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Equality of algebraic numbers given by huge symbolic expressions

asked 2020-07-24 09:38:15 -0500

creyesm1992 gravatar image

updated 2020-08-05 06:58:15 -0500

slelievre gravatar image

I calculated a matrix whose first entry is a huge numerical value:

N = 1/16*(44*(7*sqrt(2) - 10)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) + 2*(11*(7*sqrt(2) - 10)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - 10*(63*sqrt(2) - 89)*sqrt(sqrt(2) + 2) - (3*(3*sqrt(2) - 4)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - (85*sqrt(2) - 122)*sqrt(sqrt(2) + 2))*sqrt(-sqrt(2) + 2) + 2*((3*(3*sqrt(2) - 4)*sqrt(-17*sqrt(2) + 26) - 85*sqrt(2) + 122)*sqrt(-sqrt(2) + 2) - 11*(7*sqrt(2) - 10)*sqrt(-17*sqrt(2) + 26) + 630*sqrt(2) - 890)*sqrt(sqrt(sqrt(2) + 2) - 1))*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3) - 40*(63*sqrt(2) - 89)*sqrt(sqrt(2) + 2) - 4*(3*(3*sqrt(2) - 4)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - (85*sqrt(2) - 122)*sqrt(sqrt(2) + 2))*sqrt(-sqrt(2) + 2) + (22*(5*sqrt(2) - 7)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) + (11*(5*sqrt(2) - 7)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - 5*(89*sqrt(2) - 126)*sqrt(sqrt(2) + 2) - (3*(2*sqrt(2) - 3)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - (61*sqrt(2) - 85)*sqrt(sqrt(2) + 2))*sqrt(-sqrt(2) + 2) + 2*((3*(2*sqrt(2) - 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) - 11*(5*sqrt(2) - 7)*sqrt(-17*sqrt(2) + 26) + 445*sqrt(2) - 630)*sqrt(sqrt(sqrt(2) + 2) - 1))*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3) - 10*(89*sqrt(2) - 126)*sqrt(sqrt(2) + 2) - 2*(3*(2*sqrt(2) - 3)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - (61*sqrt(2) - 85)*sqrt(sqrt(2) + 2))*sqrt(-sqrt(2) + 2) + 4*((3*(2*sqrt(2) - 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) - 11*(5*sqrt(2) - 7)*sqrt(-17*sqrt(2) + 26) + 445*sqrt(2) - 630)*sqrt(sqrt(sqrt(2) + 2) - 1))*sqrt(-12*sqrt(2) - 2*sqrt(-sqrt(2) + 2) - 2*sqrt(-17*sqrt(2) + 26) + 24) + 8*((3*(3*sqrt(2) - 4)*sqrt(-17*sqrt(2) + 26) - 85*sqrt(2) + 122)*sqrt(-sqrt(2) + 2) - 11*(7*sqrt(2) - 10)*sqrt(-17*sqrt(2) + 26) + 630*sqrt(2) - 890)*sqrt(sqrt(sqrt(2) + 2) - 1))*(((sqrt(2)*sqrt(sqrt(2) + 2) - sqrt(2) - 1)*sqrt(sqrt(sqrt(2) + 2) - 1) + sqrt(sqrt(2) + 2) - 1)*(8*((5*(89*sqrt(2) - 126)*sqrt(sqrt(2) + 2) - ((61*sqrt(2) - 85)*sqrt(sqrt(2) + 2) - 3*((2*sqrt(2) - 3)*sqrt(sqrt(2) + 2) - 4*sqrt(2) + 6)*sqrt(-17*sqrt(2) + 26) - 122*sqrt(2) + 170)*sqrt(-sqrt(2) + 2) - 11*((5*sqrt(2) - 7)*sqrt(sqrt(2) + 2) - 10*sqrt(2) + 14)*sqrt(-17*sqrt(2) + 26) - 890*sqrt(2) + 1260)*sqrt(3*sqrt(2) + sqrt(-sqrt(2) + 2) - 5)*sqrt(-12*sqrt(2) - 2*sqrt(-sqrt(2) + 2) - 2*sqrt(-17*sqrt(2) + 26) + 24) + 2*(10*(63*sqrt(2) - 89)*sqrt(sqrt(2) + 2) - ((85*sqrt(2) - 122)*sqrt(sqrt(2) + 2) - 3*((3*sqrt(2) - 4)*sqrt(sqrt(2) + 2) - 6*sqrt(2) + 8)*sqrt(-17*sqrt(2) + 26) - 170*sqrt(2) + 244)*sqrt(-sqrt(2) + 2) - 11*((7*sqrt(2) - 10)*sqrt(sqrt(2) + 2) - 14*sqrt(2) + 20)*sqrt(-17*sqrt(2) + 26) - 1260*sqrt(2) + 1780)*sqrt(3*sqrt(2) + sqrt(-sqrt(2) + 2) - 5))*(1/((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) + sqrt(sqrt(2) + 2) + 1) + ((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) + 3*sqrt(2) - 5*sqrt(sqrt(2) + 2) + 8)*((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) - 3*sqrt(2) + 5*sqrt(sqrt(2) + 2) - 8)*(sqrt(sqrt(2) + 2) + 2)/(((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) + sqrt(sqrt(2) + 2) + 1)^2*((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) - sqrt(sqrt(2) + 2) - ((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) + 3*sqrt(2) - 5*sqrt(sqrt(2) + 2) + 8)*((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) - 3*sqrt(2) + 5*sqrt(sqrt(2) + 2) - 8)*(sqrt(sqrt(2) + 2) + 2)/(((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) + sqrt(sqrt(2) + 2) + 1)*(sqrt(sqrt(2) + 2) - 2)^3) - 1)*(sqrt(sqrt(2) + 2) - 2)^3))/(2*((3*(3*sqrt(2) - 4)*sqrt(-17*sqrt(2) + 26) - 85*sqrt(2) + 122)*sqrt(-sqrt(2) + 2) - 11*(7*sqrt(2) - 10)*sqrt(-17*sqrt(2) + 26) + 630*sqrt(2) - 890)*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3)*sqrt(sqrt(sqrt(2) + 2) - 1) - 2*(4896*sqrt(2) - 6923)*sqrt(sqrt(2) + 2) + (20*(79*sqrt(2) - 112)*sqrt(sqrt(2) + 2) - (7*(27*sqrt(2) - 38)*sqrt(sqrt(2) + 2) - 342*sqrt(2) + 484)*sqrt(-17*sqrt(2) + 26) - 2820*sqrt(2) + 3992)*sqrt(-sqrt(2) + 2) + (((3*(2*sqrt(2) - 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) - 11*(5*sqrt(2) - 7)*sqrt(-17*sqrt(2) + 26) + 445*sqrt(2) - 630)*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3)*sqrt(sqrt(sqrt(2) + 2) - 1) - 10*(89*sqrt(2) - 126)*sqrt(sqrt(2) + 2) + 2*((61*sqrt(2) - 85)*sqrt(sqrt(2) + 2) - 3*((2*sqrt(2) - 3)*sqrt(sqrt(2) + 2) - 2*sqrt(2) + 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) + 22*((5*sqrt(2) - 7)*sqrt(sqrt(2) + 2) - 5*sqrt(2) + 7)*sqrt(-17*sqrt(2) + 26) + 890*sqrt(2) - 1260)*sqrt(-12*sqrt(2) - 2*sqrt(-sqrt(2) + 2) - 2*sqrt(-17*sqrt(2) + 26) + 24) + 4*((319*sqrt(2) - 452)*sqrt(sqrt(2) + 2) - 561*sqrt(2) + 794)*sqrt(-17*sqrt(2) + 26) + 17064*sqrt(2) - 24132) + (44*(7*sqrt(2) - 10)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) + 2*(11*(7*sqrt(2) - 10)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - 10*(63*sqrt(2) - 89)*sqrt(sqrt(2) + 2) - (3*(3*sqrt(2) - 4)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - (85*sqrt(2) - 122)*sqrt(sqrt(2) + 2))*sqrt(-sqrt(2) + 2) + 2*((3*(3*sqrt(2) - 4)*sqrt(-17*sqrt(2) + 26) - 85*sqrt(2) + 122)*sqrt(-sqrt(2) + 2) - 11*(7*sqrt(2) - 10)*sqrt(-17*sqrt(2) + 26) + 630*sqrt(2) - 890)*sqrt(sqrt(sqrt(2) + 2) - 1))*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3) - 40*(63*sqrt(2) - 89)*sqrt(sqrt(2) + 2) - 4*(3*(3*sqrt(2) - 4)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - (85*sqrt(2) - 122)*sqrt(sqrt(2) + 2))*sqrt(-sqrt(2) + 2) + (22*(5*sqrt(2) - 7)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) + (11*(5*sqrt(2) - 7)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - 5*(89*sqrt(2) - 126)*sqrt(sqrt(2) + 2) - (3*(2*sqrt(2) - 3)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - (61*sqrt(2) - 85)*sqrt(sqrt(2) + 2))*sqrt(-sqrt(2) + 2) + 2*((3*(2*sqrt(2) - 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) - 11*(5*sqrt(2) - 7)*sqrt(-17*sqrt(2) + 26) + 445*sqrt(2) - 630)*sqrt(sqrt(sqrt(2) + 2) - 1))*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3) - 10*(89*sqrt(2) - 126)*sqrt(sqrt(2) + 2) - 2*(3*(2*sqrt(2) - 3)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - (61*sqrt(2) - 85)*sqrt(sqrt(2) + 2))*sqrt(-sqrt(2) + 2) + 4*((3*(2*sqrt(2) - 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) - 11*(5*sqrt(2) - 7)*sqrt(-17*sqrt(2) + 26) + 445*sqrt(2) - 630)*sqrt(sqrt(sqrt(2) + 2) - 1))*sqrt(-12*sqrt(2) - 2*sqrt(-sqrt(2) + 2) - 2*sqrt(-17*sqrt(2) + 26) + 24) + 8*((3*(3*sqrt(2) - 4)*sqrt(-17*sqrt(2) + 26) - 85*sqrt(2) + 122)*sqrt(-sqrt(2) + 2) - 11*(7*sqrt(2) - 10)*sqrt(-17*sqrt(2) + 26) + 630*sqrt(2) - 890)*sqrt(sqrt(sqrt(2) + 2) - 1))*((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) + 3*sqrt(2) - 5*sqrt(sqrt(2) + 2) + 8)*sqrt(sqrt(sqrt(2) + 2) + 2)/((2*((3*(3*sqrt(2) - 4)*sqrt(-17*sqrt(2) + 26) - 85*sqrt(2) + 122)*sqrt(-sqrt(2) + 2) - 11*(7*sqrt(2) - 10)*sqrt(-17*sqrt(2) + 26) + 630*sqrt(2) - 890)*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3)*sqrt(sqrt(sqrt(2) + 2) - 1) - 2*(4896*sqrt(2) - 6923)*sqrt(sqrt(2) + 2) + (20*(79*sqrt(2) - 112)*sqrt(sqrt(2) + 2) - (7*(27*sqrt(2) - 38)*sqrt(sqrt(2) + 2) - 342*sqrt(2) + 484)*sqrt(-17*sqrt(2) + 26) - 2820*sqrt(2) + 3992)*sqrt(-sqrt(2) + 2) + (((3*(2*sqrt(2) - 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) - 11*(5*sqrt(2) - 7)*sqrt(-17*sqrt(2) + 26) + 445*sqrt(2) - 630)*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3)*sqrt(sqrt(sqrt(2) + 2) - 1) - 10*(89*sqrt(2) - 126)*sqrt(sqrt(2) + 2) + 2*((61*sqrt(2) - 85)*sqrt(sqrt(2) + 2) - 3*((2*sqrt(2) - 3)*sqrt(sqrt(2) + 2) - 2*sqrt(2) + 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) + 22*((5*sqrt(2) - 7)*sqrt(sqrt(2) + 2) - 5*sqrt(2) + 7)*sqrt(-17*sqrt(2) + 26) + 890*sqrt(2) - 1260)*sqrt(-12*sqrt(2) - 2*sqrt(-sqrt(2) + 2) - 2*sqrt(-17*sqrt(2) + 26) + 24) + 4*((319*sqrt(2) - 452)*sqrt(sqrt(2) + 2) - 561*sqrt(2) + 794)*sqrt(-17*sqrt(2) + 26) + 17064*sqrt(2) - 24132)*((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) + sqrt(sqrt(2) + 2) + 1)*((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) - sqrt(sqrt(2) + 2) - ((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) + 3*sqrt(2) - 5*sqrt(sqrt(2) + 2) + 8)*((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) - 3*sqrt(2) + 5*sqrt(sqrt(2) + 2) - 8)*(sqrt(sqrt(2) + 2) + 2)/(((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) + sqrt(sqrt(2) + 2) + 1)*(sqrt(sqrt(2) + 2) - 2)^3) - 1)*(-1/4*sqrt(sqrt(2) + 2) + 1/2)^(3/2)))*sqrt(sqrt(sqrt(2) + 2) + 2)/sqrt(-1/4*sqrt(sqrt(2) + 2) + 1/2) + 2*(sqrt(sqrt(2) + 2)*(sqrt(2) - 1)*sqrt(sqrt(sqrt(2) + 2) - 1) - sqrt(2) + sqrt(sqrt(2) + 2))*(8*(44*(7*sqrt(2) - 10)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - 2*(11*(7*sqrt(2) - 10)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - 10*(63*sqrt(2) - 89)*sqrt(sqrt(2) + 2) - (3*(3*sqrt(2) - 4)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - (85*sqrt(2) - 122)*sqrt(sqrt(2) + 2))*sqrt(-sqrt(2) + 2) - 2*((3*(3*sqrt(2) - 4)*sqrt(-17*sqrt(2) + 26) - 85*sqrt(2) + 122)*sqrt(-sqrt(2) + 2) - 11*(7*sqrt(2) - 10)*sqrt(-17*sqrt(2) + 26) + 630*sqrt(2) - 890)*sqrt(sqrt(sqrt(2) + 2) - 1))*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3) - 40*(63*sqrt(2) - 89)*sqrt(sqrt(2) + 2) - 4*(3*(3*sqrt(2) - 4)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - (85*sqrt(2) - 122)*sqrt(sqrt(2) + 2))*sqrt(-sqrt(2) + 2) + (22*(5*sqrt(2) - 7)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - (11*(5*sqrt(2) - 7)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - 5*(89*sqrt(2) - 126)*sqrt(sqrt(2) + 2) - (3*(2*sqrt(2) - 3)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - (61*sqrt(2) - 85)*sqrt(sqrt(2) + 2))*sqrt(-sqrt(2) + 2) - 2*((3*(2*sqrt(2) - 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) - 11*(5*sqrt(2) - 7)*sqrt(-17*sqrt(2) + 26) + 445*sqrt(2) - 630)*sqrt(sqrt(sqrt(2) + 2) - 1))*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3) - 10*(89*sqrt(2) - 126)*sqrt(sqrt(2) + 2) - 2*(3*(2*sqrt(2) - 3)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - (61*sqrt(2) - 85)*sqrt(sqrt(2) + 2))*sqrt(-sqrt(2) + 2) - 4*((3*(2*sqrt(2) - 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) - 11*(5*sqrt(2) - 7)*sqrt(-17*sqrt(2) + 26) + 445*sqrt(2) - 630)*sqrt(sqrt(sqrt(2) + 2) - 1))*sqrt(-12*sqrt(2) - 2*sqrt(-sqrt(2) + 2) - 2*sqrt(-17*sqrt(2) + 26) + 24) - 8*((3*(3*sqrt(2) - 4)*sqrt(-17*sqrt(2) + 26) - 85*sqrt(2) + 122)*sqrt(-sqrt(2) + 2) - 11*(7*sqrt(2) - 10)*sqrt(-17*sqrt(2) + 26) + 630*sqrt(2) - 890)*sqrt(sqrt(sqrt(2) + 2) - 1))*(1/((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) + sqrt(sqrt(2) + 2) + 1) + ((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) + 3*sqrt(2) - 5*sqrt(sqrt(2) + 2) + 8)*((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) - 3*sqrt(2) + 5*sqrt(sqrt(2) + 2) - 8)*(sqrt(sqrt(2) + 2) + 2)/(((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) + sqrt(sqrt(2) + 2) + 1)^2*((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) - sqrt(sqrt(2) + 2) - ((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) + 3*sqrt(2) - 5*sqrt(sqrt(2) + 2) + 8)*((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) - 3*sqrt(2) + 5*sqrt(sqrt(2) + 2) - 8)*(sqrt(sqrt(2) + 2) + 2)/(((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) + sqrt(sqrt(2) + 2) + 1)*(sqrt(sqrt(2) + 2) - 2)^3) - 1)*(sqrt(sqrt(2) + 2) - 2)^3))/(2*((3*(3*sqrt(2) - 4)*sqrt(-17*sqrt(2) + 26) - 85*sqrt(2) + 122)*sqrt(-sqrt(2) + 2) - 11*(7*sqrt(2) - 10)*sqrt(-17*sqrt(2) + 26) + 630*sqrt(2) - 890)*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3)*sqrt(sqrt(sqrt(2) + 2) - 1) - 2*(4896*sqrt(2) - 6923)*sqrt(sqrt(2) + 2) + (20*(79*sqrt(2) - 112)*sqrt(sqrt(2) + 2) - (7*(27*sqrt(2) - 38)*sqrt(sqrt(2) + 2) - 342*sqrt(2) + 484)*sqrt(-17*sqrt(2) + 26) - 2820*sqrt(2) + 3992)*sqrt(-sqrt(2) + 2) + (((3*(2*sqrt(2) - 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) - 11*(5*sqrt(2) - 7)*sqrt(-17*sqrt(2) + 26) + 445*sqrt(2) - 630)*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3)*sqrt(sqrt(sqrt(2) + 2) - 1) - 10*(89*sqrt(2) - 126)*sqrt(sqrt(2) + 2) + 2*((61*sqrt(2) - 85)*sqrt(sqrt(2) + 2) - 3*((2*sqrt(2) - 3)*sqrt(sqrt(2) + 2) - 2*sqrt(2) + 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) + 22*((5*sqrt(2) - 7)*sqrt(sqrt(2) + 2) - 5*sqrt(2) + 7)*sqrt(-17*sqrt(2) + 26) + 890*sqrt(2) - 1260)*sqrt(-12*sqrt(2) - 2*sqrt(-sqrt(2) + 2) - 2*sqrt(-17*sqrt(2) + 26) + 24) + 4*((319*sqrt(2) - 452)*sqrt(sqrt(2) + 2) - 561*sqrt(2) + 794)*sqrt(-17*sqrt(2) + 26) + 17064*sqrt(2) - 24132) + ((5*(89*sqrt(2) - 126)*sqrt(sqrt(2) + 2) - ((61*sqrt(2) - 85)*sqrt(sqrt(2) + 2) - 3*((2*sqrt(2) - 3)*sqrt(sqrt(2) + 2) - 4*sqrt(2) + 6)*sqrt(-17*sqrt(2) + 26) - 122*sqrt(2) + 170)*sqrt(-sqrt(2) + 2) - 11*((5*sqrt(2) - 7)*sqrt(sqrt(2) + 2) - 10*sqrt(2) + 14)*sqrt(-17*sqrt(2) + 26) - 890*sqrt(2) + 1260)*sqrt(3*sqrt(2) + sqrt(-sqrt(2) + 2) - 5)*sqrt(-12*sqrt(2) - 2*sqrt(-sqrt(2) + 2) - 2*sqrt(-17*sqrt(2) + 26) + 24) + 2*(10*(63*sqrt(2) - 89)*sqrt(sqrt(2) + 2) - ((85*sqrt(2) - 122)*sqrt(sqrt(2) + 2) - 3*((3*sqrt(2) - 4)*sqrt(sqrt(2) + 2) - 6*sqrt(2) + 8)*sqrt(-17*sqrt(2) + 26) - 170*sqrt(2) + 244)*sqrt(-sqrt(2) + 2) - 11*((7*sqrt(2) - 10)*sqrt(sqrt(2) + 2) - 14*sqrt(2) + 20)*sqrt(-17*sqrt(2) + 26) - 1260*sqrt(2) + 1780)*sqrt(3*sqrt(2) + sqrt(-sqrt(2) + 2) - 5))*((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) + 3*sqrt(2) - 5*sqrt(sqrt(2) + 2) + 8)*sqrt(sqrt(sqrt(2) + 2) + 2)/((2*((3*(3*sqrt(2) - 4)*sqrt(-17*sqrt(2) + 26) - 85*sqrt(2) + 122)*sqrt(-sqrt(2) + 2) - 11*(7*sqrt(2) - 10)*sqrt(-17*sqrt(2) + 26) + 630*sqrt(2) - 890)*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3)*sqrt(sqrt(sqrt(2) + 2) - 1) - 2*(4896*sqrt(2) - 6923)*sqrt(sqrt(2) + 2) + (20*(79*sqrt(2) - 112)*sqrt(sqrt(2) + 2) - (7*(27*sqrt(2) - 38)*sqrt(sqrt(2) + 2) - 342*sqrt(2) + 484)*sqrt(-17*sqrt(2) + 26) - 2820*sqrt(2) + 3992)*sqrt(-sqrt(2) + 2) + (((3*(2*sqrt(2) - 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) - 11*(5*sqrt(2) - 7)*sqrt(-17*sqrt(2) + 26) + 445*sqrt(2) - 630)*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3)*sqrt(sqrt(sqrt(2) + 2) - 1) - 10*(89*sqrt(2) - 126)*sqrt(sqrt(2) + 2) + 2*((61*sqrt(2) - 85)*sqrt(sqrt(2) + 2) - 3*((2*sqrt(2) - 3)*sqrt(sqrt(2) + 2) - 2*sqrt(2) + 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) + 22*((5*sqrt(2) - 7)*sqrt(sqrt(2) + 2) - 5*sqrt(2) + 7)*sqrt(-17*sqrt(2) + 26) + 890*sqrt(2) - 1260)*sqrt(-12*sqrt(2) - 2*sqrt(-sqrt(2) + 2) - 2*sqrt(-17*sqrt(2) + 26) + 24) + 4*((319*sqrt(2) - 452)*sqrt(sqrt(2) + 2) - 561*sqrt(2) + 794)*sqrt(-17*sqrt(2) + 26) + 17064*sqrt(2) - 24132)*((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) + sqrt(sqrt(2) + 2) + 1)*((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) - sqrt(sqrt(2) + 2) - ((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) + 3*sqrt(2) - 5*sqrt(sqrt(2) + 2) + 8)*((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) - 3*sqrt(2) + 5*sqrt(sqrt(2) + 2) - 8)*(sqrt(sqrt(2) + 2) + 2)/(((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) + sqrt(sqrt(2) + 2) + 1)*(sqrt(sqrt(2) + 2) - 2)^3) - 1)*(-1/4*sqrt(sqrt(2) + 2) + 1/2)^(3/2)))/(sqrt(sqrt(2) + 2) - 2))/(2*((3*(3*sqrt(2) - 4)*sqrt(-17*sqrt(2) + 26) - 85*sqrt(2) + 122)*sqrt(-sqrt(2) + 2) - 11*(7*sqrt(2) - 10)*sqrt(-17*sqrt(2) + 26) + 630*sqrt(2) - 890)*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3)*sqrt(sqrt(sqrt(2) + 2) - 1) - 2*(4896*sqrt(2) - 6923)*sqrt(sqrt(2) + 2) + (20*(79*sqrt(2) - 112)*sqrt(sqrt(2) + 2) - (7*(27*sqrt(2) - 38)*sqrt(sqrt(2) + 2) - 342*sqrt(2) + 484)*sqrt(-17*sqrt(2) + 26) - 2820*sqrt(2) + 3992)*sqrt(-sqrt(2) + 2) + (((3*(2*sqrt(2) - 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) - 11*(5*sqrt(2) - 7)*sqrt(-17*sqrt(2) + 26) + 445*sqrt(2) - 630)*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3)*sqrt(sqrt(sqrt(2) + 2) - 1) - 10*(89*sqrt(2) - 126)*sqrt(sqrt(2) + 2) + 2*((61*sqrt(2) - 85)*sqrt(sqrt(2) + 2) - 3*((2*sqrt(2) - 3)*sqrt(sqrt(2) + 2) - 2*sqrt(2) + 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) + 22*((5*sqrt(2) - 7)*sqrt(sqrt(2) + 2) - 5*sqrt(2) + 7)*sqrt(-17*sqrt(2) + 26) + 890*sqrt(2) - 1260)*sqrt(-12*sqrt(2) - 2*sqrt(-sqrt(2) + 2) - 2*sqrt(-17*sqrt(2) + 26) + 24) + 4*((319*sqrt(2) - 452)*sqrt(sqrt(2) + 2) - 561*sqrt(2) + 794)*sqrt(-17*sqrt(2) + 26) + 17064*sqrt(2) - 24132) - 1/16*((5*(89*sqrt(2) - 126)*sqrt(sqrt(2) + 2) - ((61*sqrt(2) - 85)*sqrt(sqrt(2) + 2) - 3*((2*sqrt(2) - 3)*sqrt(sqrt(2) + 2) - 4*sqrt(2) + 6)*sqrt(-17*sqrt(2) + 26) - 122*sqrt(2) + 170)*sqrt(-sqrt(2) + 2) - 11*((5*sqrt(2) - 7)*sqrt(sqrt(2) + 2) - 10*sqrt(2) + 14)*sqrt(-17*sqrt(2) + 26) - 890*sqrt(2) + 1260)*sqrt(3*sqrt(2) + sqrt(-sqrt(2) + 2) - 5)*sqrt(-12*sqrt(2) - 2*sqrt(-sqrt(2) + 2) - 2*sqrt(-17*sqrt(2) + 26) + 24) + 2*(10*(63*sqrt(2) - 89)*sqrt(sqrt(2) + 2) - ((85*sqrt(2) - 122)*sqrt(sqrt(2) + 2) - 3*((3*sqrt(2) - 4)*sqrt(sqrt(2) + 2) - 6*sqrt(2) + 8)*sqrt(-17*sqrt(2) + 26) - 170*sqrt(2) + 244)*sqrt(-sqrt(2) + 2) - 11*((7*sqrt(2) - 10)*sqrt(sqrt(2) + 2) - 14*sqrt(2) + 20)*sqrt(-17*sqrt(2) + 26) - 1260*sqrt(2) + 1780)*sqrt(3*sqrt(2) + sqrt(-sqrt(2) + 2) - 5))*(((sqrt(2)*sqrt(sqrt(2) + 2) - sqrt(2) - 1)*sqrt(sqrt(sqrt(2) + 2) - 1) - sqrt(sqrt(2) + 2) + 1)*(8*(44*(7*sqrt(2) - 10)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - 2*(11*(7*sqrt(2) - 10)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - 10*(63*sqrt(2) - 89)*sqrt(sqrt(2) + 2) - (3*(3*sqrt(2) - 4)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - (85*sqrt(2) - 122)*sqrt(sqrt(2) + 2))*sqrt(-sqrt(2) + 2) - 2*((3*(3*sqrt(2) - 4)*sqrt(-17*sqrt(2) + 26) - 85*sqrt(2) + 122)*sqrt(-sqrt(2) + 2) - 11*(7*sqrt(2) - 10)*sqrt(-17*sqrt(2) + 26) + 630*sqrt(2) - 890)*sqrt(sqrt(sqrt(2) + 2) - 1))*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3) - 40*(63*sqrt(2) - 89)*sqrt(sqrt(2) + 2) - 4*(3*(3*sqrt(2) - 4)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - (85*sqrt(2) - 122)*sqrt(sqrt(2) + 2))*sqrt(-sqrt(2) + 2) + (22*(5*sqrt(2) - 7)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - (11*(5*sqrt(2) - 7)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - 5*(89*sqrt(2) - 126)*sqrt(sqrt(2) + 2) - (3*(2*sqrt(2) - 3)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - (61*sqrt(2) - 85)*sqrt(sqrt(2) + 2))*sqrt(-sqrt(2) + 2) - 2*((3*(2*sqrt(2) - 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) - 11*(5*sqrt(2) - 7)*sqrt(-17*sqrt(2) + 26) + 445*sqrt(2) - 630)*sqrt(sqrt(sqrt(2) + 2) - 1))*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3) - 10*(89*sqrt(2) - 126)*sqrt(sqrt(2) + 2) - 2*(3*(2*sqrt(2) - 3)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - (61*sqrt(2) - 85)*sqrt(sqrt(2) + 2))*sqrt(-sqrt(2) + 2) - 4*((3*(2*sqrt(2) - 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) - 11*(5*sqrt(2) - 7)*sqrt(-17*sqrt(2) + 26) + 445*sqrt(2) - 630)*sqrt(sqrt(sqrt(2) + 2) - 1))*sqrt(-12*sqrt(2) - 2*sqrt(-sqrt(2) + 2) - 2*sqrt(-17*sqrt(2) + 26) + 24) - 8*((3*(3*sqrt(2) - 4)*sqrt(-17*sqrt(2) + 26) - 85*sqrt(2) + 122)*sqrt(-sqrt(2) + 2) - 11*(7*sqrt(2) - 10)*sqrt(-17*sqrt(2) + 26) + 630*sqrt(2) - 890)*sqrt(sqrt(sqrt(2) + 2) - 1))*(1/((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) + sqrt(sqrt(2) + 2) + 1) + ((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) + 3*sqrt(2) - 5*sqrt(sqrt(2) + 2) + 8)*((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) - 3*sqrt(2) + 5*sqrt(sqrt(2) + 2) - 8)*(sqrt(sqrt(2) + 2) + 2)/(((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) + sqrt(sqrt(2) + 2) + 1)^2*((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) - sqrt(sqrt(2) + 2) - ((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) + 3*sqrt(2) - 5*sqrt(sqrt(2) + 2) + 8)*((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) - 3*sqrt(2) + 5*sqrt(sqrt(2) + 2) - 8)*(sqrt(sqrt(2) + 2) + 2)/(((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) + sqrt(sqrt(2) + 2) + 1)*(sqrt(sqrt(2) + 2) - 2)^3) - 1)*(sqrt(sqrt(2) + 2) - 2)^3))/(2*((3*(3*sqrt(2) - 4)*sqrt(-17*sqrt(2) + 26) - 85*sqrt(2) + 122)*sqrt(-sqrt(2) + 2) - 11*(7*sqrt(2) - 10)*sqrt(-17*sqrt(2) + 26) + 630*sqrt(2) - 890)*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3)*sqrt(sqrt(sqrt(2) + 2) - 1) - 2*(4896*sqrt(2) - 6923)*sqrt(sqrt(2) + 2) + (20*(79*sqrt(2) - 112)*sqrt(sqrt(2) + 2) - (7*(27*sqrt(2) - 38)*sqrt(sqrt(2) + 2) - 342*sqrt(2) + 484)*sqrt(-17*sqrt(2) + 26) - 2820*sqrt(2) + 3992)*sqrt(-sqrt(2) + 2) + (((3*(2*sqrt(2) - 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) - 11*(5*sqrt(2) - 7)*sqrt(-17*sqrt(2) + 26) + 445*sqrt(2) - 630)*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3)*sqrt(sqrt(sqrt(2) + 2) - 1) - 10*(89*sqrt(2) - 126)*sqrt(sqrt(2) + 2) + 2*((61*sqrt(2) - 85)*sqrt(sqrt(2) + 2) - 3*((2*sqrt(2) - 3)*sqrt(sqrt(2) + 2) - 2*sqrt(2) + 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) + 22*((5*sqrt(2) - 7)*sqrt(sqrt(2) + 2) - 5*sqrt(2) + 7)*sqrt(-17*sqrt(2) + 26) + 890*sqrt(2) - 1260)*sqrt(-12*sqrt(2) - 2*sqrt(-sqrt(2) + 2) - 2*sqrt(-17*sqrt(2) + 26) + 24) + 4*((319*sqrt(2) - 452)*sqrt(sqrt(2) + 2) - 561*sqrt(2) + 794)*sqrt(-17*sqrt(2) + 26) + 17064*sqrt(2) - 24132) + ((5*(89*sqrt(2) - 126)*sqrt(sqrt(2) + 2) - ((61*sqrt(2) - 85)*sqrt(sqrt(2) + 2) - 3*((2*sqrt(2) - 3)*sqrt(sqrt(2) + 2) - 4*sqrt(2) + 6)*sqrt(-17*sqrt(2) + 26) - 122*sqrt(2) + 170)*sqrt(-sqrt(2) + 2) - 11*((5*sqrt(2) - 7)*sqrt(sqrt(2) + 2) - 10*sqrt(2) + 14)*sqrt(-17*sqrt(2) + 26) - 890*sqrt(2) + 1260)*sqrt(3*sqrt(2) + sqrt(-sqrt(2) + 2) - 5)*sqrt(-12*sqrt(2) - 2*sqrt(-sqrt(2) + 2) - 2*sqrt(-17*sqrt(2) + 26) + 24) + 2*(10*(63*sqrt(2) - 89)*sqrt(sqrt(2) + 2) - ((85*sqrt(2) - 122)*sqrt(sqrt(2) + 2) - 3*((3*sqrt(2) - 4)*sqrt(sqrt(2) + 2) - 6*sqrt(2) + 8)*sqrt(-17*sqrt(2) + 26) - 170*sqrt(2) + 244)*sqrt(-sqrt(2) + 2) - 11*((7*sqrt(2) - 10)*sqrt(sqrt(2) + 2) - 14*sqrt(2) + 20)*sqrt(-17*sqrt(2) + 26) - 1260*sqrt(2) + 1780)*sqrt(3*sqrt(2) + sqrt(-sqrt(2) + 2) - 5))*((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) + 3*sqrt(2) - 5*sqrt(sqrt(2) + 2) + 8)*sqrt(sqrt(sqrt(2) + 2) + 2)/((2*((3*(3*sqrt(2) - 4)*sqrt(-17*sqrt(2) + 26) - 85*sqrt(2) + 122)*sqrt(-sqrt(2) + 2) - 11*(7*sqrt(2) - 10)*sqrt(-17*sqrt(2) + 26) + 630*sqrt(2) - 890)*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3)*sqrt(sqrt(sqrt(2) + 2) - 1) - 2*(4896*sqrt(2) - 6923)*sqrt(sqrt(2) + 2) + (20*(79*sqrt(2) - 112)*sqrt(sqrt(2) + 2) - (7*(27*sqrt(2) - 38)*sqrt(sqrt(2) + 2) - 342*sqrt(2) + 484)*sqrt(-17*sqrt(2) + 26) - 2820*sqrt(2) + 3992)*sqrt(-sqrt(2) + 2) + (((3*(2*sqrt(2) - 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) - 11*(5*sqrt(2) - 7)*sqrt(-17*sqrt(2) + 26) + 445*sqrt(2) - 630)*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3)*sqrt(sqrt(sqrt(2) + 2) - 1) - 10*(89*sqrt(2) - 126)*sqrt(sqrt(2) + 2) + 2*((61*sqrt(2) - 85)*sqrt(sqrt(2) + 2) - 3*((2*sqrt(2) - 3)*sqrt(sqrt(2) + 2) - 2*sqrt(2) + 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) + 22*((5*sqrt(2) - 7)*sqrt(sqrt(2) + 2) - 5*sqrt(2) + 7)*sqrt(-17*sqrt(2) + 26) + 890*sqrt(2) - 1260)*sqrt(-12*sqrt(2) - 2*sqrt(-sqrt(2) + 2) - 2*sqrt(-17*sqrt(2) + 26) + 24) + 4*((319*sqrt(2) - 452)*sqrt(sqrt(2) + 2) - 561*sqrt(2) + 794)*sqrt(-17*sqrt(2) + 26) + 17064*sqrt(2) - 24132)*((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) + sqrt(sqrt(2) + 2) + 1)*((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) - sqrt(sqrt(2) + 2) - ((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) + 3*sqrt(2) - 5*sqrt(sqrt(2) + 2) + 8)*((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) - 3*sqrt(2) + 5*sqrt(sqrt(2) + 2) - 8)*(sqrt(sqrt(2) + 2) + 2)/(((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) + sqrt(sqrt(2) + 2) + 1)*(sqrt(sqrt(2) + 2) - 2)^3) - 1)*(-1/4*sqrt(sqrt(2) + 2) + 1/2)^(3/2)))*sqrt(sqrt(sqrt(2) + 2) + 2)/sqrt(-1/4*sqrt(sqrt(2) + 2) + 1/2) - 2*(sqrt(sqrt(2) + 2)*(sqrt(2) - 1)*sqrt(sqrt(sqrt(2) + 2) - 1) + sqrt(2) - sqrt(sqrt(2) + 2))*(8*((5*(89*sqrt(2) - 126)*sqrt(sqrt(2) + 2) - ((61*sqrt(2) - 85)*sqrt(sqrt(2) + 2) - 3*((2*sqrt(2) - 3)*sqrt(sqrt(2) + 2) - 4*sqrt(2) + 6)*sqrt(-17*sqrt(2) + 26) - 122*sqrt(2) + 170)*sqrt(-sqrt(2) + 2) - 11*((5*sqrt(2) - 7)*sqrt(sqrt(2) + 2) - 10*sqrt(2) + 14)*sqrt(-17*sqrt(2) + 26) - 890*sqrt(2) + 1260)*sqrt(3*sqrt(2) + sqrt(-sqrt(2) + 2) - 5)*sqrt(-12*sqrt(2) - 2*sqrt(-sqrt(2) + 2) - 2*sqrt(-17*sqrt(2) + 26) + 24) + 2*(10*(63*sqrt(2) - 89)*sqrt(sqrt(2) + 2) - ((85*sqrt(2) - 122)*sqrt(sqrt(2) + 2) - 3*((3*sqrt(2) - 4)*sqrt(sqrt(2) + 2) - 6*sqrt(2) + 8)*sqrt(-17*sqrt(2) + 26) - 170*sqrt(2) + 244)*sqrt(-sqrt(2) + 2) - 11*((7*sqrt(2) - 10)*sqrt(sqrt(2) + 2) - 14*sqrt(2) + 20)*sqrt(-17*sqrt(2) + 26) - 1260*sqrt(2) + 1780)*sqrt(3*sqrt(2) + sqrt(-sqrt(2) + 2) - 5))*(1/((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) + sqrt(sqrt(2) + 2) + 1) + ((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) + 3*sqrt(2) - 5*sqrt(sqrt(2) + 2) + 8)*((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) - 3*sqrt(2) + 5*sqrt(sqrt(2) + 2) - 8)*(sqrt(sqrt(2) + 2) + 2)/(((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) + sqrt(sqrt(2) + 2) + 1)^2*((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) - sqrt(sqrt(2) + 2) - ((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) + 3*sqrt(2) - 5*sqrt(sqrt(2) + 2) + 8)*((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) - 3*sqrt(2) + 5*sqrt(sqrt(2) + 2) - 8)*(sqrt(sqrt(2) + 2) + 2)/(((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) + sqrt(sqrt(2) + 2) + 1)*(sqrt(sqrt(2) + 2) - 2)^3) - 1)*(sqrt(sqrt(2) + 2) - 2)^3))/(2*((3*(3*sqrt(2) - 4)*sqrt(-17*sqrt(2) + 26) - 85*sqrt(2) + 122)*sqrt(-sqrt(2) + 2) - 11*(7*sqrt(2) - 10)*sqrt(-17*sqrt(2) + 26) + 630*sqrt(2) - 890)*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3)*sqrt(sqrt(sqrt(2) + 2) - 1) - 2*(4896*sqrt(2) - 6923)*sqrt(sqrt(2) + 2) + (20*(79*sqrt(2) - 112)*sqrt(sqrt(2) + 2) - (7*(27*sqrt(2) - 38)*sqrt(sqrt(2) + 2) - 342*sqrt(2) + 484)*sqrt(-17*sqrt(2) + 26) - 2820*sqrt(2) + 3992)*sqrt(-sqrt(2) + 2) + (((3*(2*sqrt(2) - 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) - 11*(5*sqrt(2) - 7)*sqrt(-17*sqrt(2) + 26) + 445*sqrt(2) - 630)*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3)*sqrt(sqrt(sqrt(2) + 2) - 1) - 10*(89*sqrt(2) - 126)*sqrt(sqrt(2) + 2) + 2*((61*sqrt(2) - 85)*sqrt(sqrt(2) + 2) - 3*((2*sqrt(2) - 3)*sqrt(sqrt(2) + 2) - 2*sqrt(2) + 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) + 22*((5*sqrt(2) - 7)*sqrt(sqrt(2) + 2) - 5*sqrt(2) + 7)*sqrt(-17*sqrt(2) + 26) + 890*sqrt(2) - 1260)*sqrt(-12*sqrt(2) - 2*sqrt(-sqrt(2) + 2) - 2*sqrt(-17*sqrt(2) + 26) + 24) + 4*((319*sqrt(2) - 452)*sqrt(sqrt(2) + 2) - 561*sqrt(2) + 794)*sqrt(-17*sqrt(2) + 26) + 17064*sqrt(2) - 24132) + (44*(7*sqrt(2) - 10)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) + 2*(11*(7*sqrt(2) - 10)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - 10*(63*sqrt(2) - 89)*sqrt(sqrt(2) + 2) - (3*(3*sqrt(2) - 4)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - (85*sqrt(2) - 122)*sqrt(sqrt(2) + 2))*sqrt(-sqrt(2) + 2) + 2*((3*(3*sqrt(2) - 4)*sqrt(-17*sqrt(2) + 26) - 85*sqrt(2) + 122)*sqrt(-sqrt(2) + 2) - 11*(7*sqrt(2) - 10)*sqrt(-17*sqrt(2) + 26) + 630*sqrt(2) - 890)*sqrt(sqrt(sqrt(2) + 2) - 1))*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3) - 40*(63*sqrt(2) - 89)*sqrt(sqrt(2) + 2) - 4*(3*(3*sqrt(2) - 4)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - (85*sqrt(2) - 122)*sqrt(sqrt(2) + 2))*sqrt(-sqrt(2) + 2) + (22*(5*sqrt(2) - 7)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) + (11*(5*sqrt(2) - 7)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - 5*(89*sqrt(2) - 126)*sqrt(sqrt(2) + 2) - (3*(2*sqrt(2) - 3)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - (61*sqrt(2) - 85)*sqrt(sqrt(2) + 2))*sqrt(-sqrt(2) + 2) + 2*((3*(2*sqrt(2) - 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) - 11*(5*sqrt(2) - 7)*sqrt(-17*sqrt(2) + 26) + 445*sqrt(2) - 630)*sqrt(sqrt(sqrt(2) + 2) - 1))*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3) - 10*(89*sqrt(2) - 126)*sqrt(sqrt(2) + 2) - 2*(3*(2*sqrt(2) - 3)*sqrt(sqrt(2) + 2)*sqrt(-17*sqrt(2) + 26) - (61*sqrt(2) - 85)*sqrt(sqrt(2) + 2))*sqrt(-sqrt(2) + 2) + 4*((3*(2*sqrt(2) - 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) - 11*(5*sqrt(2) - 7)*sqrt(-17*sqrt(2) + 26) + 445*sqrt(2) - 630)*sqrt(sqrt(sqrt(2) + 2) - 1))*sqrt(-12*sqrt(2) - 2*sqrt(-sqrt(2) + 2) - 2*sqrt(-17*sqrt(2) + 26) + 24) + 8*((3*(3*sqrt(2) - 4)*sqrt(-17*sqrt(2) + 26) - 85*sqrt(2) + 122)*sqrt(-sqrt(2) + 2) - 11*(7*sqrt(2) - 10)*sqrt(-17*sqrt(2) + 26) + 630*sqrt(2) - 890)*sqrt(sqrt(sqrt(2) + 2) - 1))*((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) + 3*sqrt(2) - 5*sqrt(sqrt(2) + 2) + 8)*sqrt(sqrt(sqrt(2) + 2) + 2)/((2*((3*(3*sqrt(2) - 4)*sqrt(-17*sqrt(2) + 26) - 85*sqrt(2) + 122)*sqrt(-sqrt(2) + 2) - 11*(7*sqrt(2) - 10)*sqrt(-17*sqrt(2) + 26) + 630*sqrt(2) - 890)*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3)*sqrt(sqrt(sqrt(2) + 2) - 1) - 2*(4896*sqrt(2) - 6923)*sqrt(sqrt(2) + 2) + (20*(79*sqrt(2) - 112)*sqrt(sqrt(2) + 2) - (7*(27*sqrt(2) - 38)*sqrt(sqrt(2) + 2) - 342*sqrt(2) + 484)*sqrt(-17*sqrt(2) + 26) - 2820*sqrt(2) + 3992)*sqrt(-sqrt(2) + 2) + (((3*(2*sqrt(2) - 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) - 11*(5*sqrt(2) - 7)*sqrt(-17*sqrt(2) + 26) + 445*sqrt(2) - 630)*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3)*sqrt(sqrt(sqrt(2) + 2) - 1) - 10*(89*sqrt(2) - 126)*sqrt(sqrt(2) + 2) + 2*((61*sqrt(2) - 85)*sqrt(sqrt(2) + 2) - 3*((2*sqrt(2) - 3)*sqrt(sqrt(2) + 2) - 2*sqrt(2) + 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) + 22*((5*sqrt(2) - 7)*sqrt(sqrt(2) + 2) - 5*sqrt(2) + 7)*sqrt(-17*sqrt(2) + 26) + 890*sqrt(2) - 1260)*sqrt(-12*sqrt(2) - 2*sqrt(-sqrt(2) + 2) - 2*sqrt(-17*sqrt(2) + 26) + 24) + 4*((319*sqrt(2) - 452)*sqrt(sqrt(2) + 2) - 561*sqrt(2) + 794)*sqrt(-17*sqrt(2) + 26) + 17064*sqrt(2) - 24132)*((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) + sqrt(sqrt(2) + 2) + 1)*((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) - sqrt(sqrt(2) + 2) - ((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) + 3*sqrt(2) - 5*sqrt(sqrt(2) + 2) + 8)*((sqrt(2) + sqrt(sqrt(2) + 2))*sqrt(sqrt(sqrt(2) + 2) - 1) - 3*sqrt(2) + 5*sqrt(sqrt(2) + 2) - 8)*(sqrt(sqrt(2) + 2) + 2)/(((sqrt(sqrt(2) + 2) + 1)*sqrt(sqrt(sqrt(2) + 2) - 1) + sqrt(sqrt(2) + 2) + 1)*(sqrt(sqrt(2) + 2) - 2)^3) - 1)*(-1/4*sqrt(sqrt(2) + 2) + 1/2)^(3/2)))/(sqrt(sqrt(2) + 2) - 2))/(2*((3*(3*sqrt(2) - 4)*sqrt(-17*sqrt(2) + 26) - 85*sqrt(2) + 122)*sqrt(-sqrt(2) + 2) - 11*(7*sqrt(2) - 10)*sqrt(-17*sqrt(2) + 26) + 630*sqrt(2) - 890)*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3)*sqrt(sqrt(sqrt(2) + 2) - 1) - 2*(4896*sqrt(2) - 6923)*sqrt(sqrt(2) + 2) + (20*(79*sqrt(2) - 112)*sqrt(sqrt(2) + 2) - (7*(27*sqrt(2) - 38)*sqrt(sqrt(2) + 2) - 342*sqrt(2) + 484)*sqrt(-17*sqrt(2) + 26) - 2820*sqrt(2) + 3992)*sqrt(-sqrt(2) + 2) + (((3*(2*sqrt(2) - 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) - 11*(5*sqrt(2) - 7)*sqrt(-17*sqrt(2) + 26) + 445*sqrt(2) - 630)*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3)*sqrt(sqrt(sqrt(2) + 2) - 1) - 10*(89*sqrt(2) - 126)*sqrt(sqrt(2) + 2) + 2*((61*sqrt(2) - 85)*sqrt(sqrt(2) + 2) - 3*((2*sqrt(2) - 3)*sqrt(sqrt(2) + 2) - 2*sqrt(2) + 3)*sqrt(-17*sqrt(2) + 26) - 61*sqrt(2) + 85)*sqrt(-sqrt(2) + 2) + 22*((5*sqrt(2) - 7)*sqrt(sqrt(2) + 2) - 5*sqrt(2) + 7)*sqrt(-17*sqrt(2) + 26) + 890*sqrt(2) - 1260)*sqrt(-12*sqrt(2) - 2*sqrt(-sqrt(2) + 2) - 2*sqrt(-17*sqrt(2) + 26) + 24) + 4*((319*sqrt(2) - 452)*sqrt(sqrt(2) + 2) - 561*sqrt(2) + 794)*sqrt(-17*sqrt(2) + 26) + 17064*sqrt(2) - 24132)

I have to check if this value is equal to -(1-(abs(M))^2)^2).

where

M = -(4*(6*sqrt(2) + sqrt(-sqrt(2) + 2) + sqrt(-17*sqrt(2) + 26) - 8)*sqrt(3*sqrt(2) + sqrt(-sqrt(2) + 2) - 5) - sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3)*(-24*I*sqrt(2) - 4*I*sqrt(-sqrt(2) + 2) - 4*I*sqrt(-17*sqrt(2) + 26) + 32*I) - ((sqrt(2)*sqrt(-sqrt(2) + 2) + sqrt(2)*sqrt(-17*sqrt(2) + 26) - 8*sqrt(2) + 12)*sqrt(3*sqrt(2) + sqrt(-sqrt(2) + 2) - 5) + (I*sqrt(2)*sqrt(-sqrt(2) + 2) + I*sqrt(2)*sqrt(-17*sqrt(2) + 26) - 8*I*sqrt(2) + 12*I)*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3))*sqrt(-12*sqrt(2) - 2*sqrt(-sqrt(2) + 2) - 2*sqrt(-17*sqrt(2) + 26) + 24) - ((24*I*sqrt(2) + 4*I*sqrt(-17*sqrt(2) + 26) - 32*I)*sqrt(-sqrt(2) + 2) + 8*I*(3*sqrt(2) - 4)*sqrt(-17*sqrt(2) + 26) - 228*I*sqrt(2) + 328*I)*sqrt(sqrt(sqrt(2) + 2) - 1))/(4*(6*sqrt(2) + sqrt(-sqrt(2) + 2) + sqrt(-17*sqrt(2) + 26) - 8)*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3)*sqrt(sqrt(sqrt(2) + 2) - 1) + sqrt(3*sqrt(2) + sqrt(-sqrt(2) + 2) - 5)*(-24*I*sqrt(2) - 4*I*sqrt(-sqrt(2) + 2) - 4*I*sqrt(-17*sqrt(2) + 26) + 32*I)*sqrt(sqrt(sqrt(2) + 2) - 1) - 4*(6*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 8)*sqrt(-sqrt(2) + 2) + ((I*sqrt(2)*sqrt(-sqrt(2) + 2) + I*sqrt(2)*sqrt(-17*sqrt(2) + 26) - 8*I*sqrt(2) + 12*I)*sqrt(3*sqrt(2) + sqrt(-sqrt(2) + 2) - 5)*sqrt(sqrt(sqrt(2) + 2) - 1) - (sqrt(2)*sqrt(-sqrt(2) + 2) + sqrt(2)*sqrt(-17*sqrt(2) + 26) - 8*sqrt(2) + 12)*sqrt(3*sqrt(2) + sqrt(-17*sqrt(2) + 26) - 3)*sqrt(sqrt(sqrt(2) + 2) - 1))*sqrt(-12*sqrt(2) - 2*sqrt(-sqrt(2) + 2) - 2*sqrt(-17*sqrt(2) + 26) + 24) - 8*(3*sqrt(2) - 4)*sqrt(-17*sqrt(2) + 26) + 228*sqrt(2) - 328)

so i run the following cell:

bool(N == -(1 - abs(M)^2)^2)

Sadly it keeps loading for hours (at 6 hours I stopped the kernel), and i do not know if this last cell gives me true of false.

I want to know if there exists another way to verify equality between large symbolic expressions like above, with Sage or with other software.

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2 answers

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answered 2020-07-24 17:21:33 -0500

slelievre gravatar image

updated 2020-07-25 19:10:52 -0500

[Edited 2020-07-26 to turn vague hints into an actual answer].

Here is one way to go, with M and N as in the question.

Our solution uses algebraic numbers in AA and QQbar rather than symbolic expressions in Sage's "symbolic ring".

However we start by computing the square of the modulus of M using the symbolic ring and string manipulations.

Along the way, two of the computations take a few seconds each, but under a minute (on a 2014 MacBook Air).


To define the conjugate of M, we turn M into a string, then replace I by (-I) in that string, then convert back to a symbolic expression:

sage: Mc_str = str(M).replace('I', '(-I)')
sage: Mc = SR(Mc_str)

Compute (abs(M)^2 - 1)^2 using M*Mc for abs(M)^2:

sage: W = (M*Mc - 1)^2

Our goal is to prove that N + W is zero.

See N as a real algebraic number:

sage: N_aa = AA(N)
-0.1619085305331?

See W as an algebraic number (not known to be real yet):

sage: W_qqbar = QQbar(W)
sage: W_qqbar
0.1619085305331120? + 0.?e-16*I

Examine the imaginary part:

sage: W_imag = W_qqbar.imag()
sage: W_imag
0.?e-16

Check whether it is actually zero:

sage: W_imag.is_zero()  # takes a few seconds
True

Since its imaginary part is zero, see W as a real algebraic number:

sage: W_aa = AA(W_qqbar)

sage: W_aa
0.1619085305331121?

Compute the sum of W and N (we hope it is zero):

sage: Z_aa = W_aa + N_aa

Does it look like it might be zero:

sage: Z_aa
0.?e-13

Is it actually zero:

sage: Z_aa.is_zero()  # takes a few seconds
True

Finally:

sage: Z_aa
0

The main take-away hint is to work over AA or QQbar.

Ideally, compute in AA or QQbar (or in an appropriate number field if you know which one to use) from the start and avoid ending up with such complicated expressions.

Some ways to do this:

  • use matrix(QQbar, ...) instead of matrix(...)
  • define r = AA(2).sqrt() and use r instead of sqrt(2)
  • use appropriate tower of quadratic extensions of QQ

Regarding other software, Calcium might help. See


My initial approach was somewhat along the same lines but did not quite get us there. I stopped the final computations after several minutes or intense cpu usage.

I still leave it here for comparison.

Having defined M and N as in the question...

Define corresponding elements in QQbar or AA:

sage: m = QQbar(M)
sage: n = AA(N)
sage: p = -(1-m.abs()^2)^2

Check their values:

sage: m
0.0154952449333661? - 0.772904461325769?*I

sage: n
-0.1619085305331?
sage: p
-0.1619085305331120?

So n and p might be equal...

Check their difference:

sage: d = p - n
sage: d
0.?e-13

Looks like it might be zero.

Checking it seems to take forever:

sage: d.is_zero()  # takes too long
sage: AA(N) == -(1 - QQbar(M).abs()^2)^2  # takes too long
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Comments

You may get more precision on the difference by doing d.n(digits=100). What do you get?

Sébastien gravatar imageSébastien ( 2020-07-25 02:16:38 -0500 )edit

thanks slelievre for your advices, but i did not get how to download the calcium software. Sébastien, i want to know if the numbers are equal, not approximately equal, so I do not need more digits.

creyesm1992 gravatar imagecreyesm1992 ( 2020-07-25 06:42:52 -0500 )edit

Well, if it turns out that the two values are not equal, it could be discovered by increasing the number of digits.

Sébastien gravatar imageSébastien ( 2020-07-25 14:44:17 -0500 )edit

slelievre, you solved my problem, i am in a big debt with you, infinite thanks!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

creyesm1992 gravatar imagecreyesm1992 ( 2020-07-26 06:49:33 -0500 )edit
1

answered 2020-07-26 02:33:32 -0500

fredrik gravatar image

updated 2020-07-26 08:06:25 -0500

This should not be a difficult computation. It takes me half a minute with the "naive" QQbar implementation in Pygrim (https://github.com/fredrik-johansson/...)

fredrik@agm:~/Desktop$ time python3 verifysqrt.py 
True

real     0m29,350s
user     0m29,331s
sys      0m0,017s

Content of the script verifysqrt.py:

from pygrim import alg

def sqrt(x):
    return alg(x).sqrt()

N = ...        # copy of expression, with x/y and x^y replaced by alg(x)/y and x**y for pure python compatibility
I = alg.i()
M = ...        # as with N = ...
print(-(1 - abs(M)**2)**2 == N)

As Samuel said, Calcium (http://fredrikj.net/calcium/) is meant to let you do this kind of computation easily.

First of all, there is a "naive" QQbar implementation (qqbar_t) which is essentially a cleaned-up and slightly optimized C reimplementation of the Pygrim code. This module is already complete and if someone wanted to write a Sage interface, it should be possible today :-)

Second, the actual goal of Calcium is to have a more efficient type (ca_t) based on fast field arithmetic and relative extensions (and supporting not only algebraics but also transcendental numbers). This will potentially be more efficient for this kind of problem -- in fact, it's a very nice benchmark problem, and I'm happy that you shared it. I will save it for testing purposes :-)

The reason there is no public release of Calcium yet (with easy-to-install packages and installation documentation) is that the ca_t type is far from finished (in fact, I need to rewrite a lot of the existing code...). This could take a few months.

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Asked: 2020-07-24 09:38:15 -0500

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Last updated: Jul 26