Power

The ** (power) operator raises a number to a power, which may 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 under 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 (see Note 1 for the reasons), the power is calculated by the process of left-to-right binary reduction. 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.