- Implementation of `tan` for 16-bit floating point inputs scaled by pi.
i.e,. `tanpif16()`
- Implementation of Tanpi in MPFRWrapper for MPFR versions < 4.2
- Exhaustive tests for `tanpif16()`
Implementation of `cos` for half precision floating point inputs scaled
by pi (i.e., `cospi`), correctly rounded for all rounding modes.
---------
Co-authored-by: OverMighty <its.overmighty@gmail.com>
This PR implements the iscanonical function as part of the libc math
library.
The addition of this function is crucial for completing the
implementation of remaining math macros, as referenced in #109201
This PR implements the issignaling function as part of the libc math
library, addressing the TODO items mentioned in #110011
The addition of this function is crucial for completing the
implementation of remaining math macros, as referenced in #109201
- added all variations of ffma and fdiv
- will add all new headers into yaml for next patch
- only fsub is left then all basic operations for float is complete
---------
Co-authored-by: OverMighty <its.overmighty@gmail.com>
- fadd removed because I need to add for different input types
- finishing rest of basic operations
- noticed duplicates will remove
---------
Co-authored-by: OverMighty <its.overmighty@gmail.com>
Division-less Newton iterations algorithm for cube roots.
1. **Range reduction**
For `x = (-1)^s * 2^e * (1.m)`, we get 2 reduced arguments `x_r` and `a`
as:
```
x_r = 1.m
a = (-1)^s * 2^(e % 3) * (1.m)
```
Then `cbrt(x) = x^(1/3)` can be computed as:
```
x^(1/3) = 2^(e / 3) * a^(1/3).
```
In order to avoid division, we compute `a^(-2/3)` using Newton method
and then
multiply the results by a:
```
a^(1/3) = a * a^(-2/3).
```
2. **First approximation to a^(-2/3)**
First, we use a degree-7 minimax polynomial generated by Sollya to
approximate `x_r^(-2/3)` for `1 <= x_r < 2`.
```
p = P(x_r) ~ x_r^(-2/3),
```
with relative errors bounded by:
```
| p / x_r^(-2/3) - 1 | < 1.16 * 2^-21.
```
Then we multiply with `2^(e % 3)` from a small lookup table to get:
```
x_0 = 2^(-2*(e % 3)/3) * p
~ 2^(-2*(e % 3)/3) * x_r^(-2/3)
= a^(-2/3)
```
with relative errors:
```
| x_0 / a^(-2/3) - 1 | < 1.16 * 2^-21.
```
This step is done in double precision.
3. **First Newton iteration**
We follow the method described in:
Sibidanov, A. and Zimmermann, P., "Correctly rounded cubic root
evaluation
in double precision", https://core-math.gitlabpages.inria.fr/cbrt64.pdf
to derive multiplicative Newton iterations as below:
Let `x_n` be the nth approximation to `a^(-2/3)`. Define the n^th error
as:
```
h_n = x_n^3 * a^2 - 1
```
Then:
```
a^(-2/3) = x_n / (1 + h_n)^(1/3)
= x_n * (1 - (1/3) * h_n + (2/9) * h_n^2 - (14/81) * h_n^3 + ...)
```
using the Taylor series expansion of `(1 + h_n)^(-1/3)`.
Apply to `x_0` above:
```
h_0 = x_0^3 * a^2 - 1
= a^2 * (x_0 - a^(-2/3)) * (x_0^2 + x_0 * a^(-2/3) + a^(-4/3)),
```
it's bounded by:
```
|h_0| < 4 * 3 * 1.16 * 2^-21 * 4 < 2^-17.
```
So in the first iteration step, we use:
```
x_1 = x_0 * (1 - (1/3) * h_n + (2/9) * h_n^2 - (14/81) * h_n^3)
```
Its relative error is bounded by:
```
| x_1 / a^(-2/3) - 1 | < 35/242 * |h_0|^4 < 2^-70.
```
Then we perform Ziv's rounding test and check if the answer is exact.
This step is done in double-double precision.
4. **Second Newton iteration**
If the Ziv's rounding test from the previous step fails, we define the
error
term:
```
h_1 = x_1^3 * a^2 - 1,
```
And perform another iteration:
```
x_2 = x_1 * (1 - h_1 / 3)
```
with the relative errors exceed the precision of double-double.
We then check the Ziv's accuracy test with relative errors < 2^-102 to
compensate for rounding errors.
5. **Final iteration**
If the Ziv's accuracy test from the previous step fails, we perform
another
iteration in 128-bit precision and check for exact outputs.
Summary:
This function is used by the CUDA / HIP / OpenMP headers and exists as
an NVIDIA extension basically. This function is implemented in the C23
standard as `pown`, but for now we need to provide `powi` for backwards
compatibility. In the future this entrypoint will just be a redirect to
`pown` once that is implemented.
Fixes https://github.com/llvm/llvm-project/issues/92874
Algorithm: Let `x = (-1)^s * 2^e * (1 + m)`.
- Step 1: Range reduction: reduce the exponent with:
```
y = cbrt(x) = (-1)^s * 2^(floor(e/3)) * 2^((e % 3)/3) * (1 + m)^(1/3)
```
- Step 2: Use the first 4 bit fractional bits of `m` to look up for a
degree-7 polynomial approximation to:
```
(1 + m)^(1/3) ~ 1 + m * P(m).
```
- Step 3: Perform the multiplication:
```
2^((e % 3)/3) * (1 + m)^(1/3).
```
- Step 4: Check for exact cases to prevent rounding and clear
`FE_INEXACT` floating point exception.
- Step 5: Combine with the exponent and sign before converting down to
`float` and return.
I also fixed a comment in sinpif.cpp in the first commit. Should this be
included in this PR?
All tests were passed, including the exhaustive test.
CC: @lntue
Using the same range reduction as `sin`, `cos`, and `sincos`:
1) Reducing `x = k*pi/128 + u`, with `|u| <= pi/256`, and `u` is in
double-double.
2) Approximate `tan(u)` using degree-9 Taylor polynomial.
3) Compute
```
tan(x) ~ (sin(k*pi/128) + tan(u) * cos(k*pi/128)) / (cos(k*pi/128) - tan(u) * sin(k*pi/128))
```
using the fast double-double division algorithm in [the CORE-MATH
project](https://gitlab.inria.fr/core-math/core-math/-/blob/master/src/binary64/tan/tan.c#L1855).
4) Perform relative-error Ziv's accuracy test
5) If the accuracy tests failed, we redo the computations using 128-bit
precision `DyadicFloat`.
Fixes https://github.com/llvm/llvm-project/issues/96930
