I need to test again, but IMHO this depends on the number before the operation being a float or not.

AFAIK is exponentiation ** using an approximation algorithm and hence always producing a float.

So even if 2**53 (-1) should be a perfectly "whole number" it's stored as a float.

But that's speculation, I'm thinking of writing simple test code using only basic operations.

This could be translated to different dynamically typed languages to test their behavior.

update

I can't see how your wtf-example shows any rounding to 2**53.

Your floats have 64 bits leaving 53 for the mantissa and ~0 == 2^64-1 , hence there is no way to represent a whole number ~0+1 loss free.

Going up to 128bit floats will just replicate the situation on 32bit systems with 64bit floats.

update

OK I tried to nail down the weirdness, please note in the first example I'm constructing 2^53-1 and everything goes fine, even since ** will always produce a NV, summing them up will convert back to IV.

lanx:$ perl -MDevel::Peek -E'$x+=2**$_ for 0..52;Dump ($x);Dump($x+1)'
SV = IV(0x64c9edf22bb0) at 0x64c9edf22bc0
  REFCNT = 1
  FLAGS = (IOK,pIOK)
  IV = 9007199254740991
SV = IV(0x64c9edf838d0) at 0x64c9edf838e0
  REFCNT = 1
  FLAGS = (PADTMP,IOK,pIOK)
  IV = 9007199254740992
[download]

Now the same with 2^54-1, because the last element of the sum is the NV 2**53 which doesn't fit into the mantissa, we are stuck in NV even after adding an IV.

lanx:$ perl -MDevel::Peek -E'$x+=2**$_ for 0..53;Dump ($x);Dump($x+1);
+say "*** WTF *** " if $x == $x+1'
SV = PVNV(0x5ea09827d340) at 0x5ea0982abd00
  REFCNT = 1
  FLAGS = (NOK,pNOK)
  IV = 9007199254740991
  NV = 18014398509481984
  PV = 0
SV = NV(0x5ea09830cb18) at 0x5ea09830cb30
  REFCNT = 1
  FLAGS = (PADTMP,NOK,pNOK)
  NV = 18014398509481984
*** WTF ***
[download]

The real issue here is the ** operand, it should ideally produce an integer since 2 and 53 are integers and they fit into IV.

But even special casing the X**Y algorithm for integer results isn't trivial.

• 2**65 would still need to produce an NV
• 4**0.5 would need to be IV 2

I can't see how your wtf-example shows any rounding to 2**53.

The following examples imply rounding to 53 bits:

D:\>perl -le "print 'wtf' unless ~0 < (~0) + 2048;"
wtf

D:\>perl -le "print 'wtf' unless ~0 < (~0) + 2049;"

D:\>
[download]

Cheers,
Rob

(see update further down)

> The following examples imply rounding to 53 bits:

Again, ~0 == 2^64-1 no 53 involved.

You are out of any integer precision on "normal" 64bit systems, if you use the "Largest integer in 64-bit perl" (sic).

  DB<35> p log(~0)/log(2)
64
  DB<36> printf "%x",~0
ffffffffffffffff
  DB<37> $x=0; $x=$x*2+1, printf "2**%d-1 : %x\n",$_,$x  for 1..64
2**1-1 : 1
2**2-1 : 3
2**3-1 : 7
2**4-1 : f
2**5-1 : 1f
2**6-1 : 3f
2**7-1 : 7f
2**8-1 : ff
2**9-1 : 1ff
2**10-1 : 3ff
2**11-1 : 7ff
2**12-1 : fff
2**13-1 : 1fff
2**14-1 : 3fff
2**15-1 : 7fff
2**16-1 : ffff
2**17-1 : 1ffff
2**18-1 : 3ffff
2**19-1 : 7ffff
2**20-1 : fffff
2**21-1 : 1fffff
2**22-1 : 3fffff
2**23-1 : 7fffff
2**24-1 : ffffff
2**25-1 : 1ffffff
2**26-1 : 3ffffff
2**27-1 : 7ffffff
2**28-1 : fffffff
2**29-1 : 1fffffff
2**30-1 : 3fffffff
2**31-1 : 7fffffff
2**32-1 : ffffffff
2**33-1 : 1ffffffff
2**34-1 : 3ffffffff
2**35-1 : 7ffffffff
2**36-1 : fffffffff
2**37-1 : 1fffffffff
2**38-1 : 3fffffffff
2**39-1 : 7fffffffff
2**40-1 : ffffffffff
2**41-1 : 1ffffffffff
2**42-1 : 3ffffffffff
2**43-1 : 7ffffffffff
2**44-1 : fffffffffff
2**45-1 : 1fffffffffff
2**46-1 : 3fffffffffff
2**47-1 : 7fffffffffff
2**48-1 : ffffffffffff
2**49-1 : 1ffffffffffff
2**50-1 : 3ffffffffffff
2**51-1 : 7ffffffffffff
2**52-1 : fffffffffffff
2**53-1 : 1fffffffffffff
2**54-1 : 3fffffffffffff
2**55-1 : 7fffffffffffff
2**56-1 : ffffffffffffff
2**57-1 : 1ffffffffffffff
2**58-1 : 3ffffffffffffff
2**59-1 : 7ffffffffffffff
2**60-1 : fffffffffffffff
2**61-1 : 1fffffffffffffff
2**62-1 : 3fffffffffffffff
2**63-1 : 7fffffffffffffff
2**64-1 : ffffffffffffffff

  DB<38> Dump(~0), Dump($x), Dump($x+1)
SV = IV(0x5ec1cb040bc8) at 0x5ec1cb040bd8
  REFCNT = 1
  FLAGS = (PADTMP,IOK,READONLY,PROTECT,pIOK,IsUV)
  UV = 18446744073709551615
SV = PVNV(0x5ec1ca86f980) at 0x5ec1caeeb210
  REFCNT = 1
  FLAGS = (IOK,pIOK,pNOK,IsUV)
  UV = 18446744073709551615
  NV = 1.8446744073709552e+19
  PV = 0x5ec1cb01d8b0 "18446744073709551615"\0
  CUR = 20
  LEN = 22
SV = NV(0x5ec1cb040c50) at 0x5ec1cb040c68
  REFCNT = 1
  FLAGS = (PADTMP,NOK,pNOK)                         # <--- Integer Fla
+gs are gone
  NV = 1.8446744073709552e+19
[download]

Nothing to do with 53 bits.

update

Since we are falling back on float precision with 53 bit mantissa , we'll get ~0 == ~0 + 2048 , because 2048 = 2**11 and 53+11==64 . That's what you mean with 53bit rounding?

So your suggestion is that on 64bit engines we should use a higher float precision? Like the Quadruple-precision floating-point format?

I think I'd rather follow Nerdvana's suggestion

Cheers Rolf
_{(addicted to the Perl Programming Language :)
see Wikisyntax for the Monastery}