c² = a² + a² [right angle triangle having equal length legs]
c = sqrt(2) * a
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a = c * sin(45°) [same triangle but with trigonometric relation]
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c = sqrt(2) * c * sin(45°)
1 = sqrt(2) * sin(45°)
sqrt(2) = 1 / sin(45°)
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x = 2^log_2(x)
log_2(x) = a
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sqrt(x) = sqrt(2^a) = sqrt(2) ^ a
sqrt(2) ^ a = (1 / sin(45°)) ^ a = (1 / sin(45°)) ^ log_2(x)
sqrt(x) = (1 / sin(45°)) ^ log_2(x)
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sin(n) = 2 * sin(n/2) * cos(n/2)
sin(90°) = 2 * sin(45°) * cos(45°)
sin(90°) = 1
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sqrt(x) = (1 / sin(45°)) ^ log_2(x) = (2 * cos(45°)) ^ log_2(x) = x * (cos(45°) ^ log_2(x))
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cos(45°) = 0.707106781
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Test:
sqrt(500) = 22.360
500 * (0.707106781 ^ log_2(500)) = 22.360