Why is 7 * 0.1 ≠ 0.7 ?

	http://people.csail.mit.edu/jaffer/III/FloatingPoint
	Why is 7 * 0.1 ≠ 0.7 ?

A Scheme user asks:

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-18

Multiplying (/ #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-18

The 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.

Conclusions:

When you want compactly printed floating-point numbers, use printf or format.
Treat every arithmetic operation as potentially reducing the precision of floating-point operands.

References:

[IEEE] ANSI/IEEE Std 754-1985,
IEEE Standard for Binary Floating-Point Arithmetic.
Robert G. Burger and R. Kent Dybvig.
Printing floating-point numbers quickly and accurately.
Proceedings of the ACM SIGPLAN '96 Conference on Programming Language Design and Implementation, pages 108-116.
William Clinger.
How to read floating point numbers accurately.
Proceedings of the ACM SIGPLAN '90 Conference on Programming Language Design and Implementation, pages 92-101.
Proceedings published as SIGPLAN Notices 25(6), June 1990.

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	Radey Shouman and Aubrey Jaffer	Go Figure!