libc/src/math/generic/atan2f.cpp - llvm-project - Git at Google

 //===-- Single-precision atan2f 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/atan2f.h"
 #include "hdr/fenv_macros.h"
 #include "inv_trigf_utils.h"
 #include "src/__support/FPUtil/FEnvImpl.h"
 #include "src/__support/FPUtil/FPBits.h"
 #include "src/__support/FPUtil/PolyEval.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/FPUtil/rounding_mode.h"
 #include "src/__support/macros/config.h"
 #include "src/__support/macros/optimization.h" // LIBC_UNLIKELY

 #if defined(LIBC_MATH_HAS_SKIP_ACCURATE_PASS) &&                               \
     defined(LIBC_MATH_HAS_INTERMEDIATE_COMP_IN_FLOAT)

 // We use float-float implementation to reduce size.
 #include "src/math/generic/atan2f_float.h"

 #else

 namespace LIBC_NAMESPACE_DECL {

 namespace {

 #ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS

 // Look up tables for accurate pass:

 // atan(i/16) with i = 0..16, generated by Sollya with:
 // > for i from 0 to 16 do {
 //     a = round(atan(i/16), D, RN);
 //     b = round(atan(i/16) - a, D, RN);
 //     print("{", b, ",", a, "},");
 //   };
 constexpr fputil::DoubleDouble ATAN_I[17] = {
     {0.0, 0.0},
     {-0x1.c934d86d23f1dp-60, 0x1.ff55bb72cfdeap-5},
     {-0x1.cd37686760c17p-59, 0x1.fd5ba9aac2f6ep-4},
     {0x1.347b0b4f881cap-58, 0x1.7b97b4bce5b02p-3},
     {0x1.8ab6e3cf7afbdp-57, 0x1.f5b75f92c80ddp-3},
     {-0x1.963a544b672d8p-57, 0x1.362773707ebccp-2},
     {-0x1.c63aae6f6e918p-56, 0x1.6f61941e4def1p-2},
     {-0x1.24dec1b50b7ffp-56, 0x1.a64eec3cc23fdp-2},
     {0x1.a2b7f222f65e2p-56, 0x1.dac670561bb4fp-2},
     {-0x1.d5b495f6349e6p-56, 0x1.0657e94db30dp-1},
     {-0x1.928df287a668fp-58, 0x1.1e00babdefeb4p-1},
     {0x1.1021137c71102p-55, 0x1.345f01cce37bbp-1},
     {0x1.2419a87f2a458p-56, 0x1.4978fa3269ee1p-1},
     {0x1.0028e4bc5e7cap-57, 0x1.5d58987169b18p-1},
     {-0x1.8c34d25aadef6p-56, 0x1.700a7c5784634p-1},
     {-0x1.bf76229d3b917p-56, 0x1.819d0b7158a4dp-1},
     {0x1.1a62633145c07p-55, 0x1.921fb54442d18p-1},
 };

 // Taylor polynomial, generated by Sollya with:
 // > for i from 0 to 8 do {
 //     j = (-1)^(i + 1)/(2*i + 1);
 //     a = round(j, D, RN);
 //     b = round(j - a, D, RN);
 //     print("{", b, ",", a, "},");
 //   };
 constexpr fputil::DoubleDouble COEFFS[9] = {
     {0.0, 1.0},                                      // 1
     {-0x1.5555555555555p-56, -0x1.5555555555555p-2}, // -1/3
     {-0x1.999999999999ap-57, 0x1.999999999999ap-3},  // 1/5
     {-0x1.2492492492492p-57, -0x1.2492492492492p-3}, // -1/7
     {0x1.c71c71c71c71cp-58, 0x1.c71c71c71c71cp-4},   // 1/9
     {0x1.745d1745d1746p-59, -0x1.745d1745d1746p-4},  // -1/11
     {-0x1.3b13b13b13b14p-58, 0x1.3b13b13b13b14p-4},  // 1/13
     {-0x1.1111111111111p-60, -0x1.1111111111111p-4}, // -1/15
     {0x1.e1e1e1e1e1e1ep-61, 0x1.e1e1e1e1e1e1ep-5},   // 1/17
 };

 // Veltkamp's splitting of a double precision into hi + lo, where the hi part is
 // slightly smaller than an even split, so that the product of
 //   hi * (s1 * k + s2) is exact,
 // where:
 //   s1, s2 are single precsion,
 //   1/16 <= s1/s2 <= 1
 //   1/16 <= k <= 1 is an integer.
 // So the maximal precision of (s1 * k + s2) is:
 //   prec(s1 * k + s2) = 2 + log2(msb(s2)) - log2(lsb(k_d * s1))
 //                     = 2 + log2(msb(s1)) + 4 - log2(lsb(k_d)) - log2(lsb(s1))
 //                     = 2 + log2(lsb(s1)) + 23 + 4 - (-4) - log2(lsb(s1))
 //                     = 33.
 // Thus, the Veltkamp splitting constant is C = 2^33 + 1.
 // This is used when FMA instruction is not available.
 [[maybe_unused]] constexpr fputil::DoubleDouble split_d(double a) {
   fputil::DoubleDouble r{0.0, 0.0};
   constexpr double C = 0x1.0p33 + 1.0;
   double t1 = C * a;
   double t2 = a - t1;
   r.hi = t1 + t2;
   r.lo = a - r.hi;
   return r;
 }

 // Compute atan( num_d / den_d ) in double-double precision.
 //   num_d      = min(|x|, |y|)
 //   den_d      = max(|x|, |y|)
 //   q_d        = num_d / den_d
 //   idx, k_d   = round( 2^4 * num_d / den_d )
 //   final_sign = sign of the final result
 //   const_term = the constant term in the final expression.
 float atan2f_double_double(double num_d, double den_d, double q_d, int idx,
                            double k_d, double final_sign,
                            const fputil::DoubleDouble &const_term) {
   fputil::DoubleDouble q;
   double num_r, den_r;

   if (idx != 0) {
     // The following range reduction is accurate even without fma for
     //   1/16 <= n/d <= 1.
     // atan(n/d) - atan(idx/16) = atan((n/d - idx/16) / (1 + (n/d) * (idx/16)))
     //                          = atan((n - d*(idx/16)) / (d + n*idx/16))
     k_d *= 0x1.0p-4;
     num_r = fputil::multiply_add(k_d, -den_d, num_d); // Exact
     den_r = fputil::multiply_add(k_d, num_d, den_d);  // Exact
     q.hi = num_r / den_r;
   } else {
     // For 0 < n/d < 1/16, we just need to calculate the lower part of their
     // quotient.
     q.hi = q_d;
     num_r = num_d;
     den_r = den_d;
   }
 #ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE
   q.lo = fputil::multiply_add(q.hi, -den_r, num_r) / den_r;
 #else
   // Compute `(num_r - q.hi * den_r) / den_r` accurately without FMA
   // instructions.
   fputil::DoubleDouble q_hi_dd = split_d(q.hi);
   double t1 = fputil::multiply_add(q_hi_dd.hi, -den_r, num_r); // Exact
   double t2 = fputil::multiply_add(q_hi_dd.lo, -den_r, t1);
   q.lo = t2 / den_r;
 #endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE

   // Taylor polynomial, evaluating using Horner's scheme:
   //   P = x - x^3/3 + x^5/5 -x^7/7 + x^9/9 - x^11/11 + x^13/13 - x^15/15
   //       + x^17/17
   //     = x*(1 + x^2*(-1/3 + x^2*(1/5 + x^2*(-1/7 + x^2*(1/9 + x^2*
   //          *(-1/11 + x^2*(1/13 + x^2*(-1/15 + x^2 * 1/17))))))))
   fputil::DoubleDouble q2 = fputil::quick_mult(q, q);
   fputil::DoubleDouble p_dd =
       fputil::polyeval(q2, COEFFS[0], COEFFS[1], COEFFS[2], COEFFS[3],
                        COEFFS[4], COEFFS[5], COEFFS[6], COEFFS[7], COEFFS[8]);
   fputil::DoubleDouble r_dd =
       fputil::add(const_term, fputil::multiply_add(q, p_dd, ATAN_I[idx]));
   r_dd.hi *= final_sign;
   r_dd.lo *= final_sign;

   // Make sure the sum is normalized:
   fputil::DoubleDouble rr = fputil::exact_add(r_dd.hi, r_dd.lo);
   // Round to odd.
   uint64_t rr_bits = cpp::bit_cast<uint64_t>(rr.hi);
   if (LIBC_UNLIKELY(((rr_bits & 0xfff'ffff) == 0) && (rr.lo != 0.0))) {
     Sign hi_sign = fputil::FPBits<double>(rr.hi).sign();
     Sign lo_sign = fputil::FPBits<double>(rr.lo).sign();
     if (hi_sign == lo_sign) {
       ++rr_bits;
     } else if ((rr_bits & fputil::FPBits<double>::FRACTION_MASK) > 0) {
       --rr_bits;
     }
   }

   return static_cast<float>(cpp::bit_cast<double>(rr_bits));
 }

 #endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS

 } // 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/16 | <= 1/32.
 // In particular,
 //   idx := 2^-4 * round(2^4 * n/d)
 // Then for the fast pass, we find a polynomial approximation for:
 //   atan( n/d ) ~ atan( idx/16 ) + (n/d - idx/16) * Q(n/d - idx/16)
 // For the accurate pass, we use the addition formula:
 //   atan( n/d ) - atan( idx/16 ) = atan( (n/d - idx/16)/(1 + (n*idx)/(16*d)) )
 //                                = atan( (n - d * idx/16)/(d + n * idx/16) )
 // And finally we use Taylor polynomial to compute the RHS in the accurate pass:
 //   atan(u) ~ P(u) = u - u^3/3 + u^5/5 - u^7/7 + u^9/9 - u^11/11 + u^13/13 -
 //                      - u^15/15 + u^17/17
 // It's error in double-double precision is estimated in Sollya to be:
 // > P = x - x^3/3 + x^5/5 -x^7/7 + x^9/9 - x^11/11 + x^13/13 - x^15/15
 //       + x^17/17;
 // > dirtyinfnorm(atan(x) - P, [-2^-5, 2^-5]);
 // 0x1.aec6f...p-100
 // which is about rounding errors of double-double (2^-104).

 LLVM_LIBC_FUNCTION(float, atan2f, (float y, float x)) {
   using FPBits = typename fputil::FPBits<float>;
   constexpr double IS_NEG[2] = {1.0, -1.0};
   constexpr double PI = 0x1.921fb54442d18p1;
   constexpr double PI_LO = 0x1.1a62633145c07p-53;
   constexpr double PI_OVER_4 = 0x1.921fb54442d18p-1;
   constexpr double PI_OVER_2 = 0x1.921fb54442d18p0;
   constexpr double THREE_PI_OVER_4 = 0x1.2d97c7f3321d2p+1;
   // Adjustment for constant term:
   //   CONST_ADJ[x_sign][y_sign][recip]
   constexpr fputil::DoubleDouble CONST_ADJ[2][2][2] = {
       {{{0.0, 0.0}, {-PI_LO / 2, -PI_OVER_2}},
        {{-0.0, -0.0}, {-PI_LO / 2, -PI_OVER_2}}},
       {{{-PI_LO, -PI}, {PI_LO / 2, PI_OVER_2}},
        {{-PI_LO, -PI}, {PI_LO / 2, 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.set_sign(Sign::POS);
   y_bits.set_sign(Sign::POS);
   uint32_t x_abs = x_bits.uintval();
   uint32_t y_abs = y_bits.uintval();
   uint32_t max_abs = x_abs > y_abs ? x_abs : y_abs;
   uint32_t min_abs = x_abs <= y_abs ? x_abs : y_abs;
   float num_f = FPBits(min_abs).get_val();
   float den_f = FPBits(max_abs).get_val();
   double num_d = static_cast<double>(num_f);
   double den_d = static_cast<double>(den_f);

   if (LIBC_UNLIKELY(max_abs >= 0x7f80'0000U || num_d == 0.0)) {
     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();
     }
     double x_d = static_cast<double>(x);
     double y_d = static_cast<double>(y);
     size_t x_except = (x_d == 0.0) ? 0 : (x_abs == 0x7f80'0000 ? 2 : 1);
     size_t y_except = (y_d == 0.0) ? 0 : (y_abs == 0x7f80'0000 ? 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 double EXCEPTS[3][3][2] = {
         {{0.0, PI}, {0.0, PI}, {0.0, PI}},
         {{PI_OVER_2, PI_OVER_2}, {0.0, 0.0}, {0.0, PI}},
         {{PI_OVER_2, PI_OVER_2},
          {PI_OVER_2, PI_OVER_2},
          {PI_OVER_4, THREE_PI_OVER_4}},
     };

     double r = IS_NEG[y_sign] * EXCEPTS[y_except][x_except][x_sign];

     return static_cast<float>(r);
   }

   bool recip = x_abs < y_abs;
   double final_sign = IS_NEG[(x_sign != y_sign) != recip];
   fputil::DoubleDouble const_term = CONST_ADJ[x_sign][y_sign][recip];
   double q_d = num_d / den_d;

   double k_d = fputil::nearest_integer(q_d * 0x1.0p4);
   int idx = static_cast<int>(k_d);
   double r;

 #ifdef LIBC_MATH_HAS_SMALL_TABLES
   double p = atan_eval_no_table(num_d, den_d, k_d * 0x1.0p-4);
   r = final_sign * (p + (const_term.hi + ATAN_K_OVER_16[idx]));
 #else
   q_d = fputil::multiply_add(k_d, -0x1.0p-4, q_d);

   double p = atan_eval(q_d, idx);
   r = final_sign *
       fputil::multiply_add(q_d, p, const_term.hi + ATAN_COEFFS[idx][0]);
 #endif // LIBC_MATH_HAS_SMALL_TABLES

 #ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
   return static_cast<float>(r);
 #else
   constexpr uint32_t LOWER_ERR = 4;
   // Mask sticky bits in double precision before rounding to single precision.
   constexpr uint32_t MASK =
       mask_trailing_ones<uint32_t, fputil::FPBits<double>::SIG_LEN -
                                        FPBits::SIG_LEN - 1>();
   constexpr uint32_t UPPER_ERR = MASK - LOWER_ERR;

   uint32_t r_bits = static_cast<uint32_t>(cpp::bit_cast<uint64_t>(r)) & MASK;

   // Ziv's rounding test.
   if (LIBC_LIKELY(r_bits > LOWER_ERR && r_bits < UPPER_ERR))
     return static_cast<float>(r);

   return atan2f_double_double(num_d, den_d, q_d, idx, k_d, final_sign,
                               const_term);
 #endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
 }

 } // namespace LIBC_NAMESPACE_DECL

 #endif
	//===-- Single-precision atan2f 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/atan2f.h"
	#include "hdr/fenv_macros.h"
	#include "inv_trigf_utils.h"
	#include "src/__support/FPUtil/FEnvImpl.h"
	#include "src/__support/FPUtil/FPBits.h"
	#include "src/__support/FPUtil/PolyEval.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/FPUtil/rounding_mode.h"
	#include "src/__support/macros/config.h"
	#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY

	#if defined(LIBC_MATH_HAS_SKIP_ACCURATE_PASS) && \
	defined(LIBC_MATH_HAS_INTERMEDIATE_COMP_IN_FLOAT)

	// We use float-float implementation to reduce size.
	#include "src/math/generic/atan2f_float.h"

	#else

	namespace LIBC_NAMESPACE_DECL {

	namespace {

	#ifndef LIBC_MATH_HAS_SKIP_ACCURATE_PASS

	// Look up tables for accurate pass:

	// atan(i/16) with i = 0..16, generated by Sollya with:
	// > for i from 0 to 16 do {
	// a = round(atan(i/16), D, RN);
	// b = round(atan(i/16) - a, D, RN);
	// print("{", b, ",", a, "},");
	// };
	constexpr fputil::DoubleDouble ATAN_I[17] = {
	{0.0, 0.0},
	{-0x1.c934d86d23f1dp-60, 0x1.ff55bb72cfdeap-5},
	{-0x1.cd37686760c17p-59, 0x1.fd5ba9aac2f6ep-4},
	{0x1.347b0b4f881cap-58, 0x1.7b97b4bce5b02p-3},
	{0x1.8ab6e3cf7afbdp-57, 0x1.f5b75f92c80ddp-3},
	{-0x1.963a544b672d8p-57, 0x1.362773707ebccp-2},
	{-0x1.c63aae6f6e918p-56, 0x1.6f61941e4def1p-2},
	{-0x1.24dec1b50b7ffp-56, 0x1.a64eec3cc23fdp-2},
	{0x1.a2b7f222f65e2p-56, 0x1.dac670561bb4fp-2},
	{-0x1.d5b495f6349e6p-56, 0x1.0657e94db30dp-1},
	{-0x1.928df287a668fp-58, 0x1.1e00babdefeb4p-1},
	{0x1.1021137c71102p-55, 0x1.345f01cce37bbp-1},
	{0x1.2419a87f2a458p-56, 0x1.4978fa3269ee1p-1},
	{0x1.0028e4bc5e7cap-57, 0x1.5d58987169b18p-1},
	{-0x1.8c34d25aadef6p-56, 0x1.700a7c5784634p-1},
	{-0x1.bf76229d3b917p-56, 0x1.819d0b7158a4dp-1},
	{0x1.1a62633145c07p-55, 0x1.921fb54442d18p-1},
	};

	// Taylor polynomial, generated by Sollya with:
	// > for i from 0 to 8 do {
	// j = (-1)^(i + 1)/(2*i + 1);
	// a = round(j, D, RN);
	// b = round(j - a, D, RN);
	// print("{", b, ",", a, "},");
	// };
	constexpr fputil::DoubleDouble COEFFS[9] = {
	{0.0, 1.0}, // 1
	{-0x1.5555555555555p-56, -0x1.5555555555555p-2}, // -1/3
	{-0x1.999999999999ap-57, 0x1.999999999999ap-3}, // 1/5
	{-0x1.2492492492492p-57, -0x1.2492492492492p-3}, // -1/7
	{0x1.c71c71c71c71cp-58, 0x1.c71c71c71c71cp-4}, // 1/9
	{0x1.745d1745d1746p-59, -0x1.745d1745d1746p-4}, // -1/11
	{-0x1.3b13b13b13b14p-58, 0x1.3b13b13b13b14p-4}, // 1/13
	{-0x1.1111111111111p-60, -0x1.1111111111111p-4}, // -1/15
	{0x1.e1e1e1e1e1e1ep-61, 0x1.e1e1e1e1e1e1ep-5}, // 1/17
	};

	// Veltkamp's splitting of a double precision into hi + lo, where the hi part is
	// slightly smaller than an even split, so that the product of
	// hi * (s1 * k + s2) is exact,
	// where:
	// s1, s2 are single precsion,
	// 1/16 <= s1/s2 <= 1
	// 1/16 <= k <= 1 is an integer.
	// So the maximal precision of (s1 * k + s2) is:
	// prec(s1 * k + s2) = 2 + log2(msb(s2)) - log2(lsb(k_d * s1))
	// = 2 + log2(msb(s1)) + 4 - log2(lsb(k_d)) - log2(lsb(s1))
	// = 2 + log2(lsb(s1)) + 23 + 4 - (-4) - log2(lsb(s1))
	// = 33.
	// Thus, the Veltkamp splitting constant is C = 2^33 + 1.
	// This is used when FMA instruction is not available.
	[[maybe_unused]] constexpr fputil::DoubleDouble split_d(double a) {
	fputil::DoubleDouble r{0.0, 0.0};
	constexpr double C = 0x1.0p33 + 1.0;
	double t1 = C * a;
	double t2 = a - t1;
	r.hi = t1 + t2;
	r.lo = a - r.hi;
	return r;
	}

	// Compute atan( num_d / den_d ) in double-double precision.
	// num_d = min(\|x\|, \|y\|)
	// den_d = max(\|x\|, \|y\|)
	// q_d = num_d / den_d
	// idx, k_d = round( 2^4 * num_d / den_d )
	// final_sign = sign of the final result
	// const_term = the constant term in the final expression.
	float atan2f_double_double(double num_d, double den_d, double q_d, int idx,
	double k_d, double final_sign,
	const fputil::DoubleDouble &const_term) {
	fputil::DoubleDouble q;
	double num_r, den_r;

	if (idx != 0) {
	// The following range reduction is accurate even without fma for
	// 1/16 <= n/d <= 1.
	// atan(n/d) - atan(idx/16) = atan((n/d - idx/16) / (1 + (n/d) * (idx/16)))
	// = atan((n - d(idx/16)) / (d + nidx/16))
	k_d *= 0x1.0p-4;
	num_r = fputil::multiply_add(k_d, -den_d, num_d); // Exact
	den_r = fputil::multiply_add(k_d, num_d, den_d); // Exact
	q.hi = num_r / den_r;
	} else {
	// For 0 < n/d < 1/16, we just need to calculate the lower part of their
	// quotient.
	q.hi = q_d;
	num_r = num_d;
	den_r = den_d;
	}
	#ifdef LIBC_TARGET_CPU_HAS_FMA_DOUBLE
	q.lo = fputil::multiply_add(q.hi, -den_r, num_r) / den_r;
	#else
	// Compute `(num_r - q.hi * den_r) / den_r` accurately without FMA
	// instructions.
	fputil::DoubleDouble q_hi_dd = split_d(q.hi);
	double t1 = fputil::multiply_add(q_hi_dd.hi, -den_r, num_r); // Exact
	double t2 = fputil::multiply_add(q_hi_dd.lo, -den_r, t1);
	q.lo = t2 / den_r;
	#endif // LIBC_TARGET_CPU_HAS_FMA_DOUBLE

	// Taylor polynomial, evaluating using Horner's scheme:
	// P = x - x^3/3 + x^5/5 -x^7/7 + x^9/9 - x^11/11 + x^13/13 - x^15/15
	// + x^17/17
	// = x(1 + x^2(-1/3 + x^2(1/5 + x^2(-1/7 + x^2(1/9 + x^2
	// (-1/11 + x^2(1/13 + x^2(-1/15 + x^2 1/17))))))))
	fputil::DoubleDouble q2 = fputil::quick_mult(q, q);
	fputil::DoubleDouble p_dd =
	fputil::polyeval(q2, COEFFS[0], COEFFS[1], COEFFS[2], COEFFS[3],
	COEFFS[4], COEFFS[5], COEFFS[6], COEFFS[7], COEFFS[8]);
	fputil::DoubleDouble r_dd =
	fputil::add(const_term, fputil::multiply_add(q, p_dd, ATAN_I[idx]));
	r_dd.hi *= final_sign;
	r_dd.lo *= final_sign;

	// Make sure the sum is normalized:
	fputil::DoubleDouble rr = fputil::exact_add(r_dd.hi, r_dd.lo);
	// Round to odd.
	uint64_t rr_bits = cpp::bit_cast<uint64_t>(rr.hi);
	if (LIBC_UNLIKELY(((rr_bits & 0xfff'ffff) == 0) && (rr.lo != 0.0))) {
	Sign hi_sign = fputil::FPBits<double>(rr.hi).sign();
	Sign lo_sign = fputil::FPBits<double>(rr.lo).sign();
	if (hi_sign == lo_sign) {
	++rr_bits;
	} else if ((rr_bits & fputil::FPBits<double>::FRACTION_MASK) > 0) {
	--rr_bits;
	}
	}

	return static_cast<float>(cpp::bit_cast<double>(rr_bits));
	}

	#endif // !LIBC_MATH_HAS_SKIP_ACCURATE_PASS

	} // 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/16 \| <= 1/32.
	// In particular,
	// idx := 2^-4 * round(2^4 * n/d)
	// Then for the fast pass, we find a polynomial approximation for:
	// atan( n/d ) ~ atan( idx/16 ) + (n/d - idx/16) * Q(n/d - idx/16)
	// For the accurate pass, we use the addition formula:
	// atan( n/d ) - atan( idx/16 ) = atan( (n/d - idx/16)/(1 + (nidx)/(16d)) )
	// = atan( (n - d * idx/16)/(d + n * idx/16) )
	// And finally we use Taylor polynomial to compute the RHS in the accurate pass:
	// atan(u) ~ P(u) = u - u^3/3 + u^5/5 - u^7/7 + u^9/9 - u^11/11 + u^13/13 -
	// - u^15/15 + u^17/17
	// It's error in double-double precision is estimated in Sollya to be:
	// > P = x - x^3/3 + x^5/5 -x^7/7 + x^9/9 - x^11/11 + x^13/13 - x^15/15
	// + x^17/17;
	// > dirtyinfnorm(atan(x) - P, [-2^-5, 2^-5]);
	// 0x1.aec6f...p-100
	// which is about rounding errors of double-double (2^-104).

	LLVM_LIBC_FUNCTION(float, atan2f, (float y, float x)) {
	using FPBits = typename fputil::FPBits<float>;
	constexpr double IS_NEG[2] = {1.0, -1.0};
	constexpr double PI = 0x1.921fb54442d18p1;
	constexpr double PI_LO = 0x1.1a62633145c07p-53;
	constexpr double PI_OVER_4 = 0x1.921fb54442d18p-1;
	constexpr double PI_OVER_2 = 0x1.921fb54442d18p0;
	constexpr double THREE_PI_OVER_4 = 0x1.2d97c7f3321d2p+1;
	// Adjustment for constant term:
	// CONST_ADJ[x_sign][y_sign][recip]
	constexpr fputil::DoubleDouble CONST_ADJ[2][2][2] = {
	{{{0.0, 0.0}, {-PI_LO / 2, -PI_OVER_2}},
	{{-0.0, -0.0}, {-PI_LO / 2, -PI_OVER_2}}},
	{{{-PI_LO, -PI}, {PI_LO / 2, PI_OVER_2}},
	{{-PI_LO, -PI}, {PI_LO / 2, 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.set_sign(Sign::POS);
	y_bits.set_sign(Sign::POS);
	uint32_t x_abs = x_bits.uintval();
	uint32_t y_abs = y_bits.uintval();
	uint32_t max_abs = x_abs > y_abs ? x_abs : y_abs;
	uint32_t min_abs = x_abs <= y_abs ? x_abs : y_abs;
	float num_f = FPBits(min_abs).get_val();
	float den_f = FPBits(max_abs).get_val();
	double num_d = static_cast<double>(num_f);
	double den_d = static_cast<double>(den_f);

	if (LIBC_UNLIKELY(max_abs >= 0x7f80'0000U \|\| num_d == 0.0)) {
	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();
	}
	double x_d = static_cast<double>(x);
	double y_d = static_cast<double>(y);
	size_t x_except = (x_d == 0.0) ? 0 : (x_abs == 0x7f80'0000 ? 2 : 1);
	size_t y_except = (y_d == 0.0) ? 0 : (y_abs == 0x7f80'0000 ? 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 double EXCEPTS[3][3][2] = {
	{{0.0, PI}, {0.0, PI}, {0.0, PI}},
	{{PI_OVER_2, PI_OVER_2}, {0.0, 0.0}, {0.0, PI}},
	{{PI_OVER_2, PI_OVER_2},
	{PI_OVER_2, PI_OVER_2},
	{PI_OVER_4, THREE_PI_OVER_4}},
	};

	double r = IS_NEG[y_sign] * EXCEPTS[y_except][x_except][x_sign];

	return static_cast<float>(r);
	}

	bool recip = x_abs < y_abs;
	double final_sign = IS_NEG[(x_sign != y_sign) != recip];
	fputil::DoubleDouble const_term = CONST_ADJ[x_sign][y_sign][recip];
	double q_d = num_d / den_d;

	double k_d = fputil::nearest_integer(q_d * 0x1.0p4);
	int idx = static_cast<int>(k_d);
	double r;

	#ifdef LIBC_MATH_HAS_SMALL_TABLES
	double p = atan_eval_no_table(num_d, den_d, k_d * 0x1.0p-4);
	r = final_sign * (p + (const_term.hi + ATAN_K_OVER_16[idx]));
	#else
	q_d = fputil::multiply_add(k_d, -0x1.0p-4, q_d);

	double p = atan_eval(q_d, idx);
	r = final_sign *
	fputil::multiply_add(q_d, p, const_term.hi + ATAN_COEFFS[idx][0]);
	#endif // LIBC_MATH_HAS_SMALL_TABLES

	#ifdef LIBC_MATH_HAS_SKIP_ACCURATE_PASS
	return static_cast<float>(r);
	#else
	constexpr uint32_t LOWER_ERR = 4;
	// Mask sticky bits in double precision before rounding to single precision.
	constexpr uint32_t MASK =
	mask_trailing_ones<uint32_t, fputil::FPBits<double>::SIG_LEN -
	FPBits::SIG_LEN - 1>();
	constexpr uint32_t UPPER_ERR = MASK - LOWER_ERR;

	uint32_t r_bits = static_cast<uint32_t>(cpp::bit_cast<uint64_t>(r)) & MASK;

	// Ziv's rounding test.
	if (LIBC_LIKELY(r_bits > LOWER_ERR && r_bits < UPPER_ERR))
	return static_cast<float>(r);

	return atan2f_double_double(num_d, den_d, q_d, idx, k_d, final_sign,
	const_term);
	#endif // LIBC_MATH_HAS_SKIP_ACCURATE_PASS
	}

	} // namespace LIBC_NAMESPACE_DECL

	#endif