Implement double precision exp2 function correctly rounded for all
rounding modes. Using the same algorithm as double precision exp function in
https://reviews.llvm.org/D158551.
Reviewed By: zimmermann6
Differential Revision: https://reviews.llvm.org/D158812
Implement double precision exp function correctly rounded for all
rounding modes. Using 4 stages:
- Range reduction: reduce to `exp(x) = 2^hi * 2^mid1 * 2^mid2 * exp(lo)`.
- Use 64 + 64 LUT for 2^mid1 and 2^mid2, and use cubic Taylor polynomial to
approximate `(exp(lo) - 1) / lo` in double precision. Relative error in this
step is bounded by 1.5 * 2^-63.
- If the rounding test fails, use degree-6 Taylor polynomial to approximate
`exp(lo)` in double-double precision. Relative error in this step is bounded by
2^-99.
- If the rounding test still fails, use degree-7 Taylor polynomial to compute
`exp(lo)` in ~128-bit precision.
Reviewed By: zimmermann6
Differential Revision: https://reviews.llvm.org/D158551
Simplify the range reduction steps by choosing the reduction constants
carefully so that the reduced arguments v = r*m_x - 1 and v^2 are exact in double
precision, even without FMA instructions, and -2^-8 <= v < 2^-7. This allows the
polynomial evaluations to be parallelized more efficiently.
Reviewed By: zimmermann6
Differential Revision: https://reviews.llvm.org/D147676
Implement double precision log10 function correctly rounded for all
rounding modes. This implementation currently needs FMA instructions for
correctness.
Use 2 passes:
Fast pass:
- 1 step range reduction with a lookup table of `2^7 = 128` elements to reduce the ranges to `[-2^-7, 2^-7]`.
- Use a degree-7 minimax polynomial generated by Sollya, evaluated using a mixed of double-double and double precisions.
- Apply Ziv's test for accuracy.
Accurate pass:
- Apply 5 more range reduction steps to reduce the ranges further to [-2^-27, 2^-27].
- Use a degree-4 minimax polynomial generated by Sollya, evaluated using 192-bit precisions.
- By the result of Lefevre (add quote), this is more than enough for correct rounding to all rounding modes.
In progress: Adding detail documentations about the algorithm.
Depend on: https://reviews.llvm.org/D136799
Reviewed By: zimmermann6
Differential Revision: https://reviews.llvm.org/D139846
