in reply to Re^7: Math::BigFloat to native double?
in thread Math::BigFloat to native double?

This is what I get by adding up all the binary fractions for all the set bits in your hex/binary representation:

 0  0.50000000000000000000000000000000000000000000000000
 1  0.25000000000000000000000000000000000000000000000000
 4  0.03125000000000000000000000000000000000000000000000
 7  0.00390625000000000000000000000000000000000000000000
 12 0.00012207031250000000000000000000000000000000000000
 13 0.00006103515625000000000000000000000000000000000000
 14 0.00003051757812500000000000000000000000000000000000
 15 0.00001525878906250000000000000000000000000000000000
 16 0.00000762939453125000000000000000000000000000000000
 17 0.00000381469726562500000000000000000000000000000000
 19 0.00000095367431640625000000000000000000000000000000
 20 0.00000047683715820312500000000000000000000000000000
 22 0.00000011920928955078125000000000000000000000000000
 24 0.00000002980232238769531300000000000000000000000000
 26 0.00000000745058059692382810000000000000000000000000
 30 0.00000000046566128730773926000000000000000000000000
 34 0.00000000002910383045673370400000000000000000000000
 39 0.00000000000090949470177292824000000000000000000000
 41 0.00000000000022737367544323206000000000000000000000
 42 0.00000000000011368683772161603000000000000000000000
 44 0.00000000000002842170943040400700000000000000000000
 48 0.00000000000000177635683940025050000000000000000000
 49 0.00000000000000088817841970012523000000000000000000
 54 0.00000000000000002775557561562891400000000000000000
 58 0.00000000000000000173472347597680710000000000000000
 59 0.00000000000000000086736173798840355000000000000000
 61 0.00000000000000000021684043449710089000000000000000
 64 0.00000000000000000002710505431213761100000000000000
 65 0.00000000000000000001355252715606880500000000000000
 69 0.00000000000000000000084703294725430034000000000000
 72 0.00000000000000000000010587911840678754000000000000
 73 0.00000000000000000000005293955920339377100000000000
 77 0.00000000000000000000000330872245021211070000000000
 78 0.00000000000000000000000165436122510605530000000000
 81 0.00000000000000000000000020679515313825692000000000
 82 0.00000000000000000000000010339757656912846000000000
 86 0.00000000000000000000000000646234853557052870000000
 88 0.00000000000000000000000000161558713389263220000000
 92 0.00000000000000000000000000010097419586828951000000
 94 0.00000000000000000000000000002524354896707237800000
 95 0.00000000000000000000000000001262177448353618900000
 96 0.00000000000000000000000000000631088724176809440000
104 0.00000000000000000000000000000002465190328815661900
105 0.00000000000000000000000000000001232595164407830900
106 0.00000000000000000000000000000000616297582203915470
    0.78539816439944831161566131939647068003696135548270 <<< 
      001124476675678989811791619118619999856552441020
                         00  0 0 01  0

    0.78539816439944831161566131939647068003696135548270
  * 4                                                   
  = 3.14159265759779324646264527758588272014784542193080
    3.1415926535897932384626433832795 <<<<<<<<<<<<<<<<<<<< (my) origin
+al input.
              ^ ^^    ^^     ^^^^^^^^

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Re^9: Math::BigFloat to native double?
by syphilis (Archbishop) on Jul 13, 2015 at 12:14 UTC
    From bit 24 on, you've understated the values.
    Following, are the values yielded by Math::MPFR. It's a 107-bit string, so I've capped the decimal values I've used to represent 107 bit values.
    There's perhaps an argument for sticking to 106-bit precision, and ignoring the 107th bit - but, in any case, it makes only a minute difference to the calculations.
    Here are the calculations I've got: 
    C:\_32\pscrpt\math-mpfr>perl sum.pl
0 5e-1
1 2.5e-1
4 3.125e-2
7 3.90625e-3
12 1.220703125e-4
13 6.103515625e-5
14 3.0517578125e-5
15 1.52587890625e-5
16 7.62939453125e-6
17 3.814697265625e-6
19 9.5367431640625e-7
20 4.76837158203125e-7
22 1.1920928955078125e-7
24 2.98023223876953125e-8
26 7.450580596923828125e-9
30 4.656612873077392578125e-10
34 2.910383045673370361328125e-11
39 9.094947017729282379150390625e-13
41 2.27373675443232059478759765625e-13
42 1.136868377216160297393798828125e-13
44 2.8421709430404007434844970703125e-14
48 1.776356839400250464677810668945312e-15
49 8.881784197001252323389053344726562e-16
54 2.775557561562891351059079170227051e-17
58 1.734723475976807094411924481391907e-18
59 8.673617379884035472059622406959534e-19
61 2.168404344971008868014905601739883e-19
64 2.710505431213761085018632002174854e-20
65 1.355252715606880542509316001087427e-20
69 8.47032947254300339068322500679642e-22
72 1.058791184067875423835403125849552e-22
73 5.293955920339377119177015629247762e-23
77 3.308722450212110699485634768279851e-24
78 1.654361225106055349742817384139926e-24
81 2.067951531382569187178521730174907e-25
82 1.033975765691284593589260865087454e-25
86 6.462348535570528709932880406796585e-27
88 1.615587133892632177483220101699146e-27
92 1.009741958682895110927012563561966e-28
94 2.524354896707237777317531408904916e-29
95 1.262177448353618888658765704452458e-29
96 6.31088724176809444329382852226229e-30
104 2.465190328815661891911651766508707e-32
105 1.232595164407830945955825883254353e-32
106 6.162975822039154729779129416271767e-33

Sum =
7.853981633974483096156608458198765e-1
Sum * 4 =
3.141592653589793238462643383279506

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    It overstates the value by only 6e-33.
    And the script that I used: 
    use strict;
use warnings;
use Math::MPFR qw(:mpfr);

# Set precision to precisely 107 bits
Rmpfr_set_default_prec(107);

# Set $bin_str to the 107-bit representation of
# 3.1415926535897932384626433832795
my $bin_str = '1100100100001111110110101010001000100001011010001100001
+0001101001100010011000110011000101000101110000000111';

my $check = Math::MPFR->new(0);

for(my $i = 0; $i < 107; $i++) {
  if(substr($bin_str, $i, 1)) {
     print "$i ", Math::MPFR->new(2 ** (-($i + 1))), "\n";
     $check += 2 ** (-($i + 1));
  }
}

print "\nSum =\n$check\nSum * 4 =\n", $check * 4, "\n";

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    Cheers,
    Rob
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