CORDIC
Trigonometry 

Reference 
Laws and theorems 
Calculus 
CORDIC (for COordinate Rotation DIgital Computer),^{[1]}^{[2]}^{[3]} also known as Volder's algorithm, is a simple and efficient algorithm to calculate hyperbolic and trigonometric functions, typically converging with one digit (or bit) per iteration. CORDIC is therefore also an example of digitbydigit algorithms. CORDIC and closely related methods known as pseudomultiplication and pseudodivision or factor combining are commonly used when no hardware multiplier is available (e.g. in simple microcontrollers and FPGAs), as the only operations it requires are addition, subtraction, bitshift and table lookup. As such, they belong to the class of shiftandadd algorithms.
Contents
History
Similar mathematical techniques were published by Henry Briggs as early as 1624^{[4]}^{[5]} and Robert Flower in 1771,^{[6]} but CORDIC is optimized for lowcomplexity finitestate CPUs.
CORDIC was conceived in 1956^{[7]}^{[8]} by Jack E. Volder at the aeroelectronics department of Convair out of necessity to replace the analog resolver in the B58 bomber's navigation computer with a more accurate and performant realtime digital solution.^{[8]} Therefore, CORDIC is sometimes referred to as digital resolver.^{[9]}^{[10]}
In his research Volder was inspired by a formula in the 1946 edition of the CRC Handbook of Chemistry and Physics:
 ^{[8]}
His research led to an internal technical report proposing the CORDIC algorithm to solve sine and cosine functions and a prototypical computer implementing it.^{[7]}^{[8]} The report also discussed the possibility to compute hyperbolic coordinate rotation, logarithms and exponential functions with modified CORDIC algorithms.^{[7]}^{[8]} Utilizing CORDIC for multiplication and division was also conceived at this time.^{[8]} Based on the CORDIC principle, Dan H. Daggett, a colleague of Volder at Convair, developed conversion algorithms between binary and binarycoded decimal (BCD).^{[8]}^{[11]}
In 1958, Convair finally started to build a demonstration system to solve radar fixtaking problems named CORDIC I, completed in 1960 without Volder, who had left the company already.^{[1]}^{[8]} More universal CORDIC II models A (stationary) and B (airborne) were built and tested by Daggett and Harry Schuss in 1962.^{[8]}^{[12]}
Volder's CORDIC algorithm was first described in public in 1959,^{[1]}^{[2]}^{[8]}^{[10]}^{[13]} which caused it to be incorporated into navigation computers by companies including MartinOrlando, Computer Control, Litton, Kearfott, LearSiegler, Sperry, Raytheon, and Collins Radio soon.^{[8]}
Volder teamed up with Malcolm MacMillan to build Athena, a fixedpoint desktop calculator utilizing his binary CORDIC algorithm.^{[14]} The design was introduced to HewlettPackard in June 1965, but not accepted.^{[14]} Still, MacMillan introduced David S. Cochran (HP) to Volder's algorithm and when Cochran later met Volder he referred him to a similar approach John E. Meggitt (IBM^{[15]}) had proposed as pseudomultiplication and pseudodivision in 1961.^{[15]}^{[16]} Meggitt's method was also suggesting the use of base 10^{[15]} rather than base 2, as used by Volder's CORDIC so far. These efforts led to the ROMable logic implementation of a decimal CORDIC prototype machine inside of HewlettPackard in 1966,^{[17]}^{[16]} build by and conceptually derived from Thomas E. Osborne's prototypical Green Machine, a fourfunction, floatingpoint desktop calculator he had completed in DTL logic^{[14]} in December 1964.^{[18]} This project resulted in the public demonstration of HewlettPackard's first desktop calculator with scientific functions, the hp 9100A in March 1968, with series production starting later that year.^{[14]}^{[18]}^{[19]}^{[20]}
When Wang Laboratories found that the hp 9100A used an approach similar to the factor combining method in their earlier LOCI1^{[21]} (September 1964) and LOCI2 (January 1965)^{[22]}^{[23]} Logarithmic Computing Instrument desktop calculators,^{[24]} they unsuccessfully accused HewlettPackard of infringement of one of An Wang's patents in 1968.^{[16]}^{[25]}^{[26]}^{[27]}
John Stephen Walther at HewlettPackard generalized the algorithm into the Unified CORDIC algorithm in 1971, allowing it to calculate hyperbolic and exponential functions, logarithms, multiplications, divisions, and square roots.^{[3]}^{[28]}^{[29]}^{[30]} The CORDIC subroutines for trigonometric and hyperbolic functions could share most of their code.^{[25]} This development resulted in the first scientific handheld calculator, the HP35 in 1972.^{[25]}^{[31]}^{[32]}^{[33]}^{[34]}^{[35]}
Originally, CORDIC was implemented only using the binary numeral system and despite Meggitt suggesting the use of the decimal system for his pseudomultiplication approach, decimal CORDIC continued to remain mostly unheard of for several more years, so that Hermann Schmid and Anthony Bogacki still suggested it as a novelty as late as 1973^{[13]}^{[10]}^{[36]}^{[37]}^{[38]} and it was found only later that HewlettPackard had implemented it in 1966 already.^{[8]}^{[10]}^{[17]}^{[25]}
Decimal CORDIC became widely used in pocket calculators,^{[10]} most of which operate in binarycoded decimal (BCD) rather than binary. This change in the input and output format did not alter CORDIC's core calculation algorithms. CORDIC is particularly wellsuited for handheld calculators, in which low cost – and thus low chip gate count – is much more important than speed.
CORDIC has been implemented in the Intel 8087,^{[38]}^{[39]}^{[40]}^{[41]}^{[42]} 80287,^{[42]}^{[43]} 80387^{[42]}^{[43]} up to the 80486^{[38]} coprocessor series as well as in the Motorola 68881^{[38]}^{[39]} and 68882 for some kinds of floatingpoint instructions, mainly as a way to reduce the gate counts (and complexity) of the FPU subsystem.
Applications
CORDIC uses simple shiftadd operations for several computing tasks such as the calculation of trigonometric, hyperbolic and logarithmic functions, real and complex multiplications, division, squareroot calculation, solution of linear systems, eigenvalue estimation, singular value decomposition, QR factorization and many others. As a consequence, CORDIC has been used for applications in diverse areas such as signal and image processing, communication systems, robotics and 3D graphics apart from general scientific and technical computation.^{[44]}^{[45]}
Hardware
CORDIC is generally faster than other approaches when a hardware multiplier is not available (e.g., a microcontroller), or when the number of gates required to implement the functions it supports should be minimized (e.g., in an FPGA or ASIC).
On the other hand, when a hardware multiplier is available (e.g., in a DSP microprocessor), tablelookup methods and power series are generally faster than CORDIC. In recent years, the CORDIC algorithm has been used extensively for various biomedical applications, especially in FPGA implementations.
Software
Many older systems with integeronly CPUs have implemented CORDIC to varying extents as part of their IEEE floatingpoint libraries. As most modern generalpurpose CPUs have floatingpoint registers with common operations such as add, subtract, multiply, divide, sine, cosine, square root, log_{10}, natural log, the need to implement CORDIC in them with software is nearly nonexistent. Only microcontroller or special safety and timeconstrained software applications would need to consider using CORDIC.
Modes of operation
Rotation mode
CORDIC can be used to calculate a number of different functions. This explanation shows how to use CORDIC in rotation mode to calculate the sine and cosine of an angle, and assumes the desired angle is given in radians and represented in a fixedpoint format. To determine the sine or cosine for an angle , the y or x coordinate of a point on the unit circle corresponding to the desired angle must be found. Using CORDIC, one would start with the vector :
In the first iteration, this vector is rotated 45° counterclockwise to get the vector . Successive iterations rotate the vector in one or the other direction by sizedecreasing steps, until the desired angle has been achieved. Step size is for .
More formally, every iteration calculates a rotation, which is performed by multiplying the vector with the rotation matrix :
The rotation matrix is given by:
Using the following two trigonometric identities:
the rotation matrix becomes:
The expression for the rotated vector then becomes:
where and are the components of . Restricting the angles so that takes on the values , the multiplication with the tangent can be replaced by a division by a power of two, which is efficiently done in digital computer hardware using a bit shift. The expression then becomes:
where
and can have the values of −1 or 1, and is used to determine the direction of the rotation; if the angle is positive then is +1, otherwise it is −1.
can be ignored in the iterative process and then applied afterward with a scaling factor:
which is calculated in advance and stored in a table, or as a single constant if the number of iterations is fixed. This correction could also be made in advance, by scaling and hence saving a multiplication. Additionally it can be noted that:
 ^{[38]}
to allow further reduction of the algorithm's complexity. Some applications may avoid correcting for altogether, resulting in a processing gain :
 ^{[46]}
After a sufficient number of iterations, the vector's angle will be close to the wanted angle . For most ordinary purposes, 40 iterations (n = 40) is sufficient to obtain the correct result to the 10th decimal place.
The only task left is to determine if the rotation should be clockwise or counterclockwise at each iteration (choosing the value of ). This is done by keeping track of how much the angle was rotated at each iteration and subtracting that from the wanted angle; then in order to get closer to the wanted angle , if is positive, the rotation is clockwise, otherwise it is negative and the rotation is counterclockwise.
The values of must also be precomputed and stored. But for small angles, in fixedpoint representation, reducing table size.
As can be seen in the illustration above, the sine of the angle is the y coordinate of the final vector , while the x coordinate is the cosine value.
Vectoring mode
The rotationmode algorithm described above can rotate any vector (not only a unit vector aligned along the x axis) by an angle between –90° and +90°. Decisions on the direction of the rotation depend on being positive or negative.
The vectoringmode of operation requires a slight modification of the algorithm. It starts with a vector the x coordinate of which is positive and the y coordinate is arbitrary. Successive rotations have the goal of rotating the vector to the x axis (and therefore reducing the y coordinate to zero). At each step, the value of y determines the direction of the rotation. The final value of contains the total angle of rotation. The final value of x will be the magnitude of the original vector scaled by K. So, an obvious use of the vectoring mode is the transformation from rectangular to polar coordinates.
Implementation
Software example
The following is a MATLAB/GNU Octave implementation of CORDIC that does not rely on any transcendental functions except in the precomputation of tables. If the number of iterations n is predetermined, then the second table can be replaced by a single constant. With MATLAB's standard doubleprecision arithmetic and "format long" printout, the results increase in accuracy for n up to about 48.
function v = cordic(beta,n)
% This function computes v = [cos(beta), sin(beta)] (beta in radians)
% using n iterations. Increasing n will increase the precision.
if beta < pi/2  beta > pi/2
if beta < 0
v = cordic(beta + pi, n);
else
v = cordic(beta  pi, n);
end
v = v; % flip the sign for second or third quadrant
return
end
% Initialization of tables of constants used by CORDIC
% need a table of arctangents of negative powers of two, in radians:
% angles = atan(2.^(0:27));
angles = [ ...
0.78539816339745 0.46364760900081 0.24497866312686 0.12435499454676 ...
0.06241880999596 0.03123983343027 0.01562372862048 0.00781234106010 ...
0.00390623013197 0.00195312251648 0.00097656218956 0.00048828121119 ...
0.00024414062015 0.00012207031189 0.00006103515617 0.00003051757812 ...
0.00001525878906 0.00000762939453 0.00000381469727 0.00000190734863 ...
0.00000095367432 0.00000047683716 0.00000023841858 0.00000011920929 ...
0.00000005960464 0.00000002980232 0.00000001490116 0.00000000745058 ];
% and a table of products of reciprocal lengths of vectors [1, 2^2j]:
% Kvalues = cumprod(1./abs(1 + 1j*2.^((0:23))))
Kvalues = [ ...
0.70710678118655 0.63245553203368 0.61357199107790 0.60883391251775 ...
0.60764825625617 0.60735177014130 0.60727764409353 0.60725911229889 ...
0.60725447933256 0.60725332108988 0.60725303152913 0.60725295913894 ...
0.60725294104140 0.60725293651701 0.60725293538591 0.60725293510314 ...
0.60725293503245 0.60725293501477 0.60725293501035 0.60725293500925 ...
0.60725293500897 0.60725293500890 0.60725293500889 0.60725293500888 ];
Kn = Kvalues(min(n, length(Kvalues)));
% Initialize loop variables:
v = [1;0]; % start with 2vector cosine and sine of zero
poweroftwo = 1;
angle = angles(1);
% Iterations
for j = 0:n1;
if beta < 0
sigma = 1;
else
sigma = 1;
end
factor = sigma * poweroftwo;
% Note the matrix multiplication can be done using scaling by powers of two and addition subtraction
R = [1, factor; factor, 1];
v = R * v; % 2by2 matrix multiply
beta = beta  sigma * angle; % update the remaining angle
poweroftwo = poweroftwo / 2;
% update the angle from table, or eventually by just dividing by two
if j+2 > length(angles)
angle = angle / 2;
else
angle = angles(j+2);
end
end
% Adjust length of output vector to be [cos(beta), sin(beta)]:
v = v * Kn;
return
endfunction
The twobytwo matrix multiplication can be carried out by a pair of simple shifts and adds.
x = v[0]  sigma * (v[1] * 2^(j));
y = sigma * (v[0] * 2^(j)) + v[1];
v = [x; y];
In Java the Math class has a scalb(double x,int scale)
method to perform such a shift,^{[47]} C has the ldexp function,^{[48]} and the x86 class of processors have the fscale
floating point operation.^{[49]}
Hardware example
The number of logic gates for the implementation of a CORDIC is roughly comparable to the number required for a multiplier as both require combinations of shifts and additions. The choice for a multiplierbased or CORDICbased implementation will depend on the context. The multiplication of two complex numbers represented by their real and imaginary components (rectangular coordinates), for example, requires 4 multiplications, but could be realized by a single CORDIC operating on complex numbers represented by their polar coordinates, especially if the magnitude of the numbers is not relevant (multiplying a complex vector with a vector on the unit circle actually amounts to a rotation). CORDICs are often used in circuits for telecommunications such as digital down converters.
Related algorithms
CORDIC is part of the class of "shiftandadd" algorithms, as are the logarithm and exponential algorithms derived from Henry Briggs' work. Another shiftandadd algorithm which can be used for computing many elementary functions is the BKM algorithm, which is a generalization of the logarithm and exponential algorithms to the complex plane. For instance, BKM can be used to compute the sine and cosine of a real angle (in radians) by computing the exponential of , which is . The BKM algorithm is slightly more complex than CORDIC, but has the advantage that it does not need a scaling factor (K).
See also
References
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So far CORDIC has been known to be implemented only in binary form. But, as will be demonstrated here, the algorithm can be easily modified for a decimal system.* […] *In the meantime it has been learned that Hewlett Packard and other calculator manufacturers employ the decimal CORDIC techniques in their scientific calculators.
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John E. Meggitt B.A., 1953; Ph.D., 1958, Cambridge University. Awarded the First Smith Prize at Cambridge in 1955 and elected a Research Fellowship at Emmanuel College. […] Joined IBM British Laboratory at Hursley, Winchester in 1958. Interests include errorcorrecting codes and small microprogrammed computers.
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I even flew down to Southern California to talk with Jack Volder who had implemented the transcendental functions in the Athena machine and talked to him for about an hour. He referred me to the original papers by Meggitt where he'd gotten the pseudo division, pseudo multiplication generalized functions. […] I did quite a bit of literary research leading to some very interesting discoveries. […] I found a treatise from 1624 by Henry Briggs discussing the calculation of common logarithms, interestingly used the same pseudodivision/pseudomultiplication method that MacMillan and Volder used in Athena. […] We had purchased a LOCI2 from Wang Labs and recognized that Wang Labs LOCI II used the same algorithm to do square root as well as log and exponential. After the introduction of the 9100 our legal department got a letter from Wang saying that we had infringed on their patent. And I just sent a note back with the Briggs reference in Latin and it said, "It looks like prior art to me." We never heard another word.
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During the development of the desktop HP 9100 calculator I was responsible for developing the algorithms to fit the architecture suggested by Tom Osborne. Although the suggested methodology for the algorithms came from Malcolm McMillan I did considerable amount of reading to understand the core calculations […] Although Wang Laboratories had used similar methods of calculation, my study found prior art dated 1624 that read on their patents. […] This research enabled the adaption of the transcendental functions through the use of the algorithms to match the needs of the customer within the constraints of the hardware. This proved invaluable during the development of the HP35, […] Power series, polynomial expansions, continued fractions, and Chebyshev polynomials were all considered for the transcendental functions. All were too slow because of the number of multiplications and divisions required. The generalized algorithm that best suited the requirements of speed and programming efficiency for the HP35 was an iterative pseudodivision and pseudomultiplication method first described in 1624 by Henry Briggs in 'Arithmetica Logarithmica' and later by Volder and Meggitt. This is the same type of algorithm that was used in previous HP desktop calculators. […] The complexity of the algorithms made multilevel programming a necessity. This meant the calculator had to have subroutine capability, […] To generate a transcendental function such as ArcHyperbolicTan required several levels of subroutines. […] Chris Clare later documented this as Algorithmic State Machine (ASM) methodology. Even the simple Sine or Cosine used the Tangent routine, and then calculated the Sine from trigonometric identities. These arduous manipulations were necessary to minimize the number of unique programs and program steps […] The arithmetic instruction set was designed specifically for a decimal transcendentalfunction calculator. The basic arithmetic operations are performed by a 10's complement addersubtractor which has data paths to three of the registers that are used as working storage.
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Further reading
 Parini, Joseph A. (19660905). "DIVIC Gives Answer to Complex Navigation Questions". Electronics: 105–111. ISSN 00135070. (NB. DIVIC stands for DIgital Variable Increments Computer. Some sources erroneously refer to this as by J. M. Parini.)
 Anderson, Stanley F.; Earle, John G.; Goldschmidt, Robert Elliott; Powers, Don M. (19651101). "The IBM System/360 Model 91: FloatingPoint Execution Unit" (PDF). IBM Journal of Research and Development. Riverton, New Jersey, USA (published January 1967). 11 (1): 34–53. doi:10.1147/rd.111.0034. Retrieved 20160102.
 Liccardo, Michael A. (September 1968). An Interconnect Processor with Emphasis on CORDIC Mode Operation (M.Sc. thesis). Berkeley, CA, USA: University of California, Berkeley, Department of Electrical Engineering.
 US patent 3576983A, Cochran, David S., "Digital calculator system for computing square roots", published 19710504, issued 19710504, assigned to HewlettPackard Co. ([14])
 Chen, Tien Chi (July 1972). "Automatic Computation of Exponentials, Logarithms, Ratios, and Square Roots" (PDF). IBM Journal of Research and Development. 16 (4): 380–388. doi:10.1147/rd.164.0380. ISSN 00188646. Retrieved 20160102.
 Egbert, William E. (May 1977). "Personal Calculator Algorithms I: Square Roots" (PDF). HewlettPackard Journal. Palo Alto, California, USA: HewlettPackard. 28 (9): 22–24. Retrieved 20160102. ([15])
 Egbert, William E. (June 1977). "Personal Calculator Algorithms II: Trigonometric Functions" (PDF). HewlettPackard Journal. Palo Alto, California, USA: HewlettPackard. 28 (10): 17–20. Retrieved 20160102. ([16])
 Egbert, William E. (November 1977). "Personal Calculator Algorithms III: Inverse Trigonometric Functions" (PDF). HewlettPackard Journal. Palo Alto, California, USA: HewlettPackard. 29 (3): 22–23. Retrieved 20160102. ([17])
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External links
 Wang, Shaoyun (July 2011), CORDIC Bibliography Site
 CORDIC Bibliography Site
 BASIC Stamp, CORDIC math implementation
 CORDIC implementation in verilog
 CORDIC Vectoring with Arbitrary Target Value
 PicBasic Pro, Pic18 CORDIC math implementation
 Python CORDIC implementation
 Simple C code for fixedpoint CORDIC
 Tutorial and MATLAB Implementation – Using CORDIC to Estimate Phase of a Complex Number
 Descriptions of hardware CORDICs in Arx with testbenches in C++ and VHDL
 An Introduction to the CORDIC algorithm
 Implementation of the CORDIC Algorithm in a Digital DownConverter
 50th Anniversary of the CORDIC Algorithm
 Implementation of the CORDIC Algorithm: fixed point C code for trigonometric and hyperbolic functions, C code for test and performance verification