python3 <<'PY'
import math
a=0.25; b=1.0/12
def G(t,y3):
    y1=3*t-2.0/3
    if y1<-1e-14 or y1>b+1e-14 or y3<-1e-14 or y3>a+1e-14: return -1
    al=2*a+2*b-y1-t
    mu1=a+b-y1; mu2=b+a-t; nu1=a+b
    r1=mu1/al; r2=mu2/al; s1=nu1/al; s3=1-s1
    A=al*(math.sqrt(max(0,a*r1*s1))+math.sqrt(max(0,b*r1*s3))+math.sqrt(max(0,b*r2*s1))+math.sqrt(max(0,a*r2*s3)))
    beta=a+b-y3
    B=math.sqrt(beta)*(b+math.sqrt(max(0,a*(a-y3))))
    gam=y1+t+y3
    C=math.sqrt(gam)*(math.sqrt(max(0,b*y1))+math.sqrt(max(0,a*t))+math.sqrt(max(0,a*y3)))
    return A+B+C
def opty3(t):
    lo=0.0; hi=a
    gr=(math.sqrt(5)-1)/2
    for _ in range(100):
      c=hi-gr*(hi-lo); d=lo+gr*(hi-lo)
      if G(t,c)<G(t,d): lo=c
      else: hi=d
    return (lo+hi)/2
best=-1; arg=None
N=100000
tlo=2.0/9; thi=0.25
for i in range(N+1):
  t=tlo+(thi-tlo)*i/N
  y3=opty3(t) if i%200==0 else None
  if y3 is None: continue
  g=G(t,y3)
  if g>best: best=g; arg=(t,y3)
print("rough",best,arg)
# refine around arg using ternary on t with inner y3 opt
center=arg[0]; w=0.002
lo=max(tlo,center-w); hi=min(thi,center+w)
for _ in range(80):
  m1=lo+(hi-lo)/3; m2=hi-(hi-lo)/3
  if G(m1,opty3(m1))<G(m2,opty3(m2)): lo=m1
  else: hi=m2
t=(lo+hi)/2; y3=opty3(t)
print("refined",t,y3,G(t,y3),"y1",3*t-2/3,"cost",1-G(t,y3))
PY