[gambit-list] Accuracy of sqrt of exact numbers in Gambit

Bradley Lucier lucier at math.purdue.edu
Sun Dec 1 13:59:58 EST 2013


_num.scm has some code to deal with computing the correct inexact real 
for (sqrt x) when x is an exact number and (sqrt x) is not an exact number.

The simplest thing to do would be (sqrt (exact->inexact x)), but I 
feared that the double rounding could lead to incorrect (but very close) 
answers.  (There are also issues of unnecessary overflows and underflows 
with that simple expression.)

So I ran some random tests to see (a) whether the Gambit code always 
gave correctly-rounded answers (it did), and (b) how often it happened 
that double rounding caused (sqrt (exact->inexact x)) to give incorrect 

In just 100 tests of (sqrt x) where x is a rational with numerator and 
denominator between 1 and 100, the tests found the following examples 
where (sqrt (exact->inexact x)) gives incorrect answers.  In each row 
the numbers are: x, (sqrt (exact->inexact x)), and correctly-rounded 
(sqrt x):

    (double-rounded-result-wrong 9/70 .35856858280031806 .3585685828003181)
    (double-rounded-result-wrong 33/49 .8206518066482897 .8206518066482898)
    (double-rounded-result-wrong 87/91 .977775002771065 .9777750027710649)
    (double-rounded-result-wrong 100/67 1.2216944435630523
    (double-rounded-result-wrong 77/18 2.068278940998476 2.0682789409984763)
    (double-rounded-result-wrong 3/7 .6546536707079771 .6546536707079772)
    (double-rounded-result-wrong 48/7 2.6186146828319083 2.618614682831909)
    (double-rounded-result-wrong 20/51 .6262242910851494 .6262242910851495)
    (double-rounded-result-wrong 25/11 1.5075567228888183 1.507556722888818)
    (double-rounded-result-wrong 38/59 .8025383458814721 .8025383458814722)
    (double-rounded-result-wrong 3/7 .6546536707079771 .6546536707079772)
    (double-rounded-result-wrong 25/11 1.5075567228888183 1.507556722888818)

So even for something as simple as (sqrt 3/7) the formula (sqrt 
(exact->inexact 3/7)) gives an  unnecessarily inaccurate answer.

And for integers we have the example

    (double-rounded-result-wrong 5680341405563297078633
    7.536803968237e10 7.536803968236998e10)

Pretty cool.  Maybe not so important, but still cool!

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