utils/FPUtil/SqrtLongDoubleX86.h

//===-- Square root of x86 long double numbers ------------------*- 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_UTILS_FPUTIL_SQRT_LONG_DOUBLE_X86_H
#define LLVM_LIBC_UTILS_FPUTIL_SQRT_LONG_DOUBLE_X86_H

#include "FPBits.h"
#include "Sqrt.h"

#include "utils/CPP/TypeTraits.h"

namespace __llvm_libc {
namespace fputil {

namespace internal {

template <>
inline void normalize<long double>(int &exponent, __uint128_t &mantissa) {
  // Use binary search to shift the leading 1 bit similar to float.
  // With MantissaWidth<long double> = 63, it will take
  // ceil(log2(63)) = 6 steps checking the mantissa bits.
  constexpr int nsteps = 6; // = ceil(log2(MantissaWidth))
  constexpr __uint128_t bounds[nsteps] = {
      __uint128_t(1) << 32, __uint128_t(1) << 48, __uint128_t(1) << 56,
      __uint128_t(1) << 60, __uint128_t(1) << 62, __uint128_t(1) << 63};
  constexpr int shifts[nsteps] = {32, 16, 8, 4, 2, 1};

  for (int i = 0; i < nsteps; ++i) {
    if (mantissa < bounds[i]) {
      exponent -= shifts[i];
      mantissa <<= shifts[i];
    }
  }
}

} // namespace internal

// Correctly rounded SQRT with round to nearest, ties to even.
// Shift-and-add algorithm.
template <> inline long double sqrt<long double, 0>(long double x) {
  using UIntType = typename FPBits<long double>::UIntType;
  constexpr UIntType One = UIntType(1)
                           << int(MantissaWidth<long double>::value);

  FPBits<long double> bits(x);

  if (bits.isInfOrNaN()) {
    if (bits.getSign() && (bits.getMantissa() == 0)) {
      // sqrt(-Inf) = NaN
      return FPBits<long double>::buildNaN(One >> 1);
    } else {
      // sqrt(NaN) = NaN
      // sqrt(+Inf) = +Inf
      return x;
    }
  } else if (bits.isZero()) {
    // sqrt(+0) = +0
    // sqrt(-0) = -0
    return x;
  } else if (bits.getSign()) {
    // sqrt( negative numbers ) = NaN
    return FPBits<long double>::buildNaN(One >> 1);
  } else {
    int xExp = bits.getExponent();
    UIntType xMant = bits.getMantissa();

    // Step 1a: Normalize denormal input
    if (bits.getImplicitBit()) {
      xMant |= One;
    } else if (bits.getUnbiasedExponent() == 0) {
      internal::normalize<long double>(xExp, xMant);
    }

    // Step 1b: Make sure the exponent is even.
    if (xExp & 1) {
      --xExp;
      xMant <<= 1;
    }

    // After step 1b, x = 2^(xExp) * xMant, where xExp is even, and
    // 1 <= xMant < 4.  So sqrt(x) = 2^(xExp / 2) * y, with 1 <= y < 2.
    // Notice that the output of sqrt is always in the normal range.
    // To perform shift-and-add algorithm to find y, let denote:
    //   y(n) = 1.y_1 y_2 ... y_n, we can define the nth residue to be:
    //   r(n) = 2^n ( xMant - y(n)^2 ).
    // That leads to the following recurrence formula:
    //   r(n) = 2*r(n-1) - y_n*[ 2*y(n-1) + 2^(-n-1) ]
    // with the initial conditions: y(0) = 1, and r(0) = x - 1.
    // So the nth digit y_n of the mantissa of sqrt(x) can be found by:
    //   y_n = 1 if 2*r(n-1) >= 2*y(n - 1) + 2^(-n-1)
    //         0 otherwise.
    UIntType y = One;
    UIntType r = xMant - One;

    for (UIntType current_bit = One >> 1; current_bit; current_bit >>= 1) {
      r <<= 1;
      UIntType tmp = (y << 1) + current_bit; // 2*y(n - 1) + 2^(-n-1)
      if (r >= tmp) {
        r -= tmp;
        y += current_bit;
      }
    }

    // We compute one more iteration in order to round correctly.
    bool lsb = y & 1; // Least significant bit
    bool rb = false;  // Round bit
    r <<= 2;
    UIntType tmp = (y << 2) + 1;
    if (r >= tmp) {
      r -= tmp;
      rb = true;
    }

    // Append the exponent field.
    xExp = ((xExp >> 1) + FPBits<long double>::exponentBias);
    y |= (static_cast<UIntType>(xExp)
          << (MantissaWidth<long double>::value + 1));

    // Round to nearest, ties to even
    if (rb && (lsb || (r != 0))) {
      ++y;
    }

    // Extract output
    FPBits<long double> out(0.0L);
    out.setUnbiasedExponent(xExp);
    out.setImplicitBit(1);
    out.setMantissa((y & (One - 1)));

    return out;
  }
}

} // namespace fputil
} // namespace __llvm_libc

#endif // LLVM_LIBC_UTILS_FPUTIL_SQRT_LONG_DOUBLE_X86_H