#!/usr/bin/env python
#
def benford ( ival ):

#*****************************************************************************80
#
## BENFORD returns the Benford probability of one or more significant digits.
#
#  Discussion:
#
#    Benford's law is an empirical formula explaining the observed
#    distribution of initial digits in lists culled from newspapers,
#    tax forms, stock market prices, and so on.  It predicts the observed
#    high frequency of the initial digit 1, for instance.
#
#    Note that the probabilities of digits 1 through 9 are guaranteed
#    to add up to 1, since
#      LOG10 ( 2/1 ) + LOG10 ( 3/2) + LOG10 ( 4/3 ) + ... + LOG10 ( 10/9 )
#      = LOG10 ( 2/1 * 3/2 * 4/3 * ... * 10/9 ) = LOG10 ( 10 ) = 1.
#
#  Formula:
#
#    Prob ( First significant digits are IVAL ) =
#      LOG10 ( ( IVAL + 1 ) / IVAL ).
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    21 July 2004
#
#  Author:
#
#    John Burkardt
#
#  Reference:
#
#    Frank Benford,
#    The Law of Anomalous Numbers,
#    Proceedings of the American Philosophical Society,
#    Volume 78, pages 551-572, 1938.
#
#    T P Hill,
#    The First Digit Phenomenon,
#    American Scientist,
#    Volume 86, July/August 1998, pages 358 - 363.
#
#    R Raimi,
#    The Peculiar Distribution of First Digits,
#    Scientific American,
#    December 1969, pages 109-119.
#
#  Parameters:
#
#    Input, integer IVAL, the string of significant digits to be checked.
#    If IVAL is 1, then we are asking for the Benford probability that
#    a value will have first digit 1.  If IVAL is 123, we are asking for
#    the probability that the first three digits will be 123, and so on.
#    Note that IVAL must not be 0 or negative.
#
#    Output, real VALUE, the Benford probability that an item taken
#    from a real world distribution will have the initial digits IVAL.
#
  from sys import exit
  from math import log10

  if ( ival <= 0 ):
    print ''
    print 'BENFORD - Fatal error!'
    print '  The input argument must be positive.'
    print '  Your value was %d' % ( ival )
    exit ( 'BENFORD - Fatal error!' );
 
  value = log10 ( float ( ival + 1 ) / float ( ival ) )

  return value

def benford_test ( ):

#*****************************************************************************80
#
## BENFORD_TEST tests BENFORD.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    22 December 2014
#
#  Author:
#
#    John Burkardt
#
  print ''
  print 'BENFORD_TEST'
  print '  BENFORD(I) is the Benford probability of the'
  print '  initial digit sequence I.'
  print ''
  print '   I     BENFORD(I)'
  print ''

  for i in range ( 1, 10 ):
    print '  %2d  %12f' % ( i, benford(i) )

  print ''
  print 'BENFORD_TEST'
  print '  Normal end of execution.'

  return

if ( __name__ == '__main__' ):
  from timestamp import timestamp
  timestamp ( )
  benford_test ( )
  timestamp ( )