1. Speed up exp(x) by reducing the polynomial approximant from degree 7 to
degree 6. With exactly representable coefficients computed by the Sollya tool,
this still gives a maximum relative error of 1 ulp, i.e. faithfully rounded, for
arguments where exp(x) is a normalized float. This change results in a speedup
of about 4% for AVX2.
2. Extend the range where exp(x) returns a non-zero result to from ~[-88;88] to
~[-104;88] i.e. return denormalized values for large negative arguments instead
of zero. Compared to exp<double>(x) the denormalized results gradually decrease
in accuracy down to 0.033 relative error for arguments around x = -104 where
exp(x) is ~std::numeric<float>::denorm_min(). This is expected and acceptable.
We can't make guarantees on alignment for existing calls to `pset`,
so we should default to loading unaligned. But in that case, we should
just use `ploadu` directly. For loading constants, this load should hopefully
get optimized away.
This is causing segfaults in Google Maps.
Replace usage of `std::numeric_limits<...>::min/max_exponent` in
codebase where possible. Also replaced some other `numeric_limits`
usages in affected tests with the `NumTraits` equivalent.
The previous MR !443 failed for c++03 due to lack of `constexpr`.
Because of this, we need to keep around the `std::numeric_limits`
version in enum expressions until the switch to c++11.
Fixes#2148
Replace usage of `std::numeric_limits<...>::min/max_exponent` in
codebase. Also replaced some other `numeric_limits` usages in
affected tests with the `NumTraits` equivalent.
Fixes#2148
This is a new version of !423, which failed for MSVC.
Defined `EIGEN_OPTIMIZATION_BARRIER(X)` that uses inline assembly to
prevent operations involving `X` from crossing that barrier. Should
work on most `GNUC` compatible compilers (MSVC doesn't seem to need
this). This is a modified version adapted from what was used in
`psincos_float` and tested on more platforms
(see #1674, https://godbolt.org/z/73ezTG).
Modified `rint` to use the barrier to prevent the add/subtract rounding
trick from being optimized away.
Also fixed an edge case for large inputs that get bumped up a power of two
and ends up rounding away more than just the fractional part. If we are
over `2^digits` then just return the input. This edge case was missed in
the test since the test was comparing approximate equality, which was still
satisfied. Adding a strict equality option catches it.
The original clamping bounds on `_x` actually produce finite values:
```
exp(88.3762626647950) = 2.40614e+38 < 3.40282e+38
exp(709.437) = 1.27226e+308 < 1.79769e+308
```
so with an accurate `ldexp` implementation, `pexp` fails for large
inputs, producing finite values instead of `inf`.
This adjusts the bounds slightly outside the finite range so that
the output will overflow to +/- `inf` as expected.
