http://people.csail.mit.edu/jaffer/III/FloatingPoint | |
Why is 7 * 0.1 ≠ 0.7 ? |
Why is (* .1 7) printed as 0.7000000000000001, rather than as 0.7?Floating-point numbers in most Scheme implementations are [IEEE] double precision consisting of a 53 bit mantissa multiplied by a power of two. 0.1 cannot be exactly represented in this format; the closest double precision number is:
(/ #x1999999999999a (expt 2 56)) (- .1 (/ #x19999999999999 (expt 2 56))) ==> 13.877787807814456e-18 (- .1 (/ #x1999999999999a (expt 2 56))) ==> 0.0 (- .1 (/ #x1999999999999b (expt 2 56))) ==> -13.877787807814456e-18Multiplying (/ #x1999999999999a (expt 2 56)) by 7 yields:
(/ #xb3333333333336 (expt 2 56))The closest (normalized) IEEE double is:
(/ #x16666666666667 (expt 2 53))This number is too large to be correctly written as 0.7:
(- .7 (/ #x16666666666667 (expt 2 53))) ==> -111.02230246251565e-18 (- .7 (/ #x16666666666666 (expt 2 53))) ==> 0.0 (- .7 (/ #x16666666666665 (expt 2 53))) ==> 111.02230246251565e-18 (- .7 (* 7 .1)) ==> -111.02230246251565e-18The last calculation makes plain that this is not an issue of number syntax, but of numerical conditioning. (* .1 7) is not printed as 0.7 because it is not equal to 0.7.
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Radey Shouman and Aubrey Jaffer | Go Figure! |