1 | package Math::BigFloat;
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2 |
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3 | #
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4 | # Mike grinned. 'Two down, infinity to go' - Mike Nostrus in 'Before and After'
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5 | #
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6 |
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7 | # The following hash values are internally used:
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8 | # _e : exponent (ref to $CALC object)
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9 | # _m : mantissa (ref to $CALC object)
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10 | # _es : sign of _e
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11 | # sign : +,-,+inf,-inf, or "NaN" if not a number
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12 | # _a : accuracy
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13 | # _p : precision
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14 |
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15 | $VERSION = '1.51';
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16 | require 5.005;
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17 |
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18 | require Exporter;
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19 | @ISA = qw(Exporter Math::BigInt);
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20 |
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21 | use strict;
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22 | # $_trap_inf/$_trap_nan are internal and should never be accessed from outside
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23 | use vars qw/$AUTOLOAD $accuracy $precision $div_scale $round_mode $rnd_mode
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24 | $upgrade $downgrade $_trap_nan $_trap_inf/;
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25 | my $class = "Math::BigFloat";
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26 |
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27 | use overload
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28 | '<=>' => sub { $_[2] ?
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29 | ref($_[0])->bcmp($_[1],$_[0]) :
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30 | ref($_[0])->bcmp($_[0],$_[1])},
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31 | 'int' => sub { $_[0]->as_number() }, # 'trunc' to bigint
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32 | ;
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33 |
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34 | ##############################################################################
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35 | # global constants, flags and assorted stuff
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36 |
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37 | # the following are public, but their usage is not recommended. Use the
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38 | # accessor methods instead.
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39 |
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40 | # class constants, use Class->constant_name() to access
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41 | $round_mode = 'even'; # one of 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'
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42 | $accuracy = undef;
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43 | $precision = undef;
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44 | $div_scale = 40;
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45 |
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46 | $upgrade = undef;
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47 | $downgrade = undef;
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48 | # the package we are using for our private parts, defaults to:
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49 | # Math::BigInt->config()->{lib}
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50 | my $MBI = 'Math::BigInt::FastCalc';
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51 |
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52 | # are NaNs ok? (otherwise it dies when encountering an NaN) set w/ config()
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53 | $_trap_nan = 0;
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54 | # the same for infinity
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55 | $_trap_inf = 0;
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56 |
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57 | # constant for easier life
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58 | my $nan = 'NaN';
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59 |
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60 | my $IMPORT = 0; # was import() called yet? used to make require work
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61 |
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62 | # some digits of accuracy for blog(undef,10); which we use in blog() for speed
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63 | my $LOG_10 =
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64 | '2.3025850929940456840179914546843642076011014886287729760333279009675726097';
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65 | my $LOG_10_A = length($LOG_10)-1;
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66 | # ditto for log(2)
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67 | my $LOG_2 =
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68 | '0.6931471805599453094172321214581765680755001343602552541206800094933936220';
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69 | my $LOG_2_A = length($LOG_2)-1;
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70 | my $HALF = '0.5'; # made into an object if necc.
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71 |
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72 | ##############################################################################
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73 | # the old code had $rnd_mode, so we need to support it, too
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74 |
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75 | sub TIESCALAR { my ($class) = @_; bless \$round_mode, $class; }
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76 | sub FETCH { return $round_mode; }
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77 | sub STORE { $rnd_mode = $_[0]->round_mode($_[1]); }
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78 |
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79 | BEGIN
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80 | {
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81 | # when someone set's $rnd_mode, we catch this and check the value to see
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82 | # whether it is valid or not.
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83 | $rnd_mode = 'even'; tie $rnd_mode, 'Math::BigFloat';
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84 | }
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85 |
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86 | ##############################################################################
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87 |
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88 | {
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89 | # valid method aliases for AUTOLOAD
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90 | my %methods = map { $_ => 1 }
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91 | qw / fadd fsub fmul fdiv fround ffround fsqrt fmod fstr fsstr fpow fnorm
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92 | fint facmp fcmp fzero fnan finf finc fdec flog ffac fneg
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93 | fceil ffloor frsft flsft fone flog froot
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94 | /;
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95 | # valid method's that can be hand-ed up (for AUTOLOAD)
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96 | my %hand_ups = map { $_ => 1 }
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97 | qw / is_nan is_inf is_negative is_positive is_pos is_neg
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98 | accuracy precision div_scale round_mode fabs fnot
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99 | objectify upgrade downgrade
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100 | bone binf bnan bzero
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101 | /;
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102 |
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103 | sub method_alias { exists $methods{$_[0]||''}; }
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104 | sub method_hand_up { exists $hand_ups{$_[0]||''}; }
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105 | }
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106 |
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107 | ##############################################################################
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108 | # constructors
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109 |
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110 | sub new
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111 | {
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112 | # create a new BigFloat object from a string or another bigfloat object.
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113 | # _e: exponent
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114 | # _m: mantissa
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115 | # sign => sign (+/-), or "NaN"
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116 |
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117 | my ($class,$wanted,@r) = @_;
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118 |
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119 | # avoid numify-calls by not using || on $wanted!
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120 | return $class->bzero() if !defined $wanted; # default to 0
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121 | return $wanted->copy() if UNIVERSAL::isa($wanted,'Math::BigFloat');
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122 |
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123 | $class->import() if $IMPORT == 0; # make require work
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124 |
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125 | my $self = {}; bless $self, $class;
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126 | # shortcut for bigints and its subclasses
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127 | if ((ref($wanted)) && (ref($wanted) ne $class))
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128 | {
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129 | $self->{_m} = $wanted->as_number()->{value}; # get us a bigint copy
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130 | $self->{_e} = $MBI->_zero();
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131 | $self->{_es} = '+';
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132 | $self->{sign} = $wanted->sign();
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133 | return $self->bnorm();
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134 | }
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135 | # else: got a string
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136 |
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137 | # handle '+inf', '-inf' first
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138 | if ($wanted =~ /^[+-]?inf\z/)
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139 | {
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140 | return $downgrade->new($wanted) if $downgrade;
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141 |
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142 | $self->{sign} = $wanted; # set a default sign for bstr()
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143 | return $self->binf($wanted);
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144 | }
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145 |
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146 | # shortcut for simple forms like '12' that neither have trailing nor leading
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147 | # zeros
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148 | if ($wanted =~ /^([+-]?)([1-9][0-9]*[1-9])$/)
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149 | {
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150 | $self->{_e} = $MBI->_zero();
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151 | $self->{_es} = '+';
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152 | $self->{sign} = $1 || '+';
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153 | $self->{_m} = $MBI->_new($2);
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154 | return $self->round(@r) if !$downgrade;
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155 | }
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156 |
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157 | my ($mis,$miv,$mfv,$es,$ev) = Math::BigInt::_split($wanted);
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158 | if (!ref $mis)
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159 | {
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160 | if ($_trap_nan)
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161 | {
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162 | require Carp;
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163 | Carp::croak ("$wanted is not a number initialized to $class");
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164 | }
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165 |
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166 | return $downgrade->bnan() if $downgrade;
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167 |
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168 | $self->{_e} = $MBI->_zero();
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169 | $self->{_es} = '+';
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170 | $self->{_m} = $MBI->_zero();
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171 | $self->{sign} = $nan;
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172 | }
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173 | else
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174 | {
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175 | # make integer from mantissa by adjusting exp, then convert to int
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176 | $self->{_e} = $MBI->_new($$ev); # exponent
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177 | $self->{_es} = $$es || '+';
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178 | my $mantissa = "$$miv$$mfv"; # create mant.
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179 | $mantissa =~ s/^0+(\d)/$1/; # strip leading zeros
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180 | $self->{_m} = $MBI->_new($mantissa); # create mant.
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181 |
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182 | # 3.123E0 = 3123E-3, and 3.123E-2 => 3123E-5
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183 | if (CORE::length($$mfv) != 0)
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184 | {
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185 | my $len = $MBI->_new( CORE::length($$mfv));
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186 | ($self->{_e}, $self->{_es}) =
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187 | _e_sub ($self->{_e}, $len, $self->{_es}, '+');
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188 | }
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189 | # we can only have trailing zeros on the mantissa if $$mfv eq ''
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190 | else
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191 | {
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192 | # Use a regexp to count the trailing zeros in $$miv instead of _zeros()
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193 | # because that is faster, especially when _m is not stored in base 10.
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194 | my $zeros = 0; $zeros = CORE::length($1) if $$miv =~ /[1-9](0*)$/;
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195 | if ($zeros != 0)
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196 | {
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197 | my $z = $MBI->_new($zeros);
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198 | # turn '120e2' into '12e3'
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199 | $MBI->_rsft ( $self->{_m}, $z, 10);
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200 | ($self->{_e}, $self->{_es}) =
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201 | _e_add ( $self->{_e}, $z, $self->{_es}, '+');
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202 | }
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203 | }
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204 | $self->{sign} = $$mis;
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205 |
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206 | # for something like 0Ey, set y to 1, and -0 => +0
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207 | # Check $$miv for beeing '0' and $$mfv eq '', because otherwise _m could not
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208 | # have become 0. That's faster than to call $MBI->_is_zero().
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209 | $self->{sign} = '+', $self->{_e} = $MBI->_one()
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210 | if $$miv eq '0' and $$mfv eq '';
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211 |
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212 | return $self->round(@r) if !$downgrade;
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213 | }
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214 | # if downgrade, inf, NaN or integers go down
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215 |
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216 | if ($downgrade && $self->{_es} eq '+')
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217 | {
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218 | if ($MBI->_is_zero( $self->{_e} ))
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219 | {
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220 | return $downgrade->new($$mis . $MBI->_str( $self->{_m} ));
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221 | }
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222 | return $downgrade->new($self->bsstr());
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223 | }
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224 | $self->bnorm()->round(@r); # first normalize, then round
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225 | }
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226 |
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227 | sub copy
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228 | {
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229 | my ($c,$x);
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230 | if (@_ > 1)
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231 | {
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232 | # if two arguments, the first one is the class to "swallow" subclasses
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233 | ($c,$x) = @_;
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234 | }
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235 | else
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236 | {
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237 | $x = shift;
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238 | $c = ref($x);
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239 | }
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240 | return unless ref($x); # only for objects
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241 |
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242 | my $self = {}; bless $self,$c;
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243 |
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244 | $self->{sign} = $x->{sign};
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245 | $self->{_es} = $x->{_es};
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246 | $self->{_m} = $MBI->_copy($x->{_m});
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247 | $self->{_e} = $MBI->_copy($x->{_e});
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248 | $self->{_a} = $x->{_a} if defined $x->{_a};
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249 | $self->{_p} = $x->{_p} if defined $x->{_p};
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250 | $self;
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251 | }
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252 |
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253 | sub _bnan
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254 | {
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255 | # used by parent class bone() to initialize number to NaN
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256 | my $self = shift;
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257 |
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258 | if ($_trap_nan)
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259 | {
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260 | require Carp;
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261 | my $class = ref($self);
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262 | Carp::croak ("Tried to set $self to NaN in $class\::_bnan()");
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263 | }
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264 |
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265 | $IMPORT=1; # call our import only once
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266 | $self->{_m} = $MBI->_zero();
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267 | $self->{_e} = $MBI->_zero();
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268 | $self->{_es} = '+';
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269 | }
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270 |
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271 | sub _binf
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272 | {
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273 | # used by parent class bone() to initialize number to +-inf
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274 | my $self = shift;
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275 |
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276 | if ($_trap_inf)
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277 | {
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278 | require Carp;
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279 | my $class = ref($self);
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280 | Carp::croak ("Tried to set $self to +-inf in $class\::_binf()");
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281 | }
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282 |
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283 | $IMPORT=1; # call our import only once
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284 | $self->{_m} = $MBI->_zero();
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285 | $self->{_e} = $MBI->_zero();
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286 | $self->{_es} = '+';
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287 | }
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288 |
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289 | sub _bone
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290 | {
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291 | # used by parent class bone() to initialize number to 1
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292 | my $self = shift;
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293 | $IMPORT=1; # call our import only once
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294 | $self->{_m} = $MBI->_one();
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295 | $self->{_e} = $MBI->_zero();
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296 | $self->{_es} = '+';
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297 | }
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298 |
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299 | sub _bzero
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300 | {
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301 | # used by parent class bone() to initialize number to 0
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302 | my $self = shift;
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303 | $IMPORT=1; # call our import only once
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304 | $self->{_m} = $MBI->_zero();
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305 | $self->{_e} = $MBI->_one();
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306 | $self->{_es} = '+';
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307 | }
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308 |
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309 | sub isa
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310 | {
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311 | my ($self,$class) = @_;
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312 | return if $class =~ /^Math::BigInt/; # we aren't one of these
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313 | UNIVERSAL::isa($self,$class);
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314 | }
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315 |
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316 | sub config
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317 | {
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318 | # return (later set?) configuration data as hash ref
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319 | my $class = shift || 'Math::BigFloat';
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320 |
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321 | my $cfg = $class->SUPER::config(@_);
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322 |
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323 | # now we need only to override the ones that are different from our parent
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324 | $cfg->{class} = $class;
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325 | $cfg->{with} = $MBI;
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326 | $cfg;
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327 | }
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328 |
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329 | ##############################################################################
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330 | # string conversation
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331 |
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332 | sub bstr
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333 | {
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334 | # (ref to BFLOAT or num_str ) return num_str
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335 | # Convert number from internal format to (non-scientific) string format.
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336 | # internal format is always normalized (no leading zeros, "-0" => "+0")
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337 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
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338 |
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339 | if ($x->{sign} !~ /^[+-]$/)
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340 | {
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341 | return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
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342 | return 'inf'; # +inf
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343 | }
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344 |
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345 | my $es = '0'; my $len = 1; my $cad = 0; my $dot = '.';
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346 |
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347 | # $x is zero?
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348 | my $not_zero = !($x->{sign} eq '+' && $MBI->_is_zero($x->{_m}));
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349 | if ($not_zero)
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350 | {
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351 | $es = $MBI->_str($x->{_m});
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352 | $len = CORE::length($es);
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353 | my $e = $MBI->_num($x->{_e});
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354 | $e = -$e if $x->{_es} eq '-';
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355 | if ($e < 0)
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356 | {
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357 | $dot = '';
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358 | # if _e is bigger than a scalar, the following will blow your memory
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359 | if ($e <= -$len)
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360 | {
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361 | my $r = abs($e) - $len;
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362 | $es = '0.'. ('0' x $r) . $es; $cad = -($len+$r);
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363 | }
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364 | else
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365 | {
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366 | substr($es,$e,0) = '.'; $cad = $MBI->_num($x->{_e});
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367 | $cad = -$cad if $x->{_es} eq '-';
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368 | }
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369 | }
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370 | elsif ($e > 0)
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371 | {
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372 | # expand with zeros
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373 | $es .= '0' x $e; $len += $e; $cad = 0;
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374 | }
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375 | } # if not zero
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376 |
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377 | $es = '-'.$es if $x->{sign} eq '-';
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378 | # if set accuracy or precision, pad with zeros on the right side
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379 | if ((defined $x->{_a}) && ($not_zero))
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380 | {
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381 | # 123400 => 6, 0.1234 => 4, 0.001234 => 4
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382 | my $zeros = $x->{_a} - $cad; # cad == 0 => 12340
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383 | $zeros = $x->{_a} - $len if $cad != $len;
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384 | $es .= $dot.'0' x $zeros if $zeros > 0;
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385 | }
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386 | elsif ((($x->{_p} || 0) < 0))
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387 | {
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388 | # 123400 => 6, 0.1234 => 4, 0.001234 => 6
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389 | my $zeros = -$x->{_p} + $cad;
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390 | $es .= $dot.'0' x $zeros if $zeros > 0;
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391 | }
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392 | $es;
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393 | }
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394 |
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395 | sub bsstr
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396 | {
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397 | # (ref to BFLOAT or num_str ) return num_str
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398 | # Convert number from internal format to scientific string format.
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399 | # internal format is always normalized (no leading zeros, "-0E0" => "+0E0")
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400 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
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401 |
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402 | if ($x->{sign} !~ /^[+-]$/)
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403 | {
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404 | return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
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405 | return 'inf'; # +inf
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406 | }
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407 | my $sep = 'e'.$x->{_es};
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408 | my $sign = $x->{sign}; $sign = '' if $sign eq '+';
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409 | $sign . $MBI->_str($x->{_m}) . $sep . $MBI->_str($x->{_e});
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410 | }
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411 |
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412 | sub numify
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413 | {
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414 | # Make a number from a BigFloat object
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415 | # simple return a string and let Perl's atoi()/atof() handle the rest
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416 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
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417 | $x->bsstr();
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418 | }
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419 |
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420 | ##############################################################################
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421 | # public stuff (usually prefixed with "b")
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422 |
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423 | sub bneg
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424 | {
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425 | # (BINT or num_str) return BINT
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426 | # negate number or make a negated number from string
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427 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
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428 |
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429 | return $x if $x->modify('bneg');
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430 |
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431 | # for +0 dont negate (to have always normalized +0). Does nothing for 'NaN'
|
---|
432 | $x->{sign} =~ tr/+-/-+/ unless ($x->{sign} eq '+' && $MBI->_is_zero($x->{_m}));
|
---|
433 | $x;
|
---|
434 | }
|
---|
435 |
|
---|
436 | # tels 2001-08-04
|
---|
437 | # XXX TODO this must be overwritten and return NaN for non-integer values
|
---|
438 | # band(), bior(), bxor(), too
|
---|
439 | #sub bnot
|
---|
440 | # {
|
---|
441 | # $class->SUPER::bnot($class,@_);
|
---|
442 | # }
|
---|
443 |
|
---|
444 | sub bcmp
|
---|
445 | {
|
---|
446 | # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
|
---|
447 |
|
---|
448 | # set up parameters
|
---|
449 | my ($self,$x,$y) = (ref($_[0]),@_);
|
---|
450 | # objectify is costly, so avoid it
|
---|
451 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
|
---|
452 | {
|
---|
453 | ($self,$x,$y) = objectify(2,@_);
|
---|
454 | }
|
---|
455 |
|
---|
456 | return $upgrade->bcmp($x,$y) if defined $upgrade &&
|
---|
457 | ((!$x->isa($self)) || (!$y->isa($self)));
|
---|
458 |
|
---|
459 | if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
|
---|
460 | {
|
---|
461 | # handle +-inf and NaN
|
---|
462 | return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
|
---|
463 | return 0 if ($x->{sign} eq $y->{sign}) && ($x->{sign} =~ /^[+-]inf$/);
|
---|
464 | return +1 if $x->{sign} eq '+inf';
|
---|
465 | return -1 if $x->{sign} eq '-inf';
|
---|
466 | return -1 if $y->{sign} eq '+inf';
|
---|
467 | return +1;
|
---|
468 | }
|
---|
469 |
|
---|
470 | # check sign for speed first
|
---|
471 | return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y
|
---|
472 | return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0
|
---|
473 |
|
---|
474 | # shortcut
|
---|
475 | my $xz = $x->is_zero();
|
---|
476 | my $yz = $y->is_zero();
|
---|
477 | return 0 if $xz && $yz; # 0 <=> 0
|
---|
478 | return -1 if $xz && $y->{sign} eq '+'; # 0 <=> +y
|
---|
479 | return 1 if $yz && $x->{sign} eq '+'; # +x <=> 0
|
---|
480 |
|
---|
481 | # adjust so that exponents are equal
|
---|
482 | my $lxm = $MBI->_len($x->{_m});
|
---|
483 | my $lym = $MBI->_len($y->{_m});
|
---|
484 | # the numify somewhat limits our length, but makes it much faster
|
---|
485 | my ($xes,$yes) = (1,1);
|
---|
486 | $xes = -1 if $x->{_es} ne '+';
|
---|
487 | $yes = -1 if $y->{_es} ne '+';
|
---|
488 | my $lx = $lxm + $xes * $MBI->_num($x->{_e});
|
---|
489 | my $ly = $lym + $yes * $MBI->_num($y->{_e});
|
---|
490 | my $l = $lx - $ly; $l = -$l if $x->{sign} eq '-';
|
---|
491 | return $l <=> 0 if $l != 0;
|
---|
492 |
|
---|
493 | # lengths (corrected by exponent) are equal
|
---|
494 | # so make mantissa equal length by padding with zero (shift left)
|
---|
495 | my $diff = $lxm - $lym;
|
---|
496 | my $xm = $x->{_m}; # not yet copy it
|
---|
497 | my $ym = $y->{_m};
|
---|
498 | if ($diff > 0)
|
---|
499 | {
|
---|
500 | $ym = $MBI->_copy($y->{_m});
|
---|
501 | $ym = $MBI->_lsft($ym, $MBI->_new($diff), 10);
|
---|
502 | }
|
---|
503 | elsif ($diff < 0)
|
---|
504 | {
|
---|
505 | $xm = $MBI->_copy($x->{_m});
|
---|
506 | $xm = $MBI->_lsft($xm, $MBI->_new(-$diff), 10);
|
---|
507 | }
|
---|
508 | my $rc = $MBI->_acmp($xm,$ym);
|
---|
509 | $rc = -$rc if $x->{sign} eq '-'; # -124 < -123
|
---|
510 | $rc <=> 0;
|
---|
511 | }
|
---|
512 |
|
---|
513 | sub bacmp
|
---|
514 | {
|
---|
515 | # Compares 2 values, ignoring their signs.
|
---|
516 | # Returns one of undef, <0, =0, >0. (suitable for sort)
|
---|
517 |
|
---|
518 | # set up parameters
|
---|
519 | my ($self,$x,$y) = (ref($_[0]),@_);
|
---|
520 | # objectify is costly, so avoid it
|
---|
521 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
|
---|
522 | {
|
---|
523 | ($self,$x,$y) = objectify(2,@_);
|
---|
524 | }
|
---|
525 |
|
---|
526 | return $upgrade->bacmp($x,$y) if defined $upgrade &&
|
---|
527 | ((!$x->isa($self)) || (!$y->isa($self)));
|
---|
528 |
|
---|
529 | # handle +-inf and NaN's
|
---|
530 | if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/)
|
---|
531 | {
|
---|
532 | return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
|
---|
533 | return 0 if ($x->is_inf() && $y->is_inf());
|
---|
534 | return 1 if ($x->is_inf() && !$y->is_inf());
|
---|
535 | return -1;
|
---|
536 | }
|
---|
537 |
|
---|
538 | # shortcut
|
---|
539 | my $xz = $x->is_zero();
|
---|
540 | my $yz = $y->is_zero();
|
---|
541 | return 0 if $xz && $yz; # 0 <=> 0
|
---|
542 | return -1 if $xz && !$yz; # 0 <=> +y
|
---|
543 | return 1 if $yz && !$xz; # +x <=> 0
|
---|
544 |
|
---|
545 | # adjust so that exponents are equal
|
---|
546 | my $lxm = $MBI->_len($x->{_m});
|
---|
547 | my $lym = $MBI->_len($y->{_m});
|
---|
548 | my ($xes,$yes) = (1,1);
|
---|
549 | $xes = -1 if $x->{_es} ne '+';
|
---|
550 | $yes = -1 if $y->{_es} ne '+';
|
---|
551 | # the numify somewhat limits our length, but makes it much faster
|
---|
552 | my $lx = $lxm + $xes * $MBI->_num($x->{_e});
|
---|
553 | my $ly = $lym + $yes * $MBI->_num($y->{_e});
|
---|
554 | my $l = $lx - $ly;
|
---|
555 | return $l <=> 0 if $l != 0;
|
---|
556 |
|
---|
557 | # lengths (corrected by exponent) are equal
|
---|
558 | # so make mantissa equal-length by padding with zero (shift left)
|
---|
559 | my $diff = $lxm - $lym;
|
---|
560 | my $xm = $x->{_m}; # not yet copy it
|
---|
561 | my $ym = $y->{_m};
|
---|
562 | if ($diff > 0)
|
---|
563 | {
|
---|
564 | $ym = $MBI->_copy($y->{_m});
|
---|
565 | $ym = $MBI->_lsft($ym, $MBI->_new($diff), 10);
|
---|
566 | }
|
---|
567 | elsif ($diff < 0)
|
---|
568 | {
|
---|
569 | $xm = $MBI->_copy($x->{_m});
|
---|
570 | $xm = $MBI->_lsft($xm, $MBI->_new(-$diff), 10);
|
---|
571 | }
|
---|
572 | $MBI->_acmp($xm,$ym);
|
---|
573 | }
|
---|
574 |
|
---|
575 | sub badd
|
---|
576 | {
|
---|
577 | # add second arg (BFLOAT or string) to first (BFLOAT) (modifies first)
|
---|
578 | # return result as BFLOAT
|
---|
579 |
|
---|
580 | # set up parameters
|
---|
581 | my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_);
|
---|
582 | # objectify is costly, so avoid it
|
---|
583 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
|
---|
584 | {
|
---|
585 | ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
|
---|
586 | }
|
---|
587 |
|
---|
588 | # inf and NaN handling
|
---|
589 | if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
|
---|
590 | {
|
---|
591 | # NaN first
|
---|
592 | return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
|
---|
593 | # inf handling
|
---|
594 | if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
|
---|
595 | {
|
---|
596 | # +inf++inf or -inf+-inf => same, rest is NaN
|
---|
597 | return $x if $x->{sign} eq $y->{sign};
|
---|
598 | return $x->bnan();
|
---|
599 | }
|
---|
600 | # +-inf + something => +inf; something +-inf => +-inf
|
---|
601 | $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/;
|
---|
602 | return $x;
|
---|
603 | }
|
---|
604 |
|
---|
605 | return $upgrade->badd($x,$y,$a,$p,$r) if defined $upgrade &&
|
---|
606 | ((!$x->isa($self)) || (!$y->isa($self)));
|
---|
607 |
|
---|
608 | # speed: no add for 0+y or x+0
|
---|
609 | return $x->bround($a,$p,$r) if $y->is_zero(); # x+0
|
---|
610 | if ($x->is_zero()) # 0+y
|
---|
611 | {
|
---|
612 | # make copy, clobbering up x (modify in place!)
|
---|
613 | $x->{_e} = $MBI->_copy($y->{_e});
|
---|
614 | $x->{_es} = $y->{_es};
|
---|
615 | $x->{_m} = $MBI->_copy($y->{_m});
|
---|
616 | $x->{sign} = $y->{sign} || $nan;
|
---|
617 | return $x->round($a,$p,$r,$y);
|
---|
618 | }
|
---|
619 |
|
---|
620 | # take lower of the two e's and adapt m1 to it to match m2
|
---|
621 | my $e = $y->{_e};
|
---|
622 | $e = $MBI->_zero() if !defined $e; # if no BFLOAT?
|
---|
623 | $e = $MBI->_copy($e); # make copy (didn't do it yet)
|
---|
624 |
|
---|
625 | my $es;
|
---|
626 |
|
---|
627 | ($e,$es) = _e_sub($e, $x->{_e}, $y->{_es} || '+', $x->{_es});
|
---|
628 |
|
---|
629 | my $add = $MBI->_copy($y->{_m});
|
---|
630 |
|
---|
631 | if ($es eq '-') # < 0
|
---|
632 | {
|
---|
633 | $MBI->_lsft( $x->{_m}, $e, 10);
|
---|
634 | ($x->{_e},$x->{_es}) = _e_add($x->{_e}, $e, $x->{_es}, $es);
|
---|
635 | }
|
---|
636 | elsif (!$MBI->_is_zero($e)) # > 0
|
---|
637 | {
|
---|
638 | $MBI->_lsft($add, $e, 10);
|
---|
639 | }
|
---|
640 | # else: both e are the same, so just leave them
|
---|
641 |
|
---|
642 | if ($x->{sign} eq $y->{sign})
|
---|
643 | {
|
---|
644 | # add
|
---|
645 | $x->{_m} = $MBI->_add($x->{_m}, $add);
|
---|
646 | }
|
---|
647 | else
|
---|
648 | {
|
---|
649 | ($x->{_m}, $x->{sign}) =
|
---|
650 | _e_add($x->{_m}, $add, $x->{sign}, $y->{sign});
|
---|
651 | }
|
---|
652 |
|
---|
653 | # delete trailing zeros, then round
|
---|
654 | $x->bnorm()->round($a,$p,$r,$y);
|
---|
655 | }
|
---|
656 |
|
---|
657 | # sub bsub is inherited from Math::BigInt!
|
---|
658 |
|
---|
659 | sub binc
|
---|
660 | {
|
---|
661 | # increment arg by one
|
---|
662 | my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
|
---|
663 |
|
---|
664 | if ($x->{_es} eq '-')
|
---|
665 | {
|
---|
666 | return $x->badd($self->bone(),@r); # digits after dot
|
---|
667 | }
|
---|
668 |
|
---|
669 | if (!$MBI->_is_zero($x->{_e})) # _e == 0 for NaN, inf, -inf
|
---|
670 | {
|
---|
671 | # 1e2 => 100, so after the shift below _m has a '0' as last digit
|
---|
672 | $x->{_m} = $MBI->_lsft($x->{_m}, $x->{_e},10); # 1e2 => 100
|
---|
673 | $x->{_e} = $MBI->_zero(); # normalize
|
---|
674 | $x->{_es} = '+';
|
---|
675 | # we know that the last digit of $x will be '1' or '9', depending on the
|
---|
676 | # sign
|
---|
677 | }
|
---|
678 | # now $x->{_e} == 0
|
---|
679 | if ($x->{sign} eq '+')
|
---|
680 | {
|
---|
681 | $MBI->_inc($x->{_m});
|
---|
682 | return $x->bnorm()->bround(@r);
|
---|
683 | }
|
---|
684 | elsif ($x->{sign} eq '-')
|
---|
685 | {
|
---|
686 | $MBI->_dec($x->{_m});
|
---|
687 | $x->{sign} = '+' if $MBI->_is_zero($x->{_m}); # -1 +1 => -0 => +0
|
---|
688 | return $x->bnorm()->bround(@r);
|
---|
689 | }
|
---|
690 | # inf, nan handling etc
|
---|
691 | $x->badd($self->bone(),@r); # badd() does round
|
---|
692 | }
|
---|
693 |
|
---|
694 | sub bdec
|
---|
695 | {
|
---|
696 | # decrement arg by one
|
---|
697 | my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
|
---|
698 |
|
---|
699 | if ($x->{_es} eq '-')
|
---|
700 | {
|
---|
701 | return $x->badd($self->bone('-'),@r); # digits after dot
|
---|
702 | }
|
---|
703 |
|
---|
704 | if (!$MBI->_is_zero($x->{_e}))
|
---|
705 | {
|
---|
706 | $x->{_m} = $MBI->_lsft($x->{_m}, $x->{_e},10); # 1e2 => 100
|
---|
707 | $x->{_e} = $MBI->_zero(); # normalize
|
---|
708 | $x->{_es} = '+';
|
---|
709 | }
|
---|
710 | # now $x->{_e} == 0
|
---|
711 | my $zero = $x->is_zero();
|
---|
712 | # <= 0
|
---|
713 | if (($x->{sign} eq '-') || $zero)
|
---|
714 | {
|
---|
715 | $MBI->_inc($x->{_m});
|
---|
716 | $x->{sign} = '-' if $zero; # 0 => 1 => -1
|
---|
717 | $x->{sign} = '+' if $MBI->_is_zero($x->{_m}); # -1 +1 => -0 => +0
|
---|
718 | return $x->bnorm()->round(@r);
|
---|
719 | }
|
---|
720 | # > 0
|
---|
721 | elsif ($x->{sign} eq '+')
|
---|
722 | {
|
---|
723 | $MBI->_dec($x->{_m});
|
---|
724 | return $x->bnorm()->round(@r);
|
---|
725 | }
|
---|
726 | # inf, nan handling etc
|
---|
727 | $x->badd($self->bone('-'),@r); # does round
|
---|
728 | }
|
---|
729 |
|
---|
730 | sub DEBUG () { 0; }
|
---|
731 |
|
---|
732 | sub blog
|
---|
733 | {
|
---|
734 | my ($self,$x,$base,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
|
---|
735 |
|
---|
736 | # $base > 0, $base != 1; if $base == undef default to $base == e
|
---|
737 | # $x >= 0
|
---|
738 |
|
---|
739 | # we need to limit the accuracy to protect against overflow
|
---|
740 | my $fallback = 0;
|
---|
741 | my ($scale,@params);
|
---|
742 | ($x,@params) = $x->_find_round_parameters($a,$p,$r);
|
---|
743 |
|
---|
744 | # also takes care of the "error in _find_round_parameters?" case
|
---|
745 | return $x->bnan() if $x->{sign} ne '+' || $x->is_zero();
|
---|
746 |
|
---|
747 |
|
---|
748 | # no rounding at all, so must use fallback
|
---|
749 | if (scalar @params == 0)
|
---|
750 | {
|
---|
751 | # simulate old behaviour
|
---|
752 | $params[0] = $self->div_scale(); # and round to it as accuracy
|
---|
753 | $params[1] = undef; # P = undef
|
---|
754 | $scale = $params[0]+4; # at least four more for proper round
|
---|
755 | $params[2] = $r; # round mode by caller or undef
|
---|
756 | $fallback = 1; # to clear a/p afterwards
|
---|
757 | }
|
---|
758 | else
|
---|
759 | {
|
---|
760 | # the 4 below is empirical, and there might be cases where it is not
|
---|
761 | # enough...
|
---|
762 | $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
|
---|
763 | }
|
---|
764 |
|
---|
765 | return $x->bzero(@params) if $x->is_one();
|
---|
766 | # base not defined => base == Euler's constant e
|
---|
767 | if (defined $base)
|
---|
768 | {
|
---|
769 | # make object, since we don't feed it through objectify() to still get the
|
---|
770 | # case of $base == undef
|
---|
771 | $base = $self->new($base) unless ref($base);
|
---|
772 | # $base > 0; $base != 1
|
---|
773 | return $x->bnan() if $base->is_zero() || $base->is_one() ||
|
---|
774 | $base->{sign} ne '+';
|
---|
775 | # if $x == $base, we know the result must be 1.0
|
---|
776 | if ($x->bcmp($base) == 0)
|
---|
777 | {
|
---|
778 | $x->bone('+',@params);
|
---|
779 | if ($fallback)
|
---|
780 | {
|
---|
781 | # clear a/p after round, since user did not request it
|
---|
782 | delete $x->{_a}; delete $x->{_p};
|
---|
783 | }
|
---|
784 | return $x;
|
---|
785 | }
|
---|
786 | }
|
---|
787 |
|
---|
788 | # when user set globals, they would interfere with our calculation, so
|
---|
789 | # disable them and later re-enable them
|
---|
790 | no strict 'refs';
|
---|
791 | my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef;
|
---|
792 | my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef;
|
---|
793 | # we also need to disable any set A or P on $x (_find_round_parameters took
|
---|
794 | # them already into account), since these would interfere, too
|
---|
795 | delete $x->{_a}; delete $x->{_p};
|
---|
796 | # need to disable $upgrade in BigInt, to avoid deep recursion
|
---|
797 | local $Math::BigInt::upgrade = undef;
|
---|
798 | local $Math::BigFloat::downgrade = undef;
|
---|
799 |
|
---|
800 | # upgrade $x if $x is not a BigFloat (handle BigInt input)
|
---|
801 | if (!$x->isa('Math::BigFloat'))
|
---|
802 | {
|
---|
803 | $x = Math::BigFloat->new($x);
|
---|
804 | $self = ref($x);
|
---|
805 | }
|
---|
806 |
|
---|
807 | my $done = 0;
|
---|
808 |
|
---|
809 | # If the base is defined and an integer, try to calculate integer result
|
---|
810 | # first. This is very fast, and in case the real result was found, we can
|
---|
811 | # stop right here.
|
---|
812 | if (defined $base && $base->is_int() && $x->is_int())
|
---|
813 | {
|
---|
814 | my $i = $MBI->_copy( $x->{_m} );
|
---|
815 | $MBI->_lsft( $i, $x->{_e}, 10 ) unless $MBI->_is_zero($x->{_e});
|
---|
816 | my $int = Math::BigInt->bzero();
|
---|
817 | $int->{value} = $i;
|
---|
818 | $int->blog($base->as_number());
|
---|
819 | # if ($exact)
|
---|
820 | if ($base->as_number()->bpow($int) == $x)
|
---|
821 | {
|
---|
822 | # found result, return it
|
---|
823 | $x->{_m} = $int->{value};
|
---|
824 | $x->{_e} = $MBI->_zero();
|
---|
825 | $x->{_es} = '+';
|
---|
826 | $x->bnorm();
|
---|
827 | $done = 1;
|
---|
828 | }
|
---|
829 | }
|
---|
830 |
|
---|
831 | if ($done == 0)
|
---|
832 | {
|
---|
833 | # first calculate the log to base e (using reduction by 10 (and probably 2))
|
---|
834 | $self->_log_10($x,$scale);
|
---|
835 |
|
---|
836 | # and if a different base was requested, convert it
|
---|
837 | if (defined $base)
|
---|
838 | {
|
---|
839 | $base = Math::BigFloat->new($base) unless $base->isa('Math::BigFloat');
|
---|
840 | # not ln, but some other base (don't modify $base)
|
---|
841 | $x->bdiv( $base->copy()->blog(undef,$scale), $scale );
|
---|
842 | }
|
---|
843 | }
|
---|
844 |
|
---|
845 | # shortcut to not run through _find_round_parameters again
|
---|
846 | if (defined $params[0])
|
---|
847 | {
|
---|
848 | $x->bround($params[0],$params[2]); # then round accordingly
|
---|
849 | }
|
---|
850 | else
|
---|
851 | {
|
---|
852 | $x->bfround($params[1],$params[2]); # then round accordingly
|
---|
853 | }
|
---|
854 | if ($fallback)
|
---|
855 | {
|
---|
856 | # clear a/p after round, since user did not request it
|
---|
857 | delete $x->{_a}; delete $x->{_p};
|
---|
858 | }
|
---|
859 | # restore globals
|
---|
860 | $$abr = $ab; $$pbr = $pb;
|
---|
861 |
|
---|
862 | $x;
|
---|
863 | }
|
---|
864 |
|
---|
865 | sub _log
|
---|
866 | {
|
---|
867 | # internal log function to calculate ln() based on Taylor series.
|
---|
868 | # Modifies $x in place.
|
---|
869 | my ($self,$x,$scale) = @_;
|
---|
870 |
|
---|
871 | # in case of $x == 1, result is 0
|
---|
872 | return $x->bzero() if $x->is_one();
|
---|
873 |
|
---|
874 | # http://www.efunda.com/math/taylor_series/logarithmic.cfm?search_string=log
|
---|
875 |
|
---|
876 | # u = x-1, v = x+1
|
---|
877 | # _ _
|
---|
878 | # Taylor: | u 1 u^3 1 u^5 |
|
---|
879 | # ln (x) = 2 | --- + - * --- + - * --- + ... | x > 0
|
---|
880 | # |_ v 3 v^3 5 v^5 _|
|
---|
881 |
|
---|
882 | # This takes much more steps to calculate the result and is thus not used
|
---|
883 | # u = x-1
|
---|
884 | # _ _
|
---|
885 | # Taylor: | u 1 u^2 1 u^3 |
|
---|
886 | # ln (x) = 2 | --- + - * --- + - * --- + ... | x > 1/2
|
---|
887 | # |_ x 2 x^2 3 x^3 _|
|
---|
888 |
|
---|
889 | my ($limit,$v,$u,$below,$factor,$two,$next,$over,$f);
|
---|
890 |
|
---|
891 | $v = $x->copy(); $v->binc(); # v = x+1
|
---|
892 | $x->bdec(); $u = $x->copy(); # u = x-1; x = x-1
|
---|
893 | $x->bdiv($v,$scale); # first term: u/v
|
---|
894 | $below = $v->copy();
|
---|
895 | $over = $u->copy();
|
---|
896 | $u *= $u; $v *= $v; # u^2, v^2
|
---|
897 | $below->bmul($v); # u^3, v^3
|
---|
898 | $over->bmul($u);
|
---|
899 | $factor = $self->new(3); $f = $self->new(2);
|
---|
900 |
|
---|
901 | my $steps = 0 if DEBUG;
|
---|
902 | $limit = $self->new("1E-". ($scale-1));
|
---|
903 | while (3 < 5)
|
---|
904 | {
|
---|
905 | # we calculate the next term, and add it to the last
|
---|
906 | # when the next term is below our limit, it won't affect the outcome
|
---|
907 | # anymore, so we stop
|
---|
908 |
|
---|
909 | # calculating the next term simple from over/below will result in quite
|
---|
910 | # a time hog if the input has many digits, since over and below will
|
---|
911 | # accumulate more and more digits, and the result will also have many
|
---|
912 | # digits, but in the end it is rounded to $scale digits anyway. So if we
|
---|
913 | # round $over and $below first, we save a lot of time for the division
|
---|
914 | # (not with log(1.2345), but try log (123**123) to see what I mean. This
|
---|
915 | # can introduce a rounding error if the division result would be f.i.
|
---|
916 | # 0.1234500000001 and we round it to 5 digits it would become 0.12346, but
|
---|
917 | # if we truncated $over and $below we might get 0.12345. Does this matter
|
---|
918 | # for the end result? So we give $over and $below 4 more digits to be
|
---|
919 | # on the safe side (unscientific error handling as usual... :+D
|
---|
920 |
|
---|
921 | $next = $over->copy->bround($scale+4)->bdiv(
|
---|
922 | $below->copy->bmul($factor)->bround($scale+4),
|
---|
923 | $scale);
|
---|
924 |
|
---|
925 | ## old version:
|
---|
926 | ## $next = $over->copy()->bdiv($below->copy()->bmul($factor),$scale);
|
---|
927 |
|
---|
928 | last if $next->bacmp($limit) <= 0;
|
---|
929 |
|
---|
930 | delete $next->{_a}; delete $next->{_p};
|
---|
931 | $x->badd($next);
|
---|
932 | # calculate things for the next term
|
---|
933 | $over *= $u; $below *= $v; $factor->badd($f);
|
---|
934 | if (DEBUG)
|
---|
935 | {
|
---|
936 | $steps++; print "step $steps = $x\n" if $steps % 10 == 0;
|
---|
937 | }
|
---|
938 | }
|
---|
939 | $x->bmul($f); # $x *= 2
|
---|
940 | print "took $steps steps\n" if DEBUG;
|
---|
941 | }
|
---|
942 |
|
---|
943 | sub _log_10
|
---|
944 | {
|
---|
945 | # Internal log function based on reducing input to the range of 0.1 .. 9.99
|
---|
946 | # and then "correcting" the result to the proper one. Modifies $x in place.
|
---|
947 | my ($self,$x,$scale) = @_;
|
---|
948 |
|
---|
949 | # taking blog() from numbers greater than 10 takes a *very long* time, so we
|
---|
950 | # break the computation down into parts based on the observation that:
|
---|
951 | # blog(x*y) = blog(x) + blog(y)
|
---|
952 | # We set $y here to multiples of 10 so that $x is below 1 (the smaller $x is
|
---|
953 | # the faster it get's, especially because 2*$x takes about 10 times as long,
|
---|
954 | # so by dividing $x by 10 we make it at least factor 100 faster...)
|
---|
955 |
|
---|
956 | # The same observation is valid for numbers smaller than 0.1 (e.g. computing
|
---|
957 | # log(1) is fastest, and the farther away we get from 1, the longer it takes)
|
---|
958 | # so we also 'break' this down by multiplying $x with 10 and subtract the
|
---|
959 | # log(10) afterwards to get the correct result.
|
---|
960 |
|
---|
961 | # calculate nr of digits before dot
|
---|
962 | my $dbd = $MBI->_num($x->{_e});
|
---|
963 | $dbd = -$dbd if $x->{_es} eq '-';
|
---|
964 | $dbd += $MBI->_len($x->{_m});
|
---|
965 |
|
---|
966 | # more than one digit (e.g. at least 10), but *not* exactly 10 to avoid
|
---|
967 | # infinite recursion
|
---|
968 |
|
---|
969 | my $calc = 1; # do some calculation?
|
---|
970 |
|
---|
971 | # disable the shortcut for 10, since we need log(10) and this would recurse
|
---|
972 | # infinitely deep
|
---|
973 | if ($x->{_es} eq '+' && $MBI->_is_one($x->{_e}) && $MBI->_is_one($x->{_m}))
|
---|
974 | {
|
---|
975 | $dbd = 0; # disable shortcut
|
---|
976 | # we can use the cached value in these cases
|
---|
977 | if ($scale <= $LOG_10_A)
|
---|
978 | {
|
---|
979 | $x->bzero(); $x->badd($LOG_10);
|
---|
980 | $calc = 0; # no need to calc, but round
|
---|
981 | }
|
---|
982 | }
|
---|
983 | else
|
---|
984 | {
|
---|
985 | # disable the shortcut for 2, since we maybe have it cached
|
---|
986 | if (($MBI->_is_zero($x->{_e}) && $MBI->_is_two($x->{_m})))
|
---|
987 | {
|
---|
988 | $dbd = 0; # disable shortcut
|
---|
989 | # we can use the cached value in these cases
|
---|
990 | if ($scale <= $LOG_2_A)
|
---|
991 | {
|
---|
992 | $x->bzero(); $x->badd($LOG_2);
|
---|
993 | $calc = 0; # no need to calc, but round
|
---|
994 | }
|
---|
995 | }
|
---|
996 | }
|
---|
997 |
|
---|
998 | # if $x = 0.1, we know the result must be 0-log(10)
|
---|
999 | if ($calc != 0 && $x->{_es} eq '-' && $MBI->_is_one($x->{_e}) &&
|
---|
1000 | $MBI->_is_one($x->{_m}))
|
---|
1001 | {
|
---|
1002 | $dbd = 0; # disable shortcut
|
---|
1003 | # we can use the cached value in these cases
|
---|
1004 | if ($scale <= $LOG_10_A)
|
---|
1005 | {
|
---|
1006 | $x->bzero(); $x->bsub($LOG_10);
|
---|
1007 | $calc = 0; # no need to calc, but round
|
---|
1008 | }
|
---|
1009 | }
|
---|
1010 |
|
---|
1011 | return if $calc == 0; # already have the result
|
---|
1012 |
|
---|
1013 | # default: these correction factors are undef and thus not used
|
---|
1014 | my $l_10; # value of ln(10) to A of $scale
|
---|
1015 | my $l_2; # value of ln(2) to A of $scale
|
---|
1016 |
|
---|
1017 | # $x == 2 => 1, $x == 13 => 2, $x == 0.1 => 0, $x == 0.01 => -1
|
---|
1018 | # so don't do this shortcut for 1 or 0
|
---|
1019 | if (($dbd > 1) || ($dbd < 0))
|
---|
1020 | {
|
---|
1021 | # convert our cached value to an object if not already (avoid doing this
|
---|
1022 | # at import() time, since not everybody needs this)
|
---|
1023 | $LOG_10 = $self->new($LOG_10,undef,undef) unless ref $LOG_10;
|
---|
1024 |
|
---|
1025 | #print "x = $x, dbd = $dbd, calc = $calc\n";
|
---|
1026 | # got more than one digit before the dot, or more than one zero after the
|
---|
1027 | # dot, so do:
|
---|
1028 | # log(123) == log(1.23) + log(10) * 2
|
---|
1029 | # log(0.0123) == log(1.23) - log(10) * 2
|
---|
1030 |
|
---|
1031 | if ($scale <= $LOG_10_A)
|
---|
1032 | {
|
---|
1033 | # use cached value
|
---|
1034 | $l_10 = $LOG_10->copy(); # copy for mul
|
---|
1035 | }
|
---|
1036 | else
|
---|
1037 | {
|
---|
1038 | # else: slower, compute it (but don't cache it, because it could be big)
|
---|
1039 | # also disable downgrade for this code path
|
---|
1040 | local $Math::BigFloat::downgrade = undef;
|
---|
1041 | $l_10 = $self->new(10)->blog(undef,$scale); # scale+4, actually
|
---|
1042 | }
|
---|
1043 | $dbd-- if ($dbd > 1); # 20 => dbd=2, so make it dbd=1
|
---|
1044 | $l_10->bmul( $self->new($dbd)); # log(10) * (digits_before_dot-1)
|
---|
1045 | my $dbd_sign = '+';
|
---|
1046 | if ($dbd < 0)
|
---|
1047 | {
|
---|
1048 | $dbd = -$dbd;
|
---|
1049 | $dbd_sign = '-';
|
---|
1050 | }
|
---|
1051 | ($x->{_e}, $x->{_es}) =
|
---|
1052 | _e_sub( $x->{_e}, $MBI->_new($dbd), $x->{_es}, $dbd_sign); # 123 => 1.23
|
---|
1053 |
|
---|
1054 | }
|
---|
1055 |
|
---|
1056 | # Now: 0.1 <= $x < 10 (and possible correction in l_10)
|
---|
1057 |
|
---|
1058 | ### Since $x in the range 0.5 .. 1.5 is MUCH faster, we do a repeated div
|
---|
1059 | ### or mul by 2 (maximum times 3, since x < 10 and x > 0.1)
|
---|
1060 |
|
---|
1061 | $HALF = $self->new($HALF) unless ref($HALF);
|
---|
1062 |
|
---|
1063 | my $twos = 0; # default: none (0 times)
|
---|
1064 | my $two = $self->new(2);
|
---|
1065 | while ($x->bacmp($HALF) <= 0)
|
---|
1066 | {
|
---|
1067 | $twos--; $x->bmul($two);
|
---|
1068 | }
|
---|
1069 | while ($x->bacmp($two) >= 0)
|
---|
1070 | {
|
---|
1071 | $twos++; $x->bdiv($two,$scale+4); # keep all digits
|
---|
1072 | }
|
---|
1073 | # $twos > 0 => did mul 2, < 0 => did div 2 (never both)
|
---|
1074 | # calculate correction factor based on ln(2)
|
---|
1075 | if ($twos != 0)
|
---|
1076 | {
|
---|
1077 | $LOG_2 = $self->new($LOG_2,undef,undef) unless ref $LOG_2;
|
---|
1078 | if ($scale <= $LOG_2_A)
|
---|
1079 | {
|
---|
1080 | # use cached value
|
---|
1081 | $l_2 = $LOG_2->copy(); # copy for mul
|
---|
1082 | }
|
---|
1083 | else
|
---|
1084 | {
|
---|
1085 | # else: slower, compute it (but don't cache it, because it could be big)
|
---|
1086 | # also disable downgrade for this code path
|
---|
1087 | local $Math::BigFloat::downgrade = undef;
|
---|
1088 | $l_2 = $two->blog(undef,$scale); # scale+4, actually
|
---|
1089 | }
|
---|
1090 | $l_2->bmul($twos); # * -2 => subtract, * 2 => add
|
---|
1091 | }
|
---|
1092 |
|
---|
1093 | $self->_log($x,$scale); # need to do the "normal" way
|
---|
1094 | $x->badd($l_10) if defined $l_10; # correct it by ln(10)
|
---|
1095 | $x->badd($l_2) if defined $l_2; # and maybe by ln(2)
|
---|
1096 | # all done, $x contains now the result
|
---|
1097 | }
|
---|
1098 |
|
---|
1099 | sub blcm
|
---|
1100 | {
|
---|
1101 | # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT
|
---|
1102 | # does not modify arguments, but returns new object
|
---|
1103 | # Lowest Common Multiplicator
|
---|
1104 |
|
---|
1105 | my ($self,@arg) = objectify(0,@_);
|
---|
1106 | my $x = $self->new(shift @arg);
|
---|
1107 | while (@arg) { $x = Math::BigInt::__lcm($x,shift @arg); }
|
---|
1108 | $x;
|
---|
1109 | }
|
---|
1110 |
|
---|
1111 | sub bgcd
|
---|
1112 | {
|
---|
1113 | # (BINT or num_str, BINT or num_str) return BINT
|
---|
1114 | # does not modify arguments, but returns new object
|
---|
1115 |
|
---|
1116 | my $y = shift;
|
---|
1117 | $y = __PACKAGE__->new($y) if !ref($y);
|
---|
1118 | my $self = ref($y);
|
---|
1119 | my $x = $y->copy()->babs(); # keep arguments
|
---|
1120 |
|
---|
1121 | return $x->bnan() if $x->{sign} !~ /^[+-]$/ # x NaN?
|
---|
1122 | || !$x->is_int(); # only for integers now
|
---|
1123 |
|
---|
1124 | while (@_)
|
---|
1125 | {
|
---|
1126 | my $t = shift; $t = $self->new($t) if !ref($t);
|
---|
1127 | $y = $t->copy()->babs();
|
---|
1128 |
|
---|
1129 | return $x->bnan() if $y->{sign} !~ /^[+-]$/ # y NaN?
|
---|
1130 | || !$y->is_int(); # only for integers now
|
---|
1131 |
|
---|
1132 | # greatest common divisor
|
---|
1133 | while (! $y->is_zero())
|
---|
1134 | {
|
---|
1135 | ($x,$y) = ($y->copy(), $x->copy()->bmod($y));
|
---|
1136 | }
|
---|
1137 |
|
---|
1138 | last if $x->is_one();
|
---|
1139 | }
|
---|
1140 | $x;
|
---|
1141 | }
|
---|
1142 |
|
---|
1143 | ##############################################################################
|
---|
1144 |
|
---|
1145 | sub _e_add
|
---|
1146 | {
|
---|
1147 | # Internal helper sub to take two positive integers and their signs and
|
---|
1148 | # then add them. Input ($CALC,$CALC,('+'|'-'),('+'|'-')),
|
---|
1149 | # output ($CALC,('+'|'-'))
|
---|
1150 | my ($x,$y,$xs,$ys) = @_;
|
---|
1151 |
|
---|
1152 | # if the signs are equal we can add them (-5 + -3 => -(5 + 3) => -8)
|
---|
1153 | if ($xs eq $ys)
|
---|
1154 | {
|
---|
1155 | $x = $MBI->_add ($x, $y ); # a+b
|
---|
1156 | # the sign follows $xs
|
---|
1157 | return ($x, $xs);
|
---|
1158 | }
|
---|
1159 |
|
---|
1160 | my $a = $MBI->_acmp($x,$y);
|
---|
1161 | if ($a > 0)
|
---|
1162 | {
|
---|
1163 | $x = $MBI->_sub ($x , $y); # abs sub
|
---|
1164 | }
|
---|
1165 | elsif ($a == 0)
|
---|
1166 | {
|
---|
1167 | $x = $MBI->_zero(); # result is 0
|
---|
1168 | $xs = '+';
|
---|
1169 | }
|
---|
1170 | else # a < 0
|
---|
1171 | {
|
---|
1172 | $x = $MBI->_sub ( $y, $x, 1 ); # abs sub
|
---|
1173 | $xs = $ys;
|
---|
1174 | }
|
---|
1175 | ($x,$xs);
|
---|
1176 | }
|
---|
1177 |
|
---|
1178 | sub _e_sub
|
---|
1179 | {
|
---|
1180 | # Internal helper sub to take two positive integers and their signs and
|
---|
1181 | # then subtract them. Input ($CALC,$CALC,('+'|'-'),('+'|'-')),
|
---|
1182 | # output ($CALC,('+'|'-'))
|
---|
1183 | my ($x,$y,$xs,$ys) = @_;
|
---|
1184 |
|
---|
1185 | # flip sign
|
---|
1186 | $ys =~ tr/+-/-+/;
|
---|
1187 | _e_add($x,$y,$xs,$ys); # call add (does subtract now)
|
---|
1188 | }
|
---|
1189 |
|
---|
1190 | ###############################################################################
|
---|
1191 | # is_foo methods (is_negative, is_positive are inherited from BigInt)
|
---|
1192 |
|
---|
1193 | sub is_int
|
---|
1194 | {
|
---|
1195 | # return true if arg (BFLOAT or num_str) is an integer
|
---|
1196 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
|
---|
1197 |
|
---|
1198 | return 1 if ($x->{sign} =~ /^[+-]$/) && # NaN and +-inf aren't
|
---|
1199 | $x->{_es} eq '+'; # 1e-1 => no integer
|
---|
1200 | 0;
|
---|
1201 | }
|
---|
1202 |
|
---|
1203 | sub is_zero
|
---|
1204 | {
|
---|
1205 | # return true if arg (BFLOAT or num_str) is zero
|
---|
1206 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
|
---|
1207 |
|
---|
1208 | return 1 if $x->{sign} eq '+' && $MBI->_is_zero($x->{_m});
|
---|
1209 | 0;
|
---|
1210 | }
|
---|
1211 |
|
---|
1212 | sub is_one
|
---|
1213 | {
|
---|
1214 | # return true if arg (BFLOAT or num_str) is +1 or -1 if signis given
|
---|
1215 | my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
|
---|
1216 |
|
---|
1217 | $sign = '+' if !defined $sign || $sign ne '-';
|
---|
1218 | return 1
|
---|
1219 | if ($x->{sign} eq $sign &&
|
---|
1220 | $MBI->_is_zero($x->{_e}) && $MBI->_is_one($x->{_m}));
|
---|
1221 | 0;
|
---|
1222 | }
|
---|
1223 |
|
---|
1224 | sub is_odd
|
---|
1225 | {
|
---|
1226 | # return true if arg (BFLOAT or num_str) is odd or false if even
|
---|
1227 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
|
---|
1228 |
|
---|
1229 | return 1 if ($x->{sign} =~ /^[+-]$/) && # NaN & +-inf aren't
|
---|
1230 | ($MBI->_is_zero($x->{_e}) && $MBI->_is_odd($x->{_m}));
|
---|
1231 | 0;
|
---|
1232 | }
|
---|
1233 |
|
---|
1234 | sub is_even
|
---|
1235 | {
|
---|
1236 | # return true if arg (BINT or num_str) is even or false if odd
|
---|
1237 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
|
---|
1238 |
|
---|
1239 | return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
|
---|
1240 | return 1 if ($x->{_es} eq '+' # 123.45 is never
|
---|
1241 | && $MBI->_is_even($x->{_m})); # but 1200 is
|
---|
1242 | 0;
|
---|
1243 | }
|
---|
1244 |
|
---|
1245 | sub bmul
|
---|
1246 | {
|
---|
1247 | # multiply two numbers -- stolen from Knuth Vol 2 pg 233
|
---|
1248 | # (BINT or num_str, BINT or num_str) return BINT
|
---|
1249 |
|
---|
1250 | # set up parameters
|
---|
1251 | my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_);
|
---|
1252 | # objectify is costly, so avoid it
|
---|
1253 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
|
---|
1254 | {
|
---|
1255 | ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
|
---|
1256 | }
|
---|
1257 |
|
---|
1258 | return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
|
---|
1259 |
|
---|
1260 | # inf handling
|
---|
1261 | if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
|
---|
1262 | {
|
---|
1263 | return $x->bnan() if $x->is_zero() || $y->is_zero();
|
---|
1264 | # result will always be +-inf:
|
---|
1265 | # +inf * +/+inf => +inf, -inf * -/-inf => +inf
|
---|
1266 | # +inf * -/-inf => -inf, -inf * +/+inf => -inf
|
---|
1267 | return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
|
---|
1268 | return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
|
---|
1269 | return $x->binf('-');
|
---|
1270 | }
|
---|
1271 | # handle result = 0
|
---|
1272 | return $x->bzero() if $x->is_zero() || $y->is_zero();
|
---|
1273 |
|
---|
1274 | return $upgrade->bmul($x,$y,$a,$p,$r) if defined $upgrade &&
|
---|
1275 | ((!$x->isa($self)) || (!$y->isa($self)));
|
---|
1276 |
|
---|
1277 | # aEb * cEd = (a*c)E(b+d)
|
---|
1278 | $MBI->_mul($x->{_m},$y->{_m});
|
---|
1279 | ($x->{_e}, $x->{_es}) = _e_add($x->{_e}, $y->{_e}, $x->{_es}, $y->{_es});
|
---|
1280 |
|
---|
1281 | # adjust sign:
|
---|
1282 | $x->{sign} = $x->{sign} ne $y->{sign} ? '-' : '+';
|
---|
1283 | return $x->bnorm()->round($a,$p,$r,$y);
|
---|
1284 | }
|
---|
1285 |
|
---|
1286 | sub bdiv
|
---|
1287 | {
|
---|
1288 | # (dividend: BFLOAT or num_str, divisor: BFLOAT or num_str) return
|
---|
1289 | # (BFLOAT,BFLOAT) (quo,rem) or BFLOAT (only rem)
|
---|
1290 |
|
---|
1291 | # set up parameters
|
---|
1292 | my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_);
|
---|
1293 | # objectify is costly, so avoid it
|
---|
1294 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
|
---|
1295 | {
|
---|
1296 | ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
|
---|
1297 | }
|
---|
1298 |
|
---|
1299 | return $self->_div_inf($x,$y)
|
---|
1300 | if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
|
---|
1301 |
|
---|
1302 | # x== 0 # also: or y == 1 or y == -1
|
---|
1303 | return wantarray ? ($x,$self->bzero()) : $x if $x->is_zero();
|
---|
1304 |
|
---|
1305 | # upgrade ?
|
---|
1306 | return $upgrade->bdiv($upgrade->new($x),$y,$a,$p,$r) if defined $upgrade;
|
---|
1307 |
|
---|
1308 | # we need to limit the accuracy to protect against overflow
|
---|
1309 | my $fallback = 0;
|
---|
1310 | my (@params,$scale);
|
---|
1311 | ($x,@params) = $x->_find_round_parameters($a,$p,$r,$y);
|
---|
1312 |
|
---|
1313 | return $x if $x->is_nan(); # error in _find_round_parameters?
|
---|
1314 |
|
---|
1315 | # no rounding at all, so must use fallback
|
---|
1316 | if (scalar @params == 0)
|
---|
1317 | {
|
---|
1318 | # simulate old behaviour
|
---|
1319 | $params[0] = $self->div_scale(); # and round to it as accuracy
|
---|
1320 | $scale = $params[0]+4; # at least four more for proper round
|
---|
1321 | $params[2] = $r; # round mode by caller or undef
|
---|
1322 | $fallback = 1; # to clear a/p afterwards
|
---|
1323 | }
|
---|
1324 | else
|
---|
1325 | {
|
---|
1326 | # the 4 below is empirical, and there might be cases where it is not
|
---|
1327 | # enough...
|
---|
1328 | $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
|
---|
1329 | }
|
---|
1330 |
|
---|
1331 | my $rem; $rem = $self->bzero() if wantarray;
|
---|
1332 |
|
---|
1333 | $y = $self->new($y) unless $y->isa('Math::BigFloat');
|
---|
1334 |
|
---|
1335 | my $lx = $MBI->_len($x->{_m}); my $ly = $MBI->_len($y->{_m});
|
---|
1336 | $scale = $lx if $lx > $scale;
|
---|
1337 | $scale = $ly if $ly > $scale;
|
---|
1338 | my $diff = $ly - $lx;
|
---|
1339 | $scale += $diff if $diff > 0; # if lx << ly, but not if ly << lx!
|
---|
1340 |
|
---|
1341 | # already handled inf/NaN/-inf above:
|
---|
1342 |
|
---|
1343 | # check that $y is not 1 nor -1 and cache the result:
|
---|
1344 | my $y_not_one = !($MBI->_is_zero($y->{_e}) && $MBI->_is_one($y->{_m}));
|
---|
1345 |
|
---|
1346 | # flipping the sign of $y will also flip the sign of $x for the special
|
---|
1347 | # case of $x->bsub($x); so we can catch it below:
|
---|
1348 | my $xsign = $x->{sign};
|
---|
1349 | $y->{sign} =~ tr/+-/-+/;
|
---|
1350 |
|
---|
1351 | if ($xsign ne $x->{sign})
|
---|
1352 | {
|
---|
1353 | # special case of $x /= $x results in 1
|
---|
1354 | $x->bone(); # "fixes" also sign of $y, since $x is $y
|
---|
1355 | }
|
---|
1356 | else
|
---|
1357 | {
|
---|
1358 | # correct $y's sign again
|
---|
1359 | $y->{sign} =~ tr/+-/-+/;
|
---|
1360 | # continue with normal div code:
|
---|
1361 |
|
---|
1362 | # make copy of $x in case of list context for later reminder calculation
|
---|
1363 | if (wantarray && $y_not_one)
|
---|
1364 | {
|
---|
1365 | $rem = $x->copy();
|
---|
1366 | }
|
---|
1367 |
|
---|
1368 | $x->{sign} = $x->{sign} ne $y->sign() ? '-' : '+';
|
---|
1369 |
|
---|
1370 | # check for / +-1 ( +/- 1E0)
|
---|
1371 | if ($y_not_one)
|
---|
1372 | {
|
---|
1373 | # promote BigInts and it's subclasses (except when already a BigFloat)
|
---|
1374 | $y = $self->new($y) unless $y->isa('Math::BigFloat');
|
---|
1375 |
|
---|
1376 | # calculate the result to $scale digits and then round it
|
---|
1377 | # a * 10 ** b / c * 10 ** d => a/c * 10 ** (b-d)
|
---|
1378 | $MBI->_lsft($x->{_m},$MBI->_new($scale),10);
|
---|
1379 | $MBI->_div ($x->{_m},$y->{_m}); # a/c
|
---|
1380 |
|
---|
1381 | # correct exponent of $x
|
---|
1382 | ($x->{_e},$x->{_es}) = _e_sub($x->{_e}, $y->{_e}, $x->{_es}, $y->{_es});
|
---|
1383 | # correct for 10**scale
|
---|
1384 | ($x->{_e},$x->{_es}) = _e_sub($x->{_e}, $MBI->_new($scale), $x->{_es}, '+');
|
---|
1385 | $x->bnorm(); # remove trailing 0's
|
---|
1386 | }
|
---|
1387 | } # ende else $x != $y
|
---|
1388 |
|
---|
1389 | # shortcut to not run through _find_round_parameters again
|
---|
1390 | if (defined $params[0])
|
---|
1391 | {
|
---|
1392 | delete $x->{_a}; # clear before round
|
---|
1393 | $x->bround($params[0],$params[2]); # then round accordingly
|
---|
1394 | }
|
---|
1395 | else
|
---|
1396 | {
|
---|
1397 | delete $x->{_p}; # clear before round
|
---|
1398 | $x->bfround($params[1],$params[2]); # then round accordingly
|
---|
1399 | }
|
---|
1400 | if ($fallback)
|
---|
1401 | {
|
---|
1402 | # clear a/p after round, since user did not request it
|
---|
1403 | delete $x->{_a}; delete $x->{_p};
|
---|
1404 | }
|
---|
1405 |
|
---|
1406 | if (wantarray)
|
---|
1407 | {
|
---|
1408 | if ($y_not_one)
|
---|
1409 | {
|
---|
1410 | $rem->bmod($y,@params); # copy already done
|
---|
1411 | }
|
---|
1412 | if ($fallback)
|
---|
1413 | {
|
---|
1414 | # clear a/p after round, since user did not request it
|
---|
1415 | delete $rem->{_a}; delete $rem->{_p};
|
---|
1416 | }
|
---|
1417 | return ($x,$rem);
|
---|
1418 | }
|
---|
1419 | $x;
|
---|
1420 | }
|
---|
1421 |
|
---|
1422 | sub bmod
|
---|
1423 | {
|
---|
1424 | # (dividend: BFLOAT or num_str, divisor: BFLOAT or num_str) return reminder
|
---|
1425 |
|
---|
1426 | # set up parameters
|
---|
1427 | my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_);
|
---|
1428 | # objectify is costly, so avoid it
|
---|
1429 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
|
---|
1430 | {
|
---|
1431 | ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
|
---|
1432 | }
|
---|
1433 |
|
---|
1434 | # handle NaN, inf, -inf
|
---|
1435 | if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
|
---|
1436 | {
|
---|
1437 | my ($d,$re) = $self->SUPER::_div_inf($x,$y);
|
---|
1438 | $x->{sign} = $re->{sign};
|
---|
1439 | $x->{_e} = $re->{_e};
|
---|
1440 | $x->{_m} = $re->{_m};
|
---|
1441 | return $x->round($a,$p,$r,$y);
|
---|
1442 | }
|
---|
1443 | if ($y->is_zero())
|
---|
1444 | {
|
---|
1445 | return $x->bnan() if $x->is_zero();
|
---|
1446 | return $x;
|
---|
1447 | }
|
---|
1448 |
|
---|
1449 | return $x->bzero() if $x->is_zero()
|
---|
1450 | || ($x->is_int() &&
|
---|
1451 | # check that $y == +1 or $y == -1:
|
---|
1452 | ($MBI->_is_zero($y->{_e}) && $MBI->_is_one($y->{_m})));
|
---|
1453 |
|
---|
1454 | my $cmp = $x->bacmp($y); # equal or $x < $y?
|
---|
1455 | return $x->bzero($a,$p) if $cmp == 0; # $x == $y => result 0
|
---|
1456 |
|
---|
1457 | # only $y of the operands negative?
|
---|
1458 | my $neg = 0; $neg = 1 if $x->{sign} ne $y->{sign};
|
---|
1459 |
|
---|
1460 | $x->{sign} = $y->{sign}; # calc sign first
|
---|
1461 | return $x->round($a,$p,$r) if $cmp < 0 && $neg == 0; # $x < $y => result $x
|
---|
1462 |
|
---|
1463 | my $ym = $MBI->_copy($y->{_m});
|
---|
1464 |
|
---|
1465 | # 2e1 => 20
|
---|
1466 | $MBI->_lsft( $ym, $y->{_e}, 10)
|
---|
1467 | if $y->{_es} eq '+' && !$MBI->_is_zero($y->{_e});
|
---|
1468 |
|
---|
1469 | # if $y has digits after dot
|
---|
1470 | my $shifty = 0; # correct _e of $x by this
|
---|
1471 | if ($y->{_es} eq '-') # has digits after dot
|
---|
1472 | {
|
---|
1473 | # 123 % 2.5 => 1230 % 25 => 5 => 0.5
|
---|
1474 | $shifty = $MBI->_num($y->{_e}); # no more digits after dot
|
---|
1475 | $MBI->_lsft($x->{_m}, $y->{_e}, 10);# 123 => 1230, $y->{_m} is already 25
|
---|
1476 | }
|
---|
1477 | # $ym is now mantissa of $y based on exponent 0
|
---|
1478 |
|
---|
1479 | my $shiftx = 0; # correct _e of $x by this
|
---|
1480 | if ($x->{_es} eq '-') # has digits after dot
|
---|
1481 | {
|
---|
1482 | # 123.4 % 20 => 1234 % 200
|
---|
1483 | $shiftx = $MBI->_num($x->{_e}); # no more digits after dot
|
---|
1484 | $MBI->_lsft($ym, $x->{_e}, 10); # 123 => 1230
|
---|
1485 | }
|
---|
1486 | # 123e1 % 20 => 1230 % 20
|
---|
1487 | if ($x->{_es} eq '+' && !$MBI->_is_zero($x->{_e}))
|
---|
1488 | {
|
---|
1489 | $MBI->_lsft( $x->{_m}, $x->{_e},10); # es => '+' here
|
---|
1490 | }
|
---|
1491 |
|
---|
1492 | $x->{_e} = $MBI->_new($shiftx);
|
---|
1493 | $x->{_es} = '+';
|
---|
1494 | $x->{_es} = '-' if $shiftx != 0 || $shifty != 0;
|
---|
1495 | $MBI->_add( $x->{_e}, $MBI->_new($shifty)) if $shifty != 0;
|
---|
1496 |
|
---|
1497 | # now mantissas are equalized, exponent of $x is adjusted, so calc result
|
---|
1498 |
|
---|
1499 | $x->{_m} = $MBI->_mod( $x->{_m}, $ym);
|
---|
1500 |
|
---|
1501 | $x->{sign} = '+' if $MBI->_is_zero($x->{_m}); # fix sign for -0
|
---|
1502 | $x->bnorm();
|
---|
1503 |
|
---|
1504 | if ($neg != 0) # one of them negative => correct in place
|
---|
1505 | {
|
---|
1506 | my $r = $y - $x;
|
---|
1507 | $x->{_m} = $r->{_m};
|
---|
1508 | $x->{_e} = $r->{_e};
|
---|
1509 | $x->{_es} = $r->{_es};
|
---|
1510 | $x->{sign} = '+' if $MBI->_is_zero($x->{_m}); # fix sign for -0
|
---|
1511 | $x->bnorm();
|
---|
1512 | }
|
---|
1513 |
|
---|
1514 | $x->round($a,$p,$r,$y); # round and return
|
---|
1515 | }
|
---|
1516 |
|
---|
1517 | sub broot
|
---|
1518 | {
|
---|
1519 | # calculate $y'th root of $x
|
---|
1520 |
|
---|
1521 | # set up parameters
|
---|
1522 | my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_);
|
---|
1523 | # objectify is costly, so avoid it
|
---|
1524 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
|
---|
1525 | {
|
---|
1526 | ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
|
---|
1527 | }
|
---|
1528 |
|
---|
1529 | # NaN handling: $x ** 1/0, x or y NaN, or y inf/-inf or y == 0
|
---|
1530 | return $x->bnan() if $x->{sign} !~ /^\+/ || $y->is_zero() ||
|
---|
1531 | $y->{sign} !~ /^\+$/;
|
---|
1532 |
|
---|
1533 | return $x if $x->is_zero() || $x->is_one() || $x->is_inf() || $y->is_one();
|
---|
1534 |
|
---|
1535 | # we need to limit the accuracy to protect against overflow
|
---|
1536 | my $fallback = 0;
|
---|
1537 | my (@params,$scale);
|
---|
1538 | ($x,@params) = $x->_find_round_parameters($a,$p,$r);
|
---|
1539 |
|
---|
1540 | return $x if $x->is_nan(); # error in _find_round_parameters?
|
---|
1541 |
|
---|
1542 | # no rounding at all, so must use fallback
|
---|
1543 | if (scalar @params == 0)
|
---|
1544 | {
|
---|
1545 | # simulate old behaviour
|
---|
1546 | $params[0] = $self->div_scale(); # and round to it as accuracy
|
---|
1547 | $scale = $params[0]+4; # at least four more for proper round
|
---|
1548 | $params[2] = $r; # iound mode by caller or undef
|
---|
1549 | $fallback = 1; # to clear a/p afterwards
|
---|
1550 | }
|
---|
1551 | else
|
---|
1552 | {
|
---|
1553 | # the 4 below is empirical, and there might be cases where it is not
|
---|
1554 | # enough...
|
---|
1555 | $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
|
---|
1556 | }
|
---|
1557 |
|
---|
1558 | # when user set globals, they would interfere with our calculation, so
|
---|
1559 | # disable them and later re-enable them
|
---|
1560 | no strict 'refs';
|
---|
1561 | my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef;
|
---|
1562 | my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef;
|
---|
1563 | # we also need to disable any set A or P on $x (_find_round_parameters took
|
---|
1564 | # them already into account), since these would interfere, too
|
---|
1565 | delete $x->{_a}; delete $x->{_p};
|
---|
1566 | # need to disable $upgrade in BigInt, to avoid deep recursion
|
---|
1567 | local $Math::BigInt::upgrade = undef; # should be really parent class vs MBI
|
---|
1568 |
|
---|
1569 | # remember sign and make $x positive, since -4 ** (1/2) => -2
|
---|
1570 | my $sign = 0; $sign = 1 if $x->{sign} eq '-'; $x->{sign} = '+';
|
---|
1571 |
|
---|
1572 | my $is_two = 0;
|
---|
1573 | if ($y->isa('Math::BigFloat'))
|
---|
1574 | {
|
---|
1575 | $is_two = ($y->{sign} eq '+' && $MBI->_is_two($y->{_m}) && $MBI->_is_zero($y->{_e}));
|
---|
1576 | }
|
---|
1577 | else
|
---|
1578 | {
|
---|
1579 | $is_two = ($y == 2);
|
---|
1580 | }
|
---|
1581 |
|
---|
1582 | # normal square root if $y == 2:
|
---|
1583 | if ($is_two)
|
---|
1584 | {
|
---|
1585 | $x->bsqrt($scale+4);
|
---|
1586 | }
|
---|
1587 | elsif ($y->is_one('-'))
|
---|
1588 | {
|
---|
1589 | # $x ** -1 => 1/$x
|
---|
1590 | my $u = $self->bone()->bdiv($x,$scale);
|
---|
1591 | # copy private parts over
|
---|
1592 | $x->{_m} = $u->{_m};
|
---|
1593 | $x->{_e} = $u->{_e};
|
---|
1594 | $x->{_es} = $u->{_es};
|
---|
1595 | }
|
---|
1596 | else
|
---|
1597 | {
|
---|
1598 | # calculate the broot() as integer result first, and if it fits, return
|
---|
1599 | # it rightaway (but only if $x and $y are integer):
|
---|
1600 |
|
---|
1601 | my $done = 0; # not yet
|
---|
1602 | if ($y->is_int() && $x->is_int())
|
---|
1603 | {
|
---|
1604 | my $i = $MBI->_copy( $x->{_m} );
|
---|
1605 | $MBI->_lsft( $i, $x->{_e}, 10 ) unless $MBI->_is_zero($x->{_e});
|
---|
1606 | my $int = Math::BigInt->bzero();
|
---|
1607 | $int->{value} = $i;
|
---|
1608 | $int->broot($y->as_number());
|
---|
1609 | # if ($exact)
|
---|
1610 | if ($int->copy()->bpow($y) == $x)
|
---|
1611 | {
|
---|
1612 | # found result, return it
|
---|
1613 | $x->{_m} = $int->{value};
|
---|
1614 | $x->{_e} = $MBI->_zero();
|
---|
1615 | $x->{_es} = '+';
|
---|
1616 | $x->bnorm();
|
---|
1617 | $done = 1;
|
---|
1618 | }
|
---|
1619 | }
|
---|
1620 | if ($done == 0)
|
---|
1621 | {
|
---|
1622 | my $u = $self->bone()->bdiv($y,$scale+4);
|
---|
1623 | delete $u->{_a}; delete $u->{_p}; # otherwise it conflicts
|
---|
1624 | $x->bpow($u,$scale+4); # el cheapo
|
---|
1625 | }
|
---|
1626 | }
|
---|
1627 | $x->bneg() if $sign == 1;
|
---|
1628 |
|
---|
1629 | # shortcut to not run through _find_round_parameters again
|
---|
1630 | if (defined $params[0])
|
---|
1631 | {
|
---|
1632 | $x->bround($params[0],$params[2]); # then round accordingly
|
---|
1633 | }
|
---|
1634 | else
|
---|
1635 | {
|
---|
1636 | $x->bfround($params[1],$params[2]); # then round accordingly
|
---|
1637 | }
|
---|
1638 | if ($fallback)
|
---|
1639 | {
|
---|
1640 | # clear a/p after round, since user did not request it
|
---|
1641 | delete $x->{_a}; delete $x->{_p};
|
---|
1642 | }
|
---|
1643 | # restore globals
|
---|
1644 | $$abr = $ab; $$pbr = $pb;
|
---|
1645 | $x;
|
---|
1646 | }
|
---|
1647 |
|
---|
1648 | sub bsqrt
|
---|
1649 | {
|
---|
1650 | # calculate square root
|
---|
1651 | my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
|
---|
1652 |
|
---|
1653 | return $x->bnan() if $x->{sign} !~ /^[+]/; # NaN, -inf or < 0
|
---|
1654 | return $x if $x->{sign} eq '+inf'; # sqrt(inf) == inf
|
---|
1655 | return $x->round($a,$p,$r) if $x->is_zero() || $x->is_one();
|
---|
1656 |
|
---|
1657 | # we need to limit the accuracy to protect against overflow
|
---|
1658 | my $fallback = 0;
|
---|
1659 | my (@params,$scale);
|
---|
1660 | ($x,@params) = $x->_find_round_parameters($a,$p,$r);
|
---|
1661 |
|
---|
1662 | return $x if $x->is_nan(); # error in _find_round_parameters?
|
---|
1663 |
|
---|
1664 | # no rounding at all, so must use fallback
|
---|
1665 | if (scalar @params == 0)
|
---|
1666 | {
|
---|
1667 | # simulate old behaviour
|
---|
1668 | $params[0] = $self->div_scale(); # and round to it as accuracy
|
---|
1669 | $scale = $params[0]+4; # at least four more for proper round
|
---|
1670 | $params[2] = $r; # round mode by caller or undef
|
---|
1671 | $fallback = 1; # to clear a/p afterwards
|
---|
1672 | }
|
---|
1673 | else
|
---|
1674 | {
|
---|
1675 | # the 4 below is empirical, and there might be cases where it is not
|
---|
1676 | # enough...
|
---|
1677 | $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
|
---|
1678 | }
|
---|
1679 |
|
---|
1680 | # when user set globals, they would interfere with our calculation, so
|
---|
1681 | # disable them and later re-enable them
|
---|
1682 | no strict 'refs';
|
---|
1683 | my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef;
|
---|
1684 | my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef;
|
---|
1685 | # we also need to disable any set A or P on $x (_find_round_parameters took
|
---|
1686 | # them already into account), since these would interfere, too
|
---|
1687 | delete $x->{_a}; delete $x->{_p};
|
---|
1688 | # need to disable $upgrade in BigInt, to avoid deep recursion
|
---|
1689 | local $Math::BigInt::upgrade = undef; # should be really parent class vs MBI
|
---|
1690 |
|
---|
1691 | my $i = $MBI->_copy( $x->{_m} );
|
---|
1692 | $MBI->_lsft( $i, $x->{_e}, 10 ) unless $MBI->_is_zero($x->{_e});
|
---|
1693 | my $xas = Math::BigInt->bzero();
|
---|
1694 | $xas->{value} = $i;
|
---|
1695 |
|
---|
1696 | my $gs = $xas->copy()->bsqrt(); # some guess
|
---|
1697 |
|
---|
1698 | if (($x->{_es} ne '-') # guess can't be accurate if there are
|
---|
1699 | # digits after the dot
|
---|
1700 | && ($xas->bacmp($gs * $gs) == 0)) # guess hit the nail on the head?
|
---|
1701 | {
|
---|
1702 | # exact result, copy result over to keep $x
|
---|
1703 | $x->{_m} = $gs->{value}; $x->{_e} = $MBI->_zero(); $x->{_es} = '+';
|
---|
1704 | $x->bnorm();
|
---|
1705 | # shortcut to not run through _find_round_parameters again
|
---|
1706 | if (defined $params[0])
|
---|
1707 | {
|
---|
1708 | $x->bround($params[0],$params[2]); # then round accordingly
|
---|
1709 | }
|
---|
1710 | else
|
---|
1711 | {
|
---|
1712 | $x->bfround($params[1],$params[2]); # then round accordingly
|
---|
1713 | }
|
---|
1714 | if ($fallback)
|
---|
1715 | {
|
---|
1716 | # clear a/p after round, since user did not request it
|
---|
1717 | delete $x->{_a}; delete $x->{_p};
|
---|
1718 | }
|
---|
1719 | # re-enable A and P, upgrade is taken care of by "local"
|
---|
1720 | ${"$self\::accuracy"} = $ab; ${"$self\::precision"} = $pb;
|
---|
1721 | return $x;
|
---|
1722 | }
|
---|
1723 |
|
---|
1724 | # sqrt(2) = 1.4 because sqrt(2*100) = 1.4*10; so we can increase the accuracy
|
---|
1725 | # of the result by multipyling the input by 100 and then divide the integer
|
---|
1726 | # result of sqrt(input) by 10. Rounding afterwards returns the real result.
|
---|
1727 |
|
---|
1728 | # The following steps will transform 123.456 (in $x) into 123456 (in $y1)
|
---|
1729 | my $y1 = $MBI->_copy($x->{_m});
|
---|
1730 |
|
---|
1731 | my $length = $MBI->_len($y1);
|
---|
1732 |
|
---|
1733 | # Now calculate how many digits the result of sqrt(y1) would have
|
---|
1734 | my $digits = int($length / 2);
|
---|
1735 |
|
---|
1736 | # But we need at least $scale digits, so calculate how many are missing
|
---|
1737 | my $shift = $scale - $digits;
|
---|
1738 |
|
---|
1739 | # That should never happen (we take care of integer guesses above)
|
---|
1740 | # $shift = 0 if $shift < 0;
|
---|
1741 |
|
---|
1742 | # Multiply in steps of 100, by shifting left two times the "missing" digits
|
---|
1743 | my $s2 = $shift * 2;
|
---|
1744 |
|
---|
1745 | # We now make sure that $y1 has the same odd or even number of digits than
|
---|
1746 | # $x had. So when _e of $x is odd, we must shift $y1 by one digit left,
|
---|
1747 | # because we always must multiply by steps of 100 (sqrt(100) is 10) and not
|
---|
1748 | # steps of 10. The length of $x does not count, since an even or odd number
|
---|
1749 | # of digits before the dot is not changed by adding an even number of digits
|
---|
1750 | # after the dot (the result is still odd or even digits long).
|
---|
1751 | $s2++ if $MBI->_is_odd($x->{_e});
|
---|
1752 |
|
---|
1753 | $MBI->_lsft( $y1, $MBI->_new($s2), 10);
|
---|
1754 |
|
---|
1755 | # now take the square root and truncate to integer
|
---|
1756 | $y1 = $MBI->_sqrt($y1);
|
---|
1757 |
|
---|
1758 | # By "shifting" $y1 right (by creating a negative _e) we calculate the final
|
---|
1759 | # result, which is than later rounded to the desired scale.
|
---|
1760 |
|
---|
1761 | # calculate how many zeros $x had after the '.' (or before it, depending
|
---|
1762 | # on sign of $dat, the result should have half as many:
|
---|
1763 | my $dat = $MBI->_num($x->{_e});
|
---|
1764 | $dat = -$dat if $x->{_es} eq '-';
|
---|
1765 | $dat += $length;
|
---|
1766 |
|
---|
1767 | if ($dat > 0)
|
---|
1768 | {
|
---|
1769 | # no zeros after the dot (e.g. 1.23, 0.49 etc)
|
---|
1770 | # preserve half as many digits before the dot than the input had
|
---|
1771 | # (but round this "up")
|
---|
1772 | $dat = int(($dat+1)/2);
|
---|
1773 | }
|
---|
1774 | else
|
---|
1775 | {
|
---|
1776 | $dat = int(($dat)/2);
|
---|
1777 | }
|
---|
1778 | $dat -= $MBI->_len($y1);
|
---|
1779 | if ($dat < 0)
|
---|
1780 | {
|
---|
1781 | $dat = abs($dat);
|
---|
1782 | $x->{_e} = $MBI->_new( $dat );
|
---|
1783 | $x->{_es} = '-';
|
---|
1784 | }
|
---|
1785 | else
|
---|
1786 | {
|
---|
1787 | $x->{_e} = $MBI->_new( $dat );
|
---|
1788 | $x->{_es} = '+';
|
---|
1789 | }
|
---|
1790 | $x->{_m} = $y1;
|
---|
1791 | $x->bnorm();
|
---|
1792 |
|
---|
1793 | # shortcut to not run through _find_round_parameters again
|
---|
1794 | if (defined $params[0])
|
---|
1795 | {
|
---|
1796 | $x->bround($params[0],$params[2]); # then round accordingly
|
---|
1797 | }
|
---|
1798 | else
|
---|
1799 | {
|
---|
1800 | $x->bfround($params[1],$params[2]); # then round accordingly
|
---|
1801 | }
|
---|
1802 | if ($fallback)
|
---|
1803 | {
|
---|
1804 | # clear a/p after round, since user did not request it
|
---|
1805 | delete $x->{_a}; delete $x->{_p};
|
---|
1806 | }
|
---|
1807 | # restore globals
|
---|
1808 | $$abr = $ab; $$pbr = $pb;
|
---|
1809 | $x;
|
---|
1810 | }
|
---|
1811 |
|
---|
1812 | sub bfac
|
---|
1813 | {
|
---|
1814 | # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT
|
---|
1815 | # compute factorial number, modifies first argument
|
---|
1816 |
|
---|
1817 | # set up parameters
|
---|
1818 | my ($self,$x,@r) = (ref($_[0]),@_);
|
---|
1819 | # objectify is costly, so avoid it
|
---|
1820 | ($self,$x,@r) = objectify(1,@_) if !ref($x);
|
---|
1821 |
|
---|
1822 | return $x if $x->{sign} eq '+inf'; # inf => inf
|
---|
1823 | return $x->bnan()
|
---|
1824 | if (($x->{sign} ne '+') || # inf, NaN, <0 etc => NaN
|
---|
1825 | ($x->{_es} ne '+')); # digits after dot?
|
---|
1826 |
|
---|
1827 | # use BigInt's bfac() for faster calc
|
---|
1828 | if (! $MBI->_is_zero($x->{_e}))
|
---|
1829 | {
|
---|
1830 | $MBI->_lsft($x->{_m}, $x->{_e},10); # change 12e1 to 120e0
|
---|
1831 | $x->{_e} = $MBI->_zero(); # normalize
|
---|
1832 | $x->{_es} = '+';
|
---|
1833 | }
|
---|
1834 | $MBI->_fac($x->{_m}); # calculate factorial
|
---|
1835 | $x->bnorm()->round(@r); # norm again and round result
|
---|
1836 | }
|
---|
1837 |
|
---|
1838 | sub _pow
|
---|
1839 | {
|
---|
1840 | # Calculate a power where $y is a non-integer, like 2 ** 0.5
|
---|
1841 | my ($x,$y,$a,$p,$r) = @_;
|
---|
1842 | my $self = ref($x);
|
---|
1843 |
|
---|
1844 | # if $y == 0.5, it is sqrt($x)
|
---|
1845 | $HALF = $self->new($HALF) unless ref($HALF);
|
---|
1846 | return $x->bsqrt($a,$p,$r,$y) if $y->bcmp($HALF) == 0;
|
---|
1847 |
|
---|
1848 | # Using:
|
---|
1849 | # a ** x == e ** (x * ln a)
|
---|
1850 |
|
---|
1851 | # u = y * ln x
|
---|
1852 | # _ _
|
---|
1853 | # Taylor: | u u^2 u^3 |
|
---|
1854 | # x ** y = 1 + | --- + --- + ----- + ... |
|
---|
1855 | # |_ 1 1*2 1*2*3 _|
|
---|
1856 |
|
---|
1857 | # we need to limit the accuracy to protect against overflow
|
---|
1858 | my $fallback = 0;
|
---|
1859 | my ($scale,@params);
|
---|
1860 | ($x,@params) = $x->_find_round_parameters($a,$p,$r);
|
---|
1861 |
|
---|
1862 | return $x if $x->is_nan(); # error in _find_round_parameters?
|
---|
1863 |
|
---|
1864 | # no rounding at all, so must use fallback
|
---|
1865 | if (scalar @params == 0)
|
---|
1866 | {
|
---|
1867 | # simulate old behaviour
|
---|
1868 | $params[0] = $self->div_scale(); # and round to it as accuracy
|
---|
1869 | $params[1] = undef; # disable P
|
---|
1870 | $scale = $params[0]+4; # at least four more for proper round
|
---|
1871 | $params[2] = $r; # round mode by caller or undef
|
---|
1872 | $fallback = 1; # to clear a/p afterwards
|
---|
1873 | }
|
---|
1874 | else
|
---|
1875 | {
|
---|
1876 | # the 4 below is empirical, and there might be cases where it is not
|
---|
1877 | # enough...
|
---|
1878 | $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
|
---|
1879 | }
|
---|
1880 |
|
---|
1881 | # when user set globals, they would interfere with our calculation, so
|
---|
1882 | # disable them and later re-enable them
|
---|
1883 | no strict 'refs';
|
---|
1884 | my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef;
|
---|
1885 | my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef;
|
---|
1886 | # we also need to disable any set A or P on $x (_find_round_parameters took
|
---|
1887 | # them already into account), since these would interfere, too
|
---|
1888 | delete $x->{_a}; delete $x->{_p};
|
---|
1889 | # need to disable $upgrade in BigInt, to avoid deep recursion
|
---|
1890 | local $Math::BigInt::upgrade = undef;
|
---|
1891 |
|
---|
1892 | my ($limit,$v,$u,$below,$factor,$next,$over);
|
---|
1893 |
|
---|
1894 | $u = $x->copy()->blog(undef,$scale)->bmul($y);
|
---|
1895 | $v = $self->bone(); # 1
|
---|
1896 | $factor = $self->new(2); # 2
|
---|
1897 | $x->bone(); # first term: 1
|
---|
1898 |
|
---|
1899 | $below = $v->copy();
|
---|
1900 | $over = $u->copy();
|
---|
1901 |
|
---|
1902 | $limit = $self->new("1E-". ($scale-1));
|
---|
1903 | #my $steps = 0;
|
---|
1904 | while (3 < 5)
|
---|
1905 | {
|
---|
1906 | # we calculate the next term, and add it to the last
|
---|
1907 | # when the next term is below our limit, it won't affect the outcome
|
---|
1908 | # anymore, so we stop
|
---|
1909 | $next = $over->copy()->bdiv($below,$scale);
|
---|
1910 | last if $next->bacmp($limit) <= 0;
|
---|
1911 | $x->badd($next);
|
---|
1912 | # calculate things for the next term
|
---|
1913 | $over *= $u; $below *= $factor; $factor->binc();
|
---|
1914 |
|
---|
1915 | last if $x->{sign} !~ /^[-+]$/;
|
---|
1916 |
|
---|
1917 | #$steps++;
|
---|
1918 | }
|
---|
1919 |
|
---|
1920 | # shortcut to not run through _find_round_parameters again
|
---|
1921 | if (defined $params[0])
|
---|
1922 | {
|
---|
1923 | $x->bround($params[0],$params[2]); # then round accordingly
|
---|
1924 | }
|
---|
1925 | else
|
---|
1926 | {
|
---|
1927 | $x->bfround($params[1],$params[2]); # then round accordingly
|
---|
1928 | }
|
---|
1929 | if ($fallback)
|
---|
1930 | {
|
---|
1931 | # clear a/p after round, since user did not request it
|
---|
1932 | delete $x->{_a}; delete $x->{_p};
|
---|
1933 | }
|
---|
1934 | # restore globals
|
---|
1935 | $$abr = $ab; $$pbr = $pb;
|
---|
1936 | $x;
|
---|
1937 | }
|
---|
1938 |
|
---|
1939 | sub bpow
|
---|
1940 | {
|
---|
1941 | # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT
|
---|
1942 | # compute power of two numbers, second arg is used as integer
|
---|
1943 | # modifies first argument
|
---|
1944 |
|
---|
1945 | # set up parameters
|
---|
1946 | my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_);
|
---|
1947 | # objectify is costly, so avoid it
|
---|
1948 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
|
---|
1949 | {
|
---|
1950 | ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
|
---|
1951 | }
|
---|
1952 |
|
---|
1953 | return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
|
---|
1954 | return $x if $x->{sign} =~ /^[+-]inf$/;
|
---|
1955 |
|
---|
1956 | # -2 ** -2 => NaN
|
---|
1957 | return $x->bnan() if $x->{sign} eq '-' && $y->{sign} eq '-';
|
---|
1958 |
|
---|
1959 | # cache the result of is_zero
|
---|
1960 | my $y_is_zero = $y->is_zero();
|
---|
1961 | return $x->bone() if $y_is_zero;
|
---|
1962 | return $x if $x->is_one() || $y->is_one();
|
---|
1963 |
|
---|
1964 | my $x_is_zero = $x->is_zero();
|
---|
1965 | return $x->_pow($y,$a,$p,$r) if !$x_is_zero && !$y->is_int(); # non-integer power
|
---|
1966 |
|
---|
1967 | my $y1 = $y->as_number()->{value}; # make MBI part
|
---|
1968 |
|
---|
1969 | # if ($x == -1)
|
---|
1970 | if ($x->{sign} eq '-' && $MBI->_is_one($x->{_m}) && $MBI->_is_zero($x->{_e}))
|
---|
1971 | {
|
---|
1972 | # if $x == -1 and odd/even y => +1/-1 because +-1 ^ (+-1) => +-1
|
---|
1973 | return $MBI->_is_odd($y1) ? $x : $x->babs(1);
|
---|
1974 | }
|
---|
1975 | if ($x_is_zero)
|
---|
1976 | {
|
---|
1977 | return $x->bone() if $y_is_zero;
|
---|
1978 | return $x if $y->{sign} eq '+'; # 0**y => 0 (if not y <= 0)
|
---|
1979 | # 0 ** -y => 1 / (0 ** y) => 1 / 0! (1 / 0 => +inf)
|
---|
1980 | return $x->binf();
|
---|
1981 | }
|
---|
1982 |
|
---|
1983 | my $new_sign = '+';
|
---|
1984 | $new_sign = $MBI->_is_odd($y1) ? '-' : '+' if $x->{sign} ne '+';
|
---|
1985 |
|
---|
1986 | # calculate $x->{_m} ** $y and $x->{_e} * $y separately (faster)
|
---|
1987 | $x->{_m} = $MBI->_pow( $x->{_m}, $y1);
|
---|
1988 | $x->{_e} = $MBI->_mul ($x->{_e}, $y1);
|
---|
1989 |
|
---|
1990 | $x->{sign} = $new_sign;
|
---|
1991 | $x->bnorm();
|
---|
1992 | if ($y->{sign} eq '-')
|
---|
1993 | {
|
---|
1994 | # modify $x in place!
|
---|
1995 | my $z = $x->copy(); $x->bone();
|
---|
1996 | return $x->bdiv($z,$a,$p,$r); # round in one go (might ignore y's A!)
|
---|
1997 | }
|
---|
1998 | $x->round($a,$p,$r,$y);
|
---|
1999 | }
|
---|
2000 |
|
---|
2001 | ###############################################################################
|
---|
2002 | # rounding functions
|
---|
2003 |
|
---|
2004 | sub bfround
|
---|
2005 | {
|
---|
2006 | # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.'
|
---|
2007 | # $n == 0 means round to integer
|
---|
2008 | # expects and returns normalized numbers!
|
---|
2009 | my $x = shift; my $self = ref($x) || $x; $x = $self->new(shift) if !ref($x);
|
---|
2010 |
|
---|
2011 | my ($scale,$mode) = $x->_scale_p(@_);
|
---|
2012 | return $x if !defined $scale || $x->modify('bfround'); # no-op
|
---|
2013 |
|
---|
2014 | # never round a 0, +-inf, NaN
|
---|
2015 | if ($x->is_zero())
|
---|
2016 | {
|
---|
2017 | $x->{_p} = $scale if !defined $x->{_p} || $x->{_p} < $scale; # -3 < -2
|
---|
2018 | return $x;
|
---|
2019 | }
|
---|
2020 | return $x if $x->{sign} !~ /^[+-]$/;
|
---|
2021 |
|
---|
2022 | # don't round if x already has lower precision
|
---|
2023 | return $x if (defined $x->{_p} && $x->{_p} < 0 && $scale < $x->{_p});
|
---|
2024 |
|
---|
2025 | $x->{_p} = $scale; # remember round in any case
|
---|
2026 | delete $x->{_a}; # and clear A
|
---|
2027 | if ($scale < 0)
|
---|
2028 | {
|
---|
2029 | # round right from the '.'
|
---|
2030 |
|
---|
2031 | return $x if $x->{_es} eq '+'; # e >= 0 => nothing to round
|
---|
2032 |
|
---|
2033 | $scale = -$scale; # positive for simplicity
|
---|
2034 | my $len = $MBI->_len($x->{_m}); # length of mantissa
|
---|
2035 |
|
---|
2036 | # the following poses a restriction on _e, but if _e is bigger than a
|
---|
2037 | # scalar, you got other problems (memory etc) anyway
|
---|
2038 | my $dad = -(0+ ($x->{_es}.$MBI->_num($x->{_e}))); # digits after dot
|
---|
2039 | my $zad = 0; # zeros after dot
|
---|
2040 | $zad = $dad - $len if (-$dad < -$len); # for 0.00..00xxx style
|
---|
2041 |
|
---|
2042 | # p rint "scale $scale dad $dad zad $zad len $len\n";
|
---|
2043 | # number bsstr len zad dad
|
---|
2044 | # 0.123 123e-3 3 0 3
|
---|
2045 | # 0.0123 123e-4 3 1 4
|
---|
2046 | # 0.001 1e-3 1 2 3
|
---|
2047 | # 1.23 123e-2 3 0 2
|
---|
2048 | # 1.2345 12345e-4 5 0 4
|
---|
2049 |
|
---|
2050 | # do not round after/right of the $dad
|
---|
2051 | return $x if $scale > $dad; # 0.123, scale >= 3 => exit
|
---|
2052 |
|
---|
2053 | # round to zero if rounding inside the $zad, but not for last zero like:
|
---|
2054 | # 0.0065, scale -2, round last '0' with following '65' (scale == zad case)
|
---|
2055 | return $x->bzero() if $scale < $zad;
|
---|
2056 | if ($scale == $zad) # for 0.006, scale -3 and trunc
|
---|
2057 | {
|
---|
2058 | $scale = -$len;
|
---|
2059 | }
|
---|
2060 | else
|
---|
2061 | {
|
---|
2062 | # adjust round-point to be inside mantissa
|
---|
2063 | if ($zad != 0)
|
---|
2064 | {
|
---|
2065 | $scale = $scale-$zad;
|
---|
2066 | }
|
---|
2067 | else
|
---|
2068 | {
|
---|
2069 | my $dbd = $len - $dad; $dbd = 0 if $dbd < 0; # digits before dot
|
---|
2070 | $scale = $dbd+$scale;
|
---|
2071 | }
|
---|
2072 | }
|
---|
2073 | }
|
---|
2074 | else
|
---|
2075 | {
|
---|
2076 | # round left from the '.'
|
---|
2077 |
|
---|
2078 | # 123 => 100 means length(123) = 3 - $scale (2) => 1
|
---|
2079 |
|
---|
2080 | my $dbt = $MBI->_len($x->{_m});
|
---|
2081 | # digits before dot
|
---|
2082 | my $dbd = $dbt + ($x->{_es} . $MBI->_num($x->{_e}));
|
---|
2083 | # should be the same, so treat it as this
|
---|
2084 | $scale = 1 if $scale == 0;
|
---|
2085 | # shortcut if already integer
|
---|
2086 | return $x if $scale == 1 && $dbt <= $dbd;
|
---|
2087 | # maximum digits before dot
|
---|
2088 | ++$dbd;
|
---|
2089 |
|
---|
2090 | if ($scale > $dbd)
|
---|
2091 | {
|
---|
2092 | # not enough digits before dot, so round to zero
|
---|
2093 | return $x->bzero;
|
---|
2094 | }
|
---|
2095 | elsif ( $scale == $dbd )
|
---|
2096 | {
|
---|
2097 | # maximum
|
---|
2098 | $scale = -$dbt;
|
---|
2099 | }
|
---|
2100 | else
|
---|
2101 | {
|
---|
2102 | $scale = $dbd - $scale;
|
---|
2103 | }
|
---|
2104 | }
|
---|
2105 | # pass sign to bround for rounding modes '+inf' and '-inf'
|
---|
2106 | my $m = bless { sign => $x->{sign}, value => $x->{_m} }, 'Math::BigInt';
|
---|
2107 | $m->bround($scale,$mode);
|
---|
2108 | $x->{_m} = $m->{value}; # get our mantissa back
|
---|
2109 | $x->bnorm();
|
---|
2110 | }
|
---|
2111 |
|
---|
2112 | sub bround
|
---|
2113 | {
|
---|
2114 | # accuracy: preserve $N digits, and overwrite the rest with 0's
|
---|
2115 | my $x = shift; my $self = ref($x) || $x; $x = $self->new(shift) if !ref($x);
|
---|
2116 |
|
---|
2117 | if (($_[0] || 0) < 0)
|
---|
2118 | {
|
---|
2119 | require Carp; Carp::croak ('bround() needs positive accuracy');
|
---|
2120 | }
|
---|
2121 |
|
---|
2122 | my ($scale,$mode) = $x->_scale_a(@_);
|
---|
2123 | return $x if !defined $scale || $x->modify('bround'); # no-op
|
---|
2124 |
|
---|
2125 | # scale is now either $x->{_a}, $accuracy, or the user parameter
|
---|
2126 | # test whether $x already has lower accuracy, do nothing in this case
|
---|
2127 | # but do round if the accuracy is the same, since a math operation might
|
---|
2128 | # want to round a number with A=5 to 5 digits afterwards again
|
---|
2129 | return $x if defined $x->{_a} && $x->{_a} < $scale;
|
---|
2130 |
|
---|
2131 | # scale < 0 makes no sense
|
---|
2132 | # scale == 0 => keep all digits
|
---|
2133 | # never round a +-inf, NaN
|
---|
2134 | return $x if ($scale <= 0) || $x->{sign} !~ /^[+-]$/;
|
---|
2135 |
|
---|
2136 | # 1: never round a 0
|
---|
2137 | # 2: if we should keep more digits than the mantissa has, do nothing
|
---|
2138 | if ($x->is_zero() || $MBI->_len($x->{_m}) <= $scale)
|
---|
2139 | {
|
---|
2140 | $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale;
|
---|
2141 | return $x;
|
---|
2142 | }
|
---|
2143 |
|
---|
2144 | # pass sign to bround for '+inf' and '-inf' rounding modes
|
---|
2145 | my $m = bless { sign => $x->{sign}, value => $x->{_m} }, 'Math::BigInt';
|
---|
2146 |
|
---|
2147 | $m->bround($scale,$mode); # round mantissa
|
---|
2148 | $x->{_m} = $m->{value}; # get our mantissa back
|
---|
2149 | $x->{_a} = $scale; # remember rounding
|
---|
2150 | delete $x->{_p}; # and clear P
|
---|
2151 | $x->bnorm(); # del trailing zeros gen. by bround()
|
---|
2152 | }
|
---|
2153 |
|
---|
2154 | sub bfloor
|
---|
2155 | {
|
---|
2156 | # return integer less or equal then $x
|
---|
2157 | my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
|
---|
2158 |
|
---|
2159 | return $x if $x->modify('bfloor');
|
---|
2160 |
|
---|
2161 | return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf
|
---|
2162 |
|
---|
2163 | # if $x has digits after dot
|
---|
2164 | if ($x->{_es} eq '-')
|
---|
2165 | {
|
---|
2166 | $x->{_m} = $MBI->_rsft($x->{_m},$x->{_e},10); # cut off digits after dot
|
---|
2167 | $x->{_e} = $MBI->_zero(); # trunc/norm
|
---|
2168 | $x->{_es} = '+'; # abs e
|
---|
2169 | $MBI->_inc($x->{_m}) if $x->{sign} eq '-'; # increment if negative
|
---|
2170 | }
|
---|
2171 | $x->round($a,$p,$r);
|
---|
2172 | }
|
---|
2173 |
|
---|
2174 | sub bceil
|
---|
2175 | {
|
---|
2176 | # return integer greater or equal then $x
|
---|
2177 | my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
|
---|
2178 |
|
---|
2179 | return $x if $x->modify('bceil');
|
---|
2180 | return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf
|
---|
2181 |
|
---|
2182 | # if $x has digits after dot
|
---|
2183 | if ($x->{_es} eq '-')
|
---|
2184 | {
|
---|
2185 | $x->{_m} = $MBI->_rsft($x->{_m},$x->{_e},10); # cut off digits after dot
|
---|
2186 | $x->{_e} = $MBI->_zero(); # trunc/norm
|
---|
2187 | $x->{_es} = '+'; # abs e
|
---|
2188 | $MBI->_inc($x->{_m}) if $x->{sign} eq '+'; # increment if positive
|
---|
2189 | }
|
---|
2190 | $x->round($a,$p,$r);
|
---|
2191 | }
|
---|
2192 |
|
---|
2193 | sub brsft
|
---|
2194 | {
|
---|
2195 | # shift right by $y (divide by power of $n)
|
---|
2196 |
|
---|
2197 | # set up parameters
|
---|
2198 | my ($self,$x,$y,$n,$a,$p,$r) = (ref($_[0]),@_);
|
---|
2199 | # objectify is costly, so avoid it
|
---|
2200 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
|
---|
2201 | {
|
---|
2202 | ($self,$x,$y,$n,$a,$p,$r) = objectify(2,@_);
|
---|
2203 | }
|
---|
2204 |
|
---|
2205 | return $x if $x->modify('brsft');
|
---|
2206 | return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf
|
---|
2207 |
|
---|
2208 | $n = 2 if !defined $n; $n = $self->new($n);
|
---|
2209 | $x->bdiv($n->bpow($y),$a,$p,$r,$y);
|
---|
2210 | }
|
---|
2211 |
|
---|
2212 | sub blsft
|
---|
2213 | {
|
---|
2214 | # shift left by $y (multiply by power of $n)
|
---|
2215 |
|
---|
2216 | # set up parameters
|
---|
2217 | my ($self,$x,$y,$n,$a,$p,$r) = (ref($_[0]),@_);
|
---|
2218 | # objectify is costly, so avoid it
|
---|
2219 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
|
---|
2220 | {
|
---|
2221 | ($self,$x,$y,$n,$a,$p,$r) = objectify(2,@_);
|
---|
2222 | }
|
---|
2223 |
|
---|
2224 | return $x if $x->modify('blsft');
|
---|
2225 | return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf
|
---|
2226 |
|
---|
2227 | $n = 2 if !defined $n; $n = $self->new($n);
|
---|
2228 | $x->bmul($n->bpow($y),$a,$p,$r,$y);
|
---|
2229 | }
|
---|
2230 |
|
---|
2231 | ###############################################################################
|
---|
2232 |
|
---|
2233 | sub DESTROY
|
---|
2234 | {
|
---|
2235 | # going through AUTOLOAD for every DESTROY is costly, avoid it by empty sub
|
---|
2236 | }
|
---|
2237 |
|
---|
2238 | sub AUTOLOAD
|
---|
2239 | {
|
---|
2240 | # make fxxx and bxxx both work by selectively mapping fxxx() to MBF::bxxx()
|
---|
2241 | # or falling back to MBI::bxxx()
|
---|
2242 | my $name = $AUTOLOAD;
|
---|
2243 |
|
---|
2244 | $name =~ s/(.*):://; # split package
|
---|
2245 | my $c = $1 || $class;
|
---|
2246 | no strict 'refs';
|
---|
2247 | $c->import() if $IMPORT == 0;
|
---|
2248 | if (!method_alias($name))
|
---|
2249 | {
|
---|
2250 | if (!defined $name)
|
---|
2251 | {
|
---|
2252 | # delayed load of Carp and avoid recursion
|
---|
2253 | require Carp;
|
---|
2254 | Carp::croak ("$c: Can't call a method without name");
|
---|
2255 | }
|
---|
2256 | if (!method_hand_up($name))
|
---|
2257 | {
|
---|
2258 | # delayed load of Carp and avoid recursion
|
---|
2259 | require Carp;
|
---|
2260 | Carp::croak ("Can't call $c\-\>$name, not a valid method");
|
---|
2261 | }
|
---|
2262 | # try one level up, but subst. bxxx() for fxxx() since MBI only got bxxx()
|
---|
2263 | $name =~ s/^f/b/;
|
---|
2264 | return &{"Math::BigInt"."::$name"}(@_);
|
---|
2265 | }
|
---|
2266 | my $bname = $name; $bname =~ s/^f/b/;
|
---|
2267 | $c .= "::$name";
|
---|
2268 | *{$c} = \&{$bname};
|
---|
2269 | &{$c}; # uses @_
|
---|
2270 | }
|
---|
2271 |
|
---|
2272 | sub exponent
|
---|
2273 | {
|
---|
2274 | # return a copy of the exponent
|
---|
2275 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
|
---|
2276 |
|
---|
2277 | if ($x->{sign} !~ /^[+-]$/)
|
---|
2278 | {
|
---|
2279 | my $s = $x->{sign}; $s =~ s/^[+-]//;
|
---|
2280 | return Math::BigInt->new($s); # -inf, +inf => +inf
|
---|
2281 | }
|
---|
2282 | Math::BigInt->new( $x->{_es} . $MBI->_str($x->{_e}));
|
---|
2283 | }
|
---|
2284 |
|
---|
2285 | sub mantissa
|
---|
2286 | {
|
---|
2287 | # return a copy of the mantissa
|
---|
2288 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
|
---|
2289 |
|
---|
2290 | if ($x->{sign} !~ /^[+-]$/)
|
---|
2291 | {
|
---|
2292 | my $s = $x->{sign}; $s =~ s/^[+]//;
|
---|
2293 | return Math::BigInt->new($s); # -inf, +inf => +inf
|
---|
2294 | }
|
---|
2295 | my $m = Math::BigInt->new( $MBI->_str($x->{_m}));
|
---|
2296 | $m->bneg() if $x->{sign} eq '-';
|
---|
2297 |
|
---|
2298 | $m;
|
---|
2299 | }
|
---|
2300 |
|
---|
2301 | sub parts
|
---|
2302 | {
|
---|
2303 | # return a copy of both the exponent and the mantissa
|
---|
2304 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
|
---|
2305 |
|
---|
2306 | if ($x->{sign} !~ /^[+-]$/)
|
---|
2307 | {
|
---|
2308 | my $s = $x->{sign}; $s =~ s/^[+]//; my $se = $s; $se =~ s/^[-]//;
|
---|
2309 | return ($self->new($s),$self->new($se)); # +inf => inf and -inf,+inf => inf
|
---|
2310 | }
|
---|
2311 | my $m = Math::BigInt->bzero();
|
---|
2312 | $m->{value} = $MBI->_copy($x->{_m});
|
---|
2313 | $m->bneg() if $x->{sign} eq '-';
|
---|
2314 | ($m, Math::BigInt->new( $x->{_es} . $MBI->_num($x->{_e}) ));
|
---|
2315 | }
|
---|
2316 |
|
---|
2317 | ##############################################################################
|
---|
2318 | # private stuff (internal use only)
|
---|
2319 |
|
---|
2320 | sub import
|
---|
2321 | {
|
---|
2322 | my $self = shift;
|
---|
2323 | my $l = scalar @_;
|
---|
2324 | my $lib = ''; my @a;
|
---|
2325 | $IMPORT=1;
|
---|
2326 | for ( my $i = 0; $i < $l ; $i++)
|
---|
2327 | {
|
---|
2328 | if ( $_[$i] eq ':constant' )
|
---|
2329 | {
|
---|
2330 | # This causes overlord er load to step in. 'binary' and 'integer'
|
---|
2331 | # are handled by BigInt.
|
---|
2332 | overload::constant float => sub { $self->new(shift); };
|
---|
2333 | }
|
---|
2334 | elsif ($_[$i] eq 'upgrade')
|
---|
2335 | {
|
---|
2336 | # this causes upgrading
|
---|
2337 | $upgrade = $_[$i+1]; # or undef to disable
|
---|
2338 | $i++;
|
---|
2339 | }
|
---|
2340 | elsif ($_[$i] eq 'downgrade')
|
---|
2341 | {
|
---|
2342 | # this causes downgrading
|
---|
2343 | $downgrade = $_[$i+1]; # or undef to disable
|
---|
2344 | $i++;
|
---|
2345 | }
|
---|
2346 | elsif ($_[$i] eq 'lib')
|
---|
2347 | {
|
---|
2348 | # alternative library
|
---|
2349 | $lib = $_[$i+1] || ''; # default Calc
|
---|
2350 | $i++;
|
---|
2351 | }
|
---|
2352 | elsif ($_[$i] eq 'with')
|
---|
2353 | {
|
---|
2354 | # alternative class for our private parts()
|
---|
2355 | # XXX: no longer supported
|
---|
2356 | # $MBI = $_[$i+1] || 'Math::BigInt';
|
---|
2357 | $i++;
|
---|
2358 | }
|
---|
2359 | else
|
---|
2360 | {
|
---|
2361 | push @a, $_[$i];
|
---|
2362 | }
|
---|
2363 | }
|
---|
2364 |
|
---|
2365 | $lib =~ tr/a-zA-Z0-9,://cd; # restrict to sane characters
|
---|
2366 | # let use Math::BigInt lib => 'GMP'; use Math::BigFloat; still work
|
---|
2367 | my $mbilib = eval { Math::BigInt->config()->{lib} };
|
---|
2368 | if ((defined $mbilib) && ($MBI eq 'Math::BigInt::Calc'))
|
---|
2369 | {
|
---|
2370 | # MBI already loaded
|
---|
2371 | Math::BigInt->import('lib',"$lib,$mbilib", 'objectify');
|
---|
2372 | }
|
---|
2373 | else
|
---|
2374 | {
|
---|
2375 | # MBI not loaded, or with ne "Math::BigInt::Calc"
|
---|
2376 | $lib .= ",$mbilib" if defined $mbilib;
|
---|
2377 | $lib =~ s/^,//; # don't leave empty
|
---|
2378 |
|
---|
2379 | # replacement library can handle lib statement, but also could ignore it
|
---|
2380 |
|
---|
2381 | # Perl < 5.6.0 dies with "out of memory!" when eval() and ':constant' is
|
---|
2382 | # used in the same script, or eval inside import(). So we require MBI:
|
---|
2383 | require Math::BigInt;
|
---|
2384 | Math::BigInt->import( lib => $lib, 'objectify' );
|
---|
2385 | }
|
---|
2386 | if ($@)
|
---|
2387 | {
|
---|
2388 | require Carp; Carp::croak ("Couldn't load $lib: $! $@");
|
---|
2389 | }
|
---|
2390 | # find out which one was actually loaded
|
---|
2391 | $MBI = Math::BigInt->config()->{lib};
|
---|
2392 |
|
---|
2393 | # register us with MBI to get notified of future lib changes
|
---|
2394 | Math::BigInt::_register_callback( $self, sub { $MBI = $_[0]; } );
|
---|
2395 |
|
---|
2396 | # any non :constant stuff is handled by our parent, Exporter
|
---|
2397 | # even if @_ is empty, to give it a chance
|
---|
2398 | $self->SUPER::import(@a); # for subclasses
|
---|
2399 | $self->export_to_level(1,$self,@a); # need this, too
|
---|
2400 | }
|
---|
2401 |
|
---|
2402 | sub bnorm
|
---|
2403 | {
|
---|
2404 | # adjust m and e so that m is smallest possible
|
---|
2405 | # round number according to accuracy and precision settings
|
---|
2406 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
|
---|
2407 |
|
---|
2408 | return $x if $x->{sign} !~ /^[+-]$/; # inf, nan etc
|
---|
2409 |
|
---|
2410 | my $zeros = $MBI->_zeros($x->{_m}); # correct for trailing zeros
|
---|
2411 | if ($zeros != 0)
|
---|
2412 | {
|
---|
2413 | my $z = $MBI->_new($zeros);
|
---|
2414 | $x->{_m} = $MBI->_rsft ($x->{_m}, $z, 10);
|
---|
2415 | if ($x->{_es} eq '-')
|
---|
2416 | {
|
---|
2417 | if ($MBI->_acmp($x->{_e},$z) >= 0)
|
---|
2418 | {
|
---|
2419 | $x->{_e} = $MBI->_sub ($x->{_e}, $z);
|
---|
2420 | $x->{_es} = '+' if $MBI->_is_zero($x->{_e});
|
---|
2421 | }
|
---|
2422 | else
|
---|
2423 | {
|
---|
2424 | $x->{_e} = $MBI->_sub ( $MBI->_copy($z), $x->{_e});
|
---|
2425 | $x->{_es} = '+';
|
---|
2426 | }
|
---|
2427 | }
|
---|
2428 | else
|
---|
2429 | {
|
---|
2430 | $x->{_e} = $MBI->_add ($x->{_e}, $z);
|
---|
2431 | }
|
---|
2432 | }
|
---|
2433 | else
|
---|
2434 | {
|
---|
2435 | # $x can only be 0Ey if there are no trailing zeros ('0' has 0 trailing
|
---|
2436 | # zeros). So, for something like 0Ey, set y to 1, and -0 => +0
|
---|
2437 | $x->{sign} = '+', $x->{_es} = '+', $x->{_e} = $MBI->_one()
|
---|
2438 | if $MBI->_is_zero($x->{_m});
|
---|
2439 | }
|
---|
2440 |
|
---|
2441 | $x; # MBI bnorm is no-op, so dont call it
|
---|
2442 | }
|
---|
2443 |
|
---|
2444 | ##############################################################################
|
---|
2445 |
|
---|
2446 | sub as_hex
|
---|
2447 | {
|
---|
2448 | # return number as hexadecimal string (only for integers defined)
|
---|
2449 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
|
---|
2450 |
|
---|
2451 | return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
|
---|
2452 | return '0x0' if $x->is_zero();
|
---|
2453 |
|
---|
2454 | return $nan if $x->{_es} ne '+'; # how to do 1e-1 in hex!?
|
---|
2455 |
|
---|
2456 | my $z = $MBI->_copy($x->{_m});
|
---|
2457 | if (! $MBI->_is_zero($x->{_e})) # > 0
|
---|
2458 | {
|
---|
2459 | $MBI->_lsft( $z, $x->{_e},10);
|
---|
2460 | }
|
---|
2461 | $z = Math::BigInt->new( $x->{sign} . $MBI->_num($z));
|
---|
2462 | $z->as_hex();
|
---|
2463 | }
|
---|
2464 |
|
---|
2465 | sub as_bin
|
---|
2466 | {
|
---|
2467 | # return number as binary digit string (only for integers defined)
|
---|
2468 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
|
---|
2469 |
|
---|
2470 | return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
|
---|
2471 | return '0b0' if $x->is_zero();
|
---|
2472 |
|
---|
2473 | return $nan if $x->{_es} ne '+'; # how to do 1e-1 in hex!?
|
---|
2474 |
|
---|
2475 | my $z = $MBI->_copy($x->{_m});
|
---|
2476 | if (! $MBI->_is_zero($x->{_e})) # > 0
|
---|
2477 | {
|
---|
2478 | $MBI->_lsft( $z, $x->{_e},10);
|
---|
2479 | }
|
---|
2480 | $z = Math::BigInt->new( $x->{sign} . $MBI->_num($z));
|
---|
2481 | $z->as_bin();
|
---|
2482 | }
|
---|
2483 |
|
---|
2484 | sub as_number
|
---|
2485 | {
|
---|
2486 | # return copy as a bigint representation of this BigFloat number
|
---|
2487 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
|
---|
2488 |
|
---|
2489 | my $z = $MBI->_copy($x->{_m});
|
---|
2490 | if ($x->{_es} eq '-') # < 0
|
---|
2491 | {
|
---|
2492 | $MBI->_rsft( $z, $x->{_e},10);
|
---|
2493 | }
|
---|
2494 | elsif (! $MBI->_is_zero($x->{_e})) # > 0
|
---|
2495 | {
|
---|
2496 | $MBI->_lsft( $z, $x->{_e},10);
|
---|
2497 | }
|
---|
2498 | $z = Math::BigInt->new( $x->{sign} . $MBI->_num($z));
|
---|
2499 | $z;
|
---|
2500 | }
|
---|
2501 |
|
---|
2502 | sub length
|
---|
2503 | {
|
---|
2504 | my $x = shift;
|
---|
2505 | my $class = ref($x) || $x;
|
---|
2506 | $x = $class->new(shift) unless ref($x);
|
---|
2507 |
|
---|
2508 | return 1 if $MBI->_is_zero($x->{_m});
|
---|
2509 |
|
---|
2510 | my $len = $MBI->_len($x->{_m});
|
---|
2511 | $len += $MBI->_num($x->{_e}) if $x->{_es} eq '+';
|
---|
2512 | if (wantarray())
|
---|
2513 | {
|
---|
2514 | my $t = 0;
|
---|
2515 | $t = $MBI->_num($x->{_e}) if $x->{_es} eq '-';
|
---|
2516 | return ($len, $t);
|
---|
2517 | }
|
---|
2518 | $len;
|
---|
2519 | }
|
---|
2520 |
|
---|
2521 | 1;
|
---|
2522 | __END__
|
---|
2523 |
|
---|
2524 | =head1 NAME
|
---|
2525 |
|
---|
2526 | Math::BigFloat - Arbitrary size floating point math package
|
---|
2527 |
|
---|
2528 | =head1 SYNOPSIS
|
---|
2529 |
|
---|
2530 | use Math::BigFloat;
|
---|
2531 |
|
---|
2532 | # Number creation
|
---|
2533 | $x = Math::BigFloat->new($str); # defaults to 0
|
---|
2534 | $nan = Math::BigFloat->bnan(); # create a NotANumber
|
---|
2535 | $zero = Math::BigFloat->bzero(); # create a +0
|
---|
2536 | $inf = Math::BigFloat->binf(); # create a +inf
|
---|
2537 | $inf = Math::BigFloat->binf('-'); # create a -inf
|
---|
2538 | $one = Math::BigFloat->bone(); # create a +1
|
---|
2539 | $one = Math::BigFloat->bone('-'); # create a -1
|
---|
2540 |
|
---|
2541 | # Testing
|
---|
2542 | $x->is_zero(); # true if arg is +0
|
---|
2543 | $x->is_nan(); # true if arg is NaN
|
---|
2544 | $x->is_one(); # true if arg is +1
|
---|
2545 | $x->is_one('-'); # true if arg is -1
|
---|
2546 | $x->is_odd(); # true if odd, false for even
|
---|
2547 | $x->is_even(); # true if even, false for odd
|
---|
2548 | $x->is_pos(); # true if >= 0
|
---|
2549 | $x->is_neg(); # true if < 0
|
---|
2550 | $x->is_inf(sign); # true if +inf, or -inf (default is '+')
|
---|
2551 |
|
---|
2552 | $x->bcmp($y); # compare numbers (undef,<0,=0,>0)
|
---|
2553 | $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
|
---|
2554 | $x->sign(); # return the sign, either +,- or NaN
|
---|
2555 | $x->digit($n); # return the nth digit, counting from right
|
---|
2556 | $x->digit(-$n); # return the nth digit, counting from left
|
---|
2557 |
|
---|
2558 | # The following all modify their first argument. If you want to preserve
|
---|
2559 | # $x, use $z = $x->copy()->bXXX($y); See under L<CAVEATS> for why this is
|
---|
2560 | # neccessary when mixing $a = $b assigments with non-overloaded math.
|
---|
2561 |
|
---|
2562 | # set
|
---|
2563 | $x->bzero(); # set $i to 0
|
---|
2564 | $x->bnan(); # set $i to NaN
|
---|
2565 | $x->bone(); # set $x to +1
|
---|
2566 | $x->bone('-'); # set $x to -1
|
---|
2567 | $x->binf(); # set $x to inf
|
---|
2568 | $x->binf('-'); # set $x to -inf
|
---|
2569 |
|
---|
2570 | $x->bneg(); # negation
|
---|
2571 | $x->babs(); # absolute value
|
---|
2572 | $x->bnorm(); # normalize (no-op)
|
---|
2573 | $x->bnot(); # two's complement (bit wise not)
|
---|
2574 | $x->binc(); # increment x by 1
|
---|
2575 | $x->bdec(); # decrement x by 1
|
---|
2576 |
|
---|
2577 | $x->badd($y); # addition (add $y to $x)
|
---|
2578 | $x->bsub($y); # subtraction (subtract $y from $x)
|
---|
2579 | $x->bmul($y); # multiplication (multiply $x by $y)
|
---|
2580 | $x->bdiv($y); # divide, set $x to quotient
|
---|
2581 | # return (quo,rem) or quo if scalar
|
---|
2582 |
|
---|
2583 | $x->bmod($y); # modulus ($x % $y)
|
---|
2584 | $x->bpow($y); # power of arguments ($x ** $y)
|
---|
2585 | $x->blsft($y); # left shift
|
---|
2586 | $x->brsft($y); # right shift
|
---|
2587 | # return (quo,rem) or quo if scalar
|
---|
2588 |
|
---|
2589 | $x->blog(); # logarithm of $x to base e (Euler's number)
|
---|
2590 | $x->blog($base); # logarithm of $x to base $base (f.i. 2)
|
---|
2591 |
|
---|
2592 | $x->band($y); # bit-wise and
|
---|
2593 | $x->bior($y); # bit-wise inclusive or
|
---|
2594 | $x->bxor($y); # bit-wise exclusive or
|
---|
2595 | $x->bnot(); # bit-wise not (two's complement)
|
---|
2596 |
|
---|
2597 | $x->bsqrt(); # calculate square-root
|
---|
2598 | $x->broot($y); # $y'th root of $x (e.g. $y == 3 => cubic root)
|
---|
2599 | $x->bfac(); # factorial of $x (1*2*3*4*..$x)
|
---|
2600 |
|
---|
2601 | $x->bround($N); # accuracy: preserve $N digits
|
---|
2602 | $x->bfround($N); # precision: round to the $Nth digit
|
---|
2603 |
|
---|
2604 | $x->bfloor(); # return integer less or equal than $x
|
---|
2605 | $x->bceil(); # return integer greater or equal than $x
|
---|
2606 |
|
---|
2607 | # The following do not modify their arguments:
|
---|
2608 |
|
---|
2609 | bgcd(@values); # greatest common divisor
|
---|
2610 | blcm(@values); # lowest common multiplicator
|
---|
2611 |
|
---|
2612 | $x->bstr(); # return string
|
---|
2613 | $x->bsstr(); # return string in scientific notation
|
---|
2614 |
|
---|
2615 | $x->as_int(); # return $x as BigInt
|
---|
2616 | $x->exponent(); # return exponent as BigInt
|
---|
2617 | $x->mantissa(); # return mantissa as BigInt
|
---|
2618 | $x->parts(); # return (mantissa,exponent) as BigInt
|
---|
2619 |
|
---|
2620 | $x->length(); # number of digits (w/o sign and '.')
|
---|
2621 | ($l,$f) = $x->length(); # number of digits, and length of fraction
|
---|
2622 |
|
---|
2623 | $x->precision(); # return P of $x (or global, if P of $x undef)
|
---|
2624 | $x->precision($n); # set P of $x to $n
|
---|
2625 | $x->accuracy(); # return A of $x (or global, if A of $x undef)
|
---|
2626 | $x->accuracy($n); # set A $x to $n
|
---|
2627 |
|
---|
2628 | # these get/set the appropriate global value for all BigFloat objects
|
---|
2629 | Math::BigFloat->precision(); # Precision
|
---|
2630 | Math::BigFloat->accuracy(); # Accuracy
|
---|
2631 | Math::BigFloat->round_mode(); # rounding mode
|
---|
2632 |
|
---|
2633 | =head1 DESCRIPTION
|
---|
2634 |
|
---|
2635 | All operators (inlcuding basic math operations) are overloaded if you
|
---|
2636 | declare your big floating point numbers as
|
---|
2637 |
|
---|
2638 | $i = new Math::BigFloat '12_3.456_789_123_456_789E-2';
|
---|
2639 |
|
---|
2640 | Operations with overloaded operators preserve the arguments, which is
|
---|
2641 | exactly what you expect.
|
---|
2642 |
|
---|
2643 | =head2 Canonical notation
|
---|
2644 |
|
---|
2645 | Input to these routines are either BigFloat objects, or strings of the
|
---|
2646 | following four forms:
|
---|
2647 |
|
---|
2648 | =over 2
|
---|
2649 |
|
---|
2650 | =item *
|
---|
2651 |
|
---|
2652 | C</^[+-]\d+$/>
|
---|
2653 |
|
---|
2654 | =item *
|
---|
2655 |
|
---|
2656 | C</^[+-]\d+\.\d*$/>
|
---|
2657 |
|
---|
2658 | =item *
|
---|
2659 |
|
---|
2660 | C</^[+-]\d+E[+-]?\d+$/>
|
---|
2661 |
|
---|
2662 | =item *
|
---|
2663 |
|
---|
2664 | C</^[+-]\d*\.\d+E[+-]?\d+$/>
|
---|
2665 |
|
---|
2666 | =back
|
---|
2667 |
|
---|
2668 | all with optional leading and trailing zeros and/or spaces. Additonally,
|
---|
2669 | numbers are allowed to have an underscore between any two digits.
|
---|
2670 |
|
---|
2671 | Empty strings as well as other illegal numbers results in 'NaN'.
|
---|
2672 |
|
---|
2673 | bnorm() on a BigFloat object is now effectively a no-op, since the numbers
|
---|
2674 | are always stored in normalized form. On a string, it creates a BigFloat
|
---|
2675 | object.
|
---|
2676 |
|
---|
2677 | =head2 Output
|
---|
2678 |
|
---|
2679 | Output values are BigFloat objects (normalized), except for bstr() and bsstr().
|
---|
2680 |
|
---|
2681 | The string output will always have leading and trailing zeros stripped and drop
|
---|
2682 | a plus sign. C<bstr()> will give you always the form with a decimal point,
|
---|
2683 | while C<bsstr()> (s for scientific) gives you the scientific notation.
|
---|
2684 |
|
---|
2685 | Input bstr() bsstr()
|
---|
2686 | '-0' '0' '0E1'
|
---|
2687 | ' -123 123 123' '-123123123' '-123123123E0'
|
---|
2688 | '00.0123' '0.0123' '123E-4'
|
---|
2689 | '123.45E-2' '1.2345' '12345E-4'
|
---|
2690 | '10E+3' '10000' '1E4'
|
---|
2691 |
|
---|
2692 | Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
|
---|
2693 | C<is_nan()>) return true or false, while others (C<bcmp()>, C<bacmp()>)
|
---|
2694 | return either undef, <0, 0 or >0 and are suited for sort.
|
---|
2695 |
|
---|
2696 | Actual math is done by using the class defined with C<with => Class;> (which
|
---|
2697 | defaults to BigInts) to represent the mantissa and exponent.
|
---|
2698 |
|
---|
2699 | The sign C</^[+-]$/> is stored separately. The string 'NaN' is used to
|
---|
2700 | represent the result when input arguments are not numbers, as well as
|
---|
2701 | the result of dividing by zero.
|
---|
2702 |
|
---|
2703 | =head2 C<mantissa()>, C<exponent()> and C<parts()>
|
---|
2704 |
|
---|
2705 | C<mantissa()> and C<exponent()> return the said parts of the BigFloat
|
---|
2706 | as BigInts such that:
|
---|
2707 |
|
---|
2708 | $m = $x->mantissa();
|
---|
2709 | $e = $x->exponent();
|
---|
2710 | $y = $m * ( 10 ** $e );
|
---|
2711 | print "ok\n" if $x == $y;
|
---|
2712 |
|
---|
2713 | C<< ($m,$e) = $x->parts(); >> is just a shortcut giving you both of them.
|
---|
2714 |
|
---|
2715 | A zero is represented and returned as C<0E1>, B<not> C<0E0> (after Knuth).
|
---|
2716 |
|
---|
2717 | Currently the mantissa is reduced as much as possible, favouring higher
|
---|
2718 | exponents over lower ones (e.g. returning 1e7 instead of 10e6 or 10000000e0).
|
---|
2719 | This might change in the future, so do not depend on it.
|
---|
2720 |
|
---|
2721 | =head2 Accuracy vs. Precision
|
---|
2722 |
|
---|
2723 | See also: L<Rounding|Rounding>.
|
---|
2724 |
|
---|
2725 | Math::BigFloat supports both precision (rounding to a certain place before or
|
---|
2726 | after the dot) and accuracy (rounding to a certain number of digits). For a
|
---|
2727 | full documentation, examples and tips on these topics please see the large
|
---|
2728 | section about rounding in L<Math::BigInt>.
|
---|
2729 |
|
---|
2730 | Since things like C<sqrt(2)> or C<1 / 3> must presented with a limited
|
---|
2731 | accuracy lest a operation consumes all resources, each operation produces
|
---|
2732 | no more than the requested number of digits.
|
---|
2733 |
|
---|
2734 | If there is no gloabl precision or accuracy set, B<and> the operation in
|
---|
2735 | question was not called with a requested precision or accuracy, B<and> the
|
---|
2736 | input $x has no accuracy or precision set, then a fallback parameter will
|
---|
2737 | be used. For historical reasons, it is called C<div_scale> and can be accessed
|
---|
2738 | via:
|
---|
2739 |
|
---|
2740 | $d = Math::BigFloat->div_scale(); # query
|
---|
2741 | Math::BigFloat->div_scale($n); # set to $n digits
|
---|
2742 |
|
---|
2743 | The default value for C<div_scale> is 40.
|
---|
2744 |
|
---|
2745 | In case the result of one operation has more digits than specified,
|
---|
2746 | it is rounded. The rounding mode taken is either the default mode, or the one
|
---|
2747 | supplied to the operation after the I<scale>:
|
---|
2748 |
|
---|
2749 | $x = Math::BigFloat->new(2);
|
---|
2750 | Math::BigFloat->accuracy(5); # 5 digits max
|
---|
2751 | $y = $x->copy()->bdiv(3); # will give 0.66667
|
---|
2752 | $y = $x->copy()->bdiv(3,6); # will give 0.666667
|
---|
2753 | $y = $x->copy()->bdiv(3,6,undef,'odd'); # will give 0.666667
|
---|
2754 | Math::BigFloat->round_mode('zero');
|
---|
2755 | $y = $x->copy()->bdiv(3,6); # will also give 0.666667
|
---|
2756 |
|
---|
2757 | Note that C<< Math::BigFloat->accuracy() >> and C<< Math::BigFloat->precision() >>
|
---|
2758 | set the global variables, and thus B<any> newly created number will be subject
|
---|
2759 | to the global rounding B<immidiately>. This means that in the examples above, the
|
---|
2760 | C<3> as argument to C<bdiv()> will also get an accuracy of B<5>.
|
---|
2761 |
|
---|
2762 | It is less confusing to either calculate the result fully, and afterwards
|
---|
2763 | round it explicitely, or use the additional parameters to the math
|
---|
2764 | functions like so:
|
---|
2765 |
|
---|
2766 | use Math::BigFloat;
|
---|
2767 | $x = Math::BigFloat->new(2);
|
---|
2768 | $y = $x->copy()->bdiv(3);
|
---|
2769 | print $y->bround(5),"\n"; # will give 0.66667
|
---|
2770 |
|
---|
2771 | or
|
---|
2772 |
|
---|
2773 | use Math::BigFloat;
|
---|
2774 | $x = Math::BigFloat->new(2);
|
---|
2775 | $y = $x->copy()->bdiv(3,5); # will give 0.66667
|
---|
2776 | print "$y\n";
|
---|
2777 |
|
---|
2778 | =head2 Rounding
|
---|
2779 |
|
---|
2780 | =over 2
|
---|
2781 |
|
---|
2782 | =item ffround ( +$scale )
|
---|
2783 |
|
---|
2784 | Rounds to the $scale'th place left from the '.', counting from the dot.
|
---|
2785 | The first digit is numbered 1.
|
---|
2786 |
|
---|
2787 | =item ffround ( -$scale )
|
---|
2788 |
|
---|
2789 | Rounds to the $scale'th place right from the '.', counting from the dot.
|
---|
2790 |
|
---|
2791 | =item ffround ( 0 )
|
---|
2792 |
|
---|
2793 | Rounds to an integer.
|
---|
2794 |
|
---|
2795 | =item fround ( +$scale )
|
---|
2796 |
|
---|
2797 | Preserves accuracy to $scale digits from the left (aka significant digits)
|
---|
2798 | and pads the rest with zeros. If the number is between 1 and -1, the
|
---|
2799 | significant digits count from the first non-zero after the '.'
|
---|
2800 |
|
---|
2801 | =item fround ( -$scale ) and fround ( 0 )
|
---|
2802 |
|
---|
2803 | These are effectively no-ops.
|
---|
2804 |
|
---|
2805 | =back
|
---|
2806 |
|
---|
2807 | All rounding functions take as a second parameter a rounding mode from one of
|
---|
2808 | the following: 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'.
|
---|
2809 |
|
---|
2810 | The default rounding mode is 'even'. By using
|
---|
2811 | C<< Math::BigFloat->round_mode($round_mode); >> you can get and set the default
|
---|
2812 | mode for subsequent rounding. The usage of C<$Math::BigFloat::$round_mode> is
|
---|
2813 | no longer supported.
|
---|
2814 | The second parameter to the round functions then overrides the default
|
---|
2815 | temporarily.
|
---|
2816 |
|
---|
2817 | The C<as_number()> function returns a BigInt from a Math::BigFloat. It uses
|
---|
2818 | 'trunc' as rounding mode to make it equivalent to:
|
---|
2819 |
|
---|
2820 | $x = 2.5;
|
---|
2821 | $y = int($x) + 2;
|
---|
2822 |
|
---|
2823 | You can override this by passing the desired rounding mode as parameter to
|
---|
2824 | C<as_number()>:
|
---|
2825 |
|
---|
2826 | $x = Math::BigFloat->new(2.5);
|
---|
2827 | $y = $x->as_number('odd'); # $y = 3
|
---|
2828 |
|
---|
2829 | =head1 METHODS
|
---|
2830 |
|
---|
2831 | =head2 accuracy
|
---|
2832 |
|
---|
2833 | $x->accuracy(5); # local for $x
|
---|
2834 | CLASS->accuracy(5); # global for all members of CLASS
|
---|
2835 | # Note: This also applies to new()!
|
---|
2836 |
|
---|
2837 | $A = $x->accuracy(); # read out accuracy that affects $x
|
---|
2838 | $A = CLASS->accuracy(); # read out global accuracy
|
---|
2839 |
|
---|
2840 | Set or get the global or local accuracy, aka how many significant digits the
|
---|
2841 | results have. If you set a global accuracy, then this also applies to new()!
|
---|
2842 |
|
---|
2843 | Warning! The accuracy I<sticks>, e.g. once you created a number under the
|
---|
2844 | influence of C<< CLASS->accuracy($A) >>, all results from math operations with
|
---|
2845 | that number will also be rounded.
|
---|
2846 |
|
---|
2847 | In most cases, you should probably round the results explicitely using one of
|
---|
2848 | L<round()>, L<bround()> or L<bfround()> or by passing the desired accuracy
|
---|
2849 | to the math operation as additional parameter:
|
---|
2850 |
|
---|
2851 | my $x = Math::BigInt->new(30000);
|
---|
2852 | my $y = Math::BigInt->new(7);
|
---|
2853 | print scalar $x->copy()->bdiv($y, 2); # print 4300
|
---|
2854 | print scalar $x->copy()->bdiv($y)->bround(2); # print 4300
|
---|
2855 |
|
---|
2856 | =head2 precision()
|
---|
2857 |
|
---|
2858 | $x->precision(-2); # local for $x, round at the second digit right of the dot
|
---|
2859 | $x->precision(2); # ditto, round at the second digit left of the dot
|
---|
2860 |
|
---|
2861 | CLASS->precision(5); # Global for all members of CLASS
|
---|
2862 | # This also applies to new()!
|
---|
2863 | CLASS->precision(-5); # ditto
|
---|
2864 |
|
---|
2865 | $P = CLASS->precision(); # read out global precision
|
---|
2866 | $P = $x->precision(); # read out precision that affects $x
|
---|
2867 |
|
---|
2868 | Note: You probably want to use L<accuracy()> instead. With L<accuracy> you
|
---|
2869 | set the number of digits each result should have, with L<precision> you
|
---|
2870 | set the place where to round!
|
---|
2871 |
|
---|
2872 | =head1 Autocreating constants
|
---|
2873 |
|
---|
2874 | After C<use Math::BigFloat ':constant'> all the floating point constants
|
---|
2875 | in the given scope are converted to C<Math::BigFloat>. This conversion
|
---|
2876 | happens at compile time.
|
---|
2877 |
|
---|
2878 | In particular
|
---|
2879 |
|
---|
2880 | perl -MMath::BigFloat=:constant -e 'print 2E-100,"\n"'
|
---|
2881 |
|
---|
2882 | prints the value of C<2E-100>. Note that without conversion of
|
---|
2883 | constants the expression 2E-100 will be calculated as normal floating point
|
---|
2884 | number.
|
---|
2885 |
|
---|
2886 | Please note that ':constant' does not affect integer constants, nor binary
|
---|
2887 | nor hexadecimal constants. Use L<bignum> or L<Math::BigInt> to get this to
|
---|
2888 | work.
|
---|
2889 |
|
---|
2890 | =head2 Math library
|
---|
2891 |
|
---|
2892 | Math with the numbers is done (by default) by a module called
|
---|
2893 | Math::BigInt::Calc. This is equivalent to saying:
|
---|
2894 |
|
---|
2895 | use Math::BigFloat lib => 'Calc';
|
---|
2896 |
|
---|
2897 | You can change this by using:
|
---|
2898 |
|
---|
2899 | use Math::BigFloat lib => 'BitVect';
|
---|
2900 |
|
---|
2901 | The following would first try to find Math::BigInt::Foo, then
|
---|
2902 | Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
|
---|
2903 |
|
---|
2904 | use Math::BigFloat lib => 'Foo,Math::BigInt::Bar';
|
---|
2905 |
|
---|
2906 | Calc.pm uses as internal format an array of elements of some decimal base
|
---|
2907 | (usually 1e7, but this might be differen for some systems) with the least
|
---|
2908 | significant digit first, while BitVect.pm uses a bit vector of base 2, most
|
---|
2909 | significant bit first. Other modules might use even different means of
|
---|
2910 | representing the numbers. See the respective module documentation for further
|
---|
2911 | details.
|
---|
2912 |
|
---|
2913 | Please note that Math::BigFloat does B<not> use the denoted library itself,
|
---|
2914 | but it merely passes the lib argument to Math::BigInt. So, instead of the need
|
---|
2915 | to do:
|
---|
2916 |
|
---|
2917 | use Math::BigInt lib => 'GMP';
|
---|
2918 | use Math::BigFloat;
|
---|
2919 |
|
---|
2920 | you can roll it all into one line:
|
---|
2921 |
|
---|
2922 | use Math::BigFloat lib => 'GMP';
|
---|
2923 |
|
---|
2924 | It is also possible to just require Math::BigFloat:
|
---|
2925 |
|
---|
2926 | require Math::BigFloat;
|
---|
2927 |
|
---|
2928 | This will load the neccessary things (like BigInt) when they are needed, and
|
---|
2929 | automatically.
|
---|
2930 |
|
---|
2931 | Use the lib, Luke! And see L<Using Math::BigInt::Lite> for more details than
|
---|
2932 | you ever wanted to know about loading a different library.
|
---|
2933 |
|
---|
2934 | =head2 Using Math::BigInt::Lite
|
---|
2935 |
|
---|
2936 | It is possible to use L<Math::BigInt::Lite> with Math::BigFloat:
|
---|
2937 |
|
---|
2938 | # 1
|
---|
2939 | use Math::BigFloat with => 'Math::BigInt::Lite';
|
---|
2940 |
|
---|
2941 | There is no need to "use Math::BigInt" or "use Math::BigInt::Lite", but you
|
---|
2942 | can combine these if you want. For instance, you may want to use
|
---|
2943 | Math::BigInt objects in your main script, too.
|
---|
2944 |
|
---|
2945 | # 2
|
---|
2946 | use Math::BigInt;
|
---|
2947 | use Math::BigFloat with => 'Math::BigInt::Lite';
|
---|
2948 |
|
---|
2949 | Of course, you can combine this with the C<lib> parameter.
|
---|
2950 |
|
---|
2951 | # 3
|
---|
2952 | use Math::BigFloat with => 'Math::BigInt::Lite', lib => 'GMP,Pari';
|
---|
2953 |
|
---|
2954 | There is no need for a "use Math::BigInt;" statement, even if you want to
|
---|
2955 | use Math::BigInt's, since Math::BigFloat will needs Math::BigInt and thus
|
---|
2956 | always loads it. But if you add it, add it B<before>:
|
---|
2957 |
|
---|
2958 | # 4
|
---|
2959 | use Math::BigInt;
|
---|
2960 | use Math::BigFloat with => 'Math::BigInt::Lite', lib => 'GMP,Pari';
|
---|
2961 |
|
---|
2962 | Notice that the module with the last C<lib> will "win" and thus
|
---|
2963 | it's lib will be used if the lib is available:
|
---|
2964 |
|
---|
2965 | # 5
|
---|
2966 | use Math::BigInt lib => 'Bar,Baz';
|
---|
2967 | use Math::BigFloat with => 'Math::BigInt::Lite', lib => 'Foo';
|
---|
2968 |
|
---|
2969 | That would try to load Foo, Bar, Baz and Calc (in that order). Or in other
|
---|
2970 | words, Math::BigFloat will try to retain previously loaded libs when you
|
---|
2971 | don't specify it onem but if you specify one, it will try to load them.
|
---|
2972 |
|
---|
2973 | Actually, the lib loading order would be "Bar,Baz,Calc", and then
|
---|
2974 | "Foo,Bar,Baz,Calc", but independend of which lib exists, the result is the
|
---|
2975 | same as trying the latter load alone, except for the fact that one of Bar or
|
---|
2976 | Baz might be loaded needlessly in an intermidiate step (and thus hang around
|
---|
2977 | and waste memory). If neither Bar nor Baz exist (or don't work/compile), they
|
---|
2978 | will still be tried to be loaded, but this is not as time/memory consuming as
|
---|
2979 | actually loading one of them. Still, this type of usage is not recommended due
|
---|
2980 | to these issues.
|
---|
2981 |
|
---|
2982 | The old way (loading the lib only in BigInt) still works though:
|
---|
2983 |
|
---|
2984 | # 6
|
---|
2985 | use Math::BigInt lib => 'Bar,Baz';
|
---|
2986 | use Math::BigFloat;
|
---|
2987 |
|
---|
2988 | You can even load Math::BigInt afterwards:
|
---|
2989 |
|
---|
2990 | # 7
|
---|
2991 | use Math::BigFloat;
|
---|
2992 | use Math::BigInt lib => 'Bar,Baz';
|
---|
2993 |
|
---|
2994 | But this has the same problems like #5, it will first load Calc
|
---|
2995 | (Math::BigFloat needs Math::BigInt and thus loads it) and then later Bar or
|
---|
2996 | Baz, depending on which of them works and is usable/loadable. Since this
|
---|
2997 | loads Calc unnecc., it is not recommended.
|
---|
2998 |
|
---|
2999 | Since it also possible to just require Math::BigFloat, this poses the question
|
---|
3000 | about what libary this will use:
|
---|
3001 |
|
---|
3002 | require Math::BigFloat;
|
---|
3003 | my $x = Math::BigFloat->new(123); $x += 123;
|
---|
3004 |
|
---|
3005 | It will use Calc. Please note that the call to import() is still done, but
|
---|
3006 | only when you use for the first time some Math::BigFloat math (it is triggered
|
---|
3007 | via any constructor, so the first time you create a Math::BigFloat, the load
|
---|
3008 | will happen in the background). This means:
|
---|
3009 |
|
---|
3010 | require Math::BigFloat;
|
---|
3011 | Math::BigFloat->import ( lib => 'Foo,Bar' );
|
---|
3012 |
|
---|
3013 | would be the same as:
|
---|
3014 |
|
---|
3015 | use Math::BigFloat lib => 'Foo, Bar';
|
---|
3016 |
|
---|
3017 | But don't try to be clever to insert some operations in between:
|
---|
3018 |
|
---|
3019 | require Math::BigFloat;
|
---|
3020 | my $x = Math::BigFloat->bone() + 4; # load BigInt and Calc
|
---|
3021 | Math::BigFloat->import( lib => 'Pari' ); # load Pari, too
|
---|
3022 | $x = Math::BigFloat->bone()+4; # now use Pari
|
---|
3023 |
|
---|
3024 | While this works, it loads Calc needlessly. But maybe you just wanted that?
|
---|
3025 |
|
---|
3026 | B<Examples #3 is highly recommended> for daily usage.
|
---|
3027 |
|
---|
3028 | =head1 BUGS
|
---|
3029 |
|
---|
3030 | Please see the file BUGS in the CPAN distribution Math::BigInt for known bugs.
|
---|
3031 |
|
---|
3032 | =head1 CAVEATS
|
---|
3033 |
|
---|
3034 | =over 1
|
---|
3035 |
|
---|
3036 | =item stringify, bstr()
|
---|
3037 |
|
---|
3038 | Both stringify and bstr() now drop the leading '+'. The old code would return
|
---|
3039 | '+1.23', the new returns '1.23'. See the documentation in L<Math::BigInt> for
|
---|
3040 | reasoning and details.
|
---|
3041 |
|
---|
3042 | =item bdiv
|
---|
3043 |
|
---|
3044 | The following will probably not do what you expect:
|
---|
3045 |
|
---|
3046 | print $c->bdiv(123.456),"\n";
|
---|
3047 |
|
---|
3048 | It prints both quotient and reminder since print works in list context. Also,
|
---|
3049 | bdiv() will modify $c, so be carefull. You probably want to use
|
---|
3050 |
|
---|
3051 | print $c / 123.456,"\n";
|
---|
3052 | print scalar $c->bdiv(123.456),"\n"; # or if you want to modify $c
|
---|
3053 |
|
---|
3054 | instead.
|
---|
3055 |
|
---|
3056 | =item Modifying and =
|
---|
3057 |
|
---|
3058 | Beware of:
|
---|
3059 |
|
---|
3060 | $x = Math::BigFloat->new(5);
|
---|
3061 | $y = $x;
|
---|
3062 |
|
---|
3063 | It will not do what you think, e.g. making a copy of $x. Instead it just makes
|
---|
3064 | a second reference to the B<same> object and stores it in $y. Thus anything
|
---|
3065 | that modifies $x will modify $y (except overloaded math operators), and vice
|
---|
3066 | versa. See L<Math::BigInt> for details and how to avoid that.
|
---|
3067 |
|
---|
3068 | =item bpow
|
---|
3069 |
|
---|
3070 | C<bpow()> now modifies the first argument, unlike the old code which left
|
---|
3071 | it alone and only returned the result. This is to be consistent with
|
---|
3072 | C<badd()> etc. The first will modify $x, the second one won't:
|
---|
3073 |
|
---|
3074 | print bpow($x,$i),"\n"; # modify $x
|
---|
3075 | print $x->bpow($i),"\n"; # ditto
|
---|
3076 | print $x ** $i,"\n"; # leave $x alone
|
---|
3077 |
|
---|
3078 | =item precision() vs. accuracy()
|
---|
3079 |
|
---|
3080 | A common pitfall is to use L<precision()> when you want to round a result to
|
---|
3081 | a certain number of digits:
|
---|
3082 |
|
---|
3083 | use Math::BigFloat;
|
---|
3084 |
|
---|
3085 | Math::BigFloat->precision(4); # does not do what you think it does
|
---|
3086 | my $x = Math::BigFloat->new(12345); # rounds $x to "12000"!
|
---|
3087 | print "$x\n"; # print "12000"
|
---|
3088 | my $y = Math::BigFloat->new(3); # rounds $y to "0"!
|
---|
3089 | print "$y\n"; # print "0"
|
---|
3090 | $z = $x / $y; # 12000 / 0 => NaN!
|
---|
3091 | print "$z\n";
|
---|
3092 | print $z->precision(),"\n"; # 4
|
---|
3093 |
|
---|
3094 | Replacing L<precision> with L<accuracy> is probably not what you want, either:
|
---|
3095 |
|
---|
3096 | use Math::BigFloat;
|
---|
3097 |
|
---|
3098 | Math::BigFloat->accuracy(4); # enables global rounding:
|
---|
3099 | my $x = Math::BigFloat->new(123456); # rounded immidiately to "12350"
|
---|
3100 | print "$x\n"; # print "123500"
|
---|
3101 | my $y = Math::BigFloat->new(3); # rounded to "3
|
---|
3102 | print "$y\n"; # print "3"
|
---|
3103 | print $z = $x->copy()->bdiv($y),"\n"; # 41170
|
---|
3104 | print $z->accuracy(),"\n"; # 4
|
---|
3105 |
|
---|
3106 | What you want to use instead is:
|
---|
3107 |
|
---|
3108 | use Math::BigFloat;
|
---|
3109 |
|
---|
3110 | my $x = Math::BigFloat->new(123456); # no rounding
|
---|
3111 | print "$x\n"; # print "123456"
|
---|
3112 | my $y = Math::BigFloat->new(3); # no rounding
|
---|
3113 | print "$y\n"; # print "3"
|
---|
3114 | print $z = $x->copy()->bdiv($y,4),"\n"; # 41150
|
---|
3115 | print $z->accuracy(),"\n"; # undef
|
---|
3116 |
|
---|
3117 | In addition to computing what you expected, the last example also does B<not>
|
---|
3118 | "taint" the result with an accuracy or precision setting, which would
|
---|
3119 | influence any further operation.
|
---|
3120 |
|
---|
3121 | =back
|
---|
3122 |
|
---|
3123 | =head1 SEE ALSO
|
---|
3124 |
|
---|
3125 | L<Math::BigInt>, L<Math::BigRat> and L<Math::Big> as well as
|
---|
3126 | L<Math::BigInt::BitVect>, L<Math::BigInt::Pari> and L<Math::BigInt::GMP>.
|
---|
3127 |
|
---|
3128 | The pragmas L<bignum>, L<bigint> and L<bigrat> might also be of interest
|
---|
3129 | because they solve the autoupgrading/downgrading issue, at least partly.
|
---|
3130 |
|
---|
3131 | The package at
|
---|
3132 | L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains
|
---|
3133 | more documentation including a full version history, testcases, empty
|
---|
3134 | subclass files and benchmarks.
|
---|
3135 |
|
---|
3136 | =head1 LICENSE
|
---|
3137 |
|
---|
3138 | This program is free software; you may redistribute it and/or modify it under
|
---|
3139 | the same terms as Perl itself.
|
---|
3140 |
|
---|
3141 | =head1 AUTHORS
|
---|
3142 |
|
---|
3143 | Mark Biggar, overloaded interface by Ilya Zakharevich.
|
---|
3144 | Completely rewritten by Tels L<http://bloodgate.com> in 2001 - 2004, and still
|
---|
3145 | at it in 2005.
|
---|
3146 |
|
---|
3147 | =cut
|
---|