EXP

BASIC keyword
Keyword:	EXP
Abbreviation:	E Shift+X
Type:	Function
Token code:	189/$BD
Handling routine in BASIC ROM:	49133–57410 $BFED–E042
List of all BASIC keywords

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

Type: Numeric Function 
General Programming-Syntax: EXP(<Number>)

EXP is a mathematical function that evaluates the inverse natural LOG of the argument. That is, the argument X is applied as an exponent to e(2.71828183). So EXP(X) = e^X.

Arguments in excess of 89.0296919 result in the BASIC error ?OVERFLOW ERROR IN line. Arguments less than -89.0296919 evaluate to 0.

When the argument doesn't evaluate to a number, the BASIC error ?TYPE MISMATCH ERROR IN line occurs. When the numeric argument is absent, it displays a ?SYNTAX ERROR IN line.

Examples

PRINT EXP(0) Screen: 1
PRINT EXP(-1) Screen: 0.367879441
PRINT EXP(1) Screen: 2.71828183

Implementation

The EXP function uses the identity:

e^x = 2^{x * log₂(e)}

Let the parameter be X. EXP first calculates T = X * log₂(e). It then separates T into an integer N and a fraction TF, such that N + TF = T and 0 <= TF < 1.

It then calculates 2^TF by the polynomial approximation:

2.1498763701*10^-5 * TF⁷ + 1.4352314037*10^-4 * TF⁶ + 1.3422634825*10^-3 * TF⁵ + 9.6140170135*10^-3 * TF⁴ + 5.5505126860*10^-2 * TF³ + 0.24022638460 * TF² + 0.69314718618 * TF + 1.0

It then adds the integer N, calculated above, to the exponent of the result, effectively multiplying it by 2^N. The result is 2^T, which is equal to e^X.

The polynomial resembles the seventh-degree Maclaurin polynomial expansion Special case of 2^x:

1 + x log(2) + 1/2 log²(2) x² + 1/6 log³(2) x³ + 1/24 log⁴(2) x⁴ + 1/120 log⁵(2) x⁵ + 1/720 log⁶(2) x⁶ + 1/5040 log⁷(2) x⁷

but is most likely derived via the Remez algorithm. See POLY1#Notes_on_derivation for more details.

Accuracy

Accuracy of EXP function

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

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

EXP calls POLY2, which calls MLTPLY ($BA59), and consequently the multiply bug affects its results. A small number of input values give results that match to only 26 bits.

References

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