Two 64-bit floating point numbers are used to represent the base-10 mantissa and exponent of a real number, encoded as a single complex atom. Therefore, in this representation 3.124j2 is the number 312.4 and _3.44j0 is the number _3.44 .
Operations
plus=: 4 : 0 " 0
'x xe'=. +. x
'y ye'=. +. y
e=. xe>.ye
z=. (x*10^xe-e)+y*10^ye-e
(0~:z) * (z%10^k) j. e+k=. <.10^.|z
)
minus=: _1 1&*"1&.+. : (plus $:) " 0
times=: 4 : 0 " 0
'x xe'=. +. x
'y ye'=. +. y
z=. x*y
(0~:z) * (z%10^k) j. xe+ye+k=. <.10^.|z
)
div=: 4 : 0 " 0
'x xe'=. +. x
'y ye'=. +. y
if. 0=x do. 0 return. end.
if. 0=y do. (*x){0 _ __ return. end.
z=. x%y
(z%10^k) j. (xe-ye)+k=. <.10^.|z
)
pow=: 4 : 0 " 0
'x xe'=. +. x
'y ye'=. +. y
if. 0=x do. (*y){1 0 _ return. end.
n=. y*10^ye
assert. (0<x)+.n=<.n NB. complex number
((_1^(0>x)*.2|n) * 10^z-e) j. e=. <.z=. n*xe+10^.|x
)
log=: 10&$: : (4 : 0) " 0
'x xe'=. +. x
'y ye'=. +. y
assert. (0<x)*.0<y
(ye+10^.y) div (xe+10^.x)
)
Examples
3j4 plus 5j6
5.03j6
3j4 times 5j6
1.5j11
minus~ 3j4
0
3j4 div 5j6
6j_3
3j4 pow 12
5.31441j53
3j4 pow 1 12 123456
3j4 5.31441j53 3.03126j552727
0 pow _5 0 5
_ 1 0
div/~ i: 3
1 1.5 3 __ _3 _1.5 _1
6.66667j_1 1 2 __ _2 _1 _6.66667j_1
3.33333j_1 5j_1 1 __ _1 _5j_1 _3.33333j_1
0 0 0 0 0 0 0
_3.33333j_1 _5j_1 _1 _ 1 5j_1 3.33333j_1
_6.66667j_1 _1 _2 _ 2 1 6.66667j_1
_1 _1.5 _3 _ 3 1.5 1
An Application
The n-th Fibonacci number can be computed as <. 0.5 + (%:5) %~ phi^n . Thus:
phi=: -: 1+%:5 (%:5) div~ phi pow 2749 1.43727j574 fmt=: ":!.16 fmt (%:5) div~ phi pow 2749 1.437268955338741j574
f7 from the Fibonacci numbers page computes the result exactly:
f7=: 3 : 0
mp=. +/ .*
{.{: mp/ mp~^:(I.|.#:y) 2 2$0 1 1 1x
)
f7 2749
143726895533879176618296456715643341443476345064489177 ...
0j_20 ": f7 2749
1.43726895533879176618e574Fib 2749 is pandigital in its leading 9 digits.
Contributed by RogerHui.