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

fredrik gravatar image

updated 2020-08-23 08:51:15 -0500

Updated answer on 2020-08-23:

This is now possible with Calcium (http://fredrikj.net/calcium/). I have added an example program called huge_expr.c that performs this computation in two different ways (the C file was auto-generated from the Sage expression, since I do not yet have an expression parser in Calcium itself).

The example program is documented here: http://fredrikj.net/calcium/examples....

By default, the program runs the computation using qqbar_t arithmetic. This takes half a minute:

> build/examples/huge_expr
Evaluating N...
cpu/wall(s): 18.279 18.279
Evaluating M...
cpu/wall(s): 6.049 6.051
Evaluating E = -(1-|M|^2)^2...
cpu/wall(s): 0.595 0.595
N ~ -0.16190853053311203695842869991458578203473645660641
E ~ -0.16190853053311203695842869991458578203473645660641
Testing E = N...
cpu/wall(s): 0 0

Equal = T_TRUE

Total: cpu/wall(s): 24.927 24.93
virt/peak/res/peak(MB): 56.61 68.64 28.73 40.70

To run the computation using ca_t arithmetic instead, pass the -ca flag. This currently takes longer (three minutes), but at least it works:

> build/examples/huge_expr -ca
Evaluating N...
cpu/wall(s): 2.116 2.116
Evaluating M...
cpu/wall(s): 0.068 0.068
Evaluating E = -(1-|M|^2)^2...
cpu/wall(s): 0.043 0.043
N ~ -0.16190853053311203695842869991458578203473645660641
E ~ -0.16190853053311203695842869991458578203473645660641
Testing E = N...
cpu/wall(s): 176.235 176.242

Equal = T_TRUE

Total: cpu/wall(s): 178.465 178.472
virt/peak/res/peak(MB): 55.92 67.88 29.80 41.76

It should be possible to improve the ca_t arithmetic so that is competitive with the qqbar_t version (hopefully even faster, if it can be done using suitable number field operations). This example program will remain in Calcium as a nice test/benchmark case :-)


Old answer: this takes half a minute with the "naive" QQbar implementation in Pygrim (https://github.com/fredrik-johansson/...).

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

slelievre gravatar image

updated 2020-08-23 19:29:47 -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 Mc of M, we turn M into a string M_str, then replace I by (-I) in that string to get Mc_str, then convert back to a symbolic expression:

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

Call W the quantity (abs(M)^2 - 1)^2 computed 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

Now that its imaginary part is known to be zero, we can 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 of 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.

Trying to check whether it is actually zero 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

Happy to help. Don't hesitate to ask another question about your matrix.

There might be a way to never even get the complicated expression in the question.

Anyway this complicated expression gave us some fun and a stress test case. Thanks!

slelievre gravatar imageslelievre ( 2020-08-23 19:33:31 -0500 )edit

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

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Last updated: Aug 23