atan2f.cpp | Explore in Territory

//===-- 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 "inv_trigf_utils.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

namespace LIBC_NAMESPACE_DECL {

namespace {

// 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] = …;

// 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] = …;

// 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) { … }

// 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) { … }

} // 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)) { … }

} // namespace LIBC_NAMESPACE_DECL
llvm/libc/src/math/generic/atan2f.cpp
