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956 bytes, as of the version from 2026-11-10 01:14 (current). Recorded here, not run.
import numberdb.sage as numberdb
from sage.rings.real_arb import RealBallField
from sage.rings.complex_arb import ComplexBallField
R=RealBallField(470);C=ComplexBallField(470)
q=(-2*R.pi()).exp(); k=4*(1+sum(2*q**(n*n) for n in range(1,13)))**2/(1+sum(2*(-q)**(n*n) for n in range(1,13)))**2
def native(t,analytic):return ((C(k)+2*(2*C.pi()*t).cos())/2).arccosh(analytic=analytic)
def composed(t,analytic):
z=(C(k)+2*(2*C.pi()*t).cos())/2
return (z+(z-1).sqrt(analytic=analytic)*(z+1).sqrt(analytic=analytic)).log(analytic=analytic)
def safe(t,analytic):
z=(C(k)+2*(2*C.pi()*t).cos())/2
if analytic and not z.real()>1:return C('nan')
return z.arccosh()
for name,f in [('native',native),('composed',composed),('safe',safe)]:
print(name, C.integral(f,0,1),flush=True)
print('reciprocal control',C.integral(lambda t,a:1/(C(k)/2+(2*C.pi()*t).cos()),0,1)-1/(k*k/4-1).sqrt())
print('point function difference',native(C(1)/3,False)-composed(C(1)/3,False))