LOG

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BASIC keyword
Keyword: LOG
Abbreviation:
Type: Function
Token code: 188/$BC
Handling routine
in BASIC ROM:
47594–47655
$B9EA–BA27
List of all BASIC keywords


Remark: This article describes the BASIC-Command LOG in BASIC V2 at the Commodore 64.

Typ: Numeric Function 
General Programming-Syntax: LOG(<Numeric>)

The mathematical function LOG is a natural logarithm with the basis e(E). If the term is 0 or negative the BASIC error ?ILLEGAL QUANTITY ERROR IN line occurs. When the number is over 1.7014118342E+38 the BASIC-error ?OVERFLOW ERROR IN line appears. When the numeric argument is absent, a ?SYNTAX ERROR IN line appears.

Examples[edit]

PRINT LOG(1) Result: 0
PRINT LOG(10) Result: 2.30258509
PRINT LOG(100)/LOG(10) Result: 2
                       because you can divide by LOG(10) and get a result with the base "10". 

Implementation[edit]

Let the parameter be X. The LOG function first separates the exponent of X from the significand. That is, it calculates two numbers, N and XF, such that:

X = 2N * XF, N is an integer and 0.5 <= XF < 1.0

C programmers will recognize this separation as the frexp() function.

The natural logarithm is tricky to calculate accurately. The Taylor series converges too slowly to be useful. The BASIC implementation uses a faster approach. Given XF calculated above, it calculates:

T = 1.0 - sqrt(2.0) / (XF + sqrt(0.5))

It then applies the polynomial approximation:

0.43425594189 * T7 + 0.57658454124 * T5 + 0.96180075919 * T3 + 2.8853900731 * T - 0.5

The result of this approximation is log2(XF).

The routine then adds N to that approximation, resulting in log2(X). Finally, it multiplies by LOG(2), giving the final result of LOG(X).

The derivation of the polynomial is not publicly known.

Accuracy[edit]

Accuracy of LOG function

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

At least for the plotted domain of (0,+5], LOG mostly matches the true value to 29 bits. Accuracy degrades around X equal to 1, where LOG(X) is close to 0.

LOG 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 24 bits.


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