Implement tanf function correctly rounded for all rounding modes.
We use the range reduction that is shared with `sinf`, `cosf`, and `sincosf`:
```
k = round(x * 32/pi) and y = x * (32/pi) - k.
```
Then we use the tangent of sum formula:
```
tan(x) = tan((k + y)* pi/32) = tan((k mod 32) * pi / 32 + y * pi/32)
= (tan((k mod 32) * pi/32) + tan(y * pi/32)) / (1 - tan((k mod 32) * pi/32) * tan(y * pi/32))
```
We need to make a further reduction when `k mod 32 >= 16` due to the pole at `pi/2` of `tan(x)` function:
```
if (k mod 32 >= 16): k = k - 31, y = y - 1.0
```
And to compute the final result, we store `tan(k * pi/32)` for `k = -15..15` in a table of 32 double values,
and evaluate `tan(y * pi/32)` with a degree-11 minimax odd polynomial generated by Sollya with:
```
> P = fpminimax(tan(y * pi/32)/y, [|0, 2, 4, 6, 8, 10|], [|D...|], [0, 1.5]);
```
Performance benchmark using `perf` tool from the CORE-MATH project on Ryzen 1700:
```
$ CORE_MATH_PERF_MODE="rdtsc" ./perf.sh tanf
CORE-MATH reciprocal throughput : 18.586
System LIBC reciprocal throughput : 50.068
LIBC reciprocal throughput : 33.823
LIBC reciprocal throughput : 25.161 (with `-msse4.2` flag)
LIBC reciprocal throughput : 19.157 (with `-mfma` flag)
$ CORE_MATH_PERF_MODE="rdtsc" ./perf.sh tanf --latency
GNU libc version: 2.31
GNU libc release: stable
CORE-MATH latency : 55.630
System LIBC latency : 106.264
LIBC latency : 96.060
LIBC latency : 90.727 (with `-msse4.2` flag)
LIBC latency : 82.361 (with `-mfma` flag)
```
Reviewed By: orex
Differential Revision: https://reviews.llvm.org/D131715
Change sinf/cosf range reduction to mod pi/32 to be shared with tanf,
since polynomial approximations for tanf on subintervals of length pi/16 do not
provide enough accuracy.
Reviewed By: orex
Differential Revision: https://reviews.llvm.org/D131652
New exp2 function algorithm:
1) Improved performance: 8.176 vs 15.270 by core-math perf tool.
2) Improved accuracy. Only two special values left.
3) Lookup table size reduced twice.
Differential Revision: https://reviews.llvm.org/D129005
Change `sinf` range reduction to mod pi/16 to be shared with `cosf`.
Previously, `sinf` used range reduction `mod pi`, but this cannot be used to implement `cosf` since the minimax algorithm for `cosf` does not converge due to critical points at `pi/2`. In order to be able to share the same range reduction functions for both `sinf` and `cosf`, we change the range reduction to `mod pi/16` for the following reasons:
- The table size is sufficiently small: 32 entries for `sin(k * pi/16)` with `k = 0..31`. It could be reduced to 16 entries if we treat the final sign separately, with an extra multiplication at the end.
- The polynomials' degrees are reduced to 7/8 from 15, with extra computations to combine `sin` and `cos` with trig sum equality.
- The number of exceptional cases reduced to 2 (with FMA) and 3 (without FMA).
- The latency is reduced while maintaining similar throughput as before.
Reviewed By: zimmermann6
Differential Revision: https://reviews.llvm.org/D130629
This is a implementation of find remainder fmod function from standard libm.
The underline algorithm is developed by myself, but probably it was first
invented before.
Some features of the implementation:
1. The code is written on more-or-less modern C++.
2. One general implementation for both float and double precision numbers.
3. Spitted platform/architecture dependent and independent code and tests.
4. Tests covers 100% of the code for both float and double numbers. Tests cases with NaN/Inf etc is copied from glibc.
5. The new implementation in general 2-4 times faster for “regular” x,y values. It can be 20 times faster for x/y huge value, but can also be 2 times slower for double denormalized range (according to perf tests provided).
6. Two different implementation of division loop are provided. In some platforms division can be very time consuming operation. Depend on platform it can be 3-10 times slower than multiplication.
Performance tests:
The test is based on core-math project (https://gitlab.inria.fr/core-math/core-math). By Tue Ly suggestion I took hypot function and use it as template for fmod. Preserving all test cases.
`./check.sh <--special|--worst> fmodf` passed.
`CORE_MATH_PERF_MODE=rdtsc ./perf.sh fmodf` results are
```
GNU libc version: 2.35
GNU libc release: stable
21.166 <-- FPU
51.031 <-- current glibc
37.659 <-- this fmod version.
```
Tue Ly