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<title>llvm-project.git/libc/src/math/generic/asin.cpp, branch users/nico/python-2</title>
<subtitle>Unnamed repository; edit this file 'description' to name the repository.
</subtitle>
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<entry>
<title>[libc][math] Implement double precision acos correctly rounded for all rounding modes. (#138308)</title>
<updated>2025-05-09T03:23:09+00:00</updated>
<author>
<name>lntue</name>
<email>lntue@google.com</email>
</author>
<published>2025-05-09T03:23:09+00:00</published>
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<content type='text'>
We reduce computation of `acos` to `asin` as follow:

When `|x| &lt; 0.5`:
```math
acos(x) = \frac{\pi}{2} - asin(x).
```
For `0.5 &lt;= |x| &lt; 1`, let
```math
u = \frac{1 - \left| x \right|}{2},
```
then
```math
acos(x) = \begin{cases}
  2 \cdot asin \left( \sqrt{u} \right) &amp;, 0.5 \leq x &lt; 1 \\
  \pi - 2 \cdot asin \left( \sqrt{u} \right) &amp;, -1 &lt; x \leq 0.5 
\end{cases}
```</content>
<content type='xhtml'>
<div xmlns='http://www.w3.org/1999/xhtml'>
<pre>
We reduce computation of `acos` to `asin` as follow:

When `|x| &lt; 0.5`:
```math
acos(x) = \frac{\pi}{2} - asin(x).
```
For `0.5 &lt;= |x| &lt; 1`, let
```math
u = \frac{1 - \left| x \right|}{2},
```
then
```math
acos(x) = \begin{cases}
  2 \cdot asin \left( \sqrt{u} \right) &amp;, 0.5 \leq x &lt; 1 \\
  \pi - 2 \cdot asin \left( \sqrt{u} \right) &amp;, -1 &lt; x \leq 0.5 
\end{cases}
```</pre>
</div>
</content>
</entry>
<entry>
<title>[libc][math] Implement double precision asin correctly rounded for all rounding modes. (#134401)</title>
<updated>2025-04-25T13:55:21+00:00</updated>
<author>
<name>lntue</name>
<email>lntue@google.com</email>
</author>
<published>2025-04-25T13:55:21+00:00</published>
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<id>ade502a8c46b8393e022b4eabcdd445af91a451c</id>
<content type='text'>
Main algorithm:

The Taylor series expansion of `asin(x)` is:
```math
\begin{align*}
  asin(x) &amp;= x + x^3 / 6 + 3x^5 / 40 + ... \\
       &amp;= x \cdot P(x^2) \\
       &amp;= x \cdot P(u) &amp;\text{, where } u = x^2.
\end{align*}
```
For the fast path, we perform range reduction mod 1/64 and use degree-7
(minimax + Taylor) polynomials to approximate `P(x^2)`.

When `|x| &gt;= 0.5`, we use the transformation:
```math
  u = \frac{1 + x}{2}
```
and apply half-angle formula to reduce `asin(x)` to:
```math
\begin{align*}
  asin(x) &amp;= sign(x) \cdot \left( \frac{\pi}{2} - 2 \cdot asin(\sqrt{u}) \right) \\
       &amp;= sign(x) \cdot \left( \frac{\pi}{2} - 2 \cdot \sqrt{u} \cdot P(u) \right).
\end{align*}
```
Since `0.5 &lt;= |x| &lt;= 1`, `|u| &lt;= 0.5`. So we can reuse the polynomial
evaluation of `P(u)` when `|x| &lt; 0.5`.

For the accurate path, we redo the computations in 128-bit precision
with degree-15 (minimax + Taylor) polynomials to approximate `P(u)`.</content>
<content type='xhtml'>
<div xmlns='http://www.w3.org/1999/xhtml'>
<pre>
Main algorithm:

The Taylor series expansion of `asin(x)` is:
```math
\begin{align*}
  asin(x) &amp;= x + x^3 / 6 + 3x^5 / 40 + ... \\
       &amp;= x \cdot P(x^2) \\
       &amp;= x \cdot P(u) &amp;\text{, where } u = x^2.
\end{align*}
```
For the fast path, we perform range reduction mod 1/64 and use degree-7
(minimax + Taylor) polynomials to approximate `P(x^2)`.

When `|x| &gt;= 0.5`, we use the transformation:
```math
  u = \frac{1 + x}{2}
```
and apply half-angle formula to reduce `asin(x)` to:
```math
\begin{align*}
  asin(x) &amp;= sign(x) \cdot \left( \frac{\pi}{2} - 2 \cdot asin(\sqrt{u}) \right) \\
       &amp;= sign(x) \cdot \left( \frac{\pi}{2} - 2 \cdot \sqrt{u} \cdot P(u) \right).
\end{align*}
```
Since `0.5 &lt;= |x| &lt;= 1`, `|u| &lt;= 0.5`. So we can reuse the polynomial
evaluation of `P(u)` when `|x| &lt; 0.5`.

For the accurate path, we redo the computations in 128-bit precision
with degree-15 (minimax + Taylor) polynomials to approximate `P(u)`.</pre>
</div>
</content>
</entry>
<entry>
<title>[libc][math] Remove placeholder implementations of asin and pow.</title>
<updated>2023-04-20T05:28:16+00:00</updated>
<author>
<name>Tue Ly</name>
<email>lntue@google.com</email>
</author>
<published>2023-04-20T05:03:41+00:00</published>
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<id>f79264b5f66f54d9102d358ac00a3d8de25fada7</id>
<content type='text'>
Reviewed By: sivachandra

Differential Revision: https://reviews.llvm.org/D148781
</content>
<content type='xhtml'>
<div xmlns='http://www.w3.org/1999/xhtml'>
<pre>
Reviewed By: sivachandra

Differential Revision: https://reviews.llvm.org/D148781
</pre>
</div>
</content>
</entry>
<entry>
<title>[libc][math] Add place-holder implementation for asin to unblock demo examples.</title>
<updated>2022-10-31T21:22:12+00:00</updated>
<author>
<name>Tue Ly</name>
<email>lntue@google.com</email>
</author>
<published>2022-10-31T19:42:00+00:00</published>
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<id>97b4cc83e16abec8daa67f5dc82f71791c523f43</id>
<content type='text'>
Add a place-holder implementation for asin to unblock libc demo
examples.

Reviewed By: michaelrj

Differential Revision: https://reviews.llvm.org/D137105
</content>
<content type='xhtml'>
<div xmlns='http://www.w3.org/1999/xhtml'>
<pre>
Add a place-holder implementation for asin to unblock libc demo
examples.

Reviewed By: michaelrj

Differential Revision: https://reviews.llvm.org/D137105
</pre>
</div>
</content>
</entry>
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