Implementing expm1 function for double precision based on exp function
algorithm:
- Reduced x = log2(e) * (hi + mid1 + mid2) + lo, where:
* hi is an integer
* mid1 * 2^-6 is an integer
* mid2 * 2^-12 is an integer
* |lo| < 2^-13 + 2^-30
- Then exp(x) - 1 = 2^hi * 2^mid1 * 2^mid2 * exp(lo) - 1 ~ 2^hi *
(2^mid1 * 2^mid2 * (1 + lo * P(lo)) - 2^(-hi) )
- We evaluate fast pass with P(lo) is a degree-3 Taylor polynomial of
(e^lo - 1) / lo in double precision
- If the Ziv accuracy test fails, we use degree-6 Taylor polynomial of
(e^lo - 1) / lo in double double precision
- If the Ziv accuracy test still fails, we re-evaluate everything in
128-bit precision.
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