from fractions import Fraction
import math

# EXACT e via sum 1/k! to many terms -> a precise rational.
def e_rational(terms=400):
    s=Fraction(0); f=1
    for k in range(terms):
        if k>0: f*=k
        s+=Fraction(1,f)
    return s

def cf_of(x, n):
    out=[]
    for _ in range(n):
        a=x.numerator//x.denominator
        out.append(a)
        frac=x-a
        if frac==0: break
        x=1/frac
    return out

# float version (the lying instrument)
def cf_float(x,n):
    out=[]
    for _ in range(n):
        a=math.floor(x); out.append(a)
        if x-a<1e-15: break
        x=1/(x-a)
    return out

E=e_rational(400)
exact=cf_of(E,50)
flt=cf_float(math.e,50)

print("EXACT cf(e), 50 terms:")
print(exact)
print("\nFLOAT cf(e), 50 terms (watch it derail):")
print(flt)

# TEST the committed pattern against the EXACT terms
def predicted(i):
    if i==0: return 2
    if i%3==2: return 2*(i+1)//3
    return 1
ok=True; firstbad=None
for i,a in enumerate(exact):
    if a!=predicted(i):
        ok=False; firstbad=(i,a,predicted(i)); break
print("\nPattern holds on all %d exact terms: %s"%(len(exact),ok))
if firstbad: print("first deviation:", firstbad)

# where does the FLOAT instrument first lie vs exact?
for i in range(min(len(exact),len(flt))):
    if exact[i]!=flt[i]:
        print("float first diverges from truth at term %d: float=%s exact=%s"%(i,flt[i],exact[i]))
        break