Casio fx-CG 50 CORDIC Simulation: Approximating Sine and Cosine of Angles
Introduction - How Computers Calculate Mathematical Functions
First developed by Jack E. Volder, the Coordinate Rotation Digital Computer, better known as CORDIC, is an algorithm used to calculate many mathematical functions, including trigonometric functions, logarithms, exponentials, and hyperbolic functions. CORDIC is a fundamental algorithm, which variants of CORDIC are used in computers and calculators.
Today's focus will be a calculating sines and cosines of angles. The steps will be detailed in the next sections.
CORDIC1: Series of Arctangents
Let Θ be the angle. The first step is to build Θ as additions and subtractions of terms of arctan(1 ÷ (2^n)), starting at n = 0 and stopping at the required accuracy. Let A be the approximation.
Examples:
Θ = 45° requires only one term:
A = 45° = arctan 1
Θ ≈ 71.56505118°
A = 71.56505118° = arctan 1 + arctan 1/2
Θ ≈ 32.47119229°
A = 32.47119229° = arctan 1 - arctan 1/2 + arctan 1/4
A series to recreate Θ = 30° takes 32 terms of ±arctan (1 ÷ 2^n) from n = 0 to n = 31. (8 decimal places) This is known as coordinate rotation.
The program CORDIC1 determines the number of terms need to created to obtain Θ. Accuracy in this code is set to 5 decimal places, but can be adjusted (see the line in blue).
Casio fx-CG 50 Program Code: CORDIC1
Deg
"ANGLE"?->Θ
0->I
0->A
Lbl 0
tan^-1 (2^(-I))->T
If A<Θ
Then
A+T->A
Else
A-T->A
IfEnd
I+1->I
Abs (A-Θ)>1×10^-5=>Goto 0
ClrText
Blue Locate 1,3,"Θ: "
Blue Locate 5,3,Θ
Red Locate 1,4,"A: "
Red Locate 5,4,A
Green Locate 1,7,I
Notes:
-> is the store arrow →
=> is the jump command ⇒
For reference:
arctan 1 = 45°
arctan 1/2 ≈ 26.56505118°
arctan 1/4 ≈ 14.03624347°
arctan 1/8 ≈ 7.125016349°
arctan 1/16 ≈ 3.576334375°
arctan 1/32 ≈ 1.789910608°
CORDIC2: Calculating Sine and Cosine
Imagine the coordinate (cos Θ, sin Θ) on a unit circle. A unit circle is a circle with radius of length 1 and center located at the origin (0,0).
Set the initial angle at 0°. Then x = cos 0° = 1 and y = sin 0° = 0. The initial vector is set to be [ [ cos 0° ] [ sin 0° ] ] = [ [ 1 ] [ 0 ] ].
Approximate Θ in terms of arctangent (1 ÷ (2^n)), starting at n = 0.
Direction of Rotation
Set σ as the direction of rotation.
A rotation is positive if we add arctan(1 ÷ (2^n)) to A. For a positive rotation, set σ_i = +1.
A rotation is negative if we subtract arctan(1 ÷ (2^n)) to A. For a negative rotation, set σ_i = -1.
For example:
Θ ≈ 32.47119229°
A = 32.47119229° = arctan 1 - arctan 1/2 + arctan 1/4
Then: σ_0 = 1 (positive rotation), σ_1 = -1 (negative rotation), σ_2 = 1 (positive rotation).
Multiplying Factor
The number of iterations is also used to determine the required multiplication factor:
n = Π( 1 ÷ √(1 + 2^(-2 × I)), I = 0 to I = terms needed - 1)
For the example:
Θ ≈ 32.47119229°
Needed 3 iterations, i_0 to i_2.
Then:
n = 1 ÷ √(1 + 2^(-2 × 0)) × 1 ÷ √(1 + 2^(-2 × 1)) × 1 ÷ √(1 + 2^(-2 × 2))
= 4 × √170 ÷ 85
≈ 0.6135719911
Calculating the Next Iteration
The next iteration for x and y are:
x_i+1 = x_i - 2^(-i) × σ_i × y_i
y_i+1 = 2^(-i) × σ_i × x_i + y_i
When A is sufficiently near or equal to Θ, the cosine and sine are approximated as:
cos(A) ≈ n × x_final
sin(A) ≈ n × y_final
For the example:
Θ ≈ 32.47119229°
The approximated angle, in this case, happens to be exact angle, A = Θ. And,
cos(A) ≈ 0.8436614877
sin(A) ≈ 0.5368754922
For more details, please check out resources in the Sources section. I particularly like Oxford's A Very Short Introduction series.
Casio fx-CG 50 Program Code: CORDIC2
This algorithm is adopted from the Python example (see Wikipedia article).
Deg
"ANGLE"?->Θ
0->I
0->A
1->X
0->Y
1->N
Lbl 0
tan^-1 (2^(-I))->T
If A<Θ
Then
A+T->A
1->S
Else
A-T->A
-1->S
IfEnd
N÷√(1+2^(-2*I))->N
X-S*2^(-I)*Y->P
Y+X*S*2^(-I)->Q
P->X
Q->Y
I+1->I
Abs (A-Θ)>1×10^-5=>Goto 0
X*N->X
Y*N->Y
ClrText
Blue Locate 1,3,"Θ: "
Blue Locate 5,3,Θ
Red Locate 1,4,"A: "
Red Locate 5,4,A
Black Locate 1,5,"cos :"
Black Locate 7,5,X
Black Locate 1,6,"sin :"
Black Locate 7,6,Y
Green Locate 1,7,I
CORDIC3: Doing it without a Arctangent function
In reality, most of the time the arctangent function also has to be approximated. There are many ways to approximate, which varying accuracy. I used the fx-CG50's statistics mode to come up with a regression equation with the following lists:
x_list = sequence of 1÷(2^i) from i = 0 to i = 39
y_list = sequence of arctan(1÷(2^i)) from i = 0 to i = 39
Of the regression models the fx-CG50 offers, the best regression model is quartic regression (4th-order polynomial):
y ≈ 9.43597784 × x^4 - 22.116232 × x^3 + 0.40116676 × x^2 + 57.2790727 × x + 1.4558 × 10^-5
Remember that I am working with degree angle measurement.
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Casio fx-CG 50 Program Code: CORDIC3
Deg
"ANGLE"?->Θ
0->I
0->A
1->X
0->Y
1->N
Lbl 0
2^(-I)->K
9.43597784917955K^(4)-22.1162323028173K^(3)+
0.401166766193516K^2+57.2790727477367K+
1.45584715586876×10^-5->T
If A<Θ
Then
A+T->A
1->S
Else
A-T->A
-1->S
IfEnd
N÷√(1+2^(-2*I))->N
X-S*2^(-I)*Y->P
Y+X*S*2^(-I)->Q
P->X
Q->Y
I+1->I
Abs (A-Θ)>1×10^-5=>Goto 0
X*N->X
Y*N->Y
ClrText
Blue Locate 1,3,"Θ: "
Blue Locate 5,3,Θ
Red Locate 1,4,"A: "
Red Locate 5,4,A
Black Locate 1,5,"cos :"
Black Locate 7,5,X
Black Locate 1,6,"sin :"
Black Locate 7,6,Y
Green Locate 1,7,I
For the example:
Θ ≈ 32.47119229°
Approximating Θ within 8 decimal places (10^-5) yields these results:
A: 32.47118598
cos A: 0.8436683456
sin A: 0.5368647154
24 terms used
Sources
"CORDIC" Wikipedia. Last Edited January 11, 2024. Accessed January 12, 2024. https://en.wikipedia.org/wiki/CORDIC
Brummelen, Glen Van. Trigonometry: A Very Short Introduction Oxford University Press: Oxford, United Kingdom. 2020. ISBN 978-0-19-881431-3
Eddie
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