diff --git a/Misc/NEWS.d/next/Library/2026-07-19-19-05-00.gh-issue-154146.acospi.rst b/Misc/NEWS.d/next/Library/2026-07-19-19-05-00.gh-issue-154146.acospi.rst
new file mode 100644
index 00000000000000..5dbddbadee0bd3
--- /dev/null
+++ b/Misc/NEWS.d/next/Library/2026-07-19-19-05-00.gh-issue-154146.acospi.rst
@@ -0,0 +1,2 @@
+Fix accuracy of :func:`math.acospi` for arguments close to 1
+on platforms that do not provide ``acospi()`` in libm.
diff --git a/Modules/mathmodule.c b/Modules/mathmodule.c
index 64e5372d73d2f2..eaa1850b8aee1a 100644
--- a/Modules/mathmodule.c
+++ b/Modules/mathmodule.c
@@ -220,6 +220,12 @@ static const double logpi = 1.144729885849400174143427351353058711647;
 static double
 m_acospi(double x)
 {
+    if (x >= 0.5) {
+        /* acos(x) = 2*asin(sqrt((1 - x)/2)).  1 - x is exact here.
+           Some libms (old fdlibm derivatives) lose precision in acos(x)
+           near x = 1, while asin() is accurate for small arguments. */
+        return 2.0*asin(sqrt((1.0 - x)/2.0))/pi;
+    }
     double r = acos(x)/pi;
     if (isgreater(r, 1.0)) {
         return 1.0;