Ln(x) using repeated square root extraction

[Paul Dale]

I've got a feeling I tried something along these lines for the LN function of the 34S. The LN function has always been very slow on the 34S and I've tried several implementations to try to get both performance and accuracy. The current implementation is much faster than the original but still far slower than I'd like. I ended up with:

Code:

/* Natural logarithm.
 *
 * Take advantage of the fact that we store our numbers in the form: m * 10^e
 * so log(m * 10^e) = log(m) + e * log(10)
 * do this so that m is always in the range 0.1 <= m < 2.  However if the number
 * is already in the range 0.5 .. 1.5, this step is skipped.
 *
 * Then use the fact that ln(x^2) = 2 * ln(x) to range reduce the mantissa
 * into 1/sqrt(2) <= m < 2.
 *
 * Finally, apply the series expansion:
 *   ln(x) = 2(a+a^3/3+a^5/5+...) where a=(x-1)/(x+1)
 * which converges quickly for an argument near unity.
 */

- Pauli