mul()

static diyfp nlohmann::detail::dtoa_impl::diyfp::mul ( const diyfp &  x, const diyfp &  y  )

inlinestaticnoexcept

returns x * y

Note

The result is rounded. (Only the upper q bits are returned.)

Definition at line 15614 of file json.hpp.

    {
        static_assert(kPrecision == 64, "internal error");
        // Computes:
        //  f = round((x.f * y.f) / 2^q)
        //  e = x.e + y.e + q
        // Emulate the 64-bit * 64-bit multiplication:
        //
        // p = u * v
        //   = (u_lo + 2^32 u_hi) (v_lo + 2^32 v_hi)
        //   = (u_lo v_lo         ) + 2^32 ((u_lo v_hi         ) + (u_hi v_lo         )) + 2^64 (u_hi v_hi         )
        //   = (p0                ) + 2^32 ((p1                ) + (p2                )) + 2^64 (p3                )
        //   = (p0_lo + 2^32 p0_hi) + 2^32 ((p1_lo + 2^32 p1_hi) + (p2_lo + 2^32 p2_hi)) + 2^64 (p3                )
        //   = (p0_lo             ) + 2^32 (p0_hi + p1_lo + p2_lo                      ) + 2^64 (p1_hi + p2_hi + p3)
        //   = (p0_lo             ) + 2^32 (Q                                          ) + 2^64 (H                 )
        //   = (p0_lo             ) + 2^32 (Q_lo + 2^32 Q_hi                           ) + 2^64 (H                 )
        //
        // (Since Q might be larger than 2^32 - 1)
        //
        //   = (p0_lo + 2^32 Q_lo) + 2^64 (Q_hi + H)
        //
        // (Q_hi + H does not overflow a 64-bit int)
        //
        //   = p_lo + 2^64 p_hi
        const std::uint64_t u_lo = x.f & 0xFFFFFFFFu;
        const std::uint64_t u_hi = x.f >> 32u;
        const std::uint64_t v_lo = y.f & 0xFFFFFFFFu;
        const std::uint64_t v_hi = y.f >> 32u;
        const std::uint64_t p0 = u_lo * v_lo;
        const std::uint64_t p1 = u_lo * v_hi;
        const std::uint64_t p2 = u_hi * v_lo;
        const std::uint64_t p3 = u_hi * v_hi;
        const std::uint64_t p0_hi = p0 >> 32u;
        const std::uint64_t p1_lo = p1 & 0xFFFFFFFFu;
        const std::uint64_t p1_hi = p1 >> 32u;
        const std::uint64_t p2_lo = p2 & 0xFFFFFFFFu;
        const std::uint64_t p2_hi = p2 >> 32u;
        std::uint64_t Q = p0_hi + p1_lo + p2_lo;
        // The full product might now be computed as
        //
        // p_hi = p3 + p2_hi + p1_hi + (Q >> 32)
        // p_lo = p0_lo + (Q << 32)
        //
        // But in this particular case here, the full p_lo is not required.
        // Effectively we only need to add the highest bit in p_lo to p_hi (and
        // Q_hi + 1 does not overflow).
        Q += std::uint64_t{1} << (64u - 32u - 1u); // round, ties up
        const std::uint64_t h = p3 + p2_hi + p1_hi + (Q >> 32u);
        return {h, x.e + y.e + 64};
    }