Interval arithmetic by Robert Elton Maas:

Demonstration of using Newton's method with intervals of exact
rational numbers to compute square root of an interval approximation
to 2 which starts out very imprecise then gets more precise when needed,
i.e. "lazy evaluation" emulated manually:

* (setq ndig 2) (setq powr (expt 10 ndig))

2
*
100
* (setq nums (list (/ (+ powr powr -1) powr) (/ (+ powr powr 1) powr)))

(199/100 201/100)
* (show-interval nums)
1.99d0
2.01d0
1.99[0..K]
NIL
* (setq guess (interval-to-sqrt-firstguess+bounds nums))

3/2
* (setq rats (interval+guess-to-sqrt-bounds nums guess))

(199/150 3/2)
* (show-interval rats)
1.3266666666666667d0
1.5d0
1.3[2..K]
NIL
* (setq rats (interval-newton-sqrt nums rats))
1.3266666666666667d0
1.5d0
1.3[2..K]
[--------------------------------!!!!!!!-------------------------------]
1.4080188679245282d0
1.4221698113207548d0
1.4[0..3]
[-------------------------------!!!!!!!--------------------------------]
1.4047058823529412d0
1.4188235294117648d0
1.40[4..J]
(597/425 603/425)
* (setq rats (interval-newton-sqrt nums rats))
1.4047058823529412d0
1.4188235294117648d0
1.40[4..J]

** WARNING: New interval not included within old interval,
**  so I'll have to intersect the two to get a valid interval refinement.
[------------------------!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!]
1.4095833333333334d0
1.4188235294117648d0
1.40[9..J]
[!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!]
1.4047058823529412d0
1.4188235294117648d0
1.40[4..J]
Interval didn't shrink enough:
Old: 1.40[4..J]
New: 1.40[4..J]
Refinement (i.e. shrinkage) factor = 1.0
** RETURN to go on:

(597/425 603/425)

;;Note: Newton's method couldn't get much more accuracy above,
; given imprecise interval of the input data, so now we make the
; input data more precise then before:

* (setq ndig 5) (setq powr (expt 10 ndig))

5
*
100000
* (setq nums (list (/ (+ powr powr -1) powr) (/ (+ powr powr 1) powr)))

(199999/100000 200001/100000)
* (show-interval nums)
1.99999d0
2.00001d0
1.99999[0..K]
NIL

;;Now we just pick up the Newton's method iteration from where we left off,
; no need to re-start from the initial state. Note that because the
; previous interval was based on imprecise data, it's not nicely centered
; on sqrt(2), so the first step below will be lopsided (the midpoint of
; the old interval isn't a good guess of sqrt(2)), but the next
; step after it will be centered (the midpoint is now a good guess):

* (setq rats (interval-newton-sqrt nums rats))
1.4047058823529412d0
1.4188235294117648d0
1.40[4..J]
[-----------------------------------!!!!!!!!!!!!!!!!!!!!!!!!!----------]
1.411764705882353d0
1.41667375d0
1.41[1..7]
[----------------------------------!!!!!!!!!!!!!!!!!!!!!!!!!!----------]
1.4117576470588236d0
1.4166666666666667d0
1.41[1..7]
(599997/425000 17/12)
* (setq rats (interval-newton-sqrt nums rats))
1.4117576470588236d0
1.4166666666666667d0
1.41[1..7]
[----------------------------------!!----------------------------------]
1.4142078968100025d0
1.4142220389596813d0
1.4142[0..3]
[----------------------------------!!----------------------------------]
1.4142043674176776d0
1.4142185095320623d0
1.41420[4..J]
(10199949/7212500 10200051/7212500)
* (setq rats (interval-newton-sqrt nums rats))
1.4142043674176776d0
1.4142185095320623d0
1.41420[4..J]

** WARNING: New interval not included within old interval,
**  so I'll have to intersect the two to get a valid interval refinement.
[---------------------!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!]
1.4142086151960784d0
1.4142185095320623d0
1.41420[8..J]
[!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!]
1.4142043674176776d0
1.4142185095320623d0
1.41420[4..J]
Interval didn't shrink enough:
Old: 1.41420[4..J]
New: 1.41420[4..J]
Refinement (i.e. shrinkage) factor = 1.0
** RETURN to go on:

(10199949/7212500 10200051/7212500)

;;Once again Newton's method couldn't do any better with the given
; data, so once again we make the given data more precise and then
; pick up Newton's method where we left off:

* (setq ndig 25) (setq powr (expt 10 ndig))

25
*
10000000000000000000000000
* (setq nums (list (/ (+ powr powr -1) powr) (/ (+ powr powr 1) powr)))

(19999999999999999999999999/10000000000000000000000000
 20000000000000000000000001/10000000000000000000000000)
* (show-interval nums)
2.0d0
2.0d0
1.9999999999999999999999999[0..K]
NIL
* (setq rats (interval-newton-sqrt nums rats))
1.4142043674176776d0
1.4142185095320623d0
1.41420[4..J]
[-----------------------------------!!!!!!!!!!!!!!!!!!!!!!-------------]
1.41421143847487d0
1.41421568627451d0
1.41421[1..6]
[----------------------------------!!!!!!!!!!!!!!!!!!!!!!!-------------]
1.4142114384748696d0
1.4142156862745099d0
1.41421[1..6]
(7009386627311/4956392259753 577/408)
* (setq rats (interval-newton-sqrt nums rats))
1.4142114384748696d0
1.4142156862745099d0
1.41421[1..6]
[----------------------------------!-----------------------------------]
1.4142135623715004d0
1.4142135623746896d0
1.41421356237[1..5]
[----------------------------------!-----------------------------------]
1.4142135623728214d0
1.4142135623733687d0
1.414213562372[8..E]
(1607521/1136689
 22733780000000000000000001136689/16075210000000000000000000000000)
* (setq rats (interval-newton-sqrt nums rats))
1.4142135623728214d0
1.4142135623733687d0
1.414213562372[8..E]
[----------------------------------!-----------------------------------]
1.4142135623730951d0
1.4142135623730951d0
1.414213562373095048801688[5..8]
[----------------------------------!-----------------------------------]
1.4142135623730951d0
1.4142135623730951d0
1.4142135623730950488016886[2..H]
(36545028759379999999999998172748562031/25841237654415000000000000000000000000
 12181676253126666666666667275750479323/8613745884805000000000000000000000000)
* (setq rats (interval-newton-sqrt nums rats))
1.4142135623730951d0
1.4142135623730951d0
1.4142135623730950488016886[2..H]

** WARNING: New interval not included within old interval,
**  so I'll have to intersect the two to get a valid interval refinement.
[--------------------------!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!]
1.4142135623730951d0
1.4142135623730951d0
1.4142135623730950488016886[7..H]
[!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!]
1.4142135623730951d0
1.4142135623730951d0
1.4142135623730950488016886[2..H]
Interval didn't shrink enough:
Old: 1.4142135623730950488016886[2..H]
New: 1.4142135623730950488016886[2..H]
Refinement (i.e. shrinkage) factor = 1.0
** RETURN to go on:

(36545028759379999999999998172748562031/25841237654415000000000000000000000000
 12181676253126666666666667275750479323/8613745884805000000000000000000000000)

;;Again, exhausted the precision available in the input data, so we
; make the input data more precise again etc.:

* (setq ndig 70) (setq powr (expt 10 ndig))

70
*
10000000000000000000000000000000000000000000000000000000000000000000000
* (setq nums (list (/ (+ powr powr -1) powr) (/ (+ powr powr 1) powr)))

(19999999999999999999999999999999999999999999999999999999999999999999999/1000000
0000000000000000000000000000000000000000000000000000000000000000
 20000000000000000000000000000000000000000000000000000000000000000000001/1000000
0000000000000000000000000000000000000000000000000000000000000000)
* (show-interval nums)
2.0d0
2.0d0
1.9999999999999999999999999999999999999999999999999999999999999999999999[0..K]
NIL
* (setq rats (interval-newton-sqrt nums rats))
1.4142135623730951d0
1.4142135623730951d0
1.4142135623730950488016886[2..H]
[-----------------------------------!!!!!!!!!!!!!!!!!!!!!!!!!!!--------]
1.4142135623730951d0
1.4142135623730951d0
1.4142135623730950488016886[9..G]
[----------------------------------!!!!!!!!!!!!!!!!!!!!!!!!!!!!--------]
1.4142135623730951d0
1.4142135623730951d0
1.4142135623730950488016886[9..G]
(3654502875937999999999999999999999999999999999999999999999999999999999817274856
2031/258412376544150000000000000000000000000000000000000000000000000000000000000
00000000
 5168247530883/3654502875938)
* (setq rats (interval-newton-sqrt nums rats))
1.4142135623730951d0
1.4142135623730951d0
1.4142135623730950488016886[9..G]
[----------------------------------!-----------------------------------]
1.4142135623730951d0
1.4142135623730951d0
1.414213562373095048801688724209698078569671875376947[8..E]
[----------------------------------!!----------------------------------]
1.4142135623730951d0
1.4142135623730951d0
1.414213562373095048801688724209698078569671875376947[8..E]
(75549501860685563890373016/53421565080956452077519377
 3561437672063763471834625133333333333333333333333333333333333333333333511405216
93652150692506459/25183167286895187963457672000000000000000000000000000000000000
0000000000000000000000000000000000)
* (setq rats (interval-newton-sqrt nums rats))
1.4142135623730951d0
1.4142135623730951d0
1.414213562373095048801688724209698078569671875376947[8..E]
[----------------------------------!-----------------------------------]
1.4142135623730951d0
1.4142135623730951d0
1.414213562373095048801688724209698078569671875376948073176679737990732[3..6]
[----------------------------------!!----------------------------------]
1.4142135623730951d0
1.4142135623730951d0
1.414213562373095048801688724209698078569671875376948073176679737990732[3..7]
(116104232852194624065131557975456296/82098090374248746619236402542311697
 5473206024949916441282426836154113133333333333333333333333333333333333606993634
58082915539745467514103899/38701410950731541355043852658485432000000000000000000
0000000000000000000000000000000000000000000000000000)
* (setq rats (interval-newton-sqrt nums rats))
1.4142135623730951d0
1.4142135623730951d0
1.414213562373095048801688724209698078569671875376948073176679737990732[3..7]

** WARNING: New interval not included within old interval,
**  so I'll have to intersect the two to get a valid interval refinement.
[!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!-----------------------------------]
1.4142135623730951d0
1.4142135623730951d0
1.414213562373095048801688724209698078569671875376948073176679737990732[3..6]
[!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!]
1.4142135623730951d0
1.4142135623730951d0
1.414213562373095048801688724209698078569671875376948073176679737990732[3..7]
Interval didn't shrink enough:
Old: 1.414213562373095048801688724209698078569671875376948073176679737990732[3..
7]
New: 1.414213562373095048801688724209698078569671875376948073176679737990732[3..
7]
Refinement (i.e. shrinkage) factor = 1.0
** RETURN to go on:

(116104232852194624065131557975456296/82098090374248746619236402542311697
 5473206024949916441282426836154113133333333333333333333333333333333333606993634
58082915539745467514103899/38701410950731541355043852658485432000000000000000000
0000000000000000000000000000000000000000000000000000)
*

;;I could make the input data even more precise and continue to generate
; as many digits of sqrt(2) as we want, but I think you get the idea already.


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.