| Definitions:
denom(b) = (b^b - b^2 + b - 1) / (b - 1)^2
num(b) = (b^b(b - 2) + 1) / (b - 1)^2 Exact relation:
num(b) - (b - 2)denom(b) = b - 1 Therefore:
num(b) / denom(b) = (b - 2) + (b - 1)^3 / (b^b - b^2 + b - 1) [exact] Geometric expansion:
Let a = b^2 - b + 1.
1 / (b^b - b^2 + b - 1) = (1 / b^b) * 1 / (1 - a / b^b)
= (1 / b^b) * sum_{k>=0} (a / b^b)^k So:
num(b) / denom(b)
= (b - 2)
• (b - 1)^3 / b^b
• (b - 1)^3 * a / b^{2b}
• (b - 1)^3 * a^2 / b^{3b}
• … Practical approximation:
num(b) / denom(b) ≈ (b - 2) + (b - 1)^3 / b^b Exact error:
Let T_exact = (b - 1)^3 / (b^b - b^2 + b - 1)
Let T_approx = (b - 1)^3 / b^b Absolute error:
T_exact - T_approx
= (b - 1)^3 * (b^2 - b + 1) / [ b^b * (b^b - b^2 + b - 1) ] Relative error:
(T_exact - T_approx) / T_exact = (b^2 - b + 1) / b^b Sign:
The approximation with denominator b^b underestimates the exact value. Digit picture in base b:
(b - 1)^3 has base-b digits (b - 3), 2, (b - 1).
Dividing by b^b places those three digits starting b places after the radix point. Examples:
base 10: 8 + 9^3 / 10^10 = 8.0000000729
base 9: 7 + 8^3 / 9^9 = 7.000000628 in base 9
base 8: 6 + 7^3 / 8^8 = 6.00000527 in base 8 num(b) / denom(b) equals (b - 2) + (b - 1)^3 / (b^b - b^2 + b - 1) exactly.
Replacing the denominator by b^b gives a simple approximation with relative error exactly (b^2 - b + 1) / b^b. |