//===-- Collection of utils for atan/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_MATH_GENERIC_ATAN_UTILS_H
#define LLVM_LIBC_SRC_MATH_GENERIC_ATAN_UTILS_H

#include "src/__support/FPUtil/double_double.h"
#include "src/__support/FPUtil/multiply_add.h"
#include "src/__support/macros/config.h"

namespace LIBC_NAMESPACE_DECL {

namespace {

using DoubleDouble = fputil::DoubleDouble;

// atan(i/64) with i = 0..64, generated by Sollya with:
// > for i from 0 to 64 do {
//     a = round(atan(i/64), D, RN);
//     b = round(atan(i/64) - a, D, RN);
//     print("{", b, ",", a, "},");
//   };
constexpr fputil::DoubleDouble ATAN_I[65] = {
    {0.0, 0.0},
    {-0x1.220c39d4dff5p-61, 0x1.fff555bbb729bp-7},
    {-0x1.5ec431444912cp-60, 0x1.ffd55bba97625p-6},
    {-0x1.86ef8f794f105p-63, 0x1.7fb818430da2ap-5},
    {-0x1.c934d86d23f1dp-60, 0x1.ff55bb72cfdeap-5},
    {0x1.ac4ce285df847p-58, 0x1.3f59f0e7c559dp-4},
    {-0x1.cfb654c0c3d98p-58, 0x1.7ee182602f10fp-4},
    {0x1.f7b8f29a05987p-58, 0x1.be39ebe6f07c3p-4},
    {-0x1.cd37686760c17p-59, 0x1.fd5ba9aac2f6ep-4},
    {-0x1.b485914dacf8cp-59, 0x1.1e1fafb043727p-3},
    {0x1.61a3b0ce9281bp-57, 0x1.3d6eee8c6626cp-3},
    {-0x1.054ab2c010f3dp-58, 0x1.5c9811e3ec26ap-3},
    {0x1.347b0b4f881cap-58, 0x1.7b97b4bce5b02p-3},
    {0x1.cf601e7b4348ep-59, 0x1.9a6a8e96c8626p-3},
    {0x1.17b10d2e0e5abp-61, 0x1.b90d7529260a2p-3},
    {0x1.c648d1534597ep-57, 0x1.d77d5df205736p-3},
    {0x1.8ab6e3cf7afbdp-57, 0x1.f5b75f92c80ddp-3},
    {0x1.62e47390cb865p-56, 0x1.09dc597d86362p-2},
    {0x1.30ca4748b1bf9p-57, 0x1.18bf5a30bf178p-2},
    {-0x1.077cdd36dfc81p-56, 0x1.278372057ef46p-2},
    {-0x1.963a544b672d8p-57, 0x1.362773707ebccp-2},
    {-0x1.5d5e43c55b3bap-56, 0x1.44aa436c2af0ap-2},
    {-0x1.2566480884082p-57, 0x1.530ad9951cd4ap-2},
    {-0x1.a725715711fp-56, 0x1.614840309cfe2p-2},
    {-0x1.c63aae6f6e918p-56, 0x1.6f61941e4def1p-2},
    {0x1.69c885c2b249ap-56, 0x1.7d5604b63b3f7p-2},
    {0x1.b6d0ba3748fa8p-56, 0x1.8b24d394a1b25p-2},
    {0x1.9e6c988fd0a77p-56, 0x1.98cd5454d6b18p-2},
    {-0x1.24dec1b50b7ffp-56, 0x1.a64eec3cc23fdp-2},
    {0x1.ae187b1ca504p-56, 0x1.b3a911da65c6cp-2},
    {-0x1.cc1ce70934c34p-56, 0x1.c0db4c94ec9fp-2},
    {-0x1.a2cfa4418f1adp-56, 0x1.cde53432c1351p-2},
    {0x1.a2b7f222f65e2p-56, 0x1.dac670561bb4fp-2},
    {0x1.0e53dc1bf3435p-56, 0x1.e77eb7f175a34p-2},
    {-0x1.a3992dc382a23p-57, 0x1.f40dd0b541418p-2},
    {-0x1.b32c949c9d593p-55, 0x1.0039c73c1a40cp-1},
    {-0x1.d5b495f6349e6p-56, 0x1.0657e94db30dp-1},
    {0x1.974fa13b5404fp-58, 0x1.0c6145b5b43dap-1},
    {-0x1.2bdaee1c0ee35p-58, 0x1.1255d9bfbd2a9p-1},
    {0x1.c621cec00c301p-55, 0x1.1835a88be7c13p-1},
    {-0x1.928df287a668fp-58, 0x1.1e00babdefeb4p-1},
    {0x1.c421c9f38224ep-57, 0x1.23b71e2cc9e6ap-1},
    {-0x1.09e73b0c6c087p-56, 0x1.2958e59308e31p-1},
    {0x1.c5d5e9ff0cf8dp-55, 0x1.2ee628406cbcap-1},
    {0x1.1021137c71102p-55, 0x1.345f01cce37bbp-1},
    {-0x1.2304331d8bf46p-55, 0x1.39c391cd4171ap-1},
    {0x1.ecf8b492644fp-56, 0x1.3f13fb89e96f4p-1},
    {-0x1.f76d0163f79c8p-56, 0x1.445065b795b56p-1},
    {0x1.2419a87f2a458p-56, 0x1.4978fa3269ee1p-1},
    {0x1.4a33dbeb3796cp-55, 0x1.4e8de5bb6ec04p-1},
    {-0x1.1bb74abda520cp-55, 0x1.538f57b89061fp-1},
    {-0x1.5e5c9d8c5a95p-56, 0x1.587d81f732fbbp-1},
    {0x1.0028e4bc5e7cap-57, 0x1.5d58987169b18p-1},
    {-0x1.2b785350ee8c1p-57, 0x1.6220d115d7b8ep-1},
    {-0x1.6ea6febe8bbbap-56, 0x1.66d663923e087p-1},
    {-0x1.a80386188c50ep-55, 0x1.6b798920b3d99p-1},
    {-0x1.8c34d25aadef6p-56, 0x1.700a7c5784634p-1},
    {0x1.7b2a6165884a1p-59, 0x1.748978fba8e0fp-1},
    {0x1.406a08980374p-55, 0x1.78f6bbd5d315ep-1},
    {0x1.560821e2f3aa9p-55, 0x1.7d528289fa093p-1},
    {-0x1.bf76229d3b917p-56, 0x1.819d0b7158a4dp-1},
    {0x1.6b66e7fc8b8c3p-57, 0x1.85d69576cc2c5p-1},
    {-0x1.55b9a5e177a1bp-55, 0x1.89ff5ff57f1f8p-1},
    {-0x1.ec182ab042f61p-56, 0x1.8e17aa99cc05ep-1},
    {0x1.1a62633145c07p-55, 0x1.921fb54442d18p-1},
};

// Approximate atan(x) for |x| <= 2^-7.
// Using degree-9 Taylor polynomial:
//  P = x - x^3/3 + x^5/5 -x^7/7 + x^9/9;
// Then the absolute error is bounded by:
//   |atan(x) - P(x)| < |x|^11/11 < 2^(-7*11) / 11 < 2^-80.
// And the relative error is bounded by:
//   |(atan(x) - P(x))/atan(x)| < |x|^10 / 10 < 2^-73.
// For x = x_hi + x_lo, fully expand the polynomial and drop any terms less than
//   ulp(x_hi^3 / 3) gives us:
// P(x) ~ x_hi - x_hi^3/3 + x_hi^5/5 - x_hi^7/7 + x_hi^9/9 +
//        + x_lo * (1 - x_hi^2 + x_hi^4)
// Since p.lo is ~ x^3/3, the relative error from rounding is bounded by:
//   |(atan(x) - P(x))/atan(x)| < ulp(x^2) <= 2^(-14-52) = 2^-66.
DoubleDouble atan_eval(const DoubleDouble &x) {
  DoubleDouble p;
  p.hi = x.hi;
  double x_hi_sq = x.hi * x.hi;
  // c0 ~ x_hi^2 * 1/5 - 1/3
  double c0 = fputil::multiply_add(x_hi_sq, 0x1.999999999999ap-3,
                                   -0x1.5555555555555p-2);
  // c1 ~ x_hi^2 * 1/9 - 1/7
  double c1 = fputil::multiply_add(x_hi_sq, 0x1.c71c71c71c71cp-4,
                                   -0x1.2492492492492p-3);
  // x_hi^3
  double x_hi_3 = x_hi_sq * x.hi;
  // x_hi^4
  double x_hi_4 = x_hi_sq * x_hi_sq;
  // d0 ~ 1/3 - x_hi^2 / 5 + x_hi^4 / 7 - x_hi^6 / 9
  double d0 = fputil::multiply_add(x_hi_4, c1, c0);
  // x_lo - x_lo * x_hi^2 + x_lo * x_hi^4
  double d1 = fputil::multiply_add(x_hi_4 - x_hi_sq, x.lo, x.lo);
  // p.lo ~ -x_hi^3/3 + x_hi^5/5 - x_hi^7/7 + x_hi^9/9 +
  //        + x_lo * (1 - x_hi^2 + x_hi^4)
  p.lo = fputil::multiply_add(x_hi_3, d0, d1);
  return p;
}

} // anonymous namespace

} // namespace LIBC_NAMESPACE_DECL

#endif // LLVM_LIBC_SRC_MATH_GENERIC_ATAN_UTILS_H