in reply to Integers sometimes turn into Reals after substraction

First of all, 4/25 is periodic in binary (just like 1/3 is periodic in decimal). 

         ____________________
4/25 = 0.00101000111101011100 base 2

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This means it can't be stored exactly in a floating point number.

Sometimes, floating point number errors get canceled out or reduced to nothing. It turns out that this happens for one of the chains of operations but not the other.

That's why you want to avoid checking if two floating point numbers are equal without allowing for an error margin. Replace 

$n1 == $n2

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with 

abs($n1 - $n2) < $epsilon

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Re^2: Integers sometimes turn into Reals after substraction (error correction ?)
by LanX (Saint) on May 14, 2016 at 15:54 UTC
    Well Perl's error correction (or that of the underlying C lib) is better than it's reputation 
       
DB<186> $x1= 1000*(4/25)
 => 160

  DB<187> $x2= 4000/25
 => 160

  DB<188> $x1 == $x2
 => 1

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    What's puzzling me is this behaviour: 

      DB<205>  for (254..263) {$x = 1000 * ( $_ + 4/25 ); printf "%.20f\n"
+, $x} 

254160.00000000000000000000
255160.00000000000000000000
256160.00000000002910383046
257160.00000000002910383046
258160.00000000002910383046
259160.00000000002910383046
260160.00000000002910383046
261160.00000000002910383046
262160.00000000000000000000
263160.00000000000000000000

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    Cheers Rolf
    (addicted to the Perl Programming Language and ☆☆☆☆ :)
    Je suis Charlie!

    update

    interestingly this only seems to happen near some powers of 2

      DB<220>  for (0..20) {$e=2**$_; $x = 1000 * ( $e + 4/25 ); printf "$
+_:$e => %.20f\n", $x if ($x-int($x))} 
5:32 => 32159.99999999999636202119
8:256 => 256160.00000000002910383046
9:512 => 512159.99999999994179233909
10:1024 => 1024160.00000000011641532183
11:2048 => 2048159.99999999976716935635
15:32768 => 32768160.00000000372529029846
18:262144 => 262144159.99999997019767761230
19:524288 => 524288160.00000005960464477539
20:1048576 => 1048576159.99999988079071044922

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    update

    in hindsight this may be an effect of precision and correction of the underlying processor and usage of arithmetic units, hence machine dependent.

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      IEEE double-precision float is 53 bit float. 0.00000000002910383046 = 2**-35. With the 18 bits for the integral portion of 255160 or 256160, that's the full 53bits. So, apparently, when you take the 53bit fractional representation of 255.16 multiplied by 1000dec, round-off sets the final bit to 0; when you take 256.16 * 1000, round-off sets the final bit to 1. Now (when I have time) I'm going to have to figure out the formula to figure out which powers of two will have this error and which won't. It all depends on the 53rd bit of the product of 2**N+16/25 times 128 (same as the bits of times 1000, since the factor of 8 is a binary shift). Oh, I'm nearly there... the 125 is 0b111_1101... and that will just interact with the 53rd bit... urgh but I've got to leave now... I guess it's an exercise for later. :-)

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        Convert ieee floats into hex notation, for each of 2**n - 1 and 2**n:

        cmd.exe> perl -e "use 5.010; for(0..31){$e=2**$_; $f=$e-1; $str = join
+ '', map { sprintf '%02x ', ord } split(//, pack('dddd', $f+4/25, $e+
+4/25, ($f+4/25)*1000, ($e+4/25)*1000)); $str =~ s/(.{24})(.{24})(.{24
+})(.{24})/\1\t\2\t\3\t\4/; say qq{$_\t$str};}"
0       7b 14 ae 47 e1 7a c4 3f         8f c2 f5 28 5c 8f f2 3f       
+  00 00 00 00 00 00 64 40         00 00 00 00 00 20 92 40
1       8f c2 f5 28 5c 8f f2 3f         48 e1 7a 14 ae 47 01 40       
+  00 00 00 00 00 20 92 40         00 00 00 00 00 e0 a0 40
2       48 e1 7a 14 ae 47 09 40         a4 70 3d 0a d7 a3 10 40       
+  00 00 00 00 00 b0 a8 40         00 00 00 00 00 40 b0 40
3       a4 70 3d 0a d7 a3 1c 40         52 b8 1e 85 eb 51 20 40       
+  00 00 00 00 00 f8 bb 40         00 00 00 00 00 e0 bf 40
4       52 b8 1e 85 eb 51 2e 40         29 5c 8f c2 f5 28 30 40       
+  00 00 00 00 00 9c cd 40         00 00 00 00 00 90 cf 40
5       29 5c 8f c2 f5 28 3f 40         14 ae 47 e1 7a 14 40 40       
+  00 00 00 00 00 6e de 40         ff ff ff ff ff 67 df 40
6       14 ae 47 e1 7a 94 4f 40         0a d7 a3 70 3d 0a 50 40       
+  00 00 00 00 00 d7 ee 40         00 00 00 00 00 54 ef 40
7       0a d7 a3 70 3d ca 5f 40         85 eb 51 b8 1e 05 60 40       
+  00 00 00 00 80 0b ff 40         00 00 00 00 00 4a ff 40
8       85 eb 51 b8 1e e5 6f 40         c3 f5 28 5c 8f 02 70 40       
+  00 00 00 00 c0 25 0f 41         01 00 00 00 00 45 0f 41
9       c3 f5 28 5c 8f f2 7f 40         e1 7a 14 ae 47 01 80 40       
+  00 00 00 00 e0 32 1f 41         ff ff ff ff 7f 42 1f 41
10      e1 7a 14 ae 47 f9 8f 40         71 3d 0a d7 a3 00 90 40       
+  00 00 00 00 70 39 2f 41         01 00 00 00 40 41 2f 41
11      71 3d 0a d7 a3 fc 9f 40         b8 1e 85 eb 51 00 a0 40       
+  00 00 00 00 b8 3c 3f 41         ff ff ff ff 9f 40 3f 41
12      b8 1e 85 eb 51 fe af 40         5c 8f c2 f5 28 00 b0 40       
+  00 00 00 00 5c 3e 4f 41         00 00 00 00 50 40 4f 41
13      5c 8f c2 f5 28 ff bf 40         ae 47 e1 7a 14 00 c0 40       
+  00 00 00 00 2e 3f 5f 41         00 00 00 00 28 40 5f 41
14      ae 47 e1 7a 94 ff cf 40         d7 a3 70 3d 0a 00 d0 40       
+  00 00 00 00 97 3f 6f 41         00 00 00 00 14 40 6f 41
15      d7 a3 70 3d ca ff df 40         ec 51 b8 1e 05 00 e0 40       
+  00 00 00 80 cb 3f 7f 41         01 00 00 00 0a 40 7f 41
16      ec 51 b8 1e e5 ff ef 40         f6 28 5c 8f 02 00 f0 40       
+  00 00 00 c0 e5 3f 8f 41         00 00 00 00 05 40 8f 41
17      f6 28 5c 8f f2 ff ff 40         7b 14 ae 47 01 00 00 41       
+  00 00 00 e0 f2 3f 9f 41         00 00 00 80 02 40 9f 41
18      7b 14 ae 47 f9 ff 0f 41         3d 0a d7 a3 00 00 10 41       
+  00 00 00 70 f9 3f af 41         ff ff ff 3f 01 40 af 41
19      3d 0a d7 a3 fc ff 1f 41         1f 85 eb 51 00 00 20 41       
+  00 00 00 b8 fc 3f bf 41         01 00 00 a0 00 40 bf 41
20      1f 85 eb 51 fe ff 2f 41         8f c2 f5 28 00 00 30 41       
+  00 00 00 5c fe 3f cf 41         ff ff ff 4f 00 40 cf 41
21      8f c2 f5 28 ff ff 3f 41         48 e1 7a 14 00 00 40 41       
+  00 00 00 2e ff 3f df 41         01 00 00 28 00 40 df 41
22      48 e1 7a 94 ff ff 4f 41         a4 70 3d 0a 00 00 50 41       
+  00 00 00 97 ff 3f ef 41         00 00 00 14 00 40 ef 41
23      a4 70 3d ca ff ff 5f 41         52 b8 1e 05 00 00 60 41       
+  00 00 80 cb ff 3f ff 41         00 00 00 0a 00 40 ff 41
24      52 b8 1e e5 ff ff 6f 41         29 5c 8f 02 00 00 70 41       
+  00 00 c0 e5 ff 3f 0f 42         00 00 00 05 00 40 0f 42
25      29 5c 8f f2 ff ff 7f 41         14 ae 47 01 00 00 80 41       
+  00 00 e0 f2 ff 3f 1f 42         ff ff 7f 02 00 40 1f 42
26      14 ae 47 f9 ff ff 8f 41         0a d7 a3 00 00 00 90 41       
+  00 00 70 f9 ff 3f 2f 42         00 00 40 01 00 40 2f 42
27      0a d7 a3 fc ff ff 9f 41         85 eb 51 00 00 00 a0 41       
+  00 00 b8 fc ff 3f 3f 42         00 00 a0 00 00 40 3f 42
28      85 eb 51 fe ff ff af 41         c3 f5 28 00 00 00 b0 41       
+  00 00 5c fe ff 3f 4f 42         01 00 50 00 00 40 4f 42
29      c3 f5 28 ff ff ff bf 41         e1 7a 14 00 00 00 c0 41       
+  00 00 2e ff ff 3f 5f 42         ff ff 27 00 00 40 5f 42
30      e1 7a 94 ff ff ff cf 41         71 3d 0a 00 00 00 d0 41       
+  00 00 97 ff ff 3f 6f 42         01 00 14 00 00 40 6f 42
31      71 3d ca ff ff ff df 41         b8 1e 05 00 00 00 e0 41       
+  00 80 cb ff ff 3f 7f 42         ff ff 09 00 00 40 7f 42

n       2**n - 1                        2**n                          
+  (2**n - 1)*1000                 (2**n) * 1000

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