I suppose it depends what you mean by 'direct evaluation'.
Jer proposed a puzzle some time ago which led to an approximation method for square roots, without working them out exactly: sqrty=xn/(2x1), where x^2 is the nearest square greater than y, and n=(xy) The larger the number, the more accurate the result, so we multiply everything by say 10: (10√5 + 10√7  10√12) (10√6+10√8  10√14), so the numbers are the square roots of 500, 600, etc..
Then
sqrt500 xn/(2x1), x=23, n=29; 22.3556
sqrt600 xn/(2x1), x=25, n=25; 24.4898
sqrt700 xn/(2x1), x=27, n=29; 26.4528
sqrt800 xn/(2x1), x=29, n=41; 28.2807
sqrt1200 xn/(2x1), x=35, n=25; 34.6377
sqrt1400 xn/(2x1), x=38, n=44; 37.4133
(22.3556+24.489826.4528)(28.2807+34.637737.4133)=5.1125, so the second expression is greater.
If I was on a desert island with a hundred cannibal mathematicians and no pocket calculator, I'd use a bigger multiplier.
Edited on February 23, 2018, 11:56 pm

Posted by broll
on 20180223 23:39:30 