHeptagonalHexagonalNumberOperator(n):
return nint(
floor(
fdiv(
fmul(fsub(sqrt(5), 1),
power(fadd(2, sqrt(5)), fsub(fmul(8, n), 5))), 80)))
def getNthDecagonalPentagonalNumber( n ):
return nint( floor( fmul( fdiv( 25, 192 ),
power( fadd( sqrt( 3 ), sqrt( 2 ) ),
fsub( fmul( 8, real_int( n ) ), 6 ) ) ) ) )
def getNthDecagonalOctagonalNumber( n ):
return nint( floor( fdiv( fmul( fadd( 13, fmul( 4, sqrt( 3 ) ) ),
power( fadd( 2, sqrt( 3 ) ),
fsub( fmul( 8, real_int( n ) ), 6 ) ) ), 192 ) ) )
def getNthHexagonalPentagonalNumberOperator(n):
return nint(
ceil(
fdiv(
fmul(fsub(sqrt(3), 1),
power(fadd(2, sqrt(3)), fsub(fmul(4, n), 2))), 12)))
def duplicateDigits( n, k ):
if real( k ) < 0:
raise ValueError( "'add_digits' requires a non-negative integer for the second argument" )
return appendDigits( real( n ), fmod( n, power( 10, nint( floor( k ) ) ) ), k )
def _distance(num):
return mp.fabs(mp.fsub(num, mp.nint(num)))
def getNthOctagonalSquareNumberOperator(n):
return nint(floor(fdiv(power(fadd(2, sqrt(3)), fsub(fmul(4, n), 2)), 12)))
def mpmathRoundToInt(z):
return int(mpmath.nint(z.real))
def getNthOctagonalPentagonalNumberOperator(n):
return nint(
floor(
fdiv(
fmul(fsub(11, fprod([6, sqrt(2), power(-1, n)])),
power(fadd(1, sqrt(2)), fsub(fmul(8, n), 6))), 96)))
def getNthDecagonalHexagonalNumberOperator(n):
return nint(
floor(
fdiv(
fmul(fsub(fmul(5, sqrt(2)), 1),
power(fadd(sqrt(2), 1), fsub(fmul(8, n), 5))), 64)))
def getNthDecagonalTriangularNumber( n ):
return nint( floor( fdiv( fmul( fadd( 9, fprod( [ 4, sqrt( 2 ),
power( -1, fadd( real_int( n ), 1 ) ) ] ) ),
power( fadd( 1, sqrt( 2 ) ), fsub( fmul( 4, fadd( n, 1 ) ), 6 ) ) ), 64 ) ) )
def getNthDecagonalHexagonalNumber( n ):
return nint( floor( fdiv( fmul( fsub( fmul( 5, sqrt( 2 ) ), 1 ),
power( fadd( sqrt( 2 ), 1 ),
fsub( fmul( 8, real_int( n ) ), 5 ) ) ), 64 ) ) )
def getNthDecagonalPentagonalNumberOperator(n):
return nint(
floor(
fmul(fdiv(25, 192),
power(fadd(sqrt(3), sqrt(2)), fsub(fmul(8, n), 6)))))
def getNthNonagonalHeptagonalNumber( n ):
sqrt35 = sqrt( 35 )
return nint( floor( fdiv( fmul( fadd( 39, fmul( 4, sqrt35 ) ),
power( fadd( 6, sqrt35 ), fsub( fmul( 4, real_int( n ) ), 3 ) ) ),
560 ) ) )
def appendDigits( n, digits, digitCount ):
if real( n ) < 0:
return nint( fsub( fmul( floor( n ), power( 10, digitCount ) ), digits ) )
else:
return nint( fadd( fmul( floor( n ), power( 10, digitCount ) ), digits ) )