Number.cpp

#include <xrpl/basics/Number.h>
// Keep Number.h first to ensure it can build without hidden dependencies
#include <xrpl/basics/contract.h>
#include <xrpl/beast/utility/instrumentation.h>
 
#include <algorithm>
#include <cstddef>
#include <cstdint>
#include <iterator>
#include <limits>
#include <numeric>
#include <stdexcept>
#include <string>
#include <type_traits>
#include <utility>
 
#ifdef _MSC_VER
#pragma message("Using boost::multiprecision::uint128_t and int128_t")
#include <boost/multiprecision/cpp_int.hpp>
using uint128_t = boost::multiprecision::uint128_t;
using int128_t = boost::multiprecision::int128_t;
#else   // !defined(_MSC_VER)
using uint128_t = __uint128_t;
using int128_t = __int128_t;
#endif  // !defined(_MSC_VER)
 
namespace xrpl {
 
thread_local Number::rounding_mode Number::mode_ = Number::to_nearest;
thread_local std::reference_wrapper<MantissaRange const> Number::range_ =
    largeRange;
 
Number::rounding_mode
Number::getround()
{
    return mode_;
}
 
Number::rounding_mode
Number::setround(rounding_mode mode)
{
    return std::exchange(mode_, mode);
}
 
MantissaRange::mantissa_scale
Number::getMantissaScale()
{
    return range_.get().scale;
}
 
void
Number::setMantissaScale(MantissaRange::mantissa_scale scale)
{
    if (scale != MantissaRange::small && scale != MantissaRange::large)
        LogicError("Unknown mantissa scale");
    range_ = scale == MantissaRange::small ? smallRange : largeRange;
}
 
// Guard
 
// The Guard class is used to temporarily add extra digits of
// precision to an operation.  This enables the final result
// to be correctly rounded to the internal precision of Number.
 
template <class T>
concept UnsignedMantissa =
    std::is_unsigned_v<T> || std::is_same_v<T, uint128_t>;
 
class Number::Guard
{
    std::uint64_t digits_;   // 16 decimal guard digits
    std::uint8_t xbit_ : 1;  // has a non-zero digit been shifted off the end
    std::uint8_t sbit_ : 1;  // the sign of the guard digits
 
public:
    explicit Guard() : digits_{0}, xbit_{0}, sbit_{0}
    {
    }
 
    // set & test the sign bit
    void
    set_positive() noexcept;
    void
    set_negative() noexcept;
    bool
    is_negative() const noexcept;
 
    // add a digit
    template <class T>
    void
    push(T d) noexcept;
 
    // recover a digit
    unsigned
    pop() noexcept;
 
    // Indicate round direction:  1 is up, -1 is down, 0 is even
    // This enables the client to round towards nearest, and on
    // tie, round towards even.
    int
    round() noexcept;
 
    // Modify the result to the correctly rounded value
    template <UnsignedMantissa T>
    void
    doRoundUp(
        bool& negative,
        T& mantissa,
        int& exponent,
        internalrep const& minMantissa,
        internalrep const& maxMantissa,
        std::string location);
 
    // Modify the result to the correctly rounded value
    template <UnsignedMantissa T>
    void
    doRoundDown(
        bool& negative,
        T& mantissa,
        int& exponent,
        internalrep const& minMantissa);
 
    // Modify the result to the correctly rounded value
    void
    doRound(rep& drops, std::string location);
 
private:
    void
    doPush(unsigned d) noexcept;
 
    template <UnsignedMantissa T>
    void
    bringIntoRange(
        bool& negative,
        T& mantissa,
        int& exponent,
        internalrep const& minMantissa);
};
 
inline void
Number::Guard::set_positive() noexcept
{
    sbit_ = 0;
}
 
inline void
Number::Guard::set_negative() noexcept
{
    sbit_ = 1;
}
 
inline bool
Number::Guard::is_negative() const noexcept
{
    return sbit_ == 1;
}
 
inline void
Number::Guard::doPush(unsigned d) noexcept
{
    xbit_ = xbit_ || ((digits_ & 0x0000'0000'0000'000F) != 0);
    digits_ >>= 4;
    digits_ |= (d & 0x0000'0000'0000'000FULL) << 60;
}
 
template <class T>
inline void
Number::Guard::push(T d) noexcept
{
    doPush(static_cast<unsigned>(d));
}
 
inline unsigned
Number::Guard::pop() noexcept
{
    unsigned d = (digits_ & 0xF000'0000'0000'0000) >> 60;
    digits_ <<= 4;
    return d;
}
 
// Returns:
//     -1 if Guard is less than half
//      0 if Guard is exactly half
//      1 if Guard is greater than half
int
Number::Guard::round() noexcept
{
    auto mode = Number::getround();
 
    if (mode == towards_zero)
        return -1;
 
    if (mode == downward)
    {
        if (sbit_)
        {
            if (digits_ > 0 || xbit_)
                return 1;
        }
        return -1;
    }
 
    if (mode == upward)
    {
        if (sbit_)
            return -1;
        if (digits_ > 0 || xbit_)
            return 1;
        return -1;
    }
 
    // assume round to nearest if mode is not one of the predefined values
    if (digits_ > 0x5000'0000'0000'0000)
        return 1;
    if (digits_ < 0x5000'0000'0000'0000)
        return -1;
    if (xbit_)
        return 1;
    return 0;
}
 
template <UnsignedMantissa T>
void
Number::Guard::bringIntoRange(
    bool& negative,
    T& mantissa,
    int& exponent,
    internalrep const& minMantissa)
{
    // Bring mantissa back into the minMantissa / maxMantissa range AFTER
    // rounding
    if (mantissa < minMantissa)
    {
        mantissa *= 10;
        --exponent;
    }
    if (exponent < minExponent)
    {
        constexpr Number zero = Number{};
 
        negative = zero.negative_;
        mantissa = zero.mantissa_;
        exponent = zero.exponent_;
    }
}
 
template <UnsignedMantissa T>
void
Number::Guard::doRoundUp(
    bool& negative,
    T& mantissa,
    int& exponent,
    internalrep const& minMantissa,
    internalrep const& maxMantissa,
    std::string location)
{
    auto r = round();
    if (r == 1 || (r == 0 && (mantissa & 1) == 1))
    {
        ++mantissa;
        // Ensure mantissa after incrementing fits within both the
        // min/maxMantissa range and is a valid "rep".
        if (mantissa > maxMantissa || mantissa > maxRep)
        {
            mantissa /= 10;
            ++exponent;
        }
    }
    bringIntoRange(negative, mantissa, exponent, minMantissa);
    if (exponent > maxExponent)
        throw std::overflow_error(location);
}
 
template <UnsignedMantissa T>
void
Number::Guard::doRoundDown(
    bool& negative,
    T& mantissa,
    int& exponent,
    internalrep const& minMantissa)
{
    auto r = round();
    if (r == 1 || (r == 0 && (mantissa & 1) == 1))
    {
        --mantissa;
        if (mantissa < minMantissa)
        {
            mantissa *= 10;
            --exponent;
        }
    }
    bringIntoRange(negative, mantissa, exponent, minMantissa);
}
 
// Modify the result to the correctly rounded value
void
Number::Guard::doRound(rep& drops, std::string location)
{
    auto r = round();
    if (r == 1 || (r == 0 && (drops & 1) == 1))
    {
        if (drops >= maxRep)
        {
            static_assert(sizeof(internalrep) == sizeof(rep));
            // This should be impossible, because it's impossible to represent
            // "maxRep + 0.6" in Number, regardless of the scale. There aren't
            // enough digits available. You'd either get a mantissa of "maxRep"
            // or "(maxRep + 1) / 10", neither of which will round up when
            // converting to rep, though the latter might overflow _before_
            // rounding.
            throw std::overflow_error(location);  // LCOV_EXCL_LINE
        }
        ++drops;
    }
    if (is_negative())
        drops = -drops;
}
 
// Number
 
// Safely convert rep (int64) mantissa to internalrep (uint64). If the rep is
// negative, returns the positive value. This takes a little extra work because
// converting std::numeric_limits<std::int64_t>::min() flirts with UB, and can
// vary across compilers.
Number::internalrep
Number::externalToInternal(rep mantissa)
{
    // If the mantissa is already positive, just return it
    if (mantissa >= 0)
        return mantissa;
    // If the mantissa is negative, but fits within the positive range of rep,
    // return it negated
    if (mantissa >= -std::numeric_limits<rep>::max())
        return -mantissa;
 
    // If the mantissa doesn't fit within the positive range, convert to
    // int128_t, negate that, and cast it back down to the internalrep
    // In practice, this is only going to cover the case of
    // std::numeric_limits<rep>::min().
    int128_t temp = mantissa;
    return static_cast<internalrep>(-temp);
}
 
constexpr Number
Number::oneSmall()
{
    return Number{
        false,
        Number::smallRange.min,
        -Number::smallRange.log,
        Number::unchecked{}};
};
 
constexpr Number oneSml = Number::oneSmall();
 
constexpr Number
Number::oneLarge()
{
    return Number{
        false,
        Number::largeRange.min,
        -Number::largeRange.log,
        Number::unchecked{}};
};
 
constexpr Number oneLrg = Number::oneLarge();
 
Number
Number::one()
{
    if (&range_.get() == &smallRange)
        return oneSml;
    XRPL_ASSERT(&range_.get() == &largeRange, "Number::one() : valid range_");
    return oneLrg;
}
 
// Use the member names in this static function for now so the diff is cleaner
// TODO: Rename the function parameters to get rid of the "_" suffix
template <class T>
void
doNormalize(
    bool& negative,
    T& mantissa_,
    int& exponent_,
    MantissaRange::rep const& minMantissa,
    MantissaRange::rep const& maxMantissa)
{
    auto constexpr minExponent = Number::minExponent;
    auto constexpr maxExponent = Number::maxExponent;
    auto constexpr maxRep = Number::maxRep;
 
    using Guard = Number::Guard;
 
    constexpr Number zero = Number{};
    if (mantissa_ == 0)
    {
        mantissa_ = zero.mantissa_;
        exponent_ = zero.exponent_;
        negative = zero.negative_;
        return;
    }
    auto m = mantissa_;
    while ((m < minMantissa) && (exponent_ > minExponent))
    {
        m *= 10;
        --exponent_;
    }
    Guard g;
    if (negative)
        g.set_negative();
    while (m > maxMantissa)
    {
        if (exponent_ >= maxExponent)
            throw std::overflow_error("Number::normalize 1");
        g.push(m % 10);
        m /= 10;
        ++exponent_;
    }
    if ((exponent_ < minExponent) || (m < minMantissa))
    {
        mantissa_ = zero.mantissa_;
        exponent_ = zero.exponent_;
        negative = zero.negative_;
        return;
    }
 
    // When using the largeRange, "m" needs fit within an int64, even if
    // the final mantissa_ is going to end up larger to fit within the
    // MantissaRange. Cut it down here so that the rounding will be done while
    // it's smaller.
    //
    // Example: 9,900,000,000,000,123,456 > 9,223,372,036,854,775,807,
    //      so "m" will be modified to 990,000,000,000,012,345. Then that value
    //      will be rounded to 990,000,000,000,012,345 or
    //      990,000,000,000,012,346, depending on the rounding mode. Finally,
    //      mantissa_ will be "m*10" so it fits within the range, and end up as
    //      9,900,000,000,000,123,450 or 9,900,000,000,000,123,460.
    // mantissa() will return mantissa_ / 10, and exponent() will return
    // exponent_ + 1.
    if (m > maxRep)
    {
        if (exponent_ >= maxExponent)
            throw std::overflow_error("Number::normalize 1.5");
        g.push(m % 10);
        m /= 10;
        ++exponent_;
    }
    // Before modification, m should be within the min/max range. After
    // modification, it must be less than maxRep. In other words, the original
    // value should have been no more than maxRep * 10.
    // (maxRep * 10 > maxMantissa)
    XRPL_ASSERT_PARTS(
        m <= maxRep,
        "xrpl::doNormalize",
        "intermediate mantissa fits in int64");
    mantissa_ = m;
 
    g.doRoundUp(
        negative,
        mantissa_,
        exponent_,
        minMantissa,
        maxMantissa,
        "Number::normalize 2");
    XRPL_ASSERT_PARTS(
        mantissa_ >= minMantissa && mantissa_ <= maxMantissa,
        "xrpl::doNormalize",
        "final mantissa fits in range");
}
 
template <>
void
Number::normalize<uint128_t>(
    bool& negative,
    uint128_t& mantissa,
    int& exponent,
    internalrep const& minMantissa,
    internalrep const& maxMantissa)
{
    doNormalize(negative, mantissa, exponent, minMantissa, maxMantissa);
}
 
template <>
void
Number::normalize<unsigned long long>(
    bool& negative,
    unsigned long long& mantissa,
    int& exponent,
    internalrep const& minMantissa,
    internalrep const& maxMantissa)
{
    doNormalize(negative, mantissa, exponent, minMantissa, maxMantissa);
}
 
template <>
void
Number::normalize<unsigned long>(
    bool& negative,
    unsigned long& mantissa,
    int& exponent,
    internalrep const& minMantissa,
    internalrep const& maxMantissa)
{
    doNormalize(negative, mantissa, exponent, minMantissa, maxMantissa);
}
 
void
Number::normalize()
{
    auto const& range = range_.get();
    normalize(negative_, mantissa_, exponent_, range.min, range.max);
}
 
// Copy the number, but set a new exponent. Because the mantissa doesn't change,
// the result will be "mostly" normalized, but the exponent could go out of
// range.
Number
Number::shiftExponent(int exponentDelta) const
{
    XRPL_ASSERT_PARTS(isnormal(), "xrpl::Number::shiftExponent", "normalized");
    auto const newExponent = exponent_ + exponentDelta;
    if (newExponent >= maxExponent)
        throw std::overflow_error("Number::shiftExponent");
    if (newExponent < minExponent)
    {
        return Number{};
    }
    Number const result{negative_, mantissa_, newExponent, unchecked{}};
    XRPL_ASSERT_PARTS(
        result.isnormal(),
        "xrpl::Number::shiftExponent",
        "result is normalized");
    return result;
}
 
Number&
Number::operator+=(Number const& y)
{
    constexpr Number zero = Number{};
    if (y == zero)
        return *this;
    if (*this == zero)
    {
        *this = y;
        return *this;
    }
    if (*this == -y)
    {
        *this = zero;
        return *this;
    }
 
    XRPL_ASSERT(
        isnormal() && y.isnormal(),
        "xrpl::Number::operator+=(Number) : is normal");
    // *n = negative
    // *s = sign
    // *m = mantissa
    // *e = exponent
 
    // Need to use uint128_t, because large mantissas can overflow when added
    // together.
    bool xn = negative_;
    uint128_t xm = mantissa_;
    auto xe = exponent_;
 
    bool yn = y.negative_;
    uint128_t ym = y.mantissa_;
    auto ye = y.exponent_;
    Guard g;
    if (xe < ye)
    {
        if (xn)
            g.set_negative();
        do
        {
            g.push(xm % 10);
            xm /= 10;
            ++xe;
        } while (xe < ye);
    }
    else if (xe > ye)
    {
        if (yn)
            g.set_negative();
        do
        {
            g.push(ym % 10);
            ym /= 10;
            ++ye;
        } while (xe > ye);
    }
 
    auto const& range = range_.get();
    auto const& minMantissa = range.min;
    auto const& maxMantissa = range.max;
 
    if (xn == yn)
    {
        xm += ym;
        if (xm > maxMantissa || xm > maxRep)
        {
            g.push(xm % 10);
            xm /= 10;
            ++xe;
        }
        g.doRoundUp(
            xn, xm, xe, minMantissa, maxMantissa, "Number::addition overflow");
    }
    else
    {
        if (xm > ym)
        {
            xm = xm - ym;
        }
        else
        {
            xm = ym - xm;
            xe = ye;
            xn = yn;
        }
        while (xm < minMantissa && xm * 10 <= maxRep)
        {
            xm *= 10;
            xm -= g.pop();
            --xe;
        }
        g.doRoundDown(xn, xm, xe, minMantissa);
    }
 
    negative_ = xn;
    mantissa_ = static_cast<internalrep>(xm);
    exponent_ = xe;
    normalize();
    return *this;
}
 
// Optimization equivalent to:
// auto r = static_cast<unsigned>(u % 10);
// u /= 10;
// return r;
// Derived from Hacker's Delight Second Edition Chapter 10
// by Henry S. Warren, Jr.
static inline unsigned
divu10(uint128_t& u)
{
    // q = u * 0.75
    auto q = (u >> 1) + (u >> 2);
    // iterate towards q = u * 0.8
    q += q >> 4;
    q += q >> 8;
    q += q >> 16;
    q += q >> 32;
    q += q >> 64;
    // q /= 8 approximately == u / 10
    q >>= 3;
    // r = u - q * 10  approximately == u % 10
    auto r = static_cast<unsigned>(u - ((q << 3) + (q << 1)));
    // correction c is 1 if r >= 10 else 0
    auto c = (r + 6) >> 4;
    u = q + c;
    r -= c * 10;
    return r;
}
 
Number&
Number::operator*=(Number const& y)
{
    constexpr Number zero = Number{};
    if (*this == zero)
        return *this;
    if (y == zero)
    {
        *this = y;
        return *this;
    }
    // *n = negative
    // *s = sign
    // *m = mantissa
    // *e = exponent
 
    bool xn = negative_;
    int xs = xn ? -1 : 1;
    internalrep xm = mantissa_;
    auto xe = exponent_;
 
    bool yn = y.negative_;
    int ys = yn ? -1 : 1;
    internalrep ym = y.mantissa_;
    auto ye = y.exponent_;
 
    auto zm = uint128_t(xm) * uint128_t(ym);
    auto ze = xe + ye;
    auto zs = xs * ys;
    bool zn = (zs == -1);
    Guard g;
    if (zn)
        g.set_negative();
 
    auto const& range = range_.get();
    auto const& minMantissa = range.min;
    auto const& maxMantissa = range.max;
 
    while (zm > maxMantissa || zm > maxRep)
    {
        // The following is optimization for:
        // g.push(static_cast<unsigned>(zm % 10));
        // zm /= 10;
        g.push(divu10(zm));
        ++ze;
    }
    xm = static_cast<internalrep>(zm);
    xe = ze;
    g.doRoundUp(
        zn,
        xm,
        xe,
        minMantissa,
        maxMantissa,
        "Number::multiplication overflow : exponent is " + std::to_string(xe));
    negative_ = zn;
    mantissa_ = xm;
    exponent_ = xe;
 
    normalize();
    return *this;
}
 
Number&
Number::operator/=(Number const& y)
{
    constexpr Number zero = Number{};
    if (y == zero)
        throw std::overflow_error("Number: divide by 0");
    if (*this == zero)
        return *this;
    // n* = numerator
    // d* = denominator
    // *p = negative (positive?)
    // *s = sign
    // *m = mantissa
    // *e = exponent
 
    bool np = negative_;
    int ns = (np ? -1 : 1);
    auto nm = mantissa_;
    auto ne = exponent_;
 
    bool dp = y.negative_;
    int ds = (dp ? -1 : 1);
    auto dm = y.mantissa_;
    auto de = y.exponent_;
 
    auto const& range = range_.get();
    auto const& minMantissa = range.min;
    auto const& maxMantissa = range.max;
 
    // Shift by 10^17 gives greatest precision while not overflowing
    // uint128_t or the cast back to int64_t
    // TODO: Can/should this be made bigger for largeRange?
    // log(2^128,10) ~ 38.5
    // largeRange.log = 18, fits in 10^19
    // f can be up to 10^(38-19) = 10^19 safely
    static_assert(smallRange.log == 15);
    static_assert(largeRange.log == 18);
    bool small = Number::getMantissaScale() == MantissaRange::small;
    uint128_t const f =
        small ? 100'000'000'000'000'000 : 10'000'000'000'000'000'000ULL;
    XRPL_ASSERT_PARTS(
        f >= minMantissa * 10, "Number::operator/=", "factor expected size");
 
    // unsigned denominator
    auto const dmu = static_cast<uint128_t>(dm);
    // correctionFactor can be anything between 10 and f, depending on how much
    // extra precision we want to only use for rounding with the
    // largeRange. Three digits seems like plenty, and is more than
    // the smallRange uses.
    uint128_t const correctionFactor = 1'000;
 
    auto const numerator = uint128_t(nm) * f;
 
    auto zm = numerator / dmu;
    auto ze = ne - de - (small ? 17 : 19);
    bool zn = (ns * ds) < 0;
    if (!small)
    {
        // Virtually multiply numerator by correctionFactor. Since that would
        // overflow in the existing uint128_t, we'll do that part separately.
        // The math for this would work for small mantissas, but we need to
        // preserve existing behavior.
        //
        // Consider:
        // ((numerator * correctionFactor) / dmu) / correctionFactor
        // = ((numerator / dmu) * correctionFactor) / correctionFactor)
        //
        // But that assumes infinite precision. With integer math, this is
        // equivalent to
        //
        // = ((numerator / dmu * correctionFactor)
        //   + ((numerator % dmu) * correctionFactor) / dmu) / correctionFactor
        //
        // We have already set `mantissa_ = numerator / dmu`. Now we
        // compute `remainder = numerator % dmu`, and if it is
        // nonzero, we do the rest of the arithmetic. If it's zero, we can skip
        // it.
        auto const remainder = (numerator % dmu);
        if (remainder != 0)
        {
            zm *= correctionFactor;
            auto const correction = remainder * correctionFactor / dmu;
            zm += correction;
            // divide by 1000 by moving the exponent, so we don't lose the
            // integer value we just computed
            ze -= 3;
        }
    }
    normalize(zn, zm, ze, minMantissa, maxMantissa);
    negative_ = zn;
    mantissa_ = static_cast<internalrep>(zm);
    exponent_ = ze;
    XRPL_ASSERT_PARTS(
        isnormal(), "xrpl::Number::operator/=", "result is normalized");
 
    return *this;
}
 
Number::operator rep() const
{
    rep drops = mantissa();
    int offset = exponent();
    Guard g;
    if (drops != 0)
    {
        if (negative_)
        {
            g.set_negative();
            drops = -drops;
        }
        for (; offset < 0; ++offset)
        {
            g.push(drops % 10);
            drops /= 10;
        }
        for (; offset > 0; --offset)
        {
            if (drops > maxRep / 10)
                throw std::overflow_error("Number::operator rep() overflow");
            drops *= 10;
        }
        g.doRound(drops, "Number::operator rep() rounding overflow");
    }
    return drops;
}
 
Number
Number::truncate() const noexcept
{
    if (exponent_ >= 0 || mantissa_ == 0)
        return *this;
 
    Number ret = *this;
    while (ret.exponent_ < 0 && ret.mantissa_ != 0)
    {
        ret.exponent_ += 1;
        ret.mantissa_ /= rep(10);
    }
    // We are guaranteed that normalize() will never throw an exception
    // because exponent is either negative or zero at this point.
    ret.normalize();
    return ret;
}
 
std::string
to_string(Number const& amount)
{
    // keep full internal accuracy, but make more human friendly if possible
    constexpr Number zero = Number{};
    if (amount == zero)
        return "0";
 
    auto exponent = amount.exponent_;
    auto mantissa = amount.mantissa_;
    bool const negative = amount.negative_;
 
    // Use scientific notation for exponents that are too small or too large
    auto const rangeLog = Number::mantissaLog();
    if (((exponent != 0) &&
         ((exponent < -(rangeLog + 10)) || (exponent > -(rangeLog - 10)))))
    {
        while (mantissa != 0 && mantissa % 10 == 0 &&
               exponent < Number::maxExponent)
        {
            mantissa /= 10;
            ++exponent;
        }
        std::string ret = negative ? "-" : "";
        ret.append(std::to_string(mantissa));
        ret.append(1, 'e');
        ret.append(std::to_string(exponent));
        return ret;
    }
 
    XRPL_ASSERT(
        exponent + 43 > 0, "xrpl::to_string(Number) : minimum exponent");
 
    ptrdiff_t const pad_prefix = rangeLog + 12;
    ptrdiff_t const pad_suffix = rangeLog + 8;
 
    std::string const raw_value(std::to_string(mantissa));
    std::string val;
 
    val.reserve(raw_value.length() + pad_prefix + pad_suffix);
    val.append(pad_prefix, '0');
    val.append(raw_value);
    val.append(pad_suffix, '0');
 
    ptrdiff_t const offset(exponent + pad_prefix + rangeLog + 1);
 
    auto pre_from(val.begin());
    auto const pre_to(val.begin() + offset);
 
    auto const post_from(val.begin() + offset);
    auto post_to(val.end());
 
    // Crop leading zeroes. Take advantage of the fact that there's always a
    // fixed amount of leading zeroes and skip them.
    if (std::distance(pre_from, pre_to) > pad_prefix)
        pre_from += pad_prefix;
 
    XRPL_ASSERT(
        post_to >= post_from, "xrpl::to_string(Number) : first distance check");
 
    pre_from = std::find_if(pre_from, pre_to, [](char c) { return c != '0'; });
 
    // Crop trailing zeroes. Take advantage of the fact that there's always a
    // fixed amount of trailing zeroes and skip them.
    if (std::distance(post_from, post_to) > pad_suffix)
        post_to -= pad_suffix;
 
    XRPL_ASSERT(
        post_to >= post_from,
        "xrpl::to_string(Number) : second distance check");
 
    post_to = std::find_if(
                  std::make_reverse_iterator(post_to),
                  std::make_reverse_iterator(post_from),
                  [](char c) { return c != '0'; })
                  .base();
 
    std::string ret;
 
    if (negative)
        ret.append(1, '-');
 
    // Assemble the output:
    if (pre_from == pre_to)
        ret.append(1, '0');
    else
        ret.append(pre_from, pre_to);
 
    if (post_to != post_from)
    {
        ret.append(1, '.');
        ret.append(post_from, post_to);
    }
 
    return ret;
}
 
// Returns f^n
// Uses a log_2(n) number of multiplications
 
Number
power(Number const& f, unsigned n)
{
    if (n == 0)
        return Number::one();
    if (n == 1)
        return f;
    auto r = power(f, n / 2);
    r *= r;
    if (n % 2 != 0)
        r *= f;
    return r;
}
 
// Returns f^(1/d)
// Uses Newton–Raphson iterations until the result stops changing
// to find the non-negative root of the polynomial g(x) = x^d - f
 
// This function, and power(Number f, unsigned n, unsigned d)
// treat corner cases such as 0 roots as advised by Annex F of
// the C standard, which itself is consistent with the IEEE
// floating point standards.
 
Number
root(Number f, unsigned d)
{
    constexpr Number zero = Number{};
    auto const one = Number::one();
 
    if (f == one || d == 1)
        return f;
    if (d == 0)
    {
        if (f == -one)
            return one;
        if (abs(f) < one)
            return zero;
        throw std::overflow_error("Number::root infinity");
    }
    if (f < zero && d % 2 == 0)
        throw std::overflow_error("Number::root nan");
    if (f == zero)
        return f;
 
    // Scale f into the range (0, 1) such that f's exponent is a multiple of d
    auto e = f.exponent_ + Number::mantissaLog() + 1;
    auto const di = static_cast<int>(d);
    auto ex = [e = e, di = di]()  // Euclidean remainder of e/d
    {
        int k = (e >= 0 ? e : e - (di - 1)) / di;
        int k2 = e - k * di;
        if (k2 == 0)
            return 0;
        return di - k2;
    }();
    e += ex;
    f = f.shiftExponent(-e);  // f /= 10^e;
 
    XRPL_ASSERT_PARTS(
        f.isnormal(), "xrpl::root(Number, unsigned)", "f is normalized");
    bool neg = false;
    if (f < zero)
    {
        neg = true;
        f = -f;
    }
 
    // Quadratic least squares curve fit of f^(1/d) in the range [0, 1]
    auto const D = ((6 * di + 11) * di + 6) * di + 1;
    auto const a0 = 3 * di * ((2 * di - 3) * di + 1);
    auto const a1 = 24 * di * (2 * di - 1);
    auto const a2 = -30 * (di - 1) * di;
    Number r = ((Number{a2} * f + Number{a1}) * f + Number{a0}) / Number{D};
    if (neg)
    {
        f = -f;
        r = -r;
    }
 
    //  Newton–Raphson iteration of f^(1/d) with initial guess r
    //  halt when r stops changing, checking for bouncing on the last iteration
    Number rm1{};
    Number rm2{};
    do
    {
        rm2 = rm1;
        rm1 = r;
        r = (Number(d - 1) * r + f / power(r, d - 1)) / Number(d);
    } while (r != rm1 && r != rm2);
 
    //  return r * 10^(e/d) to reverse scaling
    auto const result = r.shiftExponent(e / di);
    XRPL_ASSERT_PARTS(
        result.isnormal(),
        "xrpl::root(Number, unsigned)",
        "result is normalized");
    return result;
}
 
Number
root2(Number f)
{
    constexpr Number zero = Number{};
    auto const one = Number::one();
 
    if (f == one)
        return f;
    if (f < zero)
        throw std::overflow_error("Number::root nan");
    if (f == zero)
        return f;
 
    // Scale f into the range (0, 1) such that f's exponent is a multiple of d
    auto e = f.exponent_ + Number::mantissaLog() + 1;
    if (e % 2 != 0)
        ++e;
    f = f.shiftExponent(-e);  // f /= 10^e;
    XRPL_ASSERT_PARTS(f.isnormal(), "xrpl::root2(Number)", "f is normalized");
 
    // Quadratic least squares curve fit of f^(1/d) in the range [0, 1]
    auto const D = 105;
    auto const a0 = 18;
    auto const a1 = 144;
    auto const a2 = -60;
    Number r = ((Number{a2} * f + Number{a1}) * f + Number{a0}) / Number{D};
 
    //  Newton–Raphson iteration of f^(1/2) with initial guess r
    //  halt when r stops changing, checking for bouncing on the last iteration
    Number rm1{};
    Number rm2{};
    do
    {
        rm2 = rm1;
        rm1 = r;
        r = (r + f / r) / Number(2);
    } while (r != rm1 && r != rm2);
 
    //  return r * 10^(e/2) to reverse scaling
    auto const result = r.shiftExponent(e / 2);
    XRPL_ASSERT_PARTS(
        result.isnormal(), "xrpl::root2(Number)", "result is normalized");
 
    return result;
}
 
// Returns f^(n/d)
 
Number
power(Number const& f, unsigned n, unsigned d)
{
    constexpr Number zero = Number{};
    auto const one = Number::one();
 
    if (f == one)
        return f;
    auto g = std::gcd(n, d);
    if (g == 0)
        throw std::overflow_error("Number::power nan");
    if (d == 0)
    {
        if (f == -one)
            return one;
        if (abs(f) < one)
            return zero;
        // abs(f) > one
        throw std::overflow_error("Number::power infinity");
    }
    if (n == 0)
        return one;
    n /= g;
    d /= g;
    if ((n % 2) == 1 && (d % 2) == 0 && f < zero)
        throw std::overflow_error("Number::power nan");
    return root(power(f, n), d);
}
 
}  // namespace xrpl