[libc][math] Refactor atan2 implementation to header-only in src/__support/math folder.

libc/shared/math.h

@@ -23,6 +23,7 @@
 #include "math/asinhf.h"
 #include "math/asinhf16.h"
 #include "math/atan.h"
+#include "math/atan2.h"
 #include "math/atanf.h"
 #include "math/atanf16.h"
 #include "math/erff.h"

libc/shared/math/atan2.h

@@ -0,0 +1,23 @@
+//===-- Shared atan2 function -----------------------------------*- C++ -*-===//
+//
+// 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
+//
+//===----------------------------------------------------------------------===//
+
+#ifndef LLVM_LIBC_SHARED_MATH_ATAN2_H
+#define LLVM_LIBC_SHARED_MATH_ATAN2_H
+
+#include "shared/libc_common.h"
+#include "src/__support/math/atan2.h"
+
+namespace LIBC_NAMESPACE_DECL {
+namespace shared {
+
+using math::atan2;
+
+} // namespace shared
+} // namespace LIBC_NAMESPACE_DECL
+
+#endif // LLVM_LIBC_SHARED_MATH_ATAN2_H

libc/src/__support/math/CMakeLists.txt

@@ -158,7 +158,7 @@ add_header_library(
   asinhf16
   HDRS
     asinhf16.h
-DEPENDS
+  DEPENDS
     .acoshf_utils
     libc.src.__support.FPUtil.fenv_impl
     libc.src.__support.FPUtil.fp_bits
@@ -176,7 +176,7 @@ add_header_library(
   atan_utils
   HDRS
     atan_utils.h
-DEPENDS
+  DEPENDS
     libc.src.__support.integer_literals
     libc.src.__support.FPUtil.double_double
     libc.src.__support.FPUtil.dyadic_float
@@ -189,7 +189,21 @@ add_header_library(
   atan
   HDRS
     atan.h
-DEPENDS
+  DEPENDS
+    .atan_utils
+    libc.src.__support.FPUtil.double_double
+    libc.src.__support.FPUtil.fenv_impl
+    libc.src.__support.FPUtil.fp_bits
+    libc.src.__support.FPUtil.multiply_add
+    libc.src.__support.FPUtil.nearest_integer
+    libc.src.__support.macros.optimization
+)
+
+add_header_library(
+  atan2
+  HDRS
+    atan2.h
+  DEPENDS
     .atan_utils
     libc.src.__support.FPUtil.double_double
     libc.src.__support.FPUtil.fenv_impl

libc/src/__support/math/atan2.h

@@ -0,0 +1,209 @@
+//===-- Implementation header for atan2 -------------------------*- C++ -*-===//
+//
+// 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
+//
+//===----------------------------------------------------------------------===//
+
+#ifndef LLVM_LIBC_SRC___SUPPORT_MATH_ATAN2_H
+#define LLVM_LIBC_SRC___SUPPORT_MATH_ATAN2_H
+
+#include "atan_utils.h"
+#include "src/__support/FPUtil/FEnvImpl.h"
+#include "src/__support/FPUtil/FPBits.h"
+#include "src/__support/FPUtil/double_double.h"
+#include "src/__support/FPUtil/multiply_add.h"
+#include "src/__support/FPUtil/nearest_integer.h"
+#include "src/__support/macros/config.h"
+#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
+
+namespace LIBC_NAMESPACE_DECL {
+
+namespace math {
+
+// There are several range reduction steps we can take for atan2(y, x) as
+// follow:
+
+// * Range reduction 1: signness
+// atan2(y, x) will return a number between -PI and PI representing the angle
+// forming by the 0x axis and the vector (x, y) on the 0xy-plane.
+// In particular, we have that:
+//   atan2(y, x) = atan( y/x )         if x >= 0 and y >= 0 (I-quadrant)
+//               = pi + atan( y/x )    if x < 0 and y >= 0  (II-quadrant)
+//               = -pi + atan( y/x )   if x < 0 and y < 0   (III-quadrant)
+//               = atan( y/x )         if x >= 0 and y < 0  (IV-quadrant)
+// Since atan function is odd, we can use the formula:
+//   atan(-u) = -atan(u)
+// to adjust the above conditions a bit further:
+//   atan2(y, x) = atan( |y|/|x| )         if x >= 0 and y >= 0 (I-quadrant)
+//               = pi - atan( |y|/|x| )    if x < 0 and y >= 0  (II-quadrant)
+//               = -pi + atan( |y|/|x| )   if x < 0 and y < 0   (III-quadrant)
+//               = -atan( |y|/|x| )        if x >= 0 and y < 0  (IV-quadrant)
+// Which can be simplified to:
+//   atan2(y, x) = sign(y) * atan( |y|/|x| )             if x >= 0
+//               = sign(y) * (pi - atan( |y|/|x| ))      if x < 0
+
+// * Range reduction 2: reciprocal
+// Now that the argument inside atan is positive, we can use the formula:
+//   atan(1/x) = pi/2 - atan(x)
+// to make the argument inside atan <= 1 as follow:
+//   atan2(y, x) = sign(y) * atan( |y|/|x|)            if 0 <= |y| <= x
+//               = sign(y) * (pi/2 - atan( |x|/|y| )   if 0 <= x < |y|
+//               = sign(y) * (pi - atan( |y|/|x| ))    if 0 <= |y| <= -x
+//               = sign(y) * (pi/2 + atan( |x|/|y| ))  if 0 <= -x < |y|
+
+// * Range reduction 3: look up table.
+// After the previous two range reduction steps, we reduce the problem to
+// compute atan(u) with 0 <= u <= 1, or to be precise:
+//   atan( n / d ) where n = min(|x|, |y|) and d = max(|x|, |y|).
+// An accurate polynomial approximation for the whole [0, 1] input range will
+// require a very large degree.  To make it more efficient, we reduce the input
+// range further by finding an integer idx such that:
+//   | n/d - idx/64 | <= 1/128.
+// In particular,
+//   idx := round(2^6 * n/d)
+// Then for the fast pass, we find a polynomial approximation for:
+//   atan( n/d ) ~ atan( idx/64 ) + (n/d - idx/64) * Q(n/d - idx/64)
+// For the accurate pass, we use the addition formula:
+//   atan( n/d ) - atan( idx/64 ) = atan( (n/d - idx/64)/(1 + (n*idx)/(64*d)) )
+//                                = atan( (n - d*(idx/64))/(d + n*(idx/64)) )
+// And for the fast pass, we use degree-9 Taylor polynomial to compute the RHS:
+//   atan(u) ~ P(u) = u - u^3/3 + u^5/5 - u^7/7 + u^9/9
+// with absolute errors bounded by:
+//   |atan(u) - P(u)| < |u|^11 / 11 < 2^-80
+// and relative errors bounded by:
+//   |(atan(u) - P(u)) / P(u)| < u^10 / 11 < 2^-73.
+
+LIBC_INLINE static constexpr double atan2(double y, double x) {
+  using namespace atan_internal;
+  using FPBits = fputil::FPBits<double>;
+
+  constexpr double IS_NEG[2] = {1.0, -1.0};
+  constexpr DoubleDouble ZERO = {0.0, 0.0};
+  constexpr DoubleDouble MZERO = {-0.0, -0.0};
+  constexpr DoubleDouble PI = {0x1.1a62633145c07p-53, 0x1.921fb54442d18p+1};
+  constexpr DoubleDouble MPI = {-0x1.1a62633145c07p-53, -0x1.921fb54442d18p+1};
+  constexpr DoubleDouble PI_OVER_2 = {0x1.1a62633145c07p-54,
+                                      0x1.921fb54442d18p0};
+  constexpr DoubleDouble MPI_OVER_2 = {-0x1.1a62633145c07p-54,
+                                       -0x1.921fb54442d18p0};
+  constexpr DoubleDouble PI_OVER_4 = {0x1.1a62633145c07p-55,
+                                      0x1.921fb54442d18p-1};
+  constexpr DoubleDouble THREE_PI_OVER_4 = {0x1.a79394c9e8a0ap-54,
+                                            0x1.2d97c7f3321d2p+1};
+  // Adjustment for constant term:
+  //   CONST_ADJ[x_sign][y_sign][recip]
+  constexpr DoubleDouble CONST_ADJ[2][2][2] = {
+      {{ZERO, MPI_OVER_2}, {MZERO, MPI_OVER_2}},
+      {{MPI, PI_OVER_2}, {MPI, PI_OVER_2}}};
+
+  FPBits x_bits(x), y_bits(y);
+  bool x_sign = x_bits.sign().is_neg();
+  bool y_sign = y_bits.sign().is_neg();
+  x_bits = x_bits.abs();
+  y_bits = y_bits.abs();
+  uint64_t x_abs = x_bits.uintval();
+  uint64_t y_abs = y_bits.uintval();
+  bool recip = x_abs < y_abs;
+  uint64_t min_abs = recip ? x_abs : y_abs;
+  uint64_t max_abs = !recip ? x_abs : y_abs;
+  unsigned min_exp = static_cast<unsigned>(min_abs >> FPBits::FRACTION_LEN);
+  unsigned max_exp = static_cast<unsigned>(max_abs >> FPBits::FRACTION_LEN);
+
+  double num = FPBits(min_abs).get_val();
+  double den = FPBits(max_abs).get_val();
+
+  // Check for exceptional cases, whether inputs are 0, inf, nan, or close to
+  // overflow, or close to underflow.
+  if (LIBC_UNLIKELY(max_exp > 0x7ffU - 128U || min_exp < 128U)) {
+    if (x_bits.is_nan() || y_bits.is_nan()) {
+      if (x_bits.is_signaling_nan() || y_bits.is_signaling_nan())
+        fputil::raise_except_if_required(FE_INVALID);
+      return FPBits::quiet_nan().get_val();
+    }
+    unsigned x_except = x == 0.0 ? 0 : (FPBits(x_abs).is_inf() ? 2 : 1);
+    unsigned y_except = y == 0.0 ? 0 : (FPBits(y_abs).is_inf() ? 2 : 1);
+
+    // Exceptional cases:
+    //   EXCEPT[y_except][x_except][x_is_neg]
+    // with x_except & y_except:
+    //   0: zero
+    //   1: finite, non-zero
+    //   2: infinity
+    constexpr DoubleDouble EXCEPTS[3][3][2] = {
+        {{ZERO, PI}, {ZERO, PI}, {ZERO, PI}},
+        {{PI_OVER_2, PI_OVER_2}, {ZERO, ZERO}, {ZERO, PI}},
+        {{PI_OVER_2, PI_OVER_2},
+         {PI_OVER_2, PI_OVER_2},
+         {PI_OVER_4, THREE_PI_OVER_4}},
+    };
+
+    if ((x_except != 1) || (y_except != 1)) {
+      DoubleDouble r = EXCEPTS[y_except][x_except][x_sign];
+      return fputil::multiply_add(IS_NEG[y_sign], r.hi, IS_NEG[y_sign] * r.lo);
+    }
+    bool scale_up = min_exp < 128U;
+    bool scale_down = max_exp > 0x7ffU - 128U;
+    // At least one input is denormal, multiply both numerator and denominator
+    // by some large enough power of 2 to normalize denormal inputs.
+    if (scale_up) {
+      num *= 0x1.0p64;
+      if (!scale_down)
+        den *= 0x1.0p64;
+    } else if (scale_down) {
+      den *= 0x1.0p-64;
+      if (!scale_up)
+        num *= 0x1.0p-64;
+    }
+
+    min_abs = FPBits(num).uintval();
+    max_abs = FPBits(den).uintval();
+    min_exp = static_cast<unsigned>(min_abs >> FPBits::FRACTION_LEN);
+    max_exp = static_cast<unsigned>(max_abs >> FPBits::FRACTION_LEN);
+  }
+
+  double final_sign = IS_NEG[(x_sign != y_sign) != recip];
+  DoubleDouble const_term = CONST_ADJ[x_sign][y_sign][recip];
+  unsigned exp_diff = max_exp - min_exp;
+  // We have the following bound for normalized n and d:
+  //   2^(-exp_diff - 1) < n/d < 2^(-exp_diff + 1).
+  if (LIBC_UNLIKELY(exp_diff > 54)) {
+    return fputil::multiply_add(final_sign, const_term.hi,
+                                final_sign * (const_term.lo + num / den));
+  }
+
+  double k = fputil::nearest_integer(64.0 * num / den);
+  unsigned idx = static_cast<unsigned>(k);
+  // k = idx / 64
+  k *= 0x1.0p-6;
+
+  // Range reduction:
+  // atan(n/d) - atan(k/64) = atan((n/d - k/64) / (1 + (n/d) * (k/64)))
+  //                        = atan((n - d * k/64)) / (d + n * k/64))
+  DoubleDouble num_k = fputil::exact_mult(num, k);
+  DoubleDouble den_k = fputil::exact_mult(den, k);
+
+  // num_dd = n - d * k
+  DoubleDouble num_dd = fputil::exact_add(num - den_k.hi, -den_k.lo);
+  // den_dd = d + n * k
+  DoubleDouble den_dd = fputil::exact_add(den, num_k.hi);
+  den_dd.lo += num_k.lo;
+
+  // q = (n - d * k) / (d + n * k)
+  DoubleDouble q = fputil::div(num_dd, den_dd);
+  // p ~ atan(q)
+  DoubleDouble p = atan_eval(q);
+
+  DoubleDouble r = fputil::add(const_term, fputil::add(ATAN_I[idx], p));
+  r.hi *= final_sign;
+  r.lo *= final_sign;
+
+  return r.hi + r.lo;
+}
+
+} // namespace math
+
+} // namespace LIBC_NAMESPACE_DECL
+
+#endif // LLVM_LIBC_SRC___SUPPORT_MATH_ATAN2_H

libc/src/math/generic/CMakeLists.txt

@@ -4079,13 +4079,7 @@ add_entrypoint_object(
   HDRS
     ../atan2.h
   DEPENDS
-    libc.src.__support.math.atan_utils
-    libc.src.__support.FPUtil.double_double
-    libc.src.__support.FPUtil.fenv_impl
-    libc.src.__support.FPUtil.fp_bits
-    libc.src.__support.FPUtil.multiply_add
-    libc.src.__support.FPUtil.nearest_integer
-    libc.src.__support.macros.optimization
+    libc.src.__support.math.atan2
 )
 
 add_entrypoint_object(
@@ -4095,7 +4089,7 @@ add_entrypoint_object(
   HDRS
     ../atan2l.h
   DEPENDS
-    .atan2
+    libc.src.__support.math.atan2
 )
 
 add_entrypoint_object(