Engel expansion: Difference between revisions

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=={{header|Nim}}==

===Task===

We use the module “rationals” from the standard library which is limited to <code>int64</code> numerators and denominators. We had to define a conversion function from string to Rational as using the provided conversion function from float to Rational gave inaccurate results.

<syntaxhighlight lang="Nim">import std/[math, rationals, strutils]

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Value: 1.414213562373095

Engel expansion: 1 3 5 5 16 18 78 102 120 144 260 968 18531 46065 63005 65105 78125

</pre>

===Stretch task===

The package “bignum” provides a “Rat” type but lacks a function to convert the string representing a real number to a <code>Rat</code>.

<syntaxhighlight lang="Nim">import std/strutils

import bignum

func engel(x: Rat): seq[Int] =

## Return the Engel expansion of rational "x".

var u = x

while u.num != 0:

let a = (u.denom + u.num - 1) div u.num

result.add a

u = u * a - 1

func toRat(s: string): Rat =

## Convert the string representation of a real to a rational.

var num = newInt(0)

var den = newInt(1)

var i = 0

var c = s[0]

while c != '.':

num = 10 * num + ord(c) - ord('0')

inc i

c = s[i]

inc i

while i < s.len:

num = 10 * num + ord(s[i]) - ord('0')

den *= 10

inc i

result = newRat(num, den)

for val in ["3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211",

"2.71828182845904523536028747135266249775724709369995957496696762772407663035354759457138217852516642743",

"1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727350138462309122970249248360558507372126441214970999358314132226659275055927558"]:

let e = engel(val.toRat)

echo "Value: ", val

echo "Engel expansion: ", e[0..29].join(" ")

echo "Number of terms: ", e.len

echo()

</syntaxhighlight>

<pre>Value: 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211

Engel expansion: 1 1 1 8 8 17 19 300 1991 2492 7236 10586 34588 63403 70637 1236467 5417668 5515697 5633167 7458122 9637848 9805775 41840855 58408380 213130873 424342175 2366457522 4109464489 21846713216 27803071890

Number of terms: 231

Value: 2.71828182845904523536028747135266249775724709369995957496696762772407663035354759457138217852516642743

Engel expansion: 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

Number of terms: 150

Value: 1.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727350138462309122970249248360558507372126441214970999358314132226659275055927558

Engel expansion: 1 3 5 5 16 18 78 102 120 144 251 363 1402 31169 88630 184655 259252 298770 4196070 38538874 616984563 1975413035 5345718057 27843871197 54516286513 334398528974 445879679626 495957494386 2450869042061 2629541150527

Number of terms: 185

</pre>