  Squeezed light enables measurements with sensitivity beyond the quantum noise
limit (QNL) for optical techniques such as spectroscopy, gravitational wave
detection, magnetometry and imaging. Precision of a measurement -- as
quantified by the variance of repeated estimates -- has also been enhanced
beyond the QNL using squeezed light. However, sub-QNL sensitivity is not
sufficient to achieve sub-QNL precision. Furthermore, demonstrations of sub-QNL
precision in estimating transmission have been limited to picowatts of probe
power. Here we demonstrate simultaneous enhancement of precision and
sensitivity to beyond the QNL for estimating modulated transmission with a
squeezed amplitude probe of 0.2 mW average (25 W peak) power, which is 8 orders
of magnitude above the power limitations of previous sub-QNL precision
measurements of transmission. Our approach enables measurements that compete
with the optical powers of current classical techniques, but have both improved
precision and sensitivity beyond the classical limit.
