Natural Logarithm

    Values are calculated by the series expansion

        ln x = 2 Sum (((x - 1) ÷ (x + 1))^i) ÷ i, i = 1, 3, 5, 7,...

    Calculation is in 25 decimal place fixed point, then result
    is rounded to 20 digits.
  
        V0  x, Sum accumulator, result
        V1  1.
        V2  2.
        V3  ((x - 1) ÷ (x + 1))²
        V4  i, initially 1.
        V5  Current term
        V6  0
        V7  100000
        V8  50000

A set decimal places to +5

    Constant and initial value number cards

N002  2.0
N004  1.0
N006 0
N001  1.0
N007 100000
N008 50000

    Scale input to use 5 more digits for computation

×
L000
L007
S000

    V3 = x - 1
-
L000
L001
S003

    V5 = x + 1

+
L000
L001
S005

    V5 = (x - 1) ÷ (x + 1)  This is first term in the summation

÷
L003
<
L005
S005'

    V3 = ((x - 1) ÷ (x + 1))²

×
L005
L005
>
S003

    V0 = 0  Clear series sum

+
L006
L006
S000

    Begin summation cycle

(?

    V10 = current_term ÷ i

÷
L005
<
L004
S010'

    V0 = V0 + V10   Add current term to summation

+
L000
L010
S000

    V5 = V5 × V3    Multiply by ((x - 1) ÷ (x + 1))²

×
L005
L003
>
S005

    i = i + 2

+
L004
L002
S004

    Test whether we've reached the final term.  A run
    up occurs when we subtract the final term from zero,
    stopping the cycle at the first zero term in the series.

-
L006
L010
)

    Multiply series sum by two

×
L000
L002
>
S000

    Add rounding constant

+
L000
L008
S000

    Scale to result

÷
L000
L007
S000'

A set decimal places to -5