Interval arithmetic by Robert Elton Maas:

Demonstration of using Newton's method with intervals
of exact rational numbers to compute bounds on square
root of an exact rational number, 2 in this case, then
feeding those imprecise bounds into a second interval
arithmetic Newton's method sqrt to yield bounds on
fourth-root of 2. Note that I manually emulate lazy
evaluation, whereby output of fourth-root-of-2 is
driving the process, and only when Newton's method for
sqrt(2) -> frrt(2) hangs do we go back and run one
iteration of 2 -> sqrt(2) before resuming the sqrt(2)
-> frrt(2) process. Typically, once this gets started,
there are three iterations of the toplevel process for
each time we go back to the sub-process:
-1- Using interval based on previous bounds sqrt(2),
which is lopsided with respect to frrt(2), getting only
a little bit more accuracy for frrt(2);
-2- Using that resultant interval, which is nicely
centered on frrt(2), doubles the number of significant
digits in a single iteration, as is expected for
Newton's method applied to exact (non-interval) input.
This exhausts the amount of information available in
our current approximation of sqrt(2), hence:
-3- Using that already-as-accurate-as-possible interval
as a starting point, Newton's method doesn't yield any
better bounds for frrt(2).

Bottom line: For each single iteration of 2 -> sqrt(2),
which does the abovementionned three iterations of
sqrt(2) -> frrt(2), the number of digits of accuracy in
both sqrt(2) and frrt(2) doubles.

* (setq two 2)
Warning:  Declaring TWO special.

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

5/3
* (setq sqrt2 (interval+guess-to-sqrt-bounds two guess))

(6/5 5/3)
* (show-interval sqrt2)
1.2d0
1.6666666666666667d0
1.[2..7]
NIL
* (setq guess (interval-to-sqrt-firstguess+bounds sqrt2))

6/5
* (setq frrt2 (interval+guess-to-sqrt-bounds sqrt2 guess))
Warning:  Declaring FRRT2 special.

(1 25/18)
* (show-interval frrt2)
1.0d0
1.3888888888888888d0
1.[0..4]
NIL
* (setq frrt2 (interval-newton-sqrt sqrt2 frrt2))
1.0d0
1.3888888888888888d0
1.[0..4]

** WARNING: New interval not included within old interval,
**  so I'll have to intersect the two to get a valid interval refinement.
[!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!]
1.0046511627906978d0
1.3888888888888888d0
1.[0..4]

** WARNING: New interval not included within old interval,
**  so I'll have to intersect the two to get a valid interval refinement.
[!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!----------]
0.9d0
1.3333333333333333d0
0.[9..E]
Interval didn't shrink enough:
Old: 1.[0..4]
New: 0.[9..E]
Refinement (i.e. shrinkage) factor = 0.8974359
** RETURN to go on:

(9/10 4/3)
* (setq sqrt2 (interval-newton-sqrt two sqrt2))
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 frrt2 (interval-newton-sqrt sqrt2 frrt2))
0.9d0
1.3333333333333333d0
0.[9..E]
[-----------------------------------!!!!!!!!!!!!!!!!!!!!!!!!!!!--------]
1.1166666666666667d0
1.279317697228145d0
1.1[1..I]
[-----------------------------------!!!!!!!!!!!!!!!!!!!!!!-------------]
1.12d0
1.25d0
1.1[2..F]
(28/25 5/4)
* (setq frrt2 (interval-newton-sqrt sqrt2 frrt2))
1.12d0
1.25d0
1.1[2..F]
[---------------------------------!!!!!!!!!!!!!!-----------------------]
1.1814345991561181d0
1.2055455093429777d0
1.1[8..B]
[-------------------------!!!!!!!!!!!!!!!!!!!--------------------------]
1.1666666666666667d0
1.2d0
1.1[6..A]
(7/6 6/5)
* (setq frrt2 (interval-newton-sqrt sqrt2 frrt2))
1.1666666666666667d0
1.2d0
1.1[6..A]

** WARNING: New interval not included within old interval,
**  so I'll have to intersect the two to get a valid interval refinement.
[----------------------------------!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!]
1.1830985915492958d0
1.2d0
1.18[3..K]
[!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!]
1.1666666666666667d0
1.2d0
1.1[6..A]
Interval didn't shrink enough:
Old: 1.1[6..A]
New: 1.1[6..A]
Refinement (i.e. shrinkage) factor = 1.0
** RETURN to go on:

(7/6 6/5)
* (setq sqrt2 (interval-newton-sqrt two sqrt2))
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 frrt2 (interval-newton-sqrt sqrt2 frrt2))
1.1666666666666667d0
1.2d0
1.1[6..A]
[-----------------------------------!!!!!!!!!!!!!!!!!!!!!!!!!----------]
1.1833333333333333d0
1.1951710261569417d0
1.18[3..G]
[-------------------------------------------!!!!!!!!!------------------]
1.1875d0
1.1909774436090226d0
1.18[7..B]
(19/16 792/665)
* (setq frrt2 (interval-newton-sqrt sqrt2 frrt2))
1.1875d0
1.1909774436090226d0
1.18[7..B]
[--------------------------------!!!-----------------------------------]
1.1891148414639938d0
1.1892387218045113d0
1.1891[1..E]
[---------------------------------!!!----------------------------------]
1.1891643709825528d0
1.1892857142857143d0
1.1891[6..J]
(1295/1089 333/280)
* (setq frrt2 (interval-newton-sqrt sqrt2 frrt2))
1.1891643709825528d0
1.1892857142857143d0
1.1891[6..J]

** WARNING: New interval not included within old interval,
**  so I'll have to intersect the two to get a valid interval refinement.
[!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!--------------------]
1.1891643709825528d0
1.1892498590116058d0
1.1891[6..F]
[!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!]
1.1891643709825528d0
1.1892857142857143d0
1.1891[6..J]
Interval didn't shrink enough:
Old: 1.1891[6..J]
New: 1.1891[6..J]
Refinement (i.e. shrinkage) factor = 1.0
** RETURN to go on:

(1295/1089 333/280)
* (setq sqrt2 (interval-newton-sqrt two sqrt2))
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 frrt2 (interval-newton-sqrt sqrt2 frrt2))
1.1891643709825528d0
1.1892857142857143d0
1.1891[6..J]
[--------------!!!!!!!!!!!!!!!!!!!!!-----------------------------------]
1.189189186093948d0
1.1892250426341335d0
1.1891[8..D]
[--------------!!!!!!!!!!!!!!!!!!!!!-----------------------------------]
1.1891891891891893d0
1.1892250426341335d0
1.1891[8..D]
(44/37 725237/609840)
* (setq frrt2 (interval-newton-sqrt sqrt2 frrt2))
1.1891891891891893d0
1.1892250426341335d0
1.1891[8..D]
[----------------------------------!-----------------------------------]
1.1892071125461368d0
1.1892071159116613d0
1.18920711[2..6]
[----------------------------------!!----------------------------------]
1.1892071135949704d0
1.1892071166902582d0
1.18920711[3..7]
(41794200/35144593 65016517/54672156)
* (setq frrt2 (interval-newton-sqrt sqrt2 frrt2))
1.1892071135949704d0
1.1892071166902582d0
1.18920711[3..7]

** WARNING: New interval not included within old interval,
**  so I'll have to intersect the two to get a valid interval refinement.
[!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!------]
1.1892071135949704d0
1.1892071164104718d0
1.18920711[3..7]
[!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!]
1.1892071135949704d0
1.1892071166902582d0
1.18920711[3..7]
Interval didn't shrink enough:
Old: 1.18920711[3..7]
New: 1.18920711[3..7]
Refinement (i.e. shrinkage) factor = 1.0
** RETURN to go on:

(41794200/35144593 65016517/54672156)
* (setq sqrt2 (interval-newton-sqrt two sqrt2))
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 frrt2 (interval-newton-sqrt sqrt2 frrt2))
1.1892071135949704d0
1.1892071166902582d0
1.18920711[3..7]
[----------------------------!!!!!!!-----------------------------------]
1.1892071148628278d0
1.1892071151426142d0
1.189207114[8..C]
[----------------------------!!!!!!!-----------------------------------]
1.1892071148628278d0
1.1892071151426142d0
1.189207114[8..C]
(19723/16585 2548777496117/2143257859512)
* (setq frrt2 (interval-newton-sqrt sqrt2 frrt2))
1.1892071148628278d0
1.1892071151426142d0
1.189207114[8..C]
[----------------------------------!-----------------------------------]
1.189207115002721d0
1.189207115002721d0
1.18920711500272106[5..8]
[----------------------------------!!----------------------------------]
1.189207115002721d0
1.189207115002721d0
1.18920711500272106[5..9]
(595083091/500403238 192254523967211219/161666140020337260)
* (setq frrt2 (interval-newton-sqrt sqrt2 frrt2))
1.189207115002721d0
1.189207115002721d0
1.18920711500272106[5..9]

** WARNING: New interval not included within old interval,
**  so I'll have to intersect the two to get a valid interval refinement.
[!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!-----------------------------------]
1.189207115002721d0
1.189207115002721d0
1.18920711500272106[5..8]
[!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!]
1.189207115002721d0
1.189207115002721d0
1.18920711500272106[5..9]
Interval didn't shrink enough:
Old: 1.18920711500272106[5..9]
New: 1.18920711500272106[5..9]
Refinement (i.e. shrinkage) factor = 1.0
** RETURN to go on:

(595083091/500403238 192254523967211219/161666140020337260)
* (setq sqrt2 (interval-newton-sqrt two sqrt2))
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 frrt2 (interval-newton-sqrt sqrt2 frrt2))
1.189207115002721d0
1.189207115002721d0
1.18920711500272106[5..9]
[--------------!!!!!!!!!!!!!!!!!!!!!-----------------------------------]
1.189207115002721d0
1.189207115002721d0
1.189207115002721066[2..D]
[------------------!!!!!!!!!!!!!---------------------------------------]
1.189207115002721d0
1.189207115002721d0
1.189207115002721066[4..A]
(2274536277529549176893349600/1912649402139125892120383317
 1619692117/1361993295)
* (setq frrt2 (interval-newton-sqrt sqrt2 frrt2))
1.189207115002721d0
1.189207115002721d0
1.189207115002721066[4..A]
[----------------------------------!!----------------------------------]
1.189207115002721d0
1.189207115002721d0
1.189207115002721066717499970560475915[0..A]
[----------------------------------!!----------------------------------]
1.189207115002721d0
1.189207115002721d0
1.189207115002721066717499970560475914[9..I]
(299498316123706112289466061790835440/251847060402948244751717109090306863
 1270663741710149377716766509546084053/1068496585396936451410720445191516320)
* (setq frrt2 (interval-newton-sqrt sqrt2 frrt2))
1.189207115002721d0
1.189207115002721d0
1.189207115002721066717499970560475914[9..I]

** WARNING: New interval not included within old interval,
**  so I'll have to intersect the two to get a valid interval refinement.
[!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!--------]
1.189207115002721d0
1.189207115002721d0
1.189207115002721066717499970560475914[9..H]
[!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!]
1.189207115002721d0
1.189207115002721d0
1.189207115002721066717499970560475914[9..I]
Interval didn't shrink enough:
Old: 1.189207115002721066717499970560475914[9..I]
New: 1.189207115002721066717499970560475914[9..I]
Refinement (i.e. shrinkage) factor = 1.0
** RETURN to go on:

(299498316123706112289466061790835440/251847060402948244751717109090306863
 1270663741710149377716766509546084053/1068496585396936451410720445191516320)
* (setq sqrt2 (interval-newton-sqrt two sqrt2))
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 frrt2 (interval-newton-sqrt sqrt2 frrt2))
1.189207115002721d0
1.189207115002721d0
1.189207115002721066717499970560475914[9..I]
[--------------------------!!!!!!!!!-----------------------------------]
1.189207115002721d0
1.189207115002721d0
1.1892071150027210667174999705604759152[4..F]
[--------------------------!!!!!!!!!-----------------------------------]
1.189207115002721d0
1.189207115002721d0
1.1892071150027210667174999705604759152[4..F]
(1279632423908100378/1076038318106743253
 3000982989783584766648703080035715273154122059853866453/25235158383463910840206
28887098319534366393801994936640)
* (setq frrt2 (interval-newton-sqrt sqrt2 frrt2))
1.189207115002721d0
1.189207115002721d0
1.1892071150027210667174999705604759152[4..F]
[----------------------------------!-----------------------------------]
1.189207115002721d0
1.189207115002721d0
1.1892071150027210667174999705604759152929720924638174130190022247194666681[1..H
]
[----------------------------------!!----------------------------------]
1.189207115002721d0
1.189207115002721d0
1.189207115002721066717499970560475915292972092463817413019002224719466668[1..4]
(6927190033166918493802262940973184057900719395991019997445680205991285760/58250
49266671321711790234283065296581938907575618691020556942654301800551
 102047146812920690074653712342908837225279420585866904907483771451727887/858110
79941854529184372734495921648466822925030403629960954911932604810)
* (setq frrt2 (interval-newton-sqrt sqrt2 frrt2))
1.189207115002721d0
1.189207115002721d0
1.189207115002721066717499970560475915292972092463817413019002224719466668[1..4]

** WARNING: New interval not included within old interval,
**  so I'll have to intersect the two to get a valid interval refinement.
[!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!------------------------------]
1.189207115002721d0
1.189207115002721d0
1.1892071150027210667174999705604759152929720924638174130190022247194666681[8..I
]
[!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!]
1.189207115002721d0
1.189207115002721d0
1.189207115002721066717499970560475915292972092463817413019002224719466668[1..4]
Interval didn't shrink enough:
Old: 1.189207115002721066717499970560475915292972092463817413019002224719466668[
1..4]
New: 1.189207115002721066717499970560475915292972092463817413019002224719466668[
1..4]
Refinement (i.e. shrinkage) factor = 1.0
** RETURN to go on:

(6927190033166918493802262940973184057900719395991019997445680205991285760/58250
49266671321711790234283065296581938907575618691020556942654301800551
 102047146812920690074653712342908837225279420585866904907483771451727887/858110
79941854529184372734495921648466822925030403629960954911932604810)
*

Would you like me to set up this software as a
Web-server-side application so that you-all can play
with it?

Urgent note: I'm in desperate financial circumstances.
I desperately need a source of income to avoid becoming
homeless. Please refer me to somebody who has money to
hire me for writing software.