[libc-commits] [libc] [libc][math][c23] Add acospif16() function (PR #132754)

[via libc-commits]

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@@ -0,0 +1,146 @@
+//===-- Half-precision acospif16(x) function ------------------------------===//
+//
+// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
+// See https://llvm.org/LICENSE.txt for license information.
+// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception.
+//
+//
+//===----------------------------------------------------------------------===//
+
+#include "src/math/acospif16.h"
+#include "hdr/errno_macros.h"
+#include "hdr/fenv_macros.h"
+#include "src/__support/FPUtil/FEnvImpl.h"
+#include "src/__support/FPUtil/FPBits.h"
+#include "src/__support/FPUtil/PolyEval.h"
+#include "src/__support/FPUtil/cast.h"
+#include "src/__support/FPUtil/except_value_utils.h"
+#include "src/__support/FPUtil/multiply_add.h"
+#include "src/__support/FPUtil/sqrt.h"
+#include "src/__support/macros/optimization.h" // remove unnecessary includes
+
+namespace LIBC_NAMESPACE_DECL {
+
+// Generated by Sollya using the following command:
+// > round(2/pi, SG, RN);
+// > round(1/pi, SG, RN);
+static constexpr float TWO_DIV_PI = 0x1.45f306p-1f;
+static constexpr float PI_INV = 0x1.45f306p-2f;
+
+#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
+static constexpr size_t N_EXCEPTS = 2;
+
+LLVM_LIBC_FUNCTION(float16, acospif16, (float16 x)) {
+  using FPBits = fputil::FPBits<float16>;
+  FPBits xbits(x);
+
+  uint16_t x_u = xbits.uintval();
+  uint16_t x_abs = x_u & 0x7fff;
+  uint16_t x_sign = x_u >> 15;
+
+  // |x| > 0x1p0, |x| > 1, or x is NaN.
+  if (LIBC_UNLIKELY(x_abs > 0x3c00)) {
+    // acospif16(NaN) = NaN
+    if (xbits.is_nan()) {
+      if (xbits.is_signaling_nan()) {
+        fputil::raise_except_if_required(FE_INVALID);
+        return FPBits::quiet_nan().get_val();
+      }
+
+      return x;
+    }
+
+    // 1 < |x| <= +/-inf
+    fputil::raise_except_if_required(FE_INVALID);
+    fputil::set_errno_if_required(EDOM);
+
+    return FPBits::quiet_nan().get_val();
+  }
+
+  // TODO: handling for exception values
+
+  // |x| == 0x1p0, x is 1 or -1
+  // if x is (-)1, return 1
+  // if x is (+)1, return 0
+  if (LIBC_UNLIKELY(x_abs == 0x3c00))
+    return fputil::cast<float16>(x_sign ? 1.0f : 0.0f);
+
+  float xf = x;
+  float xsq = xf * xf;
+
+  // |x| <= 0x1p-1, |x| <= 0.5
+  if (x_abs <= 0x3800) {
+    // if x is 0, return 0.5
+    if (LIBC_UNLIKELY(x_abs == 0))
+      return fputil::cast<float16>(0.5f);
+
+    // Note that: acos(x) = pi/2 + asin(-x) = pi/2 - asin(x), then
+    //            acospi(x) = 0.5 - asin(x)*(INVERSE_PI)
+    // Degree-6 minimax polynomial of asin(x) generated by Sollya with:
+    // > P = fpminimax(asin(x)/(x), [|0, 2, 4, 6, 8|], [|SG...|], [0, 0.5]);
+    float xf_divided = xf * PI_INV;
+    float interm =
+        fputil::polyeval(xsq, 0x1.000002p0f, 0x1.554c2ap-3f, 0x1.3541ccp-4f,
+                         0x1.43b2d6p-5f, 0x1.a0d73ep-5f);
+
+    // Same as acos(x), but devided the expression with pi
+    return fputil::cast<float16>(
+        fputil::multiply_add(-xf_divided, interm, 0.5f));
+  }
+
+  // When |x| > 0.5, assume that 0.5 < |x| <= 1
+  //
+  // Step-by-step range-reduction proof:
+  // 1:  Let y = asin(x), such that, x = sin(y)
+  // 2:  From complimentary angle identity:
+  //       x = sin(y) = cos(pi/2 - y)
+  // 3:  Let z = pi/2 - y, such that x = cos(z)
+  // 4:  From double angle formula; cos(2A) = 1 - 2 * sin^2(A):
+  //       z = 2A, z/2 = A
+  //       cos(z) = 1 - 2 * sin^2(z/2)
+  // 5:  Make sin(z/2) subject of the formula:
+  //       sin(z/2) = sqrt((1 - cos(z))/2)
+  // 6:  Recall [3]; x = cos(z). Therefore:
+  //       sin(z/2) = sqrt((1 - x)/2)
+  // 7:  Let u = (1 - x)/2
+  // 8:  Therefore:
+  //       asin(sqrt(u)) = z/2
+  //       2 * asin(sqrt(u)) = z
+  // 9:  Recall [3]; z = pi/2 - y. Therefore:
+  //       y = pi/2 - z
+  //       y = pi/2 - 2 * asin(sqrt(u))
+  // 10: Recall [1], y = asin(x). Therefore:
+  //       asin(x) = pi/2 - 2 * asin(sqrt(u))
+  // 11: Recall that: acos(x) = pi/2 + asin(-x) = pi/2 - asin(x)
+  //     Therefore:
+  //       acos(x) = pi/2 - (pi/2 - 2 * asin(sqrt(u)))
+  //       acos(x) = 2 * asin(sqrt(u))
+  //
+  // THE RANGE REDUCTION, HOW?
+  // 12: Recall [7], u = (1 - x)/2
+  // 13: Since 0.5 < x <= 1, therefore:
+  //       0 <= u <= 0.25 and 0 <= sqrt(u) <= 0.5
+  //
+  // Hence, we can reuse the same [0, 0.5] domain polynomial approximation for
+  // Step [11] as `sqrt(u)` is in range.
+  // When -1 < x <= -0.5, the identity:
+  //       acos(x) = pi - acos(-x)
+  // allows us to compute for the negative x value (lhs)
+  // with a positive x value instead (rhs).
+
+  float xf_abs = (xf < 0 ? -xf : xf);
+  float u = fputil::multiply_add(-0.5f, xf_abs, 0.5f);
+  float sqrt_u = fputil::sqrt<float>(u);
+
+  // Degree-6 minimax polynomial of asin(x) generated by Sollya with:
+  // > P = fpminimax(asin(x)/(x), [|0, 2, 4, 6, 8|], [|SG...|], [0, 0.5]);
+  float asin_sqrt_u =
+      sqrt_u * fputil::polyeval(u, 0x1.000002p0f, 0x1.554c2ap-3f,
----------------
lntue wrote:

if this polynomial is the same as the one from the other branch, use a constexpr array to share between them.

https://github.com/llvm/llvm-project/pull/132754