CORDIC: Difference between revisions
m (→{{header|Julia}}: cleanup) |
|||
Line 59:
# and a table of products of reciprocal lengths of vectors [1, 2^-2j]:
# Kvalues = cumprod(1./sqrt(1 + 1j*2.^(-(0:23))))
Kvalues =
0.70710678118655, 0.63245553203368, 0.61357199107790, 0.60883391251775,
0.60764825625617, 0.60735177014130, 0.60727764409353, 0.60725911229889,
Line 65:
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(iteration, length(Kvalues))]
|
Revision as of 03:50, 9 July 2023
- Introduction
CORDIC is the name of an algorithm for calculating trigonometric, logarithmic and hyperbolic functions, named after its first application on an airborne computer (COordinate Rotation DIgital Computer) in 1959. Unlike a Taylor expansion or polynomial approximation, it converges rapidly on machines with low computing and memory capacities: to calculate a tangent with 10 significant digits, it requires only 6 floating-point constants, and only additions, subtractions and digit shifts in its iterative part.
It is valid for angle values between 0 and π/2 only, but whatever the value of an angle, the calculation of its tangent can always be reduced to that of an angle between 0 and π/2, using trigonometric identities. Similarly, once you know the tangent, you can easily calculate the sine or cosine.
- Pseudo code
constant θ[n] = arctan 10^(-n) // or simply 10^(-n) depending on floating point precision constant epsilon = 10^-12 function tan(alpha) // 0 < alpha <= π/2 x = 1 ; y = 0 ; k = 0 while precision < alpha while alpha < θ[k] k++ end loop alpha -= θ[k] x2 = x - 10^(-k)*y y2 = y + 10^(-k)*x x = x2 ; y = y2 end loop return (y/x) end function
- Task
- Implement the CORDIC algorithm, using only the 4 arithmetic operations and right shifts in the main loop if possible.
- Use your implementation to calculate the cosine of the following angles, expressed in radians: -9, 0, 1.5 and 6
Julia
""" Modified from MATLAB example code at en.wikipedia.org/wiki/CORDIC """
using Printf
"""
Compute v = [cos(alpha), sin(alpha)] (alpha in radians).
Increasing the iteration value will increase the precision.
"""
function cordic(alpha, iteration = 24)
# Fix for the Wikipedia's MATLAB code bug in cosine when |θ| > 2π
newsgn = isodd(Int(floor(alpha / 2π))) ? 1 : -1
alpha < -π/2 && return newsgn * cordic(alpha + π, iteration)
alpha > π/2 && return newsgn * cordic(alpha - π, iteration)
# 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./sqrt(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(iteration, length(Kvalues))]
# Initialize loop variables:
v = [1, 0] # start with 2-vector cosine and sine of zero
poweroftwo = 1
angle = angles[1]
# Iterations
for j = 0:iteration-1
if alpha < 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 # 2-by-2 matrix multiply
alpha -= sigma * angle # update the remaining angle
poweroftwo /= 2
# update the angle from table, or eventually by just dividing by two
if j + 2 > length(angles)
angle /= 2
else
angle = angles[j + 2]
end
end
# Adjust length of output vector to be [cos(alpha), sin(alpha)]:
v .*= Kn
return v
end
function test_cordic()
println(" x sin(x) diff. sine cos(x) diff. cosine ")
for θ in -90:15:90
cosθ, sinθ = cordic(deg2rad(θ))
@printf("%+05.1f° %+.8f (%+.8f) %+.8f (%+.8f)\n",
θ, sinθ, sinθ - sind(θ), cosθ, cosθ - cosd(θ))
end
println("\nx(radians) sin(x) diff. sine cos(x) diff. cosine ")
for θr in [-9, 0, 1.5, 6]
cosθ, sinθ = cordic(θr)
@printf("%+3.1f %+.8f (%+.8f) %+.8f (%+.8f)\n",
θr, sinθ, sinθ - sin(θr), cosθ, cosθ - cos(θr))
end
end
test_cordic()
- Output:
x sin(x) diff. sine cos(x) diff. cosine -90.0° -1.00000000 (+0.00000000) -0.00000007 (-0.00000007) -75.0° -0.96592585 (-0.00000003) +0.25881895 (-0.00000009) -60.0° -0.86602545 (-0.00000005) +0.49999992 (-0.00000008) -45.0° -0.70710684 (-0.00000006) +0.70710672 (-0.00000006) -30.0° -0.49999992 (+0.00000008) +0.86602545 (+0.00000005) -15.0° -0.25881895 (+0.00000009) +0.96592585 (+0.00000003) +00.0° -0.00000007 (-0.00000007) +1.00000000 (-0.00000000) +15.0° +0.25881895 (-0.00000009) +0.96592585 (+0.00000003) +30.0° +0.49999992 (-0.00000008) +0.86602545 (+0.00000005) +45.0° +0.70710684 (+0.00000006) +0.70710672 (-0.00000006) +60.0° +0.86602545 (+0.00000005) +0.49999992 (-0.00000008) +75.0° +0.96592585 (+0.00000003) +0.25881895 (-0.00000009) +90.0° +1.00000000 (-0.00000000) -0.00000007 (-0.00000007) x(radians) sin(x) diff. sine cos(x) diff. cosine -9.0 -0.41211842 (+0.00000006) -0.91113029 (-0.00000003) +0.0 -0.00000007 (-0.00000007) +1.00000000 (-0.00000000) +1.5 +0.99749499 (+0.00000000) +0.07073719 (-0.00000002) +6.0 -0.27941552 (-0.00000002) +0.96017028 (-0.00000001)
RPL
≪ RAD { } 1 DO SWAP OVER ATAN + SWAP 10 / UNTIL DUP DUP TAN == END @ memorize constants until precision limit is reached DROP 'THN' STO THN SIZE →STR " constants in memory." * 1E-12 'EPSILON' STO ≫ ≫ 'INIT' STO ≪ IF DUP THEN 1 SWAP START 10 / NEXT @ shift one digit right ELSE DROP END ≫ 'SR10' STO ≪ IF THN SIZE OVER 1 + < THEN 1 SWAP SR10 @ get arctan(θ[k]) from memory ELSE THN SWAP 1 + GET END @ arctan(θ[k]) ≈ θ[k] ≫ '→THK' STO ≪ → alpha ≪ 0 1 0 @ initialize y, x and k WHILE alpha EPSILON > REPEAT WHILE DUP →THK alpha > REPEAT 1 + END 'alpha' OVER →THK STO- DUP2 SR10 4 PICK + 4 ROLLD ROT OVER SR10 ROT SWAP - SWAP END DROP / ≫ ≫ '→TAN' STO ≪ 1 CF '2*π' →NUM MOD IF DUP π / 2 * →NUM IP THEN { ≪ π SWAP 1 SF ≫ ≪ π 1 SF ≫ ≪ '2*π' SWAP ≫ } @ corrections for angles > π/2 LASTARG GET EVAL →NUM - @ apply correction according to quadrant END →TAN SQ 1 + √ INV IF 1 FS? THEN NEG END ≫ '→COS' STO ≪ INIT { -9 0 1.5 6 } { } 1 3 PICK SIZE FOR j OVER j GET →COS + NEXT SWAP DROP ≫ 'TASK' STO
- Output:
2: "6 constants in memory." 1: { -.91113026188 1 7.07372016661E-2 .960170286655 }