Arithmetic operation rules: additional operators

Power

The ** (power) operator raises a number to a power, which can be positive, negative, or 0. The power must be a whole number. (The second term in the operation must be a whole number and is rounded to DIGITS digits, if necessary, as described in Numbers used directly by REXX.) If negative, the absolute value of the power is used, and then the result is inverted (divided into 1). For calculating the power, the number is effectively multiplied by itself for the number of times expressed by the power, and finally trailing zeros are removed (as though the result were divided by 1).

In practice, the power is calculated by the process of left-to-right binary reduction (see 1 for the reason). For a**n: n is converted to binary, and a temporary accumulator is set to 1. If n = 0 the initial calculation is complete. (Thus, a**0 = 1 for all a, including 0**0.) Otherwise each bit (starting at the first nonzero bit) is inspected from left to right. If the current bit is 1, the accumulator is multiplied by a. If all bits have now been inspected, the initial calculation is complete; otherwise the accumulator is squared and the next bit is inspected for multiplication. When the initial calculation is complete, the temporary result is divided into 1 if the power was negative.

The multiplications and division are done under the arithmetic operation rules, using a precision of DIGITS + L + 1 digits. L is the length in digits of the integer part of the whole number n (that is, excluding any decimal part, as though the built-in function TRUNC(n) had been used). Finally, the result is rounded to NUMERIC DIGITS digits, if necessary, and insignificant trailing zeros are removed.

Integer division

The % (integer divide) operator divides two numbers and returns the integer part of the result. The result returned is defined to be that which would result from repeatedly subtracting the divisor from the dividend while the dividend is larger than the divisor. During this subtraction, the absolute values of both the dividend and the divisor are used: the sign of the final result is the same as that which would result from regular division.

The result returned has no fractional part (that is, no decimal point or zeros following it). If the result cannot be expressed as a whole number, the operation is in error and will fail; that is, the result must not have more digits than the current setting of NUMERIC DIGITS. For example, 10000000000%3 requires 10 digits for the result (3333333333) and would, therefore, fail if NUMERIC DIGITS 9 were in effect. Note that this operator may not give the same result as truncating regular division (which could be affected by rounding).

Reminder

The // (remainder) operator returns the remainder from integer division and is defined as being the residue of the dividend after the operation of calculating integer division as previously described. The sign of the remainder, if nonzero, is the same as that of the original dividend.

This operation fails under the same conditions as integer division (that is, if integer division on the same two terms would fail, the remainder cannot be calculated).

Examples of additional operators

/* Again with:  Numeric digits 5 */
2**3       ->     8
2**-3      ->     0.125
1.7**8     ->    69.758
2%3        ->     0
2.1//3     ->     2.1
10%3       ->     3
10//3      ->     1
-10//3     ->    -1
10.2//1    ->     0.2
10//0.3    ->     0.1
3.6//1.3   ->     1.0