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kmat = matrix(SR,[(4, 8, 4, 0, 8, 0), (5, 16, 4, 2, 12, 2), (1/3*sqrt(3) + 3, 8/3*sqrt(3) + 8, sqrt(3) + 1, 2, 4*sqrt(3) + 4, 10/9*sqrt(3)), (sqrt(2) + 2, 6*sqrt(2) + 4, 2*sqrt(2) + 2, 2, 6*sqrt(2) + 4, 2), (6, 24, 6, 4, 24, 4)])
kkernel = kmat.right_kernel().basis()[0]
k6 = kkernel[5]*(12*sqrt(2)+20)

show(k6)
-18 \, \frac{{(3 \, \sqrt{2} + 5)}}{{(\frac{{(4 \, \frac{{(\sqrt{3} - 2)}}{{(3 \, \sqrt{3} - 5)}} - 1)} {(5 \, \frac{{(2 \, \sqrt{3} - 3)} \sqrt{2}}{{(3 \, \sqrt{3} - 5)}} + 6 \, \sqrt{2} - 9)}}{{(5 \, \frac{{(\sqrt{3} - 2)} \sqrt{2}}{{(3 \, \sqrt{3} - 5)}} - 2 \, \sqrt{2} + 3)}} - \frac{{(\frac{{(\sqrt{3} - 2)} {(5 \, \frac{{(2 \, \sqrt{3} - 3)} \sqrt{2}}{{(3 \, \sqrt{3} - 5)}} + 6 \, \sqrt{2} - 9)}}{{(3 \, \sqrt{3} - 5)} {(5 \, \frac{{(\sqrt{3} - 2)} \sqrt{2}}{{(3 \, \sqrt{3} - 5)}} - 2 \, \sqrt{2} + 3)}} - \frac{{(2 \, \sqrt{3} - 3)}}{{(3 \, \sqrt{3} - 5)}})} {(2 \, \frac{{(4 \, \frac{{(\sqrt{3} - 2)}}{{(3 \, \sqrt{3} - 5)}} - 1)} {(5 \, \frac{{(\sqrt{3} - 1)} \sqrt{2}}{{(3 \, \sqrt{3} - 5)}} - \sqrt{2})}}{{(5 \, \frac{{(\sqrt{3} - 2)} \sqrt{2}}{{(3 \, \sqrt{3} - 5)}} - 2 \, \sqrt{2} + 3)}} - 8 \, \frac{{(\sqrt{3} - 1)}}{{(3 \, \sqrt{3} - 5)}} + 1)}}{{(2 \, \frac{{(\sqrt{3} - 2)} {(5 \, \frac{{(\sqrt{3} - 1)} \sqrt{2}}{{(3 \, \sqrt{3} - 5)}} - \sqrt{2})}}{{(3 \, \sqrt{3} - 5)} {(5 \, \frac{{(\sqrt{3} - 2)} \sqrt{2}}{{(3 \, \sqrt{3} - 5)}} - 2 \, \sqrt{2} + 3)}} - 2 \, \frac{{(\sqrt{3} - 1)}}{{(3 \, \sqrt{3} - 5)}} + 1)}} - 4 \, \frac{{(2 \, \sqrt{3} - 3)}}{{(3 \, \sqrt{3} - 5)}} - 3)}}
N(k6)
139.406088405362
RIF(k6)
139.406088406?
k2 = k6.simplify_rational()

show(k2)
-12 \, \frac{{(3 \, {(28960879780288 \, \sqrt{2} + 49414857768735)} \sqrt{3} - 150485145634059 \, \sqrt{2} - 256767132912716)}}{{(3 \, {(40774373203317 \, \sqrt{2} - 90564557902141)} \sqrt{3} - 211869858104760 \, \sqrt{2} + 470587246954565)}}
N(k2)
6.00000000000000
float(k2)
6.0
RIF(k2)
[-infinity .. +infinity]
RealIntervalField(90)(k2)
2.?e2
RealIntervalField(110)(k2)
139.4061?
RealField(110)(k2)
139.40609168254414230900933332163
k2.n()
6.00000000000000
N??

def M(s):
    prec = 
    a = RealIntervalField(s)

a = RealIntervalField(100)(5).sin(); a
-0.958924274663138468893154406156?
b = RealField(100)(5).sin(); b
-0.95892427466313846889315440616
timeit('a/a')
625 loops, best of 3: 594 ns per loop
timeit('b/b')
625 loops, best of 3: 339 ns per loop
%python
1/5.0
0.20000000000000001
K.<a,b> = QQ[sqrt(2),sqrt(3)]
k6sub = str(k6).replace('sqrt(2)','a').replace('sqrt(3)','b')

R = RealField(200)
print sage_eval(k6sub, {'a':R(sqrt(2)), 'b':R(sqrt(3))})
print sage_eval(k6sub, {'a':R(sqrt(2)), 'b':-R(sqrt(3))})
print sage_eval(k6sub, {'a':-R(sqrt(2)), 'b':R(sqrt(3))})
print sage_eval(k6sub, {'a':-R(sqrt(2)), 'b':-R(sqrt(3))})
139.40608840536196048491518637312057325404516252967448043667 32.961443962785172150376004734203122988717118740680715514872 18.947740667117622082072881921090831955895826607499222173340 0.68472696473524528263592697158547180134189212214558187511936
k6sub
'-18*(3*a + 5)/((4*(b - 2)/(3*b - 5) - 1)*(5*(2*b - 3)*a/(3*b - 5) + 6*a - 9)/(5*(b - 2)*a/(3*b - 5) - 2*a + 3) - ((b - 2)*(5*(2*b - 3)*a/(3*b - 5) + 6*a - 9)/((3*b - 5)*(5*(b - 2)*a/(3*b - 5) - 2*a + 3)) - (2*b - 3)/(3*b - 5))*(2*(4*(b - 2)/(3*b - 5) - 1)*(5*(b - 1)*a/(3*b - 5) - a)/(5*(b - 2)*a/(3*b - 5) - 2*a + 3) - 8*(b - 1)/(3*b - 5) + 1)/(2*(b - 2)*(5*(b - 1)*a/(3*b - 5) - a)/((3*b - 5)*(5*(b - 2)*a/(3*b - 5) - 2*a + 3)) - 2*(b - 1)/(3*b - 5) + 1) - 4*(2*b - 3)/(3*b - 5) - 3)'
R = RIF
print sage_eval(k6sub, {'a':R(sqrt(2)), 'b':R(sqrt(3))})
print sage_eval(k6sub, {'a':R(sqrt(2)), 'b':-R(sqrt(3))})
print sage_eval(k6sub, {'a':-R(sqrt(2)), 'b':R(sqrt(3))})
print sage_eval(k6sub, {'a':-R(sqrt(2)), 'b':-R(sqrt(3))})
139.406088406? 32.9614439627852? 18.9477406672? 0.68472696473525?
var('a,b')
print sage_eval(k6sub, {'a':a, 'b':b}).simplify_rational()
print sage_eval(k6sub, {'a':a, 'b':-b}).simplify_rational()
print sage_eval(k6sub, {'a':-a, 'b':b}).simplify_rational()
print sage_eval(k6sub, {'a':-a, 'b':-b}).simplify_rational()
-6*((3*a^2 + 14*a + 15)*b - 27*a - 45)/(5*(a - 3)*b - 9*a + 27) -6*((3*a^2 + 14*a + 15)*b + 27*a + 45)/(5*(a - 3)*b + 9*a - 27) 6*((3*a^2 - 14*a + 15)*b + 27*a - 45)/(5*(a + 3)*b - 9*a - 27) 6*((3*a^2 - 14*a + 15)*b - 27*a + 45)/(5*(a + 3)*b + 9*a + 27)
t = sage_eval(k6sub, {'a':a, 'b':b}); show(t)
-18 \, \frac{{(3 \, a + 5)}}{{(\frac{{(4 \, \frac{{(b - 2)}}{{(3 \, b - 5)}} - 1)} {(5 \, \frac{{(2 \, b - 3)} a}{{(3 \, b - 5)}} + 6 \, a - 9)}}{{(5 \, \frac{{(b - 2)} a}{{(3 \, b - 5)}} - 2 \, a + 3)}} - \frac{{(\frac{{(b - 2)} {(5 \, \frac{{(2 \, b - 3)} a}{{(3 \, b - 5)}} + 6 \, a - 9)}}{{(3 \, b - 5)} {(5 \, \frac{{(b - 2)} a}{{(3 \, b - 5)}} - 2 \, a + 3)}} - \frac{{(2 \, b - 3)}}{{(3 \, b - 5)}})} {(2 \, \frac{{(4 \, \frac{{(b - 2)}}{{(3 \, b - 5)}} - 1)} {(5 \, \frac{{(b - 1)} a}{{(3 \, b - 5)}} - a)}}{{(5 \, \frac{{(b - 2)} a}{{(3 \, b - 5)}} - 2 \, a + 3)}} - 8 \, \frac{{(b - 1)}}{{(3 \, b - 5)}} + 1)}}{{(2 \, \frac{{(b - 2)} {(5 \, \frac{{(b - 1)} a}{{(3 \, b - 5)}} - a)}}{{(3 \, b - 5)} {(5 \, \frac{{(b - 2)} a}{{(3 \, b - 5)}} - 2 \, a + 3)}} - 2 \, \frac{{(b - 1)}}{{(3 \, b - 5)}} + 1)}} - 4 \, \frac{{(2 \, b - 3)}}{{(3 \, b - 5)}} - 3)}}
s = t.subs(a=0); s
-90/((8*(b - 1)/(3*b - 5) - 1)*(3*(b - 2)/(3*b - 5) + (2*b - 3)/(3*b - 5))/(2*(b - 1)/(3*b - 5) - 1) - 12*(b - 2)/(3*b - 5) - 4*(2*b - 3)/(3*b - 5))
s.rational_simplify()
30*(b - 3)/(5*b - 9)
s
-90/((8*(b - 1)/(3*b - 5) - 1)*(3*(b - 2)/(3*b - 5) + (2*b - 3)/(3*b - 5))/(2*(b - 1)/(3*b - 5) - 1) - 12*(b - 2)/(3*b - 5) - 4*(2*b - 3)/(3*b - 5))
tt = maxima(t); tt
-18*(3*a+5)/(-(2*(4*(b-2)/(3*b-5)-1)*(5*a*(b-1)/(3*b-5)-a)/(5*a*(b-2)/(3*b-5)-2*a+3)-8*(b-1)/(3*b-5)+1)*((b-2)*(5*a*(2*b-3)/(3*b-5)+6*a-9)/((3*b-5)*(5*a*(b-2)/(3*b-5)-2*a+3))-(2*b-3)/(3*b-5))/(2*(b-2)*(5*a*(b-1)/(3*b-5)-a)/((3*b-5)*(5*a*(b-2)/(3*b-5)-2*a+3))-2*(b-1)/(3*b-5)+1)+(4*(b-2)/(3*b-5)-1)*(5*a*(2*b-3)/(3*b-5)+6*a-9)/(5*a*(b-2)/(3*b-5)-2*a+3)-4*(2*b-3)/(3*b-5)-3)
t3 = tt.ratsimp()

t3 == tt
False
var('a,b')
k6sub = str(k6).replace('sqrt(2)','a').replace('sqrt(3)','b')
k6sub = SR(k6sub)
k6sub = k6sub.simplify_rational()

k6sub
-6*((3*a^2 + 14*a + 15)*b - 27*a - 45)/(5*(a - 3)*b - 9*a + 27)
N(k6sub.subs({a:sqrt(2),b:sqrt(3)}))
139.406088405362
import sympy as S

k6sym = SR(str(S.simplify(k6)).replace('**','^'))
N(k6sym)
24.0000000000000
str(S.simplify(k6))
'(138547369928226569306810105702630817792 - 79990361323576271425605785776028909568*3**(1/2) - 46880459490862055388948437803092148224*6**(1/2) + 81199337720347661076591536186660487168*2**(1/2))/(21160102232101453527406358772028145664 - 12216790719783775192193044336522690560*3**(1/2) - 9526794205348280942683794601751347200*2**(1/2) + 5500297198971996875456559024297738240*6**(1/2))'
float(k6._sympy_())
139.40608840536197
s = k6._sympy_()

S.simplify(s)
(138547369928226569306810105702630817792 - 79990361323576271425605785776028909568*3**(1/2) - 46880459490862055388948437803092148224*6**(1/2) + 81199337720347661076591536186660487168*2**(1/2))/(21160102232101453527406358772028145664 - 12216790719783775192193044336522690560*3**(1/2) - 9526794205348280942683794601751347200*2**(1/2) + 5500297198971996875456559024297738240*6**(1/2))
float(S.simplify(s))
139.40608840536197
t = SR(str(S.simplify(s))); t
-6*(141632780782977203307379047456*sqrt(2) - 139524011256242413558935167631*sqrt(3) - 81771724111127143285911706233*sqrt(6) + 241662676371623798097925496989)/(99703253238173338992730229400*sqrt(2) + 127855577924477483722629670995*sqrt(3) - 57563700096140803955418873480*sqrt(6) - 221452356996276745867174342403)
N(t)
24.0000000000000
type(t)
<type 'sage.symbolic.expression.Expression'>
t
-6*(141632780782977203307379047456*sqrt(2) - 139524011256242413558935167631*sqrt(3) - 81771724111127143285911706233*sqrt(6) + 241662676371623798097925496989)/(99703253238173338992730229400*sqrt(2) + 127855577924477483722629670995*sqrt(3) - 57563700096140803955418873480*sqrt(6) - 221452356996276745867174342403)
R = RealIntervalField(1000)
ts = str(SR(str(t).replace('sqrt(2)','a').replace('sqrt(3)','b').replace('sqrt(6)','a*b')))
print sage_eval(ts, {'a':R(sqrt(2)), 'b':R(sqrt(3))})
print sage_eval(ts, {'a':R(sqrt(2)), 'b':-R(sqrt(3))})
print sage_eval(ts, {'a':-R(sqrt(2)), 'b':R(sqrt(3))})
print sage_eval(ts, {'a':-R(sqrt(2)), 'b':-R(sqrt(3))})
139.4060884053619604849151863731205732540451625296744804366691116611432146199511259552798296864185589393739958047514803763361593743564908654346331352005325000971229048820332406228588163727888551962655317646911782553037013623191262947425423342397621268429637069170591565334768? 32.9614439627851721503760047342031229887171187406807155148715941903563392170026541426410971672740859883167518917177836630097817726594489440481494256428640347991911190780520613859543809542733950350053276212250602114186934625491042540527976063668458898819521188613869961370965743896367633355488823613763? 18.947740667117622082072881921090831955895826607499222173339939599126373988765675379641433036842389784948545286203160932742758649063471423802586463234864951249888451466449760331588277150850193100972008786553179353215922557105074873338643832724411569896809536100109529266038562? 0.68472696473524528263592697158547180134189212214558187511935454937407217428054452243764010946496528736070701732757502791130020392058876671463097592173851385379752457346493765959852552208755666775713182753058218006168261802669457786601622666898041337827463812144431806338807993542521582903860740188591?