3

```
# approximate sin and cos using continued fraction expansion
# and module decimal to set higher precision
# convergence cutoff = 5 gives module math precision
# tested with Python27 by vegaseat
import decimal as dc
# for comparison
import math
try:
from sympy.mpmath import *
# set precision
mp.dps = 60
print(type(sin(0.5)))
print(sin(0.5))
print(cos(0.5))
print('-'*62)
except:
print("Install module SymPy from http://code.google.com/p/sympy")
def cont_fraction(r, cutoff=5):
"""
continued fraction expansion
higher convergence cutoff gives higher precision
but of course makes the approximation slower
"""
t = dc.Decimal(4)*cutoff + 2
for k in range(cutoff, 0, -1):
t = 4*k - 2 + dc.Decimal(r)/t
return t
def get_sin(x, cutoff=5):
"""returns an approximation of math.sin(x)"""
r = dc.Decimal(-x)*dc.Decimal(x)
s = cont_fraction(r, cutoff)
return 2*dc.Decimal(x)*s/(s*s - r)
def get_cos(x, cutoff=5):
"""returns an approximation of math.cos(x)"""
r = dc.Decimal(-x)*dc.Decimal(x)
s = cont_fraction(r, cutoff)
return (s*s + r)/(s*s - r)
# set decimal precision
dc.getcontext().prec = 60
# x is angle in radians
x = 0.5
print(type(get_sin(x)))
print(get_sin(x, 20))
print(math.sin(x))
# and some cosine approximations different cutoff values
print('-'*62)
print(get_cos(x, 15))
print(get_cos(x, 20))
print(get_cos(x, 25))
print(math.cos(x))
'''
<class 'sympy.mpmath.ctx_mp_python.mpf'>
0.479425538604203000273287935215571388081803367940600675188617
0.877582561890372716116281582603829651991645197109744052997611
--------------------------------------------------------------
<class 'decimal.Decimal'>
0.479425538604203000273287935215571388081803367940600675188618
0.479425538604
--------------------------------------------------------------
0.877582561890372716116281582603829651991645197109744053008284
0.877582561890372716116281582603829651991645197109744052997610
0.877582561890372716116281582603829651991645197109744052997610
0.87758256189
'''
```

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