(************** Content-type: application/mathematica **************
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Notebook[{
Cell["\<\
Set the range of the table.  1 to 1.6 corresponds to the first 12 pages of \
Babbage's 7 figure logarithms (6000 entries)\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(minlog\  = \ 1\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`1\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(maxlog\  = \ 1.600\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`1.6`\)], "Output"]
}, Open  ]],

Cell["\<\
Generate a table at large stepsize over the range.  This can compared to the \
table as a cross check and will be used to find a polynomial fit\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(\(Logtable = 
      Table[{N[x, 40], N[Log[10, x], 40]}, {x, minlog, \ maxlog, 
          1/100}]\)\(\[IndentingNewLine]\)
    \)\)], "Input"],

Cell[BoxData[
    FormBox[
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            {"1.14`40", 
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            {"1.1999999999999999999999999999999999999999999999998632`40", 
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            {"1.21`40", 
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            {"1.22`40", 
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              "0.1846914308175988031302257008385676484078251068064381`40"},
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              "0.187520720836463066676155124560153461958464468252527`40"},
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          ColumnAlignments->{Decimal}], "\[NoBreak]", ")"}], 
      TraditionalForm]], "Output"]
}, Open  ]],

Cell["Fit a 7th order polynomial in x", "Text"],

Cell[BoxData[
    \(x =. \)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(polyFit\  = \ 
      N[Fit[Logtable, \ Table[x^i, {i, 0, 7}], \ x], 40]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`0.0108390982779167179460439166798121548415755047166854`\
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      0.534494971187486156520502493437286940743382135435`32.2269\ x\^5 - 
      1.434145531777479444021509532153320106551224478853`32.3304\ x\^4 + 
      2.453558071956004998329895965439358452177378427013`32.4614\ x\^3 - 
      2.822704707505721359402681145482395659701851485002`32.6303\ x\^2 + 
      2.396372773620486572553703684060062605788745208409`32.924\ x - 
      1.023562894789998411947104258419257122446837120631`33.289\)], "Output"]
}, Open  ]],

Cell["\<\
Evaluate this polynomial for the same set of points as in the original table \
used for the fit\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(calcLogs = \ 
      polyFit /. x \[Rule] 
          Table[N[x, 40], {x, minlog, \ maxlog, 1/100}]\)], "Input"],

Cell[BoxData[
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      0.0899051105310137844996228133222650846521779`30.1952, 
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      0.0969100138055878097035102717981203566615005`30.2037, 
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      0.1673173359969411107220880931788995537019584`30.1871, 
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      0.1789769503932546041845294497679056028106106`30.1724, 
      0.181843590518184973873697996682621561762272`30.1684, 
      0.184691432386164210731475924702526541027853`30.1643, 
      0.1875207209930174362429812918586454020738393`30.1601, 
      0.1903316966860367927135401763611914475449063`30.1558, 
      0.1931245953205340932439994655519036197220903`30.1514, 
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      0.2013971233784759192451440155967123531360095`30.1376, 
      0.2041199880674216139635509758422010978557822`30.1329}\)], "Output"]
}, Open  ]],

Cell["\<\
Now compute maximum absolute error of the fit over this range\
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Max[
      Abs[calcLogs - 
          Table[Log[10, x], {x, minlog, \ maxlog, 1/100}]]]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`6.552686475564072868641218008814857406417`23.35*^-9\)],\
 "Output"]
}, Open  ]],

Cell["\<\
This shows we have one to two guard digits to ensure correct rounding

Now compute the first entries of the table, from the fitted polynomial at the \
desired spacing. \
\>", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(y = 
      Table[N[y, 40], {y, \ minlog, minlog + 9/10000, 1/10000}]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`{1.`40, 1.0001`40, 1.0002`40, 1.0003`40, 1.0004`40, 
      1.0005`40, 1.0006`40, 1.0007`40, 1.0008`40, 1.0009`40}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Diffs[0]\  = \ N[polyFit /. x \[Rule] y, 40]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`{6.552686475564072868641218008814857406417`23.35*^-9, 
      0.000043433716210888984467294301023705607372159688`27.1713, 
      0.000086856538410250333100724809604439031164258787`27.4721, 
      0.000130275020149733084358878571465122550394609493`27.6481, 
      0.000173689162294258698533962319863526618231643638`27.7729, 
      0.000217098965708496709360995004069968743616993011`27.8696, 
      0.000260504431256864811526083843084773723378543357`27.9486, 
      0.000303905559803528948146658008719036109716381004`28.0154, 
      0.000347302352212403398223665400944216983086730077`28.0733, 
      0.000390694809347150864065737978416107100510828896`28.1242}\)], "Output"]
}, Open  ]],

Cell["Compute the differences", "Text"],

Cell[BoxData[
    \(Do[Diffs[i] = 
        Drop[Diffs[i - 1], 1] - \(Diffs[i - 1]\)[\([Range[1, 10 - i]]\)], {i, 
        7}]\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Diffs[1]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`{0.000043427163524413420394425659805696792514753272`26.\
8703, 0.000043422822199361348633430508580733423792099098`26.8701, 
      0.000043418481739482751258153761860683519230350706`26.8699, 
      0.000043414142144525614175083748398404067837034145`26.8697, 
      0.000043409803414238010827032684206442125385349373`26.8696, 
      0.000043405465548368102165088839014804979761550346`26.8694, 
      0.000043401128546664136620574165634262386337837647`26.8692, 
      0.000043396792408874450077007392225180873370349073`26.869, 
      0.00004339245713474746584207257747189011742409882`26.8689}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Diffs[2]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`{\(-4.341325052071760995151224963368722654173`22.569*^-\
9\), \(-4.340459878597375276746720049904561748392`22.5688*^-9\), \
\(-4.339594957137083070013462279451393316561`22.5686*^-9\), \
\(-4.338730287603348051064191961942451684772`22.5684*^-9\), \
\(-4.337865869908661943845191637145623799027`22.5682*^-9\), \
\(-4.337001703965544514673380542593423712699`22.5679*^-9\), \
\(-4.336137789686543566773409081512967488574`22.5677*^-9\), \
\(-4.335274126984234934814753290755946250253`22.5675*^-9\)}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Diffs[3]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`{8.65173474385718404504913464160905781`18.5674*^-13, 
      8.64921460292206733257770453168431832`18.5672*^-13, 
      8.64669533735018949270317508941631788`18.5669*^-13, 
      8.64417694686107219000324796827885746`18.5666*^-13, 
      8.64165943117429171811094552200086328`18.5664*^-13, 
      8.63914279000947899971461080456224125`18.5661*^-13, 
      8.63662702308631958655790757021238321`18.5659*^-13}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Diffs[4]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`{\(-2.52014093511671247143010992473949`14.7306*^-16\), \
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\(-2.51839048911730269992712113746042`14.7301*^-16\), \
\(-2.51751568678047189230244627799418`14.7298*^-16\), \
\(-2.51664116481271839633471743862203`14.7295*^-16\), \
\(-2.51576692315941315670323434985804`14.7292*^-16\)}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Diffs[5]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`{8.7536323887259690066765673905`10.9703*^-20, 
      8.7508276053717460232113054002`10.97*^-20, 
      8.7480233683080762467485946624`10.9698*^-20, 
      8.7452196775349596772883937215`10.9695*^-20, 
      8.7424165330523963148308876399`10.9692*^-20}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Diffs[6]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`{\(-2.8047833542229834652619903`7.1749*^-23\), \
\(-2.8042370636697764627107377`7.1747*^-23\), \
\(-2.8036907731165694602009409`7.1745*^-23\), \
\(-2.8031444825633624575060816`7.1743*^-23\)}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(Diffs[7]\)], "Input"],

Cell[BoxData[
    \(TraditionalForm\`{5.462905532070025512526`3.1634*^-27, 
      5.462905532070025097968`3.1632*^-27, 
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}, Open  ]],

Cell["\<\
Note how the higher order differences become very small.  Hence we need a \
large number of digits in fixed point to retain adequate precision

Compute integer representations of the initial settings by scaling up by \
10^27.  The result column has 5 added to the 8th place so all results are \
automatically rounded to 7 digits. Negative differences are converted to 10's \
complement form. Note that because of the pipeline in the difference engine, \
earlier differences have to be taken from later rows of the table.\
\>", "Text"],

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Now compute an actual table as a final cross check  These numbers should \
match exactly the beinning of Babbage's 7 figure table. Note that the \
calculation is pipelined in the same way the physical difference engine is\
\>", "Text"],

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