<s>
CORDIC	O
(	O
for	O
"	O
coordinate	B-General_Concept
rotation	I-General_Concept
digital	O
computer	O
"	O
)	O
,	O
also	O
known	O
as	O
Volder	O
's	O
algorithm	O
,	O
or	O
:	O
Digit-by-digit	O
method	O
Circular	O
CORDIC	O
(	O
Jack	O
E	O
.	O
Volder	O
)	O
,	O
Linear	O
CORDIC	O
,	O
Hyperbolic	O
CORDIC	O
(	O
John	O
Stephen	O
Walther	O
)	O
,	O
and	O
Generalized	O
Hyperbolic	O
CORDIC	O
(	O
GH	O
CORDIC	O
)	O
(	O
Yuanyong	O
Luo	O
et	O
al	O
.	O
</s>
<s>
CORDIC	O
and	O
closely	O
related	O
methods	O
known	O
as	O
pseudo-multiplication	O
and	O
pseudo-division	O
or	O
factor	B-Algorithm
combining	I-Algorithm
are	O
commonly	O
used	O
when	O
no	O
hardware	O
multiplier	O
is	O
available	O
(	O
e.g.	O
</s>
<s>
in	O
simple	O
microcontrollers	B-Architecture
and	O
FPGAs	B-Architecture
)	O
,	O
as	O
the	O
only	O
operations	O
it	O
requires	O
are	O
additions	O
,	O
subtractions	O
,	O
bitshift	O
and	O
lookup	B-Data_Structure
tables	I-Data_Structure
.	O
</s>
<s>
As	O
such	O
,	O
they	O
all	O
belong	O
to	O
the	O
class	O
of	O
shift-and-add	B-Algorithm
algorithms	I-Algorithm
.	O
</s>
<s>
In	O
computer	O
science	O
,	O
CORDIC	O
is	O
often	O
used	O
to	O
implement	O
floating-point	B-Algorithm
arithmetic	I-Algorithm
when	O
the	O
target	O
platform	O
lacks	O
hardware	O
multiply	O
for	O
cost	O
or	O
space	O
reasons	O
.	O
</s>
<s>
The	O
report	O
also	O
discussed	O
the	O
possibility	O
to	O
compute	O
hyperbolic	O
coordinate	B-General_Concept
rotation	I-General_Concept
,	O
logarithms	O
and	O
exponential	O
functions	O
with	O
modified	O
CORDIC	O
algorithms	O
.	O
</s>
<s>
These	O
efforts	O
led	O
to	O
the	O
ROMable	O
logic	O
implementation	O
of	O
a	O
decimal	O
CORDIC	O
prototype	O
machine	O
inside	O
of	O
Hewlett-Packard	O
in	O
1966	O
,	O
built	O
by	O
and	O
conceptually	O
derived	O
from	O
Thomas	O
E	O
.	O
Osborne	O
's	O
prototypical	O
Green	O
Machine	O
,	O
a	O
four-function	O
,	O
floating-point	B-Algorithm
desktop	O
calculator	O
he	O
had	O
completed	O
in	O
DTL	B-General_Concept
logic	O
in	O
December	O
1964	O
.	O
</s>
<s>
When	O
Wang	O
Laboratories	O
found	O
that	O
the	O
hp9100A	O
used	O
an	O
approach	O
similar	O
to	O
the	O
factor	B-Algorithm
combining	I-Algorithm
method	O
in	O
their	O
earlier	O
LOCI-1	O
(	O
September	O
1964	O
)	O
and	O
LOCI-2	O
(	O
January	O
1965	O
)	O
Logarithmic	O
Computing	O
Instrument	O
desktop	O
calculators	O
,	O
they	O
unsuccessfully	O
accused	O
Hewlett-Packard	O
of	O
infringement	O
of	O
one	O
of	O
An	O
Wang	O
's	O
patents	O
in	O
1968	O
.	O
</s>
<s>
This	O
development	O
resulted	O
in	O
the	O
first	O
scientific	O
handheld	O
calculator	O
,	O
the	O
HP-35	B-Device
in	O
1972	O
.	O
</s>
<s>
CORDIC	O
has	O
been	O
implemented	O
in	O
the	O
ARM-based	B-Architecture
STM32G4	O
,	O
Intel	B-Device
8087	I-Device
,	O
80287	O
,	O
80387	O
up	O
to	O
the	O
80486	B-General_Concept
coprocessor	O
series	O
as	O
well	O
as	O
in	O
the	O
Motorola	B-General_Concept
68881	I-General_Concept
and	O
68882	B-General_Concept
for	O
some	O
kinds	O
of	O
floating-point	B-Algorithm
instructions	O
,	O
mainly	O
as	O
a	O
way	O
to	O
reduce	O
the	O
gate	O
counts	O
(	O
and	O
complexity	O
)	O
of	O
the	O
FPU	B-General_Concept
sub-system	O
.	O
</s>
<s>
As	O
a	O
consequence	O
,	O
CORDIC	O
has	O
been	O
used	O
for	O
applications	O
in	O
diverse	O
areas	O
such	O
as	O
signal	O
and	O
image	B-Algorithm
processing	I-Algorithm
,	O
communication	O
systems	O
,	O
robotics	O
and	O
3D	O
graphics	O
apart	O
from	O
general	O
scientific	O
and	O
technical	O
computation	O
.	O
</s>
<s>
CORDIC	O
was	O
used	O
to	O
implement	O
the	O
Intel	B-Device
8087	I-Device
math	B-General_Concept
coprocessor	I-General_Concept
in	O
1980	O
,	O
avoiding	O
the	O
need	O
to	O
implement	O
hardware	O
multiplication	O
.	O
</s>
<s>
CORDIC	O
is	O
generally	O
faster	O
than	O
other	O
approaches	O
when	O
a	O
hardware	O
multiplier	O
is	O
not	O
available	O
(	O
e.g.	O
,	O
a	O
microcontroller	B-Architecture
)	O
,	O
or	O
when	O
the	O
number	O
of	O
gates	O
required	O
to	O
implement	O
the	O
functions	O
it	O
supports	O
should	O
be	O
minimized	O
(	O
e.g.	O
,	O
in	O
an	O
FPGA	B-Architecture
or	O
ASIC	O
)	O
.	O
</s>
<s>
In	O
fact	O
,	O
CORDIC	O
is	O
a	O
standard	O
drop-in	O
IP	B-Architecture
in	O
FPGA	B-Architecture
development	O
applications	O
such	O
as	O
Vivado	O
for	O
Xilinx	O
,	O
while	O
a	O
power	O
series	O
implementation	O
is	O
not	O
due	O
to	O
the	O
specificity	O
of	O
such	O
an	O
IP	B-Architecture
,	O
i.e.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
when	O
a	O
hardware	O
multiplier	O
is	O
available	O
(	O
e.g.	O
,	O
in	O
a	O
DSP	B-General_Concept
microprocessor	O
)	O
,	O
table-lookup	O
methods	O
and	O
power	O
series	O
are	O
generally	O
faster	O
than	O
CORDIC	O
.	O
</s>
<s>
In	O
recent	O
years	O
,	O
the	O
CORDIC	O
algorithm	O
has	O
been	O
used	O
extensively	O
for	O
various	O
biomedical	O
applications	O
,	O
especially	O
in	O
FPGA	B-Architecture
implementations	O
.	O
</s>
<s>
The	O
STM32G4	O
series	O
and	O
certain	O
STM32H7	O
series	O
of	O
MCUs	O
implement	O
a	O
CORDIC	O
module	O
to	O
accelerate	O
computations	O
in	O
various	O
mixed	O
signal	O
applications	O
such	O
as	O
graphics	O
for	O
human-machine	B-Application
interface	I-Application
and	O
field	O
oriented	O
control	O
of	O
motors	O
.	O
</s>
<s>
For	O
example	O
,	O
most	O
of	O
the	O
performance	O
difference	O
compared	O
to	O
the	O
ARM	O
implementation	O
is	O
due	O
to	O
the	O
overhead	O
of	O
the	O
interpolation	O
algorithm	O
,	O
which	O
achieves	O
full	O
floating	B-Algorithm
point	I-Algorithm
precision	O
(	O
24	O
bits	O
)	O
and	O
can	O
likely	O
achieve	O
relative	O
error	O
to	O
that	O
precision	O
.	O
</s>
<s>
Other	O
means	O
of	O
polynomial	O
approximation	O
,	O
such	O
as	O
minimax	B-Algorithm
optimization	O
,	O
may	O
be	O
used	O
to	O
control	O
both	O
kinds	O
of	O
error	O
.	O
</s>
<s>
Many	O
older	O
systems	O
with	O
integer-only	O
CPUs	O
have	O
implemented	O
CORDIC	O
to	O
varying	O
extents	O
as	O
part	O
of	O
their	O
IEEE	O
floating-point	B-Algorithm
libraries	O
.	O
</s>
<s>
As	O
most	O
modern	O
general-purpose	O
CPUs	O
have	O
floating-point	B-Algorithm
registers	O
with	O
common	O
operations	O
such	O
as	O
add	O
,	O
subtract	O
,	O
multiply	O
,	O
divide	O
,	O
sine	O
,	O
cosine	O
,	O
square	O
root	O
,	O
log10	O
,	O
natural	O
log	O
,	O
the	O
need	O
to	O
implement	O
CORDIC	O
in	O
them	O
with	O
software	O
is	O
nearly	O
non-existent	O
.	O
</s>
<s>
Only	O
microcontroller	B-Architecture
or	O
special	O
safety	O
and	O
time-constrained	O
software	O
applications	O
would	O
need	O
to	O
consider	O
using	O
CORDIC	O
.	O
</s>
<s>
More	O
formally	O
,	O
every	O
iteration	O
calculates	O
a	O
rotation	O
,	O
which	O
is	O
performed	O
by	O
multiplying	O
the	O
vector	O
with	O
the	O
rotation	B-Algorithm
matrix	I-Algorithm
:	O
</s>
<s>
The	O
following	O
is	O
a	O
MATLAB/GNU	O
Octave	O
implementation	O
of	O
CORDIC	O
that	O
does	O
not	O
rely	O
on	O
any	O
transcendental	O
functions	O
except	O
in	O
the	O
precomputation	O
of	O
tables	O
.	O
</s>
<s>
With	O
MATLAB	B-Language
's	O
standard	O
double-precision	O
arithmetic	O
and	O
"	O
format	O
long	O
"	O
printout	O
,	O
the	O
results	O
increase	O
in	O
accuracy	O
for	O
n	O
up	O
to	O
about	O
48	O
.	O
</s>
<s>
In	O
Java	O
the	O
Math	O
class	O
has	O
a	O
scalb(double x,int scale )	O
method	O
to	O
perform	O
such	O
a	O
shift	O
,	O
C	O
has	O
the	O
ldexp	O
function	O
,	O
and	O
the	O
x86	O
class	O
of	O
processors	O
have	O
the	O
fscale	O
floating	B-Algorithm
point	I-Algorithm
operation	O
.	O
</s>
<s>
CORDICs	O
are	O
often	O
used	O
in	O
circuits	O
for	O
telecommunications	O
such	O
as	O
digital	B-Algorithm
down	I-Algorithm
converters	I-Algorithm
.	O
</s>
<s>
Another	O
shift-and-add	B-Algorithm
algorithm	I-Algorithm
which	O
can	O
be	O
used	O
for	O
computing	O
many	O
elementary	O
functions	O
is	O
the	O
BKM	B-Algorithm
algorithm	I-Algorithm
,	O
which	O
is	O
a	O
generalization	O
of	O
the	O
logarithm	O
and	O
exponential	O
algorithms	O
to	O
the	O
complex	O
plane	O
.	O
</s>
<s>
The	O
BKM	B-Algorithm
algorithm	I-Algorithm
is	O
slightly	O
more	O
complex	O
than	O
CORDIC	O
,	O
but	O
has	O
the	O
advantage	O
that	O
it	O
does	O
not	O
need	O
a	O
scaling	O
factor	O
(	O
K	O
)	O
.	O
</s>
