src/math/generic/cbrt.cpp

//===-- Implementation of cbrt 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/cbrt.h"
#include "hdr/fenv_macros.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/dyadic_float.h"
#include "src/__support/FPUtil/multiply_add.h"
#include "src/__support/common.h"
#include "src/__support/integer_literals.h"
#include "src/__support/macros/config.h"
#include "src/__support/macros/optimization.h" // LIBC_UNLIKELY

#if ((LIBC_MATH & LIBC_MATH_SKIP_ACCURATE_PASS) != 0)
#define LIBC_MATH_CBRT_SKIP_ACCURATE_PASS
#endif

namespace LIBC_NAMESPACE_DECL {

using DoubleDouble = fputil::DoubleDouble;
using Float128 = fputil::DyadicFloat<128>;

namespace {

// Initial approximation of x^(-2/3) for 1 <= x < 2.
// Polynomial generated by Sollya with:
// > P = fpminimax(x^(-2/3), 7, [|D...|], [1, 2]);
// > dirtyinfnorm(P/x^(-2/3) - 1, [1, 2]);
// 0x1.28...p-21
double intial_approximation(double x) {
  constexpr double COEFFS[8] = {
      0x1.bc52aedead5c6p1,  -0x1.b52bfebf110b3p2,  0x1.1d8d71d53d126p3,
      -0x1.de2db9e81cf87p2, 0x1.0154ca06153bdp2,   -0x1.5973c66ee6da7p0,
      0x1.07bf6ac832552p-2, -0x1.5e53d9ce41cb8p-6,
  };

  double x_sq = x * x;

  double c0 = fputil::multiply_add(x, COEFFS[1], COEFFS[0]);
  double c1 = fputil::multiply_add(x, COEFFS[3], COEFFS[2]);
  double c2 = fputil::multiply_add(x, COEFFS[5], COEFFS[4]);
  double c3 = fputil::multiply_add(x, COEFFS[7], COEFFS[6]);

  double x_4 = x_sq * x_sq;
  double d0 = fputil::multiply_add(x_sq, c1, c0);
  double d1 = fputil::multiply_add(x_sq, c3, c2);

  return fputil::multiply_add(x_4, d1, d0);
}

// Get the error term for Newton iteration:
//   h(x) = x^3 * a^2 - 1,
#ifdef LIBC_TARGET_CPU_HAS_FMA
double get_error(const DoubleDouble &x_3, const DoubleDouble &a_sq) {
  return fputil::multiply_add(x_3.hi, a_sq.hi, -1.0) +
         fputil::multiply_add(x_3.lo, a_sq.hi, x_3.hi * a_sq.lo);
}
#else
double get_error(const DoubleDouble &x_3, const DoubleDouble &a_sq) {
  DoubleDouble x_3_a_sq = fputil::quick_mult(a_sq, x_3);
  return (x_3_a_sq.hi - 1.0) + x_3_a_sq.lo;
}
#endif

} // anonymous namespace

// Correctly rounded cbrt algorithm:
//
// === Step 1 - Range reduction ===
// For x = (-1)^s * 2^e * (1.m), we get 2 reduced arguments x_r and a as:
//   x_r = 1.m
//   a   = (-1)^s * 2^(e % 3) * (1.m)
// Then cbrt(x) = x^(1/3) can be computed as:
//   x^(1/3) = 2^(e / 3) * a^(1/3).
//
// In order to avoid division, we compute a^(-2/3) using Newton method and then
// multiply the results by a:
//   a^(1/3) = a * a^(-2/3).
//
// === Step 2 - First approximation to a^(-2/3) ===
// First, we use a degree-7 minimax polynomial generated by Sollya to
// approximate x_r^(-2/3) for 1 <= x_r < 2.
//   p = P(x_r) ~ x_r^(-2/3),
// with relative errors bounded by:
//   | p / x_r^(-2/3) - 1 | < 1.16 * 2^-21.
//
// Then we multiply with 2^(e % 3) from a small lookup table to get:
//   x_0 = 2^(-2*(e % 3)/3) * p
//       ~ 2^(-2*(e % 3)/3) * x_r^(-2/3)
//       = a^(-2/3)
// With relative errors:
//   | x_0 / a^(-2/3) - 1 | < 1.16 * 2^-21.
// This step is done in double precision.
//
// === Step 3 - First Newton iteration ===
// We follow the method described in:
//   Sibidanov, A. and Zimmermann, P., "Correctly rounded cubic root evaluation
//   in double precision", https://core-math.gitlabpages.inria.fr/cbrt64.pdf
// to derive multiplicative Newton iterations as below:
// Let x_n be the nth approximation to a^(-2/3).  Define the n^th error as:
//   h_n = x_n^3 * a^2 - 1
// Then:
//   a^(-2/3) = x_n / (1 + h_n)^(1/3)
//            = x_n * (1 - (1/3) * h_n + (2/9) * h_n^2 - (14/81) * h_n^3 + ...)
// using the Taylor series expansion of (1 + h_n)^(-1/3).
//
// Apply to x_0 above:
//   h_0 = x_0^3 * a^2 - 1
//       = a^2 * (x_0 - a^(-2/3)) * (x_0^2 + x_0 * a^(-2/3) + a^(-4/3)),
// it's bounded by:
//   |h_0| < 4 * 3 * 1.16 * 2^-21 * 4 < 2^-17.
// So in the first iteration step, we use:
//   x_1 = x_0 * (1 - (1/3) * h_n + (2/9) * h_n^2 - (14/81) * h_n^3)
// Its relative error is bounded by:
//   | x_1 / a^(-2/3) - 1 | < 35/242 * |h_0|^4 < 2^-70.
// Then we perform Ziv's rounding test and check if the answer is exact.
// This step is done in double-double precision.
//
// === Step 4 - Second Newton iteration ===
// If the Ziv's rounding test from the previous step fails, we define the error
// term:
//   h_1 = x_1^3 * a^2 - 1,
// And perform another iteration:
//   x_2 = x_1 * (1 - h_1 / 3)
// with the relative errors exceed the precision of double-double.
// We then check the Ziv's accuracy test with relative errors < 2^-102 to
// compensate for rounding errors.
//
// === Step 5 - Final iteration ===
// If the Ziv's accuracy test from the previous step fails, we perform another
// iteration in 128-bit precision and check for exact outputs.
//
// TODO: It is possible to replace this costly computation step with special
// exceptional handling, similar to what was done in the CORE-MATH project:
// https://gitlab.inria.fr/core-math/core-math/-/blob/master/src/binary64/cbrt/cbrt.c

LLVM_LIBC_FUNCTION(double, cbrt, (double x)) {
  using FPBits = fputil::FPBits<double>;

  uint64_t x_abs = FPBits(x).abs().uintval();

  unsigned exp_bias_correction = 682; // 1023 * 2/3

  if (LIBC_UNLIKELY(x_abs < FPBits::min_normal().uintval() ||
                    x_abs >= FPBits::inf().uintval())) {
    if (x_abs == 0 || x_abs >= FPBits::inf().uintval())
      // x is 0, Inf, or NaN.
      return x;

    // x is non-zero denormal number.
    // Normalize x.
    x *= 0x1.0p60;
    exp_bias_correction -= 20;
  }

  FPBits x_bits(x);

  // When using biased exponent of x in double precision,
  //   x_e = real_exponent_of_x + 1023
  // Then:
  //   x_e / 3 = real_exponent_of_x / 3 + 1023/3
  //           = real_exponent_of_x / 3 + 341
  // So to make it the correct biased exponent of x^(1/3), we add
  //   1023 - 341 = 682
  // to the quotient x_e / 3.
  unsigned x_e = static_cast<unsigned>(x_bits.get_biased_exponent());
  unsigned out_e = (x_e / 3 + exp_bias_correction);
  unsigned shift_e = x_e % 3;

  // Set x_r = 1.mantissa
  double x_r =
      FPBits(x_bits.get_mantissa() |
             (static_cast<uint64_t>(FPBits::EXP_BIAS) << FPBits::FRACTION_LEN))
          .get_val();

  // Set a = (-1)^x_sign * 2^(x_e % 3) * (1.mantissa)
  uint64_t a_bits = x_bits.uintval() & 0x800F'FFFF'FFFF'FFFF;
  a_bits |=
      (static_cast<uint64_t>(shift_e + static_cast<unsigned>(FPBits::EXP_BIAS))
       << FPBits::FRACTION_LEN);
  double a = FPBits(a_bits).get_val();

  // Initial approximation of x_r^(-2/3).
  double p = intial_approximation(x_r);

  // Look up for 2^(-2*n/3) used for first approximation step.
  constexpr double EXP2_M2_OVER_3[3] = {1.0, 0x1.428a2f98d728bp-1,
                                        0x1.965fea53d6e3dp-2};

  // x0 is an initial approximation of a^(-2/3) for 1 <= |a| < 8.
  // Relative error: < 1.16 * 2^(-21).
  double x0 = static_cast<double>(EXP2_M2_OVER_3[shift_e] * p);

  // First iteration in double precision.
  DoubleDouble a_sq = fputil::exact_mult(a, a);

  // h0 = x0^3 * a^2 - 1
  DoubleDouble x0_sq = fputil::exact_mult(x0, x0);
  DoubleDouble x0_3 = fputil::quick_mult(x0, x0_sq);

  double h0 = get_error(x0_3, a_sq);

#ifdef LIBC_MATH_CBRT_SKIP_ACCURATE_PASS
  constexpr double REL_ERROR = 0;
#else
  constexpr double REL_ERROR = 0x1.0p-51;
#endif // LIBC_MATH_CBRT_SKIP_ACCURATE_PASS

  // Taylor polynomial of (1 + h)^(-1/3):
  //   (1 + h)^(-1/3) = 1 - h/3 + 2 h^2 / 9 - 14 h^3 / 81 + ...
  constexpr double ERR_COEFFS[3] = {
      -0x1.5555555555555p-2 - REL_ERROR, // -1/3 - relative_error
      0x1.c71c71c71c71cp-3,              // 2/9
      -0x1.61f9add3c0ca4p-3,             // -14/81
  };
  // e0 = -14 * h^2 / 81 + 2 * h / 9 - 1/3 - relative_error.
  double e0 = fputil::polyeval(h0, ERR_COEFFS[0], ERR_COEFFS[1], ERR_COEFFS[2]);
  double x0_h0 = x0 * h0;

  // x1 = x0 (1 - h0/3 + 2 h0^2 / 9 - 14 h0^3 / 81)
  // x1 approximate a^(-2/3) with relative errors bounded by:
  //   | x1 / a^(-2/3) - 1 | < (34/243) h0^4 < h0 * REL_ERROR
  DoubleDouble x1_dd{x0_h0 * e0, x0};

  // r1 = x1 * a ~ a^(-2/3) * a = a^(1/3).
  DoubleDouble r1 = fputil::quick_mult(a, x1_dd);

  // Lambda function to update the exponent of the result.
  auto update_exponent = [=](double r) -> double {
    uint64_t r_m = FPBits(r).uintval() - 0x3FF0'0000'0000'0000;
    // Adjust exponent and sign.
    uint64_t r_bits =
        r_m + (static_cast<uint64_t>(out_e) << FPBits::FRACTION_LEN);
    return FPBits(r_bits).get_val();
  };

#ifdef LIBC_MATH_CBRT_SKIP_ACCURATE_PASS
  // TODO: We probably don't need to use double-double if accurate tests and
  // passes are skipped.
  return update_exponent(r1.hi + r1.lo);
#else
  // Accurate checks and passes.
  double r1_lower = r1.hi + r1.lo;
  double r1_upper =
      r1.hi + fputil::multiply_add(x0_h0, 2.0 * REL_ERROR * a, r1.lo);

  // Ziv's accuracy test.
  if (LIBC_LIKELY(r1_upper == r1_lower)) {
    // Test for exact outputs.
    // Check if lower (52 - 17 = 35) bits are 0's.
    if (LIBC_UNLIKELY((FPBits(r1_lower).uintval() & 0x0000'0007'FFFF'FFFF) ==
                      0)) {
      double r1_err = (r1_lower - r1.hi) - r1.lo;
      if (FPBits(r1_err).abs().get_val() < 0x1.0p69)
        fputil::clear_except_if_required(FE_INEXACT);
    }

    return update_exponent(r1_lower);
  }

  // Accuracy test failed, perform another Newton iteration.
  double x1 = x1_dd.hi + (e0 + REL_ERROR) * x0_h0;

  // Second iteration in double-double precision.
  // h1 = x1^3 * a^2 - 1.
  DoubleDouble x1_sq = fputil::exact_mult(x1, x1);
  DoubleDouble x1_3 = fputil::quick_mult(x1, x1_sq);
  double h1 = get_error(x1_3, a_sq);

  // e1 = -x1*h1/3.
  double e1 = h1 * (x1 * -0x1.5555555555555p-2);
  // x2 = x1*(1 - h1/3) = x1 + e1 ~ a^(-2/3) with relative errors < 2^-101.
  DoubleDouble x2 = fputil::exact_add(x1, e1);
  // r2 = a * x2 ~ a * a^(-2/3) = a^(1/3) with relative errors < 2^-100.
  DoubleDouble r2 = fputil::quick_mult(a, x2);

  double r2_upper = r2.hi + fputil::multiply_add(a, 0x1.0p-102, r2.lo);
  double r2_lower = r2.hi + fputil::multiply_add(a, -0x1.0p-102, r2.lo);

  // Ziv's accuracy test.
  if (LIBC_LIKELY(r2_upper == r2_lower))
    return update_exponent(r2_upper);

  // TODO: Investigate removing float128 and just list exceptional cases.
  // Apply another Newton iteration with ~126-bit accuracy.
  Float128 x2_f128 = fputil::quick_add(Float128(x2.hi), Float128(x2.lo));
  // x2^3
  Float128 x2_3 =
      fputil::quick_mul(fputil::quick_mul(x2_f128, x2_f128), x2_f128);
  // a^2
  Float128 a_sq_f128 = fputil::quick_mul(Float128(a), Float128(a));
  // x2^3 * a^2
  Float128 x2_3_a_sq = fputil::quick_mul(x2_3, a_sq_f128);
  // h2 = x2^3 * a^2 - 1
  Float128 h2_f128 = fputil::quick_add(x2_3_a_sq, Float128(-1.0));
  double h2 = static_cast<double>(h2_f128);
  // t2 = 1 - h2 / 3
  Float128 t2 =
      fputil::quick_add(Float128(1.0), Float128(h2 * (-0x1.5555555555555p-2)));
  // x3 = x2 * (1 - h2 / 3) ~ a^(-2/3)
  Float128 x3 = fputil::quick_mul(x2_f128, t2);
  // r3 = a * x3 ~ a * a^(-2/3) = a^(1/3)
  Float128 r3 = fputil::quick_mul(Float128(a), x3);

  // Check for exact cases:
  Float128::MantissaType rounding_bits =
      r3.mantissa & 0x0000'0000'0000'03FF'FFFF'FFFF'FFFF'FFFF_u128;

  double result = static_cast<double>(r3);
  if ((rounding_bits < 0x0000'0000'0000'0000'0000'0000'0000'000F_u128) ||
      (rounding_bits >= 0x0000'0000'0000'03FF'FFFF'FFFF'FFFF'FFF0_u128)) {
    // Output is exact.
    r3.mantissa &= 0xFFFF'FFFF'FFFF'FFFF'FFFF'FFFF'FFFF'FFF0_u128;

    if (rounding_bits >= 0x0000'0000'0000'03FF'FFFF'FFFF'FFFF'FFF0_u128) {
      Float128 tmp{r3.sign, r3.exponent - 123,
                   0x8000'0000'0000'0000'0000'0000'0000'0000_u128};
      Float128 r4 = fputil::quick_add(r3, tmp);
      result = static_cast<double>(r4);
    } else {
      result = static_cast<double>(r3);
    }

    fputil::clear_except_if_required(FE_INEXACT);
  }

  return update_exponent(result);
#endif // LIBC_MATH_CBRT_SKIP_ACCURATE_PASS
}

} // namespace LIBC_NAMESPACE_DECL