How Can We Implement a Power Function for Both Integer and Non-Integer Exponents?

Implementing a Power Function with Non-Integer Exponents

The task of calculating real-valued exponents poses a challenge beyond the capabilities of standard library functions like pow(). This article delves into the intricate process of creating a custom function that handles both integer and fractional powers.

Negative Exponents

Addressing negative exponents is straightforward. Negative exponents simply represent the reciprocal of positive exponents. For instance, 2^-21 is equivalent to 1/2^21.

Fractional Exponents

Fractional exponents introduce a layer of complexity. A fractional exponent is essentially a root. Exploiting this relationship, we can leverage the decomposition of the exponent into its integer and rational parts.

Implementation Details

1. Extract Integer Part: Isolate the integer portion of the exponent, using integer division (e.g., 4.5 = 4 integer part).
2. Calculate Integer Power: Employ a loop to compute the integer power (e.g., 2^4 = 16).
3. Extract Fractional Part: Determine the rational portion of the exponent (e.g., 4.5 = 0.5 fractional part).
4. Calculate Fractional Power: Resort to an iterative approximation algorithm, such as bisection or Newton's method, to calculate the root (e.g., sqrt(2) ≈ 1.41421).
5. Combine Results: Multiply the integer power and the root to obtain the final result (e.g., 2^4.5 = 16 * 1.41421 ≈ 22.62741).
6. Apply Inverse (Optional): If the original exponent was negative, invert the final result to obtain the correct value (e.g., 2^-3.5 ≈ 0.03475).

Example:

Consider the calculation of 2^-3.5. Decomposing the exponent, we have -3 integer part and -0.5 fractional part. We compute 2^-3 = 1/8, calculate sqrt(2) ≈ 1.41421, and multiply to obtain -3.5 exponent ≈ 1/8 * 1.41421 ≈ 0.03475, representing the inverse of the positive exponent power.

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