sqrt.c running 
find sqrt(x=2) y_next = (x/y+y)/2
expect quadradic convergence 
i=0, y= 1.500000000000000, err=0.0857864 
i=1, y= 1.416666666666667, err=0.0024531 
i=2, y= 1.414215686274510, err=2.1239e-06 
i=3, y= 1.414213562374690, err=1.59472e-12 
i=4, y= 1.414213562373095, err=0 
i=5, y= 1.414213562373095, err=0 
i=6, y= 1.414213562373095, err=0 
i=7, y= 1.414213562373095, err=0 
i=8, y= 1.414213562373095, err=0 
find sqrt(x=1e+06) y_next = (x/y+y)/2
i=0, y=500000.500000000000000, err=499000 
i=1, y=250001.249998999992386, err=249001 
i=2, y=125002.624989500051015, err=124003 
i=3, y=62505.312410752128926, err=61505.3 
i=4, y=31260.655525445279636, err=30260.7 
i=5, y=15646.322308953220272, err=14646.3 
i=6, y=7855.117545897352102, err=6855.12 
i=7, y=3991.211544161647453, err=2991.21 
i=8, y=2120.881016068179179, err=1120.88 
i=9, y=1296.191592706878509, err=296.192 
i=10, y=1033.841239244203507, err=33.8412 
i=11, y=1000.553871053944704, err=0.553871 
i=12, y=1000.000153301662863, err=0.000153302 
i=13, y=1000.000000000011710, err=1.17097e-11 
i=14, y=1000.000000000000000, err=0 
i=15, y=1000.000000000000000, err=0 
i=16, y=1000.000000000000000, err=0 
i=17, y=1000.000000000000000, err=0 
i=18, y=1000.000000000000000, err=0 
i=19, y=1000.000000000000000, err=0 
find sqrt(x=1e+06) y_next = (x/y+y)/2
xp=x/2^2n, y=sqrt(xp)*2^n 
find sqrt(0.953674) 
i=0, y= 0.976837158203125, err=0.000274658 
i=1, y= 0.976562538612950, err=3.86129e-08 
i=2, y= 0.976562500000001, err=7.77156e-16 
i=3, y= 0.976562500000000, err=0 
i=4, y= 0.976562500000000, err=0 
i=5, y= 0.976562500000000, err=0 
i=6, y= 0.976562500000000, err=0 
i=7, y= 0.976562500000000, err=0 
i=8, y= 0.976562500000000, err=0 
y=y*2^10=1000.000000000000000, err=0