浮動小数点演算と無誤差変換と高精度計算のおはなし

1. quintia
2008/11/22

2. p 2008 11
( )
pD.E.Knuth:
The Art of Computer Programming,
Vol.2, 1969.

3. p
p ?
pKnuth

4. =
p !
6
p !1 2))
fl !
fl !
p 0
fl( )

5. ?
p
fl( (a + b) + c ) ≠ fl( a + (b + c) )
p fl( )
!

6. Knuth
p a1+a2+a3
The Art of Computer Programming 2
3 p217

7. p
p
p
pKnuth

8. p ?
?
a + b = fl( a + b ) + x
p x ?
p ?

9. p (D.E.Knuth, 1969)
fl( (a ! ((a + b) ! (a ! b)))
+ (b ! (a ! b)) )
p
T.J.Dekker, (1971)

10. fl( (a ((a + b) (a b)))
+ (b (a b)) )
p 1 0
( )
a a b+a b+b a+b = 0
p 0 !

11. fl( (a ((a + b) (a b)))
+ (b (a b)) )
0 )
0( 2 )

12. Knuth (
p a1+a2+a3 )
The Art of Computer Programming 2
3 p217

13. 6 2 1
( )
(
)
2
p 0 0 0

14. pJava
pJava
pDelphi

15. Java
public static void main(String[] args) {
float a = 1;
float b = (float)Math.pow(2, 127);
System.out.println("b: "+b);
float u = a + b;
float c = a - b;
float v = (a-(u-c)) + (b-c);
System.out.println("u: "+u);
System.out.println("v: "+v);
}
b: 1.7014118E38
u: 1.7014118E38
v: NaN
NaN !

16. Java
pFloat Double
p POSITIVE_INFINITY ( )
p NEGATIVE_INIFINITY( )
p NaN( )
p+0.0 -0.0
p1.0 / +0.0 > POSITIVE_INFINITY
p1.0 / -0.0 > NEGATIVE_INFINITY
p0.0 / 0.0 > NaN

17. Delphi
p !!
var
a,b,c,u,v: Single;
begin
a:=1.0;
b:=Power(2.0, 60.0);
u:=a+b;
c:=a-b;
v:=(a-(u-c))+(b-c);
ShowMessage(Format('%f , %f', [u, v]));
end;
: 1.15292150460684698E18 , 1.00