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   1  =head1 NAME
   2  
   3  perlnumber - semantics of numbers and numeric operations in Perl
   4  
   5  =head1 SYNOPSIS
   6  
   7      $n = 1234;            # decimal integer
   8      $n = 0b1110011;        # binary integer
   9      $n = 01234;            # octal integer
  10      $n = 0x1234;        # hexadecimal integer
  11      $n = 12.34e-56;        # exponential notation
  12      $n = "-12.34e56";        # number specified as a string
  13      $n = "1234";        # number specified as a string
  14  
  15  =head1 DESCRIPTION
  16  
  17  This document describes how Perl internally handles numeric values.
  18  
  19  Perl's operator overloading facility is completely ignored here.  Operator
  20  overloading allows user-defined behaviors for numbers, such as operations
  21  over arbitrarily large integers, floating points numbers with arbitrary
  22  precision, operations over "exotic" numbers such as modular arithmetic or
  23  p-adic arithmetic, and so on.  See L<overload> for details.
  24  
  25  =head1 Storing numbers
  26  
  27  Perl can internally represent numbers in 3 different ways: as native
  28  integers, as native floating point numbers, and as decimal strings.
  29  Decimal strings may have an exponential notation part, as in C<"12.34e-56">.
  30  I<Native> here means "a format supported by the C compiler which was used
  31  to build perl".
  32  
  33  The term "native" does not mean quite as much when we talk about native
  34  integers, as it does when native floating point numbers are involved.
  35  The only implication of the term "native" on integers is that the limits for
  36  the maximal and the minimal supported true integral quantities are close to
  37  powers of 2.  However, "native" floats have a most fundamental
  38  restriction: they may represent only those numbers which have a relatively
  39  "short" representation when converted to a binary fraction.  For example,
  40  0.9 cannot be represented by a native float, since the binary fraction
  41  for 0.9 is infinite:
  42  
  43    binary0.1110011001100...
  44  
  45  with the sequence C<1100> repeating again and again.  In addition to this
  46  limitation,  the exponent of the binary number is also restricted when it
  47  is represented as a floating point number.  On typical hardware, floating
  48  point values can store numbers with up to 53 binary digits, and with binary
  49  exponents between -1024 and 1024.  In decimal representation this is close
  50  to 16 decimal digits and decimal exponents in the range of -304..304.
  51  The upshot of all this is that Perl cannot store a number like
  52  12345678901234567 as a floating point number on such architectures without
  53  loss of information.
  54  
  55  Similarly, decimal strings can represent only those numbers which have a
  56  finite decimal expansion.  Being strings, and thus of arbitrary length, there
  57  is no practical limit for the exponent or number of decimal digits for these
  58  numbers.  (But realize that what we are discussing the rules for just the
  59  I<storage> of these numbers.  The fact that you can store such "large" numbers
  60  does not mean that the I<operations> over these numbers will use all
  61  of the significant digits.
  62  See L<"Numeric operators and numeric conversions"> for details.)
  63  
  64  In fact numbers stored in the native integer format may be stored either
  65  in the signed native form, or in the unsigned native form.  Thus the limits
  66  for Perl numbers stored as native integers would typically be -2**31..2**32-1,
  67  with appropriate modifications in the case of 64-bit integers.  Again, this
  68  does not mean that Perl can do operations only over integers in this range:
  69  it is possible to store many more integers in floating point format.
  70  
  71  Summing up, Perl numeric values can store only those numbers which have
  72  a finite decimal expansion or a "short" binary expansion.
  73  
  74  =head1 Numeric operators and numeric conversions
  75  
  76  As mentioned earlier, Perl can store a number in any one of three formats,
  77  but most operators typically understand only one of those formats.  When
  78  a numeric value is passed as an argument to such an operator, it will be
  79  converted to the format understood by the operator.
  80  
  81  Six such conversions are possible:
  82  
  83    native integer        --> native floating point    (*)
  84    native integer        --> decimal string
  85    native floating_point --> native integer        (*)
  86    native floating_point --> decimal string        (*)
  87    decimal string        --> native integer
  88    decimal string        --> native floating point    (*)
  89  
  90  These conversions are governed by the following general rules:
  91  
  92  =over 4
  93  
  94  =item *
  95  
  96  If the source number can be represented in the target form, that
  97  representation is used.
  98  
  99  =item *
 100  
 101  If the source number is outside of the limits representable in the target form,
 102  a representation of the closest limit is used.  (I<Loss of information>)
 103  
 104  =item *
 105  
 106  If the source number is between two numbers representable in the target form,
 107  a representation of one of these numbers is used.  (I<Loss of information>)
 108  
 109  =item *
 110  
 111  In C<< native floating point --> native integer >> conversions the magnitude
 112  of the result is less than or equal to the magnitude of the source.
 113  (I<"Rounding to zero".>)
 114  
 115  =item *
 116  
 117  If the C<< decimal string --> native integer >> conversion cannot be done
 118  without loss of information, the result is compatible with the conversion
 119  sequence C<< decimal_string --> native_floating_point --> native_integer >>.
 120  In particular, rounding is strongly biased to 0, though a number like
 121  C<"0.99999999999999999999"> has a chance of being rounded to 1.
 122  
 123  =back
 124  
 125  B<RESTRICTION>: The conversions marked with C<(*)> above involve steps
 126  performed by the C compiler.  In particular, bugs/features of the compiler
 127  used may lead to breakage of some of the above rules.
 128  
 129  =head1 Flavors of Perl numeric operations
 130  
 131  Perl operations which take a numeric argument treat that argument in one
 132  of four different ways: they may force it to one of the integer/floating/
 133  string formats, or they may behave differently depending on the format of
 134  the operand.  Forcing a numeric value to a particular format does not
 135  change the number stored in the value.
 136  
 137  All the operators which need an argument in the integer format treat the
 138  argument as in modular arithmetic, e.g., C<mod 2**32> on a 32-bit
 139  architecture.  C<sprintf "%u", -1> therefore provides the same result as
 140  C<sprintf "%u", ~0>.
 141  
 142  =over 4
 143  
 144  =item Arithmetic operators
 145  
 146  The binary operators C<+> C<-> C<*> C</> C<%> C<==> C<!=> C<E<gt>> C<E<lt>>
 147  C<E<gt>=> C<E<lt>=> and the unary operators C<-> C<abs> and C<--> will
 148  attempt to convert arguments to integers.  If both conversions are possible
 149  without loss of precision, and the operation can be performed without
 150  loss of precision then the integer result is used.  Otherwise arguments are
 151  converted to floating point format and the floating point result is used.
 152  The caching of conversions (as described above) means that the integer
 153  conversion does not throw away fractional parts on floating point numbers.
 154  
 155  =item ++
 156  
 157  C<++> behaves as the other operators above, except that if it is a string
 158  matching the format C</^[a-zA-Z]*[0-9]*\z/> the string increment described
 159  in L<perlop> is used.
 160  
 161  =item Arithmetic operators during C<use integer>
 162  
 163  In scopes where C<use integer;> is in force, nearly all the operators listed
 164  above will force their argument(s) into integer format, and return an integer
 165  result.  The exceptions, C<abs>, C<++> and C<-->, do not change their
 166  behavior with C<use integer;>
 167  
 168  =item Other mathematical operators
 169  
 170  Operators such as C<**>, C<sin> and C<exp> force arguments to floating point
 171  format.
 172  
 173  =item Bitwise operators
 174  
 175  Arguments are forced into the integer format if not strings.
 176  
 177  =item Bitwise operators during C<use integer>
 178  
 179  forces arguments to integer format. Also shift operations internally use
 180  signed integers rather than the default unsigned.
 181  
 182  =item Operators which expect an integer
 183  
 184  force the argument into the integer format.  This is applicable
 185  to the third and fourth arguments of C<sysread>, for example.
 186  
 187  =item Operators which expect a string
 188  
 189  force the argument into the string format.  For example, this is
 190  applicable to C<printf "%s", $value>.
 191  
 192  =back
 193  
 194  Though forcing an argument into a particular form does not change the
 195  stored number, Perl remembers the result of such conversions.  In
 196  particular, though the first such conversion may be time-consuming,
 197  repeated operations will not need to redo the conversion.
 198  
 199  =head1 AUTHOR
 200  
 201  Ilya Zakharevich C<ilya@math.ohio-state.edu>
 202  
 203  Editorial adjustments by Gurusamy Sarathy <gsar@ActiveState.com>
 204  
 205  Updates for 5.8.0 by Nicholas Clark <nick@ccl4.org>
 206  
 207  =head1 SEE ALSO
 208  
 209  L<overload>, L<perlop>