Math2.org Math Tables: Recursive Formulas for B / A

Recursive Formulas for B / A

Explicit form:

Find B/A where B and A are real numbers and B > 0

Recursive form:

Convert B and A to scientific notation base 2 (C++ has function "frexp" for this). Note: mantissa is ³ .5 and < 1.

Let

a = mantissa of A
b = mantissa of B
exp = exponent of b - exponent of a
x₀ = 1
x_n+1 = x_n(2 - a x_n)

Reiterate x until desired precision reached. Result only has to be close, not perfect. I suggest about 5 times.

y₀ = b x_n
y_i+1 = y_i + x_n(b - a y_i)

Reiterate y until desired precision reached.

B / A = y_i * 2^exp

Example: 314.51 / 5.6789

Written in base-2 scientific notation:

B = 0.61357421875 * 2⁹
A = 0.7098625 * 2³
exp = 9 - 3 = 6

iteration	value
x0	1
x1	1.2901375
x2	1.398740976607287
x3	1.408652781763906
x4	1.408723516847958 (will use this value for x)
iteration	value
y0	0.864356431284738
y1	0.864356433463227
y2	0.864356433463227
y3	0.864356433463227
y4	0.864356433463227
y5	0.864356433463227

B / A = 0.864356433463227 * 2⁶ =
55.318811741710538

55.318811741710538 (true value)

Source: Jeff Yates, et al.