[libc][math] Implement a fast pass for atan2f128 with 1ULP error using DyadicFloat<128>.

libc/config/linux/aarch64/entrypoints.txt

@@ -758,6 +758,7 @@ endif()
 if(LIBC_TYPES_HAS_FLOAT128)
   list(APPEND TARGET_LIBM_ENTRYPOINTS
     # math.h C23 _Float128 entrypoints
+    libc.src.math.atan2f128
     libc.src.math.canonicalizef128
     libc.src.math.ceilf128
     libc.src.math.copysignf128

libc/config/linux/riscv/entrypoints.txt

@@ -639,6 +639,7 @@ endif()
 if(LIBC_TYPES_HAS_FLOAT128)
   list(APPEND TARGET_LIBM_ENTRYPOINTS
     # math.h C23 _Float128 entrypoints
+    libc.src.math.atan2f128
     libc.src.math.canonicalizef128
     libc.src.math.ceilf128
     libc.src.math.copysignf128

libc/config/linux/x86_64/entrypoints.txt

@@ -773,6 +773,7 @@ endif()
 if(LIBC_TYPES_HAS_FLOAT128)
   list(APPEND TARGET_LIBM_ENTRYPOINTS
     # math.h C23 _Float128 entrypoints
+    libc.src.math.atan2f128
     libc.src.math.canonicalizef128
     libc.src.math.ceilf128
     libc.src.math.copysignf128

libc/docs/headers/math/index.rst

@@ -263,7 +263,7 @@ Higher Math Functions
 +-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
 | atan      | |check|          | 1 ULP           |                        |                      |                        | 7.12.4.3               | F.10.1.3                   |
 +-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
-| atan2     | |check|          | 1 ULP           |                        |                      |                        | 7.12.4.4               | F.10.1.4                   |
+| atan2     | |check|          | 1 ULP           |                        |                      | 1 ULP                  | 7.12.4.4               | F.10.1.4                   |
 +-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+
 | atan2pi   |                  |                 |                        |                      |                        | 7.12.4.11              | F.10.1.11                  |
 +-----------+------------------+-----------------+------------------------+----------------------+------------------------+------------------------+----------------------------+

libc/include/math.yaml

@@ -86,6 +86,14 @@ functions:
     arguments:
       - type: long double
       - type: long double
+  - name: atan2f128
+    standards:
+      - stdc
+    return_type: float128
+    arguments:
+      - type: float128
+      - type: float128
+    guard: LIBC_TYPES_HAS_FLOAT128
   - name: atanf
     standards:
       - stdc

libc/src/__support/FPUtil/dyadic_float.h

@@ -104,7 +104,7 @@ template <size_t Bits> struct DyadicFloat {
     normalize();
   }
 
-  LIBC_INLINE constexpr DyadicFloat(Sign s, int e, MantissaType m)
+  LIBC_INLINE constexpr DyadicFloat(Sign s, int e, const MantissaType &m)
       : sign(s), exponent(e), mantissa(m) {
     normalize();
   }

libc/src/math/CMakeLists.txt

@@ -61,6 +61,7 @@ add_math_entrypoint_object(atanf)
 add_math_entrypoint_object(atan2)
 add_math_entrypoint_object(atan2f)
 add_math_entrypoint_object(atan2l)
+add_math_entrypoint_object(atan2f128)
 
 add_math_entrypoint_object(atanh)
 add_math_entrypoint_object(atanhf)

libc/src/math/atan2f128.h

@@ -0,0 +1,21 @@
+//===-- Implementation header for atan2f128 ---------------------*- 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_MATH_ATAN2F128_H
+#define LLVM_LIBC_SRC_MATH_ATAN2F128_H
+
+#include "src/__support/macros/config.h"
+#include "src/__support/macros/properties/types.h"
+
+namespace LIBC_NAMESPACE_DECL {
+
+float128 atan2f128(float128 x, float128 y);
+
+} // namespace LIBC_NAMESPACE_DECL
+
+#endif // LLVM_LIBC_SRC_MATH_ATAN2F128_H

libc/src/math/generic/CMakeLists.txt

@@ -4080,8 +4080,11 @@ add_header_library(
   HDRS
     atan_utils.h
   DEPENDS
+    libc.src.__support.integer_literals
     libc.src.__support.FPUtil.double_double
+    libc.src.__support.FPUtil.dyadic_float
     libc.src.__support.FPUtil.multiply_add
+    libc.src.__support.FPUtil.polyeval
     libc.src.__support.macros.optimization
 )
 
@@ -4163,6 +4166,23 @@ add_entrypoint_object(
     .atan2
 )
 
+add_entrypoint_object(
+  atan2f128
+  SRCS
+    atan2f128.cpp
+  HDRS
+    ../atan2f128.h
+  DEPENDS
+    .atan_utils
+    libc.src.__support.integer_literals
+    libc.src.__support.FPUtil.dyadic_float
+    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.macros.properties.types
+)
+
 add_entrypoint_object(
   scalbln
   SRCS

libc/src/math/generic/atan2f128.cpp

@@ -0,0 +1,203 @@
+//===-- Quad-precision atan2 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/atan2f128.h"
+#include "atan_utils.h"
+#include "src/__support/FPUtil/FPBits.h"
+#include "src/__support/FPUtil/dyadic_float.h"
+#include "src/__support/FPUtil/multiply_add.h"
+#include "src/__support/FPUtil/nearest_integer.h"
+#include "src/__support/integer_literals.h"
+#include "src/__support/macros/config.h"
+#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY
+#include "src/__support/macros/properties/types.h"
+#include "src/__support/uint128.h"
+
+namespace LIBC_NAMESPACE_DECL {
+
+namespace {
+
+using Float128 = fputil::DyadicFloat<128>;
+
+static constexpr Float128 ZERO = {Sign::POS, 0, 0_u128};
+static constexpr Float128 MZERO = {Sign::NEG, 0, 0_u128};
+static constexpr Float128 PI = {Sign::POS, -126,
+                                0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128};
+static constexpr Float128 MPI = {Sign::NEG, -126,
+                                 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128};
+static constexpr Float128 PI_OVER_2 = {
+    Sign::POS, -127, 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128};
+static constexpr Float128 MPI_OVER_2 = {
+    Sign::NEG, -127, 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128};
+static constexpr Float128 PI_OVER_4 = {
+    Sign::POS, -128, 0xc90fdaa2'2168c234'c4c6628b'80dc1cd1_u128};
+static constexpr Float128 THREE_PI_OVER_4 = {
+    Sign::POS, -128, 0x96cbe3f9'990e91a7'9394c9e8'a0a5159d_u128};
+
+// Adjustment for constant term:
+//   CONST_ADJ[x_sign][y_sign][recip]
+static constexpr Float128 CONST_ADJ[2][2][2] = {
+    {{ZERO, MPI_OVER_2}, {MZERO, MPI_OVER_2}},
+    {{MPI, PI_OVER_2}, {MPI, PI_OVER_2}}};
+
+} // anonymous namespace
+
+// 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-13 minimax polynomial to compute the
+// RHS:
+//   atan(u) ~ P(u) = u - c_3 * u^3 + c_5 * u^5 - c_7 * u^7 + c_9 *u^9 -
+//                    - c_11 * u^11 + c_13 * u^13
+// with absolute errors bounded by:
+//   |atan(u) - P(u)| < 2^-121
+// and relative errors bounded by:
+//   |(atan(u) - P(u)) / P(u)| < 2^-114.
+
+LLVM_LIBC_FUNCTION(float128, atan2f128, (float128 y, float128 x)) {
+  using FPBits = fputil::FPBits<float128>;
+  using Float128 = fputil::DyadicFloat<128>;
+
+  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();
+  UInt128 x_abs = x_bits.uintval();
+  UInt128 y_abs = y_bits.uintval();
+  bool recip = x_abs < y_abs;
+  UInt128 min_abs = recip ? x_abs : y_abs;
+  UInt128 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);
+
+  Float128 num(FPBits(min_abs).get_val());
+  Float128 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 >= 0x7fffU || min_exp == 0U)) {
+    if (x_bits.is_nan() || y_bits.is_nan())
+      return FPBits::quiet_nan().get_val();
+    unsigned x_except = x == 0 ? 0 : (FPBits(x_abs).is_inf() ? 2 : 1);
+    unsigned y_except = y == 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 Float128 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)) {
+      Float128 r = EXCEPTS[y_except][x_except][x_sign];
+      if (y_sign)
+        r.sign = (r.sign == Sign::POS) ? Sign::NEG : Sign::POS;
+      return static_cast<float128>(r);
+    }
+  }
+
+  bool final_sign = ((x_sign != y_sign) != recip);
+  Float128 const_term = CONST_ADJ[x_sign][y_sign][recip];
+  int exp_diff = den.exponent - num.exponent;
+  // 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 > FPBits::FRACTION_LEN + 2)) {
+    if (final_sign)
+      const_term.sign = (const_term.sign == Sign::POS) ? Sign::NEG : Sign::POS;
+    return static_cast<float128>(const_term);
+  }
+
+  // Take 24 leading bits of num and den to convert to float for fast division.
+  // We also multiply the numerator by 64 using integer addition directly to the
+  // exponent field.
+  float num_f =
+      cpp::bit_cast<float>(static_cast<uint32_t>(num.mantissa >> 104) +
+                           (6U << fputil::FPBits<float>::FRACTION_LEN));
+  float den_f = cpp::bit_cast<float>(
+      static_cast<uint32_t>(den.mantissa >> 104) +
+      (static_cast<uint32_t>(exp_diff) << fputil::FPBits<float>::FRACTION_LEN));
+
+  float k = fputil::nearest_integer(num_f / den_f);
+  unsigned idx = static_cast<unsigned>(k);
+
+  // k_f128 = idx / 64
+  Float128 k_f128(Sign::POS, -6, Float128::MantissaType(idx));
+
+  // Range reduction:
+  // atan(n/d) - atan(k) = atan((n/d - k/64) / (1 + (n/d) * (k/64)))
+  //                     = atan((n - d * k/64)) / (d + n * k/64))
+  // num_f128 = n - d * k/64
+  Float128 num_f128 = fputil::multiply_add(den, -k_f128, num);
+  // den_f128 = d + n * k/64
+  Float128 den_f128 = fputil::multiply_add(num, k_f128, den);
+
+  // q = (n - d * k) / (d + n * k)
+  Float128 q = fputil::quick_mul(num_f128, fputil::approx_reciprocal(den_f128));
+  // p ~ atan(q)
+  Float128 p = atan_eval(q);
+
+  Float128 r =
+      fputil::quick_add(const_term, fputil::quick_add(ATAN_I_F128[idx], p));
+  if (final_sign)
+    r.sign = (r.sign == Sign::POS) ? Sign::NEG : Sign::POS;
+
+  return static_cast<float128>(r);
+}
+
+} // namespace LIBC_NAMESPACE_DECL