ATN

BASIC keyword
Keyword:	ATN
Abbreviation:	A Shift+T
Type:	Function
Token code:	192/$C1
Handling routine in BASIC ROM:	58126–58234 $E30E–E37A
List of all BASIC keywords

Remark: This article describes the BASIC command ATN in BASIC V2 at the Commodore 64.

Type: Numeric Function General Programming-Syntax: ATN(<numeric-expression>)

The numerical function ATN is a mathematical function that returns the arc tangent of a numeric value (the inverse function of TAN). The resulting value is the angle in radians of the given tangent. To convert radians into degrees use the formula (rad*180/π) where rad is the angle in radians.

If the term given in the parentheses is non-numerical, the BASIC error ?TYPE MISMATCH ERROR occurrs. If the argument expression is not in the range of a floating point value the execution yields the BASIC error ?OVERFLOW ERROR. Omitting the numeric argument or to many arguments results in a ?SYNTAX ERROR.

Accuracy

Accuracy of ATN function

The graph to the right was obtained by comparing results from the BASIC ATN function to values computed on a modern x86-type PC.

At least for the plotted domain of [-5,+5], ATN matches the true value to 30 bits. No part of the domain proves to be problematic.

ATN calls POLY1, which calls MLTPLY ($BA59), and consequently the multiply bug affects its results. A small number of input values give results that match to only 25 bits.

Implementation details

Let the parameter be X.

The inverse tangent is an odd function; that is, ATN(-X) = -ATN(X). The BASIC routine uses this property to simplify its calculations a bit. It saves the sign, and then computes AX = |X|.

If AX > 1, ATN uses the identity ATN(X) = π/2 - ATN(1/X) to limit AX to the interval [0, 1].

It uses this polynomial approximation to calculate ATN(AX) for 0 <= AX <= 1:

-6.8479391189*10^-4 * T²³ + 4.8509421558*10^-3 * T²¹ - 1.6111701843*10^-2 * T¹⁹ + 3.4209638048*10^-2 * T¹⁷ - 5.4279132761*10^-2 * T¹⁵ + 7.2457196540*10^-2 * T¹³

- 8.9802395378*10^-2 * T¹¹ + 0.11093241343 * T⁹ - 0.14283980767 * T⁷ + 0.19999912049 * T⁵ - 0.33333331568 * T³ + T

and modifies the result according to the identities described above.

The derivation of the polynomial is not publicly known.

Examples

PRINT ATN(1)

Output: 0.785398163

PRINT ATN(1)*4

Output: 3.1415266
The result is equivalent to π, but differs slightly from π caused by a wrong pressicion in the value of this constant

BASIC V2.0 (second release) Commands

ABS | AND | ASC | ATN | CHR$ | CLOSE | CLR | CMD | CONT | COS | DATA | DEF | DIM | END | EXP | FN | FOR | FRE | GET | GET# | GOSUB | GOTO | IF | INPUT | INPUT# | INT | LEFT$ | LEN | LET | LIST | LOAD | LOG | MID$ | NEW | NEXT | NOT | ON | OPEN | OR | PEEK | π | POKE | POS | PRINT | PRINT# | READ | REM | RESTORE | RETURN | RIGHT$ | RND | RUN | SAVE | SGN | SIN | SPC | SQR | STATUS/ST | STEP | STOP | STR$ | SYS | TAB | TAN | THEN | TIME/TI | TIME$/TI$ | TO | USR | VAL | VERIFY | WAIT