Interval arithmetic by Robert Elton Maas:

Demonstration of using Newton's method with intervals of exact
rational numbers to compute square root of an exact rational:

* (setq nums 2)

2
* (setq guess (interval-to-sqrt-firstguess+bounds nums))

5/3
* (setq rats (interval+guess-to-sqrt-bounds nums guess))

(6/5 5/3)
* (show-interval rats)
1.2d0
1.6666666666666667d0
1.[2..7]
NIL
* (setq rats (interval-newton-sqrt nums rats))
1.2d0
1.6666666666666667d0
1.[2..7]
[-----------------------------!!!!!!-----------------------------------]
1.3953488372093024d0
1.4333333333333333d0
1.3[9..E]
[------------------------------!!!!!-----------------------------------]
1.4d0
1.4285714285714286d0
1.4[0..3]
(7/5 10/7)
* (setq rats (interval-newton-sqrt nums rats))
1.4d0
1.4285714285714286d0
1.4[0..3]
[----------------------------------!-----------------------------------]
1.4141414141414141d0
1.4142857142857144d0
1.4141[4..J]
[----------------------------------!-----------------------------------]
1.4141414141414141d0
1.4142857142857144d0
1.4141[4..J]
(140/99 99/70)
* (setq rats (interval-newton-sqrt nums rats))
1.4141414141414141d0
1.4142857142857144d0
1.4141[4..J]
[----------------------------------!-----------------------------------]
1.4142135605326258d0
1.4142135642135643d0
1.41421356[0..5]
[----------------------------------!-----------------------------------]
1.4142135605326258d0
1.4142135642135643d0
1.41421356[0..5]
(27720/19601 19601/13860)
* (setq rats (interval-newton-sqrt nums rats))
1.4142135605326258d0
1.4142135642135643d0
1.41421356[0..5]
[----------------------------------!-----------------------------------]
1.4142135623730951d0
1.4142135623730951d0
1.41421356237309504[7..A]
[----------------------------------!-----------------------------------]
1.4142135623730951d0
1.4142135623730951d0
1.41421356237309504[7..A]
(1086679440/768398401 768398401/543339720)
* (setq rats (interval-newton-sqrt nums rats))
1.4142135623730951d0
1.4142135623730951d0
1.41421356237309504[7..A]
[----------------------------------!-----------------------------------]
1.4142135623730951d0
1.4142135623730951d0
1.414213562373095048801688724209698078[0..B]
[----------------------------------!-----------------------------------]
1.4142135623730951d0
1.4142135623730951d0
1.414213562373095048801688724209698078[0..B]
(1670005488191150880/1180872205318713601 1180872205318713601/835002744095575440)
* (setq rats (interval-newton-sqrt nums rats))
1.4142135623730951d0
1.4142135623730951d0
1.414213562373095048801688724209698078[0..B]
[----------------------------------!-----------------------------------]
1.4142135623730951d0
1.4142135623730951d0
1.414213562373095048801688724209698078569671875376948073176679737990732478[3..6]
[----------------------------------!-----------------------------------]
1.4142135623730951d0
1.4142135623730951d0
1.414213562373095048801688724209698078569671875376948073176679737990732478[3..6]
(3944126127469278527968910146598237760/2788918330588564181308597538924774401
 2788918330588564181308597538924774401/1972063063734639263984455073299118880)
* (setq rats (interval-newton-sqrt nums rats))
1.4142135623730951d0
1.4142135623730951d0
1.414213562373095048801688724209698078569671875376948073176679737990732478[3..6]
[----------------------------------!-----------------------------------]
1.4142135623730951d0
1.4142135623730951d0
1.414213562373095048801688724209698078569671875376948073176679737990732478462107
03885038753432764157273501384623091229702492483605585073721264412149[4..B]
[----------------------------------!-----------------------------------]
1.4142135623730951d0
1.4142135623730951d0
1.414213562373095048801688724209698078569671875376948073176679737990732478462107
03885038753432764157273501384623091229702492483605585073721264412149[4..B]
(21999691310104717527439432627183280059647288103291440870635684784319163520/1555
6130909385807535224779842639686625468648065798177627126514337489817601
 15556130909385807535224779842639686625468648065798177627126514337489817601/1099
9845655052358763719716313591640029823644051645720435317842392159581760)
* (setq rats (interval-newton-sqrt nums rats))
1.4142135623730951d0
1.4142135623730951d0
1.414213562373095048801688724209698078569671875376948073176679737990732478462107
03885038753432764157273501384623091229702492483605585073721264412149[4..B]
[----------------------------------!-----------------------------------]
1.4142135623730951d0
1.4142135623730951d0
1.414213562373095048801688724209698078569671875376948073176679737990732478462107
03885038753432764157273501384623091229702492483605585073721264412149709993583141
32226659275055927557999505011527820605714701095599716059702745345968620147285174
1864088919860955232923048430871432145083976260362799524[8..F]
[----------------------------------!-----------------------------------]
1.4142135623730951d0
1.4142135623730951d0
1.414213562373095048801688724209698078569671875376948073176679737990732478462107
03885038753432764157273501384623091229702492483605585073721264412149709993583141
32226659275055927557999505011527820605714701095599716059702745345968620147285174
1864088919860955232923048430871432145083976260362799524[8..F]
(6844601559721326940707442446581772372247199832750373430509512436961221356798680
77853932463689356406733902637826335391498271632950349602667386231040/48398641773
98970226568389107997019805362340446631167408482804585370251674736955856570734973
19288990606925212316394779012897026242645624712498790401
 4839864177398970226568389107997019805362340446631167408482804585370251674736955
85657073497319288990606925212316394779012897026242645624712498790401/34223007798
60663470353721223290886186123599916375186715254756218480610678399340389269662318
44678203366951318913167695749135816475174801333693115520)
* (setq rats (interval-newton-sqrt nums rats))
1.4142135623730951d0
1.4142135623730951d0
1.414213562373095048801688724209698078569671875376948073176679737990732478462107
03885038753432764157273501384623091229702492483605585073721264412149709993583141
32226659275055927557999505011527820605714701095599716059702745345968620147285174
1864088919860955232923048430871432145083976260362799524[8..F]
[----------------------------------!-----------------------------------]
1.4142135623730951d0
1.4142135623730951d0
1.414213562373095048801688724209698078569671875376948073176679737990732478462107
03885038753432764157273501384623091229702492483605585073721264412149709993583141
32226659275055927557999505011527820605714701095599716059702745345968620147285174
18640889198609552329230484308714321450839762603627995251407989687253396546331808
82964062061525835239505474575028775996172983557522033753185701135437460340849884
71603868999706990048150305440277903164542478230684929369186215805784631115966687
13013015618568987237235288509264861249497715421833420428568606014682472077143585
48741556570696776537202264854[1..8]
[----------------------------------!-----------------------------------]
1.4142135623730951d0
1.4142135623730951d0
1.414213562373095048801688724209698078569671875376948073176679737990732478462107
03885038753432764157273501384623091229702492483605585073721264412149709993583141
32226659275055927557999505011527820605714701095599716059702745345968620147285174
18640889198609552329230484308714321450839762603627995251407989687253396546331808
82964062061525835239505474575028775996172983557522033753185701135437460340849884
71603868999706990048150305440277903164542478230684929369186215805784631115966687
13013015618568987237235288509264861249497715421833420428568606014682472077143585
48741556570696776537202264854[1..8]
(6625388379492873719657464203235861724388425699463729098813625658205135204798087
99452620384130568587935908319621683690583446195109261195831496607850082170348760
98985298538108162966191442800154575284638294858326260324771330491621659556989311
6654612982285472053095300800060620585243416564640494080/468485705113396214873500
34820943104991070986082580321494861873507456352961028657647552611031131318428206
05826180052401657397941258739732688514005380277132191061725997102684695643814554
39513977138838473553831781520213282206409296900451790979927412200913195334021599
318597103692473541776259481601
 4684857051133962148735003482094310499107098608258032149486187350745635296102865
76475526110311313184282060582618005240165739794125873973268851400538027713219106
17259971026846956438145543951397713883847355383178152021328220640929690045179097
9927412200913195334021599318597103692473541776259481601/331269418974643685982873
21016179308621942128497318645494068128291025676023990439972631019206528429396795
41598108418452917230975546305979157483039250410851743804949264926905408148309572
14000772876423191474291631301623856652458108297784946558327306491142736026547650
400030310292621708282320247040)
*

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Web-server-side application so that you-all can play
with it?

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I desperately need a source of income to avoid becoming
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