fixup! fixup! fixup! fixup! Address review feedback from @copilot

- Update explanations.
- Use saver conversions between signed and unsigned.

include/xrpl/basics/Number.h

@@ -9,6 +9,10 @@
 #include <ostream>
 #include <string>
 
+#ifdef _MSC_VER
+#include <boost/multiprecision/cpp_int.hpp>
+#endif  // !defined(_MSC_VER)
+
 namespace xrpl {
 
 class Number;
@@ -16,18 +20,37 @@ class Number;
 std::string
 to_string(Number const& amount);
 
+/** Returns a rough estimate of log10(value).
+ *
+ * The return value is a pair (log, rem), where log is the estimated log10,
+ * and rem is value divided by 10^log. If rem is 1, then value is an exact
+ * power of ten, and log is the exact log10(value).
+ *
+ * This function only works for positive values.
+ */
+template <typename T>
+constexpr std::pair<int, T>
+logTenEstimate(T value)
+{
+    int log = 0;
+    T remainder = value;
+    while (value >= 10)
+    {
+        if (value % 10 == 0)
+            remainder = remainder / 10;
+        value /= 10;
+        ++log;
+    }
+    return {log, remainder};
+}
+
 template <typename T>
 constexpr std::optional<int>
 logTen(T value)
 {
-    int log = 0;
+    auto const est = logTenEstimate(value);
-    while (value >= 10 && value % 10 == 0)
+    if (est.second == 1)
-    {
+        return est.first;
-        value /= 10;
-        ++log;
-    }
-    if (value == 1)
-        return log;
     return std::nullopt;
 }
 
@@ -41,12 +64,10 @@ isPowerOfTen(T value)
 /** MantissaRange defines a range for the mantissa of a normalized Number.
  *
  * The mantissa is in the range [min, max], where
- * * min is a power of 10, and
- * * max = min * 10 - 1.
  *
  * The mantissa_scale enum indicates whether the range is "small" or "large".
  * This intentionally restricts the number of MantissaRanges that can be
- * instantiated to two: one for each scale.
+ * used to two: one for each scale.
  *
  * The "small" scale is based on the behavior of STAmount for IOUs. It has a min
  * value of 10^15, and a max value of 10^16-1. This was sufficient for
@@ -60,8 +81,8 @@ isPowerOfTen(T value)
  * "large" scale.
  *
  * The "large" scale is intended to represent all values that can be represented
- * by an STAmount - IOUs, XRP, and MPTs. It has a min value of 10^18, and a max
+ * by an STAmount - IOUs, XRP, and MPTs. It has a min value of 2^63/10+1
- * value of 10^19-1.
+ * (truncated), and a max value of 2^63-1.
  *
  * Note that if the mentioned amendments are eventually retired, this class
  * should be left in place, but the "small" scale option should be removed. This
@@ -73,25 +94,50 @@ struct MantissaRange
     enum mantissa_scale { small, large };
 
     explicit constexpr MantissaRange(mantissa_scale scale_)
-        : min(getMin(scale_)), max(min * 10 - 1), log(logTen(min).value_or(-1)), scale(scale_)
+        : max(getMax(scale_))
+        , min(computeMin(max))
+        , referenceMin(getReferenceMin(scale_, min))
+        , log(computeLog(min))
+        , scale(scale_)
     {
+        // Since this is constexpr, if any of these throw, it won't compile
+        if (min * 10 <= max)
+            throw std::out_of_range("min * 10 <= max");
+        if (max / 10 >= min)
+            throw std::out_of_range("max / 10 >= min");
+        if ((min - 1) * 10 > max)
+            throw std::out_of_range("(min - 1) * 10 > max");
+        // This is a little hacky
+        if ((max + 10) / 10 < min)
+            throw std::out_of_range("(max + 10) / 10 < min");
     }
 
-    rep min;
+    // Explicitly delete copy and move operations
+    MantissaRange(MantissaRange const&) = delete;
+    MantissaRange(MantissaRange&&) = delete;
+    MantissaRange&
+    operator=(MantissaRange const&) = delete;
+    MantissaRange&
+    operator=(MantissaRange&&) = delete;
+
     rep max;
+    rep min;
+    // This is not a great name. Used to determine if mantissas are in range,
+    // but have fewer digits than max
+    rep referenceMin;
     int log;
     mantissa_scale scale;
 
 private:
     static constexpr rep
-    getMin(mantissa_scale scale_)
+    getMax(mantissa_scale scale)
     {
-        switch (scale_)
+        switch (scale)
         {
             case small:
-                return 1'000'000'000'000'000ULL;
+                return 9'999'999'999'999'999ULL;
             case large:
-                return 1'000'000'000'000'000'000ULL;
+                return std::numeric_limits<std::int64_t>::max();
             default:
                 // Since this can never be called outside a non-constexpr
                 // context, this throw assures that the build fails if an
@@ -99,19 +145,59 @@ private:
                 throw std::runtime_error("Unknown mantissa scale");
         }
     }
+
+    static constexpr rep
+    computeMin(rep max)
+    {
+        return max / 10 + 1;
+    }
+
+    static constexpr rep
+    getReferenceMin(mantissa_scale scale, rep min)
+    {
+        switch (scale)
+        {
+            case large:
+                return 1'000'000'000'000'000'000ULL;
+            default:
+                if (isPowerOfTen(min))
+                    return min;
+                throw std::runtime_error("Unknown/bad mantissa scale");
+        }
+    }
+
+    static constexpr rep
+    computeLog(rep min)
+    {
+        auto const estimate = logTenEstimate(min);
+        return estimate.first + (estimate.second == 1 ? 0 : 1);
+    }
 };
 
 // Like std::integral, but only 64-bit integral types.
 template <class T>
 concept Integral64 = std::is_same_v<T, std::int64_t> || std::is_same_v<T, std::uint64_t>;
 
+namespace detail {
+#ifdef _MSC_VER
+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)
+
+template <class T>
+concept UnsignedMantissa = std::is_unsigned_v<T> || std::is_same_v<T, uint128_t>;
+}  // namespace detail
+
 /** Number is a floating point type that can represent a wide range of values.
  *
  * It can represent all values that can be represented by an STAmount -
  * regardless of asset type - XRPAmount, MPTAmount, and IOUAmount, with at least
  * as much precision as those types require.
  *
- * ---- Internal Representation ----
+ * ---- Internal Operational Representation ----
  *
  * Internally, Number is represented with three values:
  *   1. a bool sign flag,
@@ -126,15 +212,21 @@ concept Integral64 = std::is_same_v<T, std::int64_t> || std::is_same_v<T, std::u
  *
  * A non-zero mantissa is (almost) always normalized, meaning it and the
  * exponent are grown or shrunk until the mantissa is in the range
- * [MantissaRange.min, MantissaRange.max].
+ * [MantissaRange.referenceMin, MantissaRange.referenceMin * 10 - 1].
+ *
+ * This internal representation is only used during some operations to ensure
+ * that the mantissa is a known, predictable size. The class itself stores the
+ * values using the external representation described below.
  *
  * Note:
  *   1. Normalization can be disabled by using the "unchecked" ctor tag. This
  *      should only be used at specific conversion points, some constexpr
  *      values, and in unit tests.
- *   2. The max of the "large" range, 10^19-1, is the largest 10^X-1 value that
+ *   2. Unlike MantissaRange.min, referenceMin is always an exact power of 10,
- *      fits in an unsigned 64-bit number. (10^19-1 < 2^64-1 and
+ *      so a mantissa in the internal representation will always have a
- *      10^20-1 > 2^64-1). This avoids under- and overflows.
+ *      consistent number of digits.
+ *   3. The functions toInternal() and fromInternal() are used to convert
+ *      between the two representations.
  *
  * ---- External Interface ----
  *
@@ -147,13 +239,12 @@ concept Integral64 = std::is_same_v<T, std::int64_t> || std::is_same_v<T, std::u
  * represent the full range of valid XRP and MPT integer values accurately.
  *
  * Note:
- *   1. 2^63-1 is between 10^18 and 10^19-1, which are the limits of the "large"
+ *   1. The "large" mantissa range is (2^63/10+1) to 2^63-1. 2^63-1 is between
- *      mantissa range.
+ *      10^18 and 10^19-1, and (2^63/10+1) is between 10^17 and 10^18-1. Thus,
+ *      the mantissa may have 18 or 19 digits. This value will be modified to
+ *      always have 19 digits before some operations to ensure consistency.
  *   2. The functions mantissa() and exponent() return the external view of the
- *      Number value, specifically using a signed 63-bit mantissa. This may
+ *      Number value, specifically using a signed 63-bit mantissa.
- *      require altering the internal representation to fit into that range
- *      before the value is returned. The interface guarantees consistency of
- *      the two values.
  *   3. Number cannot represent -2^63 (std::numeric_limits<std::int64_t>::min())
  *      as an exact integer, but it doesn't need to, because all asset values
  *      on-ledger are non-negative. This is due to implementation details of
@@ -208,8 +299,7 @@ class Number
     using rep = std::int64_t;
     using internalrep = MantissaRange::rep;
 
-    bool negative_{false};
+    rep mantissa_{0};
-    internalrep mantissa_{0};
     int exponent_{std::numeric_limits<int>::lowest()};
 
 public:
@@ -217,10 +307,6 @@ public:
     constexpr static int minExponent = -32768;
     constexpr static int maxExponent = 32768;
 
-    constexpr static internalrep maxRep = std::numeric_limits<rep>::max();
-    static_assert(maxRep == 9'223'372'036'854'775'807);
-    static_assert(-maxRep == std::numeric_limits<rep>::min() + 1);
-
     // May need to make unchecked private
     struct unchecked
     {
@@ -294,7 +380,7 @@ public:
     friend constexpr bool
     operator==(Number const& x, Number const& y) noexcept
     {
-        return x.negative_ == y.negative_ && x.mantissa_ == y.mantissa_ && x.exponent_ == y.exponent_;
+        return x.mantissa_ == y.mantissa_ && x.exponent_ == y.exponent_;
     }
 
     friend constexpr bool
@@ -308,8 +394,8 @@ public:
     {
         // If the two amounts have different signs (zero is treated as positive)
         // then the comparison is true iff the left is negative.
-        bool const lneg = x.negative_;
+        bool const lneg = x.mantissa_ < 0;
-        bool const rneg = y.negative_;
+        bool const rneg = y.mantissa_ < 0;
 
         if (lneg != rneg)
             return lneg;
@@ -337,7 +423,7 @@ public:
     constexpr int
     signum() const noexcept
     {
-        return negative_ ? -1 : (mantissa_ ? 1 : 0);
+        return mantissa_ < 0 ? -1 : (mantissa_ ? 1 : 0);
     }
 
     Number
@@ -376,6 +462,9 @@ public:
     friend Number
     root2(Number f);
 
+    friend Number
+    power(Number const& f, unsigned n, unsigned d);
+
     // Thread local rounding control.  Default is to_nearest
     enum rounding_mode { to_nearest, towards_zero, downward, upward };
     static rounding_mode
@@ -440,22 +529,39 @@ private:
     static_assert(isPowerOfTen(smallRange.min));
     static_assert(smallRange.min == 1'000'000'000'000'000LL);
     static_assert(smallRange.max == 9'999'999'999'999'999LL);
+    static_assert(smallRange.referenceMin == smallRange.min);
     static_assert(smallRange.log == 15);
-    static_assert(smallRange.min < maxRep);
-    static_assert(smallRange.max < maxRep);
     constexpr static MantissaRange largeRange{MantissaRange::large};
-    static_assert(isPowerOfTen(largeRange.min));
+    static_assert(!isPowerOfTen(largeRange.min));
-    static_assert(largeRange.min == 1'000'000'000'000'000'000ULL);
+    static_assert(largeRange.min == 922'337'203'685'477'581ULL);
-    static_assert(largeRange.max == internalrep(9'999'999'999'999'999'999ULL));
+    static_assert(largeRange.max == internalrep(9'223'372'036'854'775'807ULL));
+    static_assert(largeRange.max == std::numeric_limits<rep>::max());
+    static_assert(largeRange.referenceMin == 1'000'000'000'000'000'000ULL);
     static_assert(largeRange.log == 18);
-    static_assert(largeRange.min < maxRep);
+    // There are 2 values that will not fit in largeRange without some extra
-    static_assert(largeRange.max > maxRep);
+    // work
+    // * 9223372036854775808
+    // * 9223372036854775809
+    // They both end up < min, but with a leftover. If they round up, everything
+    // will be fine. If they don't, we'll need to bring them up into range.
+    // Guard::bringIntoRange handles this situation.
 
     // The range for the mantissa when normalized.
     // Use reference_wrapper to avoid making copies, and prevent accidentally
     // changing the values inside the range.
     static thread_local std::reference_wrapper<MantissaRange const> range_;
 
+    // And one is needed because it needs to choose between oneSmall and
+    // oneLarge based on the current range
+    static Number
+    one(MantissaRange const& range);
+
+    static Number
+    root(MantissaRange const& range, Number f, unsigned d);
+
+    void
+    normalize(MantissaRange const& range);
+
     void
     normalize();
 
@@ -478,11 +584,14 @@ private:
     friend void
     doNormalize(
         bool& negative,
-        T& mantissa_,
+        T& mantissa,
-        int& exponent_,
+        int& exponent,
         MantissaRange::rep const& minMantissa,
         MantissaRange::rep const& maxMantissa);
 
+    bool
+    isnormal(MantissaRange const& range) const noexcept;
+
     bool
     isnormal() const noexcept;
 
@@ -492,18 +601,64 @@ private:
     Number
     shiftExponent(int exponentDelta) const;
 
-    // Safely convert rep (int64) mantissa to internalrep (uint64). If the rep
+    // Safely return the absolute value of a rep (int64) mantissa as an internalrep (uint64).
-    // 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.
     static internalrep
     externalToInternal(rep mantissa);
 
+    /** Breaks down the number into components, potentially de-normalizing it.
+     *
+     * Ensures that the mantissa always has range_.log + 1 digits.
+     *
+     */
+    template <detail::UnsignedMantissa Rep = internalrep>
+    std::tuple<bool, Rep, int>
+    toInternal(MantissaRange const& range) const;
+
+    /** Breaks down the number into components, potentially de-normalizing it.
+     *
+     * Ensures that the mantissa always has range_.log + 1 digits.
+     *
+     */
+    template <detail::UnsignedMantissa Rep = internalrep>
+    std::tuple<bool, Rep, int>
+    toInternal() const;
+
+    /** Rebuilds the number from components.
+     *
+     * If "expectNormal" is true, the values are expected to be normalized - all
+     * in their valid ranges.
+     *
+     * If "expectNormal" is false, the values are expected to be "near
+     * normalized", meaning that the mantissa has to be modified at most once to
+     * bring it back into range.
+     *
+     */
+    template <bool expectNormal = true, detail::UnsignedMantissa Rep = internalrep>
+    void
+    fromInternal(bool negative, Rep mantissa, int exponent, MantissaRange const* pRange);
+
+    /** Rebuilds the number from components.
+     *
+     * If "expectNormal" is true, the values are expected to be normalized - all
+     * in their valid ranges.
+     *
+     * If "expectNormal" is false, the values are expected to be "near
+     * normalized", meaning that the mantissa has to be modified at most once to
+     * bring it back into range.
+     *
+     */
+    template <bool expectNormal = true, detail::UnsignedMantissa Rep = internalrep>
+    void
+    fromInternal(bool negative, Rep mantissa, int exponent);
+
     class Guard;
+
+public:
+    constexpr static internalrep largestMantissa = largeRange.max;
 };
 
 inline constexpr Number::Number(bool negative, internalrep mantissa, int exponent, unchecked) noexcept
-    : negative_(negative), mantissa_{mantissa}, exponent_{exponent}
+    : mantissa_{negative ? -static_cast<rep>(mantissa) : static_cast<rep>(mantissa)}, exponent_{exponent}
 {
 }
 
@@ -514,12 +669,6 @@ inline constexpr Number::Number(internalrep mantissa, int exponent, unchecked) n
 
 constexpr static Number numZero{};
 
-inline Number::Number(bool negative, internalrep mantissa, int exponent, normalized)
-    : Number(negative, mantissa, exponent, unchecked{})
-{
-    normalize();
-}
-
 inline Number::Number(internalrep mantissa, int exponent, normalized) : Number(false, mantissa, exponent, normalized{})
 {
 }
@@ -541,17 +690,7 @@ inline Number::Number(rep mantissa) : Number{mantissa, 0}
 inline constexpr Number::rep
 Number::mantissa() const noexcept
 {
-    auto m = mantissa_;
+    return mantissa_;
-    if (m > maxRep)
-    {
-        XRPL_ASSERT_PARTS(
-            !isnormal() || (m % 10 == 0 && m / 10 <= maxRep),
-            "xrpl::Number::mantissa",
-            "large normalized mantissa has no remainder");
-        m /= 10;
-    }
-    auto const sign = negative_ ? -1 : 1;
-    return sign * static_cast<Number::rep>(m);
 }
 
 /** Returns the exponent of the external view of the Number.
@@ -562,16 +701,7 @@ Number::mantissa() const noexcept
 inline constexpr int
 Number::exponent() const noexcept
 {
-    auto e = exponent_;
+    return exponent_;
-    if (mantissa_ > maxRep)
-    {
-        XRPL_ASSERT_PARTS(
-            !isnormal() || (mantissa_ % 10 == 0 && mantissa_ / 10 <= maxRep),
-            "xrpl::Number::exponent",
-            "large normalized mantissa has no remainder");
-        ++e;
-    }
-    return e;
 }
 
 inline constexpr Number
@@ -586,7 +716,7 @@ Number::operator-() const noexcept
     if (mantissa_ == 0)
         return Number{};
     auto x = *this;
-    x.negative_ = !x.negative_;
+    x.mantissa_ = -x.mantissa_;
     return x;
 }
 
@@ -667,39 +797,55 @@ Number::min() noexcept
 inline Number
 Number::max() noexcept
 {
-    return Number{false, std::min(range_.get().max, maxRep), maxExponent, unchecked{}};
+    return Number{false, range_.get().max, maxExponent, unchecked{}};
 }
 
 inline Number
 Number::lowest() noexcept
 {
-    return Number{true, std::min(range_.get().max, maxRep), maxExponent, unchecked{}};
+    return Number{true, range_.get().max, maxExponent, unchecked{}};
+}
+
+inline bool
+Number::isnormal(MantissaRange const& range) const noexcept
+{
+    auto const abs_m = externalToInternal(mantissa_);
+
+    return *this == Number{} ||
+        (range.min <= abs_m && abs_m <= range.max &&  //
+         minExponent <= exponent_ && exponent_ <= maxExponent);
 }
 
 inline bool
 Number::isnormal() const noexcept
 {
-    MantissaRange const& range = range_;
+    return isnormal(range_);
-    auto const abs_m = mantissa_;
-    return *this == Number{} ||
-        (range.min <= abs_m && abs_m <= range.max && (abs_m <= maxRep || abs_m % 10 == 0) && minExponent <= exponent_ &&
-         exponent_ <= maxExponent);
 }
 
 template <Integral64 T>
 std::pair<T, int>
 Number::normalizeToRange(T minMantissa, T maxMantissa) const
 {
-    bool negative = negative_;
+    bool negative = mantissa_ < 0;
-    internalrep mantissa = mantissa_;
+    internalrep mantissa = externalToInternal(mantissa_);
     int exponent = exponent_;
 
     if constexpr (std::is_unsigned_v<T>)
+    {
         XRPL_ASSERT_PARTS(!negative, "xrpl::Number::normalizeToRange", "Number is non-negative for unsigned range.");
+        // To avoid logical errors in release builds, throw if the Number is
+        // negative for an unsigned range.
+        if (negative)
+            throw std::runtime_error(
+                "Number::normalizeToRange: Number is negative for "
+                "unsigned range.");
+    }
     Number::normalize(negative, mantissa, exponent, minMantissa, maxMantissa);
 
-    auto const sign = negative ? -1 : 1;
+    // Cast mantissa to signed type first (if T is a signed type) to avoid
-    return std::make_pair(static_cast<T>(sign * mantissa), exponent);
+    // unsigned integer overflow when multiplying by negative sign
+    T signedMantissa = negative ? -static_cast<T>(mantissa) : static_cast<T>(mantissa);
+    return std::make_pair(signedMantissa, exponent);
 }
 
 inline constexpr Number

include/xrpl/protocol/Protocol.h

@@ -231,7 +231,7 @@ std::size_t constexpr maxMPTokenMetadataLength = 1024;
 
 /** The maximum amount of MPTokenIssuance */
 std::uint64_t constexpr maxMPTokenAmount = 0x7FFF'FFFF'FFFF'FFFFull;
-static_assert(Number::maxRep >= maxMPTokenAmount);
+static_assert(Number::largestMantissa >= maxMPTokenAmount);
 
 /** The maximum length of Data payload */
 std::size_t constexpr maxDataPayloadLength = 256;

include/xrpl/protocol/STAmount.h

@@ -521,6 +521,7 @@ STAmount::fromNumber(A const& a, Number const& number)
         return STAmount{asset, intValue, 0, negative};
     }
 
+    XRPL_ASSERT_PARTS(working.signum() >= 0, "xrpl::STAmount::fromNumber", "non-negative Number to normalize");
     auto const [mantissa, exponent] = working.normalizeToRange(cMinValue, cMaxValue);
 
     return STAmount{asset, mantissa, exponent, negative};

include/xrpl/protocol/SystemParameters.h

@@ -23,7 +23,7 @@ systemName()
 /** Number of drops in the genesis account. */
 constexpr XRPAmount INITIAL_XRP{100'000'000'000 * DROPS_PER_XRP};
 static_assert(INITIAL_XRP.drops() == 100'000'000'000'000'000);
-static_assert(Number::maxRep >= INITIAL_XRP.drops());
+static_assert(Number::largestMantissa >= INITIAL_XRP.drops());
 
 /** Returns true if the amount does not exceed the initial XRP in existence. */
 inline bool

src/libxrpl/basics/Number.cpp

@@ -11,18 +11,16 @@
 #include <numeric>
 #include <stdexcept>
 #include <string>
+#include <string_view>
 #include <type_traits>
 #include <utility>
 
 #ifdef _MSC_VER
 #pragma message("Using boost::multiprecision::uint128_t and int128_t")
-#include <boost/multiprecision/cpp_int.hpp>
+#endif
-using uint128_t = boost::multiprecision::uint128_t;
+
-using int128_t = boost::multiprecision::int128_t;
+using uint128_t = xrpl::detail::uint128_t;
-#else   // !defined(_MSC_VER)
+using int128_t = xrpl::detail::int128_t;
-using uint128_t = __uint128_t;
-using int128_t = __int128_t;
-#endif  // !defined(_MSC_VER)
 
 namespace xrpl {
 
@@ -61,9 +59,6 @@ Number::setMantissaScale(MantissaRange::mantissa_scale scale)
 // 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
@@ -99,7 +94,7 @@ public:
     round() noexcept;
 
     // Modify the result to the correctly rounded value
-    template <UnsignedMantissa T>
+    template <detail::UnsignedMantissa T>
     void
     doRoundUp(
         bool& negative,
@@ -107,22 +102,22 @@ public:
         int& exponent,
         internalrep const& minMantissa,
         internalrep const& maxMantissa,
-        std::string location);
+        std::string_view location);
 
     // Modify the result to the correctly rounded value
-    template <UnsignedMantissa T>
+    template <detail::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);
+    doRound(rep& drops, std::string_view location);
 
 private:
     void
     doPush(unsigned d) noexcept;
 
-    template <UnsignedMantissa T>
+    template <detail::UnsignedMantissa T>
     void
     bringIntoRange(bool& negative, T& mantissa, int& exponent, internalrep const& minMantissa);
 };
@@ -209,7 +204,7 @@ Number::Guard::round() noexcept
     return 0;
 }
 
-template <UnsignedMantissa T>
+template <detail::UnsignedMantissa T>
 void
 Number::Guard::bringIntoRange(bool& negative, T& mantissa, int& exponent, internalrep const& minMantissa)
 {
@@ -224,13 +219,13 @@ Number::Guard::bringIntoRange(bool& negative, T& mantissa, int& exponent, intern
     {
         constexpr Number zero = Number{};
 
-        negative = zero.negative_;
+        negative = false;
         mantissa = zero.mantissa_;
         exponent = zero.exponent_;
     }
 }
 
-template <UnsignedMantissa T>
+template <detail::UnsignedMantissa T>
 void
 Number::Guard::doRoundUp(
     bool& negative,
@@ -238,7 +233,7 @@ Number::Guard::doRoundUp(
     int& exponent,
     internalrep const& minMantissa,
     internalrep const& maxMantissa,
-    std::string location)
+    std::string_view location)
 {
     auto r = round();
     if (r == 1 || (r == 0 && (mantissa & 1) == 1))
@@ -246,7 +241,7 @@ Number::Guard::doRoundUp(
         ++mantissa;
         // Ensure mantissa after incrementing fits within both the
         // min/maxMantissa range and is a valid "rep".
-        if (mantissa > maxMantissa || mantissa > maxRep)
+        if (mantissa > maxMantissa)
         {
             mantissa /= 10;
             ++exponent;
@@ -254,10 +249,10 @@ Number::Guard::doRoundUp(
     }
     bringIntoRange(negative, mantissa, exponent, minMantissa);
     if (exponent > maxExponent)
-        throw std::overflow_error(location);
+        throw std::overflow_error(std::string{location});
 }
 
-template <UnsignedMantissa T>
+template <detail::UnsignedMantissa T>
 void
 Number::Guard::doRoundDown(bool& negative, T& mantissa, int& exponent, internalrep const& minMantissa)
 {
@@ -276,21 +271,22 @@ Number::Guard::doRoundDown(bool& negative, T& mantissa, int& exponent, internalr
 
 // Modify the result to the correctly rounded value
 void
-Number::Guard::doRound(rep& drops, std::string location)
+Number::Guard::doRound(rep& drops, std::string_view location)
 {
     auto r = round();
     if (r == 1 || (r == 0 && (drops & 1) == 1))
     {
-        if (drops >= maxRep)
+        auto const& range = range_.get();
+        if (drops >= range.max)
         {
             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
+            // "largestMantissa + 0.6" in Number, regardless of the scale. There aren't
-            // enough digits available. You'd either get a mantissa of "maxRep"
+            // enough digits available. You'd either get a mantissa of "largestMantissa "
-            // or "(maxRep + 1) / 10", neither of which will round up when
+            // or "largestMantissa / 10 + 1", 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
+            throw std::overflow_error(std::string{location});  // LCOV_EXCL_LINE
         }
         ++drops;
     }
@@ -310,23 +306,126 @@ 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
+    // Cast to unsigned before negating to avoid undefined behavior
-    // int128_t, negate that, and cast it back down to the internalrep
+    // when v == INT64_MIN (negating INT64_MIN in signed is UB)
-    // In practice, this is only going to cover the case of
+    return -static_cast<internalrep>(mantissa);
-    // std::numeric_limits<rep>::min().
+}
-    int128_t temp = mantissa;
+
-    return static_cast<internalrep>(-temp);
+/** Breaks down the number into components, potentially de-normalizing it.
+ *
+ * Ensures that the mantissa always has range_.log + 1 digits.
+ *
+ */
+template <detail::UnsignedMantissa Rep>
+std::tuple<bool, Rep, int>
+Number::toInternal(MantissaRange const& range) const
+{
+    auto exponent = exponent_;
+    bool const negative = mantissa_ < 0;
+    // It should be impossible for mantissa_ to be INT64_MIN, but use externalToInternal just in case.
+    Rep mantissa = static_cast<Rep>(externalToInternal(mantissa_));
+
+    auto const referenceMin = range.referenceMin;
+    auto const minMantissa = range.min;
+
+    if (mantissa != 0 && mantissa >= minMantissa && mantissa < referenceMin)
+    {
+        // Ensure the mantissa has the correct number of digits
+        mantissa *= 10;
+        --exponent;
+        XRPL_ASSERT_PARTS(
+            mantissa >= referenceMin && mantissa < referenceMin * 10,
+            "xrpl::Number::toInternal()",
+            "Number is within reference range and has 'log' digits");
+    }
+
+    return {negative, mantissa, exponent};
+}
+
+/** Breaks down the number into components, potentially de-normalizing it.
+ *
+ * Ensures that the mantissa always has exactly range_.log + 1 digits.
+ *
+ */
+template <detail::UnsignedMantissa Rep>
+std::tuple<bool, Rep, int>
+Number::toInternal() const
+{
+    return toInternal(range_);
+}
+
+/** Rebuilds the number from components.
+ *
+ * If "expectNormal" is true, the values are expected to be normalized - all
+ * in their valid ranges.
+ *
+ * If "expectNormal" is false, the values are expected to be "near
+ * normalized", meaning that the mantissa has to be modified at most once to
+ * bring it back into range.
+ *
+ */
+template <bool expectNormal, detail::UnsignedMantissa Rep>
+void
+Number::fromInternal(bool negative, Rep mantissa, int exponent, MantissaRange const* pRange)
+{
+    if constexpr (std::is_same_v<std::bool_constant<expectNormal>, std::false_type>)
+    {
+        if (!pRange)
+            throw std::runtime_error("Missing range to Number::fromInternal!");
+        auto const& range = *pRange;
+
+        auto const maxMantissa = range.max;
+        auto const minMantissa = range.min;
+
+        XRPL_ASSERT_PARTS(mantissa >= minMantissa, "xrpl::Number::fromInternal", "mantissa large enough");
+
+        if (mantissa > maxMantissa || mantissa < minMantissa)
+        {
+            normalize(negative, mantissa, exponent, range.min, maxMantissa);
+        }
+
+        XRPL_ASSERT_PARTS(
+            mantissa >= minMantissa && mantissa <= maxMantissa, "xrpl::Number::fromInternal", "mantissa in range");
+    }
+
+    // mantissa is unsigned, but it might not be uint64
+    mantissa_ = static_cast<rep>(static_cast<internalrep>(mantissa));
+    if (negative)
+        mantissa_ = -mantissa_;
+    exponent_ = exponent;
+
+    XRPL_ASSERT_PARTS(
+        (pRange && isnormal(*pRange)) || isnormal(), "xrpl::Number::fromInternal", "Number is normalized");
+}
+
+/** Rebuilds the number from components.
+ *
+ * If "expectNormal" is true, the values are expected to be normalized - all in
+ * their valid ranges.
+ *
+ * If "expectNormal" is false, the values are expected to be "near normalized",
+ * meaning that the mantissa has to be modified at most once to bring it back
+ * into range.
+ *
+ */
+template <bool expectNormal, detail::UnsignedMantissa Rep>
+void
+Number::fromInternal(bool negative, Rep mantissa, int exponent)
+{
+    MantissaRange const* pRange = nullptr;
+    if constexpr (std::is_same_v<std::bool_constant<expectNormal>, std::false_type>)
+    {
+        pRange = &Number::range_.get();
+    }
+
+    fromInternal(negative, mantissa, exponent, pRange);
 }
 
 constexpr Number
 Number::oneSmall()
 {
-    return Number{false, Number::smallRange.min, -Number::smallRange.log, Number::unchecked{}};
+    return Number{false, Number::smallRange.referenceMin, -Number::smallRange.log, Number::unchecked{}};
 };
 
 constexpr Number oneSml = Number::oneSmall();
@@ -334,101 +433,84 @@ constexpr Number oneSml = Number::oneSmall();
 constexpr Number
 Number::oneLarge()
 {
-    return Number{false, Number::largeRange.min, -Number::largeRange.log, Number::unchecked{}};
+    return Number{false, Number::largeRange.referenceMin, -Number::largeRange.log, Number::unchecked{}};
 };
 
 constexpr Number oneLrg = Number::oneLarge();
 
 Number
-Number::one()
+Number::one(MantissaRange const& range)
 {
-    if (&range_.get() == &smallRange)
+    if (&range == &smallRange)
         return oneSml;
-    XRPL_ASSERT(&range_.get() == &largeRange, "Number::one() : valid range_");
+    XRPL_ASSERT(&range == &largeRange, "Number::one() : valid range");
     return oneLrg;
 }
 
+Number
+Number::one()
+{
+    return one(range_);
+}
+
 // 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_,
+    T& mantissa,
-    int& exponent_,
+    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)
+    if (mantissa == 0 || (mantissa < minMantissa && exponent <= minExponent))
     {
-        mantissa_ = zero.mantissa_;
+        mantissa = zero.mantissa_;
-        exponent_ = zero.exponent_;
+        exponent = zero.exponent_;
-        negative = zero.negative_;
+        negative = false;
         return;
     }
-    auto m = mantissa_;
+
-    while ((m < minMantissa) && (exponent_ > minExponent))
+    auto m = mantissa;
+    while ((m < minMantissa) && (exponent > minExponent))
     {
         m *= 10;
-        --exponent_;
+        --exponent;
     }
     Guard g;
     if (negative)
         g.set_negative();
     while (m > maxMantissa)
     {
-        if (exponent_ >= maxExponent)
+        if (exponent >= maxExponent)
             throw std::overflow_error("Number::normalize 1");
         g.push(m % 10);
         m /= 10;
-        ++exponent_;
+        ++exponent;
     }
-    if ((exponent_ < minExponent) || (m < minMantissa))
+    if ((exponent < minExponent) || (m == 0))
     {
-        mantissa_ = zero.mantissa_;
+        mantissa = zero.mantissa_;
-        exponent_ = zero.exponent_;
+        exponent = zero.exponent_;
-        negative = zero.negative_;
+        negative = false;
         return;
     }
 
-    // When using the largeRange, "m" needs fit within an int64, even if
+    XRPL_ASSERT_PARTS(m <= maxMantissa, "xrpl::doNormalize", "intermediate mantissa fits in int64");
-    // the final mantissa_ is going to end up larger to fit within the
+    mantissa = m;
-    // MantissaRange. Cut it down here so that the rounding will be done while
+
-    // it's smaller.
+    g.doRoundUp(negative, mantissa, exponent, minMantissa, maxMantissa, "Number::normalize 2");
-    //
-    // 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");
+        mantissa >= minMantissa && mantissa <= maxMantissa, "xrpl::doNormalize", "final mantissa fits in range");
+    XRPL_ASSERT_PARTS(
+        exponent >= minExponent && exponent <= maxExponent, "xrpl::doNormalize", "final exponent fits in range");
 }
 
 template <>
@@ -467,11 +549,20 @@ Number::normalize<unsigned long>(
     doNormalize(negative, mantissa, exponent, minMantissa, maxMantissa);
 }
 
+void
+Number::normalize(MantissaRange const& range)
+{
+    auto [negative, mantissa, exponent] = toInternal(range);
+
+    normalize(negative, mantissa, exponent, range.min, range.max);
+
+    fromInternal(negative, mantissa, exponent, &range);
+}
+
 void
 Number::normalize()
 {
-    auto const& range = range_.get();
+    normalize(range_);
-    normalize(negative_, mantissa_, exponent_, range.min, range.max);
 }
 
 // Copy the number, but set a new exponent. Because the mantissa doesn't change,
@@ -481,21 +572,33 @@ Number
 Number::shiftExponent(int exponentDelta) const
 {
     XRPL_ASSERT_PARTS(isnormal(), "xrpl::Number::shiftExponent", "normalized");
-    auto const newExponent = exponent_ + exponentDelta;
+
-    if (newExponent >= maxExponent)
+    Number result = *this;
+
+    result.exponent_ += exponentDelta;
+
+    if (result.exponent_ >= maxExponent)
         throw std::overflow_error("Number::shiftExponent");
-    if (newExponent < minExponent)
+    if (result.exponent_ < 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(bool negative, internalrep mantissa, int exponent, normalized)
+{
+    auto const& range = range_.get();
+    normalize(negative, mantissa, exponent, range.min, range.max);
+    fromInternal(negative, mantissa, exponent, &range);
+}
+
 Number&
 Number::operator+=(Number const& y)
 {
+    auto const& range = range_.get();
+
     constexpr Number zero = Number{};
     if (y == zero)
         return *this;
@@ -510,7 +613,7 @@ Number::operator+=(Number const& y)
         return *this;
     }
 
-    XRPL_ASSERT(isnormal() && y.isnormal(), "xrpl::Number::operator+=(Number) : is normal");
+    XRPL_ASSERT(isnormal(range) && y.isnormal(range), "xrpl::Number::operator+=(Number) : is normal");
     // *n = negative
     // *s = sign
     // *m = mantissa
@@ -518,13 +621,10 @@ Number::operator+=(Number const& y)
 
     // Need to use uint128_t, because large mantissas can overflow when added
     // together.
-    bool xn = negative_;
+    auto [xn, xm, xe] = toInternal<uint128_t>(range);
-    uint128_t xm = mantissa_;
+
-    auto xe = exponent_;
+    auto [yn, ym, ye] = y.toInternal<uint128_t>(range);
 
-    bool yn = y.negative_;
-    uint128_t ym = y.mantissa_;
-    auto ye = y.exponent_;
     Guard g;
     if (xe < ye)
     {
@@ -549,14 +649,13 @@ Number::operator+=(Number const& y)
         } 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)
+        if (xm > maxMantissa)
         {
             g.push(xm % 10);
             xm /= 10;
@@ -576,7 +675,7 @@ Number::operator+=(Number const& y)
             xe = ye;
             xn = yn;
         }
-        while (xm < minMantissa && xm * 10 <= maxRep)
+        while (xm < minMantissa)
         {
             xm *= 10;
             xm -= g.pop();
@@ -585,10 +684,8 @@ Number::operator+=(Number const& y)
         g.doRoundDown(xn, xm, xe, minMantissa);
     }
 
-    negative_ = xn;
+    normalize(xn, xm, xe, minMantissa, maxMantissa);
-    mantissa_ = static_cast<internalrep>(xm);
+    fromInternal(xn, xm, xe, &range);
-    exponent_ = xe;
-    normalize();
     return *this;
 }
 
@@ -623,6 +720,8 @@ divu10(uint128_t& u)
 Number&
 Number::operator*=(Number const& y)
 {
+    auto const& range = range_.get();
+
     constexpr Number zero = Number{};
     if (*this == zero)
         return *this;
@@ -636,15 +735,11 @@ Number::operator*=(Number const& y)
     // *m = mantissa
     // *e = exponent
 
-    bool xn = negative_;
+    auto [xn, xm, xe] = toInternal(range);
     int xs = xn ? -1 : 1;
-    internalrep xm = mantissa_;
-    auto xe = exponent_;
 
-    bool yn = y.negative_;
+    auto [yn, ym, ye] = y.toInternal(range);
     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;
@@ -654,11 +749,10 @@ Number::operator*=(Number const& y)
     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)
+    while (zm > maxMantissa)
     {
         // The following is optimization for:
         // g.push(static_cast<unsigned>(zm % 10));
@@ -670,17 +764,17 @@ Number::operator*=(Number const& y)
     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();
+    normalize(zn, xm, xe, minMantissa, maxMantissa);
+    fromInternal(zn, xm, xe, &range);
     return *this;
 }
 
 Number&
 Number::operator/=(Number const& y)
 {
+    auto const& range = range_.get();
+
     constexpr Number zero = Number{};
     if (y == zero)
         throw std::overflow_error("Number: divide by 0");
@@ -693,17 +787,12 @@ Number::operator/=(Number const& y)
     // *m = mantissa
     // *e = exponent
 
-    bool np = negative_;
+    auto [np, nm, ne] = toInternal(range);
     int ns = (np ? -1 : 1);
-    auto nm = mantissa_;
-    auto ne = exponent_;
 
-    bool dp = y.negative_;
+    auto [dp, dm, de] = y.toInternal(range);
     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;
 
@@ -715,7 +804,7 @@ Number::operator/=(Number const& y)
     // 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;
+    bool small = range.scale == 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");
 
@@ -765,10 +854,8 @@ Number::operator/=(Number const& y)
         }
     }
     normalize(zn, zm, ze, minMantissa, maxMantissa);
-    negative_ = zn;
+    fromInternal(zn, zm, ze, &range);
-    mantissa_ = static_cast<internalrep>(zm);
+    XRPL_ASSERT_PARTS(isnormal(range), "xrpl::Number::operator/=", "result is normalized");
-    exponent_ = ze;
-    XRPL_ASSERT_PARTS(isnormal(), "xrpl::Number::operator/=", "result is normalized");
 
     return *this;
 }
@@ -780,10 +867,10 @@ Number::operator rep() const
     Guard g;
     if (drops != 0)
     {
-        if (negative_)
+        if (drops < 0)
         {
             g.set_negative();
-            drops = -drops;
+            drops = externalToInternal(drops);
         }
         for (; offset < 0; ++offset)
         {
@@ -792,7 +879,7 @@ Number::operator rep() const
         }
         for (; offset > 0; --offset)
         {
-            if (drops > maxRep / 10)
+            if (drops >= largeRange.min)
                 throw std::overflow_error("Number::operator rep() overflow");
             drops *= 10;
         }
@@ -822,19 +909,21 @@ Number::truncate() const noexcept
 std::string
 to_string(Number const& amount)
 {
+    auto const& range = Number::range_.get();
+
     // keep full internal accuracy, but make more human friendly if possible
     constexpr Number zero = Number{};
     if (amount == zero)
         return "0";
 
-    auto exponent = amount.exponent_;
+    // The mantissa must have a set number of decimal places for this to work
-    auto mantissa = amount.mantissa_;
+    auto [negative, mantissa, exponent] = amount.toInternal(range);
-    bool const negative = amount.negative_;
 
     // Use scientific notation for exponents that are too small or too large
-    auto const rangeLog = Number::mantissaLog();
+    auto const rangeLog = range.log;
-    if (((exponent != 0) && ((exponent < -(rangeLog + 10)) || (exponent > -(rangeLog - 10)))))
+    if (((exponent != 0 && amount.exponent() != 0) && ((exponent < -(rangeLog + 10)) || (exponent > -(rangeLog - 10)))))
     {
+        // Remove trailing zeroes from the mantissa.
         while (mantissa != 0 && mantissa % 10 == 0 && exponent < Number::maxExponent)
         {
             mantissa /= 10;
@@ -842,8 +931,11 @@ to_string(Number const& amount)
         }
         std::string ret = negative ? "-" : "";
         ret.append(std::to_string(mantissa));
-        ret.append(1, 'e');
+        if (exponent != 0)
-        ret.append(std::to_string(exponent));
+        {
+            ret.append(1, 'e');
+            ret.append(std::to_string(exponent));
+        }
         return ret;
     }
 
@@ -925,20 +1017,11 @@ power(Number const& f, unsigned n)
     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)
+Number::root(MantissaRange const& range, Number f, unsigned d)
 {
     constexpr Number zero = Number{};
-    auto const one = Number::one();
+    auto const one = Number::one(range);
 
     if (f == one || d == 1)
         return f;
@@ -955,21 +1038,28 @@ root(Number f, unsigned d)
     if (f == zero)
         return f;
 
-    // Scale f into the range (0, 1) such that f's exponent is a multiple of d
+    auto const [e, di] = [&]() {
-    auto e = f.exponent_ + Number::mantissaLog() + 1;
+        auto const [negative, mantissa, exponent] = f.toInternal(range);
-    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");
+        // Scale f into the range (0, 1) such that the scale change (e) is a
+        // multiple of the root (d)
+        auto e = exponent + range.log + 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;
+        return std::make_tuple(e, di);
+    }();
+
+    XRPL_ASSERT_PARTS(e % di == 0, "xrpl::root(Number, unsigned)", "e is divisible by d");
+    XRPL_ASSERT_PARTS(f.isnormal(range), "xrpl::root(Number, unsigned)", "f is normalized");
     bool neg = false;
     if (f < zero)
     {
@@ -1002,15 +1092,32 @@ root(Number f, unsigned d)
 
     //  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");
+    XRPL_ASSERT_PARTS(result.isnormal(range), "xrpl::root(Number, unsigned)", "result is normalized");
     return result;
 }
 
+// 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)
+{
+    auto const& range = Number::range_.get();
+    return Number::root(range, f, d);
+}
+
 Number
 root2(Number f)
 {
+    auto const& range = Number::range_.get();
     constexpr Number zero = Number{};
-    auto const one = Number::one();
+    auto const one = Number::one(range);
 
     if (f == one)
         return f;
@@ -1019,12 +1126,18 @@ root2(Number f)
     if (f == zero)
         return f;
 
-    // Scale f into the range (0, 1) such that f's exponent is a multiple of d
+    auto const e = [&]() {
-    auto e = f.exponent_ + Number::mantissaLog() + 1;
+        auto const [negative, mantissa, exponent] = f.toInternal(range);
-    if (e % 2 != 0)
+
-        ++e;
+        // Scale f into the range (0, 1) such that f's exponent is a
-    f = f.shiftExponent(-e);  // f /= 10^e;
+        // multiple of d
-    XRPL_ASSERT_PARTS(f.isnormal(), "xrpl::root2(Number)", "f is normalized");
+        auto e = exponent + range.log + 1;
+        if (e % 2 != 0)
+            ++e;
+        f = f.shiftExponent(-e);  // f /= 10^e;
+        return e;
+    }();
+    XRPL_ASSERT_PARTS(f.isnormal(range), "xrpl::root2(Number)", "f is normalized");
 
     // Quadratic least squares curve fit of f^(1/d) in the range [0, 1]
     auto const D = 105;
@@ -1046,7 +1159,7 @@ root2(Number f)
 
     //  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");
+    XRPL_ASSERT_PARTS(result.isnormal(range), "xrpl::root2(Number)", "result is normalized");
 
     return result;
 }
@@ -1056,8 +1169,10 @@ root2(Number f)
 Number
 power(Number const& f, unsigned n, unsigned d)
 {
+    auto const& range = Number::range_.get();
+
     constexpr Number zero = Number{};
-    auto const one = Number::one();
+    auto const one = Number::one(range);
 
     if (f == one)
         return f;
@@ -1079,7 +1194,7 @@ power(Number const& f, unsigned n, unsigned d)
     d /= g;
     if ((n % 2) == 1 && (d % 2) == 0 && f < zero)
         throw std::overflow_error("Number::power nan");
-    return root(power(f, n), d);
+    return Number::root(range, power(f, n), d);
 }
 
 }  // namespace xrpl

src/test/basics/Number_test.cpp

@@ -32,9 +32,10 @@ public:
     test_limits()
     {
         auto const scale = Number::getMantissaScale();
-        testcase << "test_limits " << to_string(scale);
-        bool caught = false;
         auto const minMantissa = Number::minMantissa();
+
+        testcase << "test_limits " << to_string(scale) << ", " << minMantissa;
+        bool caught = false;
         try
         {
             Number x = Number{false, minMantissa * 10, 32768, Number::normalized{}};
@@ -58,8 +59,9 @@ public:
             __LINE__);
         test(Number{false, minMantissa, -32769, Number::normalized{}}, Number{}, __LINE__);
         test(
+            // Use 1501 to force rounding up
             Number{false, minMantissa, 32000, Number::normalized{}} * 1'000 +
-                Number{false, 1'500, 32000, Number::normalized{}},
+                Number{false, 1'501, 32000, Number::normalized{}},
             Number{false, minMantissa + 2, 32003, Number::normalized{}},
             __LINE__);
         // 9,223,372,036,854,775,808
@@ -159,8 +161,8 @@ public:
                 {Number{true, 9'999'999'999'999'999'999ULL, -37, Number::normalized{}},
                  Number{1'000'000'000'000'000'000, -18},
                  Number{false, 9'999'999'999'999'999'990ULL, -19, Number::normalized{}}},
-                {Number{Number::maxRep}, Number{6, -1}, Number{Number::maxRep / 10, 1}},
+                {Number{Number::largestMantissa}, Number{6, -1}, Number{Number::largestMantissa / 10, 1}},
-                {Number{Number::maxRep - 1}, Number{1, 0}, Number{Number::maxRep}},
+                {Number{Number::largestMantissa - 1}, Number{1, 0}, Number{Number::largestMantissa}},
                 // Test extremes
                 {
                     // Each Number operand rounds up, so the actual mantissa is
@@ -170,11 +172,18 @@ public:
                     Number{2, 19},
                 },
                 {
-                    // Does not round. Mantissas are going to be > maxRep, so if
+                    // Does not round. Mantissas are going to be >
-                    // added together as uint64_t's, the result will overflow.
+                    // largestMantissa, so if added together as uint64_t's, the
-                    // With addition using uint128_t, there's no problem. After
+                    // result will overflow. With addition using uint128_t,
-                    // normalizing, the resulting mantissa ends up less than
+                    // there's no problem. After normalizing, the resulting
-                    // maxRep.
+                    // mantissa ends up less than largestMantissa.
+                    Number{false, Number::largestMantissa, 0, Number::normalized{}},
+                    Number{false, Number::largestMantissa, 0, Number::normalized{}},
+                    Number{false, Number::largestMantissa * 2, 0, Number::normalized{}},
+                },
+                {
+                    // These mantissas round down, so adding them together won't
+                    // have any consequences.
                     Number{false, 9'999'999'999'999'999'990ULL, 0, Number::normalized{}},
                     Number{false, 9'999'999'999'999'999'990ULL, 0, Number::normalized{}},
                     Number{false, 1'999'999'999'999'999'998ULL, 1, Number::normalized{}},
@@ -261,12 +270,14 @@ public:
                 {Number{1'000'000'000'000'000'001, -18},
                  Number{1'000'000'000'000'000'000, -18},
                  Number{1'000'000'000'000'000'000, -36}},
-                {Number{Number::maxRep}, Number{6, -1}, Number{Number::maxRep - 1}},
+                {Number{Number::largestMantissa}, Number{6, -1}, Number{Number::largestMantissa - 1}},
-                {Number{false, Number::maxRep + 1, 0, Number::normalized{}},
+                {Number{false, Number::largestMantissa + 1, 0, Number::normalized{}},
                  Number{1, 0},
-                 Number{Number::maxRep / 10 + 1, 1}},
+                 Number{Number::largestMantissa / 10 + 1, 1}},
-                {Number{false, Number::maxRep + 1, 0, Number::normalized{}}, Number{3, 0}, Number{Number::maxRep}},
+                {Number{false, Number::largestMantissa + 1, 0, Number::normalized{}},
-                {power(2, 63), Number{3, 0}, Number{Number::maxRep}},
+                 Number{3, 0},
+                 Number{Number::largestMantissa}},
+                {power(2, 63), Number{3, 0}, Number{Number::largestMantissa}},
             });
         auto test = [this](auto const& c) {
             for (auto const& [x, y, z] : c)
@@ -289,14 +300,15 @@ public:
         auto const scale = Number::getMantissaScale();
         testcase << "test_mul " << to_string(scale);
 
-        using Case = std::tuple<Number, Number, Number>;
+        // Case: Factor 1, Factor 2, Expected product, Line number
+        using Case = std::tuple<Number, Number, Number, int>;
         auto test = [this](auto const& c) {
-            for (auto const& [x, y, z] : c)
+            for (auto const& [x, y, z, line] : c)
             {
                 auto const result = x * y;
                 std::stringstream ss;
                 ss << x << " * " << y << " = " << result << ". Expected: " << z;
-                BEAST_EXPECTS(result == z, ss.str());
+                BEAST_EXPECTS(result == z, ss.str() + " line: " + std::to_string(line));
             }
         };
         auto tests = [&](auto const& cSmall, auto const& cLarge) {
@@ -306,48 +318,83 @@ public:
                 test(cLarge);
         };
         auto const maxMantissa = Number::maxMantissa();
+        auto const maxInternalMantissa =
+            static_cast<std::uint64_t>(static_cast<std::int64_t>(power(10, Number::mantissaLog()))) * 10 - 1;
 
         saveNumberRoundMode save{Number::setround(Number::to_nearest)};
         {
             auto const cSmall = std::to_array<Case>({
-                {Number{7}, Number{8}, Number{56}},
+                {Number{7}, Number{8}, Number{56}, __LINE__},
-                {Number{1414213562373095, -15}, Number{1414213562373095, -15}, Number{2000000000000000, -15}},
+                {Number{1414213562373095, -15}, Number{1414213562373095, -15}, Number{2000000000000000, -15}, __LINE__},
-                {Number{-1414213562373095, -15}, Number{1414213562373095, -15}, Number{-2000000000000000, -15}},
+                {Number{-1414213562373095, -15},
-                {Number{-1414213562373095, -15}, Number{-1414213562373095, -15}, Number{2000000000000000, -15}},
+                 Number{1414213562373095, -15},
-                {Number{3214285714285706, -15}, Number{3111111111111119, -15}, Number{1000000000000000, -14}},
+                 Number{-2000000000000000, -15},
-                {Number{1000000000000000, -32768}, Number{1000000000000000, -32768}, Number{0}},
+                 __LINE__},
+                {Number{-1414213562373095, -15},
+                 Number{-1414213562373095, -15},
+                 Number{2000000000000000, -15},
+                 __LINE__},
+                {Number{3214285714285706, -15}, Number{3111111111111119, -15}, Number{1000000000000000, -14}, __LINE__},
+                {Number{1000000000000000, -32768}, Number{1000000000000000, -32768}, Number{0}, __LINE__},
                 // Maximum mantissa range
-                {Number{9'999'999'999'999'999, 0}, Number{9'999'999'999'999'999, 0}, Number{9'999'999'999'999'998, 16}},
+                {Number{9'999'999'999'999'999, 0},
+                 Number{9'999'999'999'999'999, 0},
+                 Number{9'999'999'999'999'998, 16},
+                 __LINE__},
             });
             auto const cLarge = std::to_array<Case>({
                 // Note that items with extremely large mantissas need to be
                 // calculated, because otherwise they overflow uint64. Items
                 // from C with larger mantissa
-                {Number{7}, Number{8}, Number{56}},
+                {Number{7}, Number{8}, Number{56}, __LINE__},
-                {Number{1414213562373095, -15}, Number{1414213562373095, -15}, Number{1999999999999999862, -18}},
+                {Number{1414213562373095, -15},
-                {Number{-1414213562373095, -15}, Number{1414213562373095, -15}, Number{-1999999999999999862, -18}},
+                 Number{1414213562373095, -15},
-                {Number{-1414213562373095, -15}, Number{-1414213562373095, -15}, Number{1999999999999999862, -18}},
+                 Number{1999999999999999862, -18},
+                 __LINE__},
+                {Number{-1414213562373095, -15},
+                 Number{1414213562373095, -15},
+                 Number{-1999999999999999862, -18},
+                 __LINE__},
+                {Number{-1414213562373095, -15},
+                 Number{-1414213562373095, -15},
+                 Number{1999999999999999862, -18},
+                 __LINE__},
                 {Number{3214285714285706, -15},
                  Number{3111111111111119, -15},
-                 Number{false, 9'999'999'999'999'999'579ULL, -18, Number::normalized{}}},
+                 Number{false, 9'999'999'999'999'999'579ULL, -18, Number::normalized{}},
-                {Number{1000000000000000000, -32768}, Number{1000000000000000000, -32768}, Number{0}},
+                 __LINE__},
+                {Number{1000000000000000000, -32768}, Number{1000000000000000000, -32768}, Number{0}, __LINE__},
                 // Items from cSmall expanded for the larger mantissa,
                 // except duplicates. Sadly, it looks like sqrt(2)^2 != 2
                 // with higher precision
-                {Number{1414213562373095049, -18}, Number{1414213562373095049, -18}, Number{2000000000000000001, -18}},
+                {Number{1414213562373095049, -18},
+                 Number{1414213562373095049, -18},
+                 Number{2000000000000000001, -18},
+                 __LINE__},
                 {Number{-1414213562373095048, -18},
                  Number{1414213562373095048, -18},
-                 Number{-1999999999999999998, -18}},
+                 Number{-1999999999999999998, -18},
+                 __LINE__},
                 {Number{-1414213562373095048, -18},
                  Number{-1414213562373095049, -18},
-                 Number{1999999999999999999, -18}},
+                 Number{1999999999999999999, -18},
-                {Number{3214285714285714278, -18}, Number{3111111111111111119, -18}, Number{10, 0}},
+                 __LINE__},
-                // Maximum mantissa range - rounds up to 1e19
+                {Number{3214285714285714278, -18}, Number{3111111111111111119, -18}, Number{10, 0}, __LINE__},
+                // Maximum internal mantissa range - rounds up to 1e19
+                {Number{false, maxInternalMantissa, 0, Number::normalized{}},
+                 Number{false, maxInternalMantissa, 0, Number::normalized{}},
+                 Number{1, 38},
+                 __LINE__},
+                // Maximum actual mantissa range - same as int64 range
                 {Number{false, maxMantissa, 0, Number::normalized{}},
                  Number{false, maxMantissa, 0, Number::normalized{}},
-                 Number{1, 38}},
+                 Number{85'070'591'730'234'615'85, 19},
+                 __LINE__},
                 // Maximum int64 range
-                {Number{Number::maxRep, 0}, Number{Number::maxRep, 0}, Number{85'070'591'730'234'615'85, 19}},
+                {Number{Number::largestMantissa, 0},
+                 Number{Number::largestMantissa, 0},
+                 Number{85'070'591'730'234'615'85, 19},
+                 __LINE__},
             });
             tests(cSmall, cLarge);
         }
@@ -355,44 +402,78 @@ public:
         testcase << "test_mul " << to_string(Number::getMantissaScale()) << " towards_zero";
         {
             auto const cSmall = std::to_array<Case>(
-                {{Number{7}, Number{8}, Number{56}},
+                {{Number{7}, Number{8}, Number{56}, __LINE__},
-                 {Number{1414213562373095, -15}, Number{1414213562373095, -15}, Number{1999999999999999, -15}},
+                 {Number{1414213562373095, -15},
-                 {Number{-1414213562373095, -15}, Number{1414213562373095, -15}, Number{-1999999999999999, -15}},
+                  Number{1414213562373095, -15},
-                 {Number{-1414213562373095, -15}, Number{-1414213562373095, -15}, Number{1999999999999999, -15}},
+                  Number{1999999999999999, -15},
-                 {Number{3214285714285706, -15}, Number{3111111111111119, -15}, Number{9999999999999999, -15}},
+                  __LINE__},
-                 {Number{1000000000000000, -32768}, Number{1000000000000000, -32768}, Number{0}}});
+                 {Number{-1414213562373095, -15},
+                  Number{1414213562373095, -15},
+                  Number{-1999999999999999, -15},
+                  __LINE__},
+                 {Number{-1414213562373095, -15},
+                  Number{-1414213562373095, -15},
+                  Number{1999999999999999, -15},
+                  __LINE__},
+                 {Number{3214285714285706, -15},
+                  Number{3111111111111119, -15},
+                  Number{9999999999999999, -15},
+                  __LINE__},
+                 {Number{1000000000000000, -32768}, Number{1000000000000000, -32768}, Number{0}, __LINE__}});
             auto const cLarge = std::to_array<Case>(
                 // Note that items with extremely large mantissas need to be
                 // calculated, because otherwise they overflow uint64. Items
                 // from C with larger mantissa
                 {
-                    {Number{7}, Number{8}, Number{56}},
+                    {Number{7}, Number{8}, Number{56}, __LINE__},
-                    {Number{1414213562373095, -15}, Number{1414213562373095, -15}, Number{1999999999999999861, -18}},
+                    {Number{1414213562373095, -15},
-                    {Number{-1414213562373095, -15}, Number{1414213562373095, -15}, Number{-1999999999999999861, -18}},
+                     Number{1414213562373095, -15},
-                    {Number{-1414213562373095, -15}, Number{-1414213562373095, -15}, Number{1999999999999999861, -18}},
+                     Number{1999999999999999861, -18},
+                     __LINE__},
+                    {Number{-1414213562373095, -15},
+                     Number{1414213562373095, -15},
+                     Number{-1999999999999999861, -18},
+                     __LINE__},
+                    {Number{-1414213562373095, -15},
+                     Number{-1414213562373095, -15},
+                     Number{1999999999999999861, -18},
+                     __LINE__},
                     {Number{3214285714285706, -15},
                      Number{3111111111111119, -15},
-                     Number{false, 9999999999999999579ULL, -18, Number::normalized{}}},
+                     Number{false, 9999999999999999579ULL, -18, Number::normalized{}},
-                    {Number{1000000000000000000, -32768}, Number{1000000000000000000, -32768}, Number{0}},
+                     __LINE__},
+                    {Number{1000000000000000000, -32768}, Number{1000000000000000000, -32768}, Number{0}, __LINE__},
                     // Items from cSmall expanded for the larger mantissa,
                     // except duplicates. Sadly, it looks like sqrt(2)^2 != 2
                     // with higher precision
-                    {Number{1414213562373095049, -18}, Number{1414213562373095049, -18}, Number{2, 0}},
+                    {Number{1414213562373095049, -18}, Number{1414213562373095049, -18}, Number{2, 0}, __LINE__},
                     {Number{-1414213562373095048, -18},
                      Number{1414213562373095048, -18},
-                     Number{-1999999999999999997, -18}},
+                     Number{-1999999999999999997, -18},
+                     __LINE__},
                     {Number{-1414213562373095048, -18},
                      Number{-1414213562373095049, -18},
-                     Number{1999999999999999999, -18}},
+                     Number{1999999999999999999, -18},
-                    {Number{3214285714285714278, -18}, Number{3111111111111111119, -18}, Number{10, 0}},
+                     __LINE__},
-                    // Maximum mantissa range - rounds down to maxMantissa/10e1
+                    {Number{3214285714285714278, -18}, Number{3111111111111111119, -18}, Number{10, 0}, __LINE__},
+                    // Maximum internal mantissa range - rounds down to
+                    // maxMantissa/10e1
                     // 99'999'999'999'999'999'800'000'000'000'000'000'100
+                    {Number{false, maxInternalMantissa, 0, Number::normalized{}},
+                     Number{false, maxInternalMantissa, 0, Number::normalized{}},
+                     Number{false, maxInternalMantissa / 10 - 1, 20, Number::normalized{}},
+                     __LINE__},
+                    // Maximum actual mantissa range - same as int64
                     {Number{false, maxMantissa, 0, Number::normalized{}},
                      Number{false, maxMantissa, 0, Number::normalized{}},
-                     Number{false, maxMantissa / 10 - 1, 20, Number::normalized{}}},
+                     Number{85'070'591'730'234'615'84, 19},
+                     __LINE__},
                     // Maximum int64 range
                     // 85'070'591'730'234'615'847'396'907'784'232'501'249
-                    {Number{Number::maxRep, 0}, Number{Number::maxRep, 0}, Number{85'070'591'730'234'615'84, 19}},
+                    {Number{Number::largestMantissa, 0},
+                     Number{Number::largestMantissa, 0},
+                     Number{85'070'591'730'234'615'84, 19},
+                     __LINE__},
                 });
             tests(cSmall, cLarge);
         }
@@ -400,44 +481,78 @@ public:
         testcase << "test_mul " << to_string(Number::getMantissaScale()) << " downward";
         {
             auto const cSmall = std::to_array<Case>(
-                {{Number{7}, Number{8}, Number{56}},
+                {{Number{7}, Number{8}, Number{56}, __LINE__},
-                 {Number{1414213562373095, -15}, Number{1414213562373095, -15}, Number{1999999999999999, -15}},
+                 {Number{1414213562373095, -15},
-                 {Number{-1414213562373095, -15}, Number{1414213562373095, -15}, Number{-2000000000000000, -15}},
+                  Number{1414213562373095, -15},
-                 {Number{-1414213562373095, -15}, Number{-1414213562373095, -15}, Number{1999999999999999, -15}},
+                  Number{1999999999999999, -15},
-                 {Number{3214285714285706, -15}, Number{3111111111111119, -15}, Number{9999999999999999, -15}},
+                  __LINE__},
-                 {Number{1000000000000000, -32768}, Number{1000000000000000, -32768}, Number{0}}});
+                 {Number{-1414213562373095, -15},
+                  Number{1414213562373095, -15},
+                  Number{-2000000000000000, -15},
+                  __LINE__},
+                 {Number{-1414213562373095, -15},
+                  Number{-1414213562373095, -15},
+                  Number{1999999999999999, -15},
+                  __LINE__},
+                 {Number{3214285714285706, -15},
+                  Number{3111111111111119, -15},
+                  Number{9999999999999999, -15},
+                  __LINE__},
+                 {Number{1000000000000000, -32768}, Number{1000000000000000, -32768}, Number{0}, __LINE__}});
             auto const cLarge = std::to_array<Case>(
                 // Note that items with extremely large mantissas need to be
                 // calculated, because otherwise they overflow uint64. Items
                 // from C with larger mantissa
                 {
-                    {Number{7}, Number{8}, Number{56}},
+                    {Number{7}, Number{8}, Number{56}, __LINE__},
-                    {Number{1414213562373095, -15}, Number{1414213562373095, -15}, Number{1999999999999999861, -18}},
+                    {Number{1414213562373095, -15},
-                    {Number{-1414213562373095, -15}, Number{1414213562373095, -15}, Number{-1999999999999999862, -18}},
+                     Number{1414213562373095, -15},
-                    {Number{-1414213562373095, -15}, Number{-1414213562373095, -15}, Number{1999999999999999861, -18}},
+                     Number{1999999999999999861, -18},
+                     __LINE__},
+                    {Number{-1414213562373095, -15},
+                     Number{1414213562373095, -15},
+                     Number{-1999999999999999862, -18},
+                     __LINE__},
+                    {Number{-1414213562373095, -15},
+                     Number{-1414213562373095, -15},
+                     Number{1999999999999999861, -18},
+                     __LINE__},
                     {Number{3214285714285706, -15},
                      Number{3111111111111119, -15},
-                     Number{false, 9'999'999'999'999'999'579ULL, -18, Number::normalized{}}},
+                     Number{false, 9'999'999'999'999'999'579ULL, -18, Number::normalized{}},
-                    {Number{1000000000000000000, -32768}, Number{1000000000000000000, -32768}, Number{0}},
+                     __LINE__},
+                    {Number{1000000000000000000, -32768}, Number{1000000000000000000, -32768}, Number{0}, __LINE__},
                     // Items from cSmall expanded for the larger mantissa,
                     // except duplicates. Sadly, it looks like sqrt(2)^2 != 2
                     // with higher precision
-                    {Number{1414213562373095049, -18}, Number{1414213562373095049, -18}, Number{2, 0}},
+                    {Number{1414213562373095049, -18}, Number{1414213562373095049, -18}, Number{2, 0}, __LINE__},
                     {Number{-1414213562373095048, -18},
                      Number{1414213562373095048, -18},
-                     Number{-1999999999999999998, -18}},
+                     Number{-1999999999999999998, -18},
+                     __LINE__},
                     {Number{-1414213562373095048, -18},
                      Number{-1414213562373095049, -18},
-                     Number{1999999999999999999, -18}},
+                     Number{1999999999999999999, -18},
-                    {Number{3214285714285714278, -18}, Number{3111111111111111119, -18}, Number{10, 0}},
+                     __LINE__},
-                    // Maximum mantissa range - rounds down to maxMantissa/10e1
+                    {Number{3214285714285714278, -18}, Number{3111111111111111119, -18}, Number{10, 0}, __LINE__},
+                    // Maximum internal mantissa range - rounds down to
+                    // maxMantissa/10-1
                     // 99'999'999'999'999'999'800'000'000'000'000'000'100
+                    {Number{false, maxInternalMantissa, 0, Number::normalized{}},
+                     Number{false, maxInternalMantissa, 0, Number::normalized{}},
+                     Number{false, maxInternalMantissa / 10 - 1, 20, Number::normalized{}},
+                     __LINE__},
+                    // Maximum mantissa range - same as int64
                     {Number{false, maxMantissa, 0, Number::normalized{}},
                      Number{false, maxMantissa, 0, Number::normalized{}},
-                     Number{false, maxMantissa / 10 - 1, 20, Number::normalized{}}},
+                     Number{85'070'591'730'234'615'84, 19},
+                     __LINE__},
                     // Maximum int64 range
                     // 85'070'591'730'234'615'847'396'907'784'232'501'249
-                    {Number{Number::maxRep, 0}, Number{Number::maxRep, 0}, Number{85'070'591'730'234'615'84, 19}},
+                    {Number{Number::largestMantissa, 0},
+                     Number{Number::largestMantissa, 0},
+                     Number{85'070'591'730'234'615'84, 19},
+                     __LINE__},
                 });
             tests(cSmall, cLarge);
         }
@@ -445,44 +560,80 @@ public:
         testcase << "test_mul " << to_string(Number::getMantissaScale()) << " upward";
         {
             auto const cSmall = std::to_array<Case>(
-                {{Number{7}, Number{8}, Number{56}},
+                {{Number{7}, Number{8}, Number{56}, __LINE__},
-                 {Number{1414213562373095, -15}, Number{1414213562373095, -15}, Number{2000000000000000, -15}},
+                 {Number{1414213562373095, -15},
-                 {Number{-1414213562373095, -15}, Number{1414213562373095, -15}, Number{-1999999999999999, -15}},
+                  Number{1414213562373095, -15},
-                 {Number{-1414213562373095, -15}, Number{-1414213562373095, -15}, Number{2000000000000000, -15}},
+                  Number{2000000000000000, -15},
-                 {Number{3214285714285706, -15}, Number{3111111111111119, -15}, Number{1000000000000000, -14}},
+                  __LINE__},
-                 {Number{1000000000000000, -32768}, Number{1000000000000000, -32768}, Number{0}}});
+                 {Number{-1414213562373095, -15},
+                  Number{1414213562373095, -15},
+                  Number{-1999999999999999, -15},
+                  __LINE__},
+                 {Number{-1414213562373095, -15},
+                  Number{-1414213562373095, -15},
+                  Number{2000000000000000, -15},
+                  __LINE__},
+                 {Number{3214285714285706, -15},
+                  Number{3111111111111119, -15},
+                  Number{1000000000000000, -14},
+                  __LINE__},
+                 {Number{1000000000000000, -32768}, Number{1000000000000000, -32768}, Number{0}, __LINE__}});
             auto const cLarge = std::to_array<Case>(
                 // Note that items with extremely large mantissas need to be
                 // calculated, because otherwise they overflow uint64. Items
                 // from C with larger mantissa
                 {
-                    {Number{7}, Number{8}, Number{56}},
+                    {Number{7}, Number{8}, Number{56}, __LINE__},
-                    {Number{1414213562373095, -15}, Number{1414213562373095, -15}, Number{1999999999999999862, -18}},
+                    {Number{1414213562373095, -15},
-                    {Number{-1414213562373095, -15}, Number{1414213562373095, -15}, Number{-1999999999999999861, -18}},
+                     Number{1414213562373095, -15},
-                    {Number{-1414213562373095, -15}, Number{-1414213562373095, -15}, Number{1999999999999999862, -18}},
+                     Number{1999999999999999862, -18},
-                    {Number{3214285714285706, -15}, Number{3111111111111119, -15}, Number{999999999999999958, -17}},
+                     __LINE__},
-                    {Number{1000000000000000000, -32768}, Number{1000000000000000000, -32768}, Number{0}},
+                    {Number{-1414213562373095, -15},
+                     Number{1414213562373095, -15},
+                     Number{-1999999999999999861, -18},
+                     __LINE__},
+                    {Number{-1414213562373095, -15},
+                     Number{-1414213562373095, -15},
+                     Number{1999999999999999862, -18},
+                     __LINE__},
+                    {Number{3214285714285706, -15},
+                     Number{3111111111111119, -15},
+                     Number{999999999999999958, -17},
+                     __LINE__},
+                    {Number{1000000000000000000, -32768}, Number{1000000000000000000, -32768}, Number{0}, __LINE__},
                     // Items from cSmall expanded for the larger mantissa,
                     // except duplicates. Sadly, it looks like sqrt(2)^2 != 2
                     // with higher precision
                     {Number{1414213562373095049, -18},
                      Number{1414213562373095049, -18},
-                     Number{2000000000000000001, -18}},
+                     Number{2000000000000000001, -18},
+                     __LINE__},
                     {Number{-1414213562373095048, -18},
                      Number{1414213562373095048, -18},
-                     Number{-1999999999999999997, -18}},
+                     Number{-1999999999999999997, -18},
-                    {Number{-1414213562373095048, -18}, Number{-1414213562373095049, -18}, Number{2, 0}},
+                     __LINE__},
+                    {Number{-1414213562373095048, -18}, Number{-1414213562373095049, -18}, Number{2, 0}, __LINE__},
                     {Number{3214285714285714278, -18},
                      Number{3111111111111111119, -18},
-                     Number{1000000000000000001, -17}},
+                     Number{1000000000000000001, -17},
-                    // Maximum mantissa range - rounds up to minMantissa*10
+                     __LINE__},
-                    // 1e19*1e19=1e38
+                    // Maximum internal mantissa range - rounds up to
+                    // minMantissa*10 1e19*1e19=1e38
+                    {Number{false, maxInternalMantissa, 0, Number::normalized{}},
+                     Number{false, maxInternalMantissa, 0, Number::normalized{}},
+                     Number{1, 38},
+                     __LINE__},
+                    // Maximum mantissa range - same as int64
                     {Number{false, maxMantissa, 0, Number::normalized{}},
                      Number{false, maxMantissa, 0, Number::normalized{}},
-                     Number{1, 38}},
+                     Number{85'070'591'730'234'615'85, 19},
+                     __LINE__},
                     // Maximum int64 range
                     // 85'070'591'730'234'615'847'396'907'784'232'501'249
-                    {Number{Number::maxRep, 0}, Number{Number::maxRep, 0}, Number{85'070'591'730'234'615'85, 19}},
+                    {Number{Number::largestMantissa, 0},
+                     Number{Number::largestMantissa, 0},
+                     Number{85'070'591'730'234'615'85, 19},
+                     __LINE__},
                 });
             tests(cSmall, cLarge);
         }
@@ -697,6 +848,9 @@ public:
         };
         */
 
+        auto const maxInternalMantissa =
+            static_cast<std::uint64_t>(static_cast<std::int64_t>(power(10, Number::mantissaLog()))) * 10 - 1;
+
         auto const cSmall = std::to_array<Case>(
             {{Number{2}, 2, Number{1414213562373095049, -18}},
              {Number{2'000'000}, 2, Number{1414213562373095049, -15}},
@@ -708,14 +862,14 @@ public:
              {Number{0}, 5, Number{0}},
              {Number{5625, -4}, 2, Number{75, -2}}});
         auto const cLarge = std::to_array<Case>({
-            {Number{false, Number::maxMantissa() - 9, -1, Number::normalized{}},
+            {Number{false, maxInternalMantissa - 9, -1, Number::normalized{}},
              2,
              Number{false, 999'999'999'999'999'999, -9, Number::normalized{}}},
-            {Number{false, Number::maxMantissa() - 9, 0, Number::normalized{}},
+            {Number{false, maxInternalMantissa - 9, 0, Number::normalized{}},
              2,
              Number{false, 3'162'277'660'168'379'330, -9, Number::normalized{}}},
-            {Number{Number::maxRep}, 2, Number{false, 3'037'000'499'976049692, -9, Number::normalized{}}},
+            {Number{Number::largestMantissa}, 2, Number{false, 3'037'000'499'976049692, -9, Number::normalized{}}},
-            {Number{Number::maxRep}, 4, Number{false, 55'108'98747006743627, -14, Number::normalized{}}},
+            {Number{Number::largestMantissa}, 4, Number{false, 55'108'98747006743627, -14, Number::normalized{}}},
         });
         test(cSmall);
         if (Number::getMantissaScale() != MantissaRange::small)
@@ -762,6 +916,8 @@ public:
             }
         };
 
+        auto const maxInternalMantissa = power(10, Number::mantissaLog()) * 10 - 1;
+
         auto const cSmall = std::to_array<Number>({
             Number{2},
             Number{2'000'000},
@@ -771,7 +927,10 @@ public:
             Number{5, -1},
             Number{0},
             Number{5625, -4},
-            Number{Number::maxRep},
+            Number{Number::largestMantissa},
+            maxInternalMantissa,
+            Number{Number::minMantissa(), 0, Number::unchecked{}},
+            Number{Number::maxMantissa(), 0, Number::unchecked{}},
         });
         test(cSmall);
         bool caught = false;
@@ -1113,16 +1272,16 @@ public:
             case MantissaRange::large:
                 // Test the edges
                 // ((exponent < -(28)) || (exponent > -(8)))))
-                test(Number::min(), "1e-32750");
+                test(Number::min(), "922337203685477581e-32768");
                 test(Number::max(), "9223372036854775807e32768");
                 test(Number::lowest(), "-9223372036854775807e32768");
                 {
                     NumberRoundModeGuard mg(Number::towards_zero);
 
                     auto const maxMantissa = Number::maxMantissa();
-                    BEAST_EXPECT(maxMantissa == 9'999'999'999'999'999'999ULL);
+                    BEAST_EXPECT(maxMantissa == 9'223'372'036'854'775'807ULL);
-                    test(Number{false, maxMantissa, 0, Number::normalized{}}, "9999999999999999990");
+                    test(Number{false, maxMantissa, 0, Number::normalized{}}, "9223372036854775807");
-                    test(Number{true, maxMantissa, 0, Number::normalized{}}, "-9999999999999999990");
+                    test(Number{true, maxMantissa, 0, Number::normalized{}}, "-9223372036854775807");
 
                     test(Number{std::numeric_limits<std::int64_t>::max(), 0}, "9223372036854775807");
                     test(-(Number{std::numeric_limits<std::int64_t>::max(), 0}), "-9223372036854775807");
@@ -1300,7 +1459,7 @@ public:
             Number const initalXrp{INITIAL_XRP};
             BEAST_EXPECT(initalXrp.exponent() > 0);
 
-            Number const maxInt64{Number::maxRep};
+            Number const maxInt64{Number::largestMantissa};
             BEAST_EXPECT(maxInt64.exponent() > 0);
             // 85'070'591'730'234'615'865'843'651'857'942'052'864 - 38 digits
             BEAST_EXPECT((power(maxInt64, 2) == Number{85'070'591'730'234'62, 22}));
@@ -1317,20 +1476,198 @@ public:
             Number const initalXrp{INITIAL_XRP};
             BEAST_EXPECT(initalXrp.exponent() <= 0);
 
-            Number const maxInt64{Number::maxRep};
+            Number const maxInt64{Number::largestMantissa};
             BEAST_EXPECT(maxInt64.exponent() <= 0);
             // 85'070'591'730'234'615'847'396'907'784'232'501'249 - 38 digits
             BEAST_EXPECT((power(maxInt64, 2) == Number{85'070'591'730'234'615'85, 19}));
 
             NumberRoundModeGuard mg(Number::towards_zero);
 
-            auto const maxMantissa = Number::maxMantissa();
+            {
-            Number const max = Number{false, maxMantissa, 0, Number::normalized{}};
+                auto const maxInternalMantissa =
-            BEAST_EXPECT(max.mantissa() == maxMantissa / 10);
+                    static_cast<std::uint64_t>(static_cast<std::int64_t>(power(10, Number::mantissaLog()))) * 10 - 1;
-            BEAST_EXPECT(max.exponent() == 1);
+
-            // 99'999'999'999'999'999'800'000'000'000'000'000'100 - also 38
+                // Rounds down to fit under 2^63
-            // digits
+                Number const max = Number{false, maxInternalMantissa, 0, Number::normalized{}};
-            BEAST_EXPECT((power(max, 2) == Number{false, maxMantissa / 10 - 1, 20, Number::normalized{}}));
+                // No alterations by the accessors
+                BEAST_EXPECT(max.mantissa() == maxInternalMantissa / 10);
+                BEAST_EXPECT(max.exponent() == 1);
+                // 99'999'999'999'999'999'800'000'000'000'000'000'100 - also 38
+                // digits
+                BEAST_EXPECT((power(max, 2) == Number{false, maxInternalMantissa / 10 - 1, 20, Number::normalized{}}));
+            }
+
+            {
+                auto const maxMantissa = Number::maxMantissa();
+                Number const max = Number{false, maxMantissa, 0, Number::normalized{}};
+                // No alterations by the accessors
+                BEAST_EXPECT(max.mantissa() == maxMantissa);
+                BEAST_EXPECT(max.exponent() == 0);
+                // 85'070'591'730'234'615'847'396'907'784'232'501'249 - also 38
+                // digits
+                BEAST_EXPECT((power(max, 2) == Number{false, 85'070'591'730'234'615'84, 19, Number::normalized{}}));
+            }
+        }
+    }
+
+    void
+    testNormalizeToRange()
+    {
+        // Test edge-cases of normalizeToRange
+        auto const scale = Number::getMantissaScale();
+        testcase << "normalizeToRange " << to_string(scale);
+
+        auto test = [this](
+                        Number const& n,
+                        auto const rangeMin,
+                        auto const rangeMax,
+                        auto const expectedMantissa,
+                        auto const expectedExponent,
+                        auto const line) {
+            auto const normalized = n.normalizeToRange(rangeMin, rangeMax);
+            BEAST_EXPECTS(
+                normalized.first == expectedMantissa,
+                "Number " + to_string(n) + " scaled to " + std::to_string(rangeMax) +
+                    ". Expected mantissa:" + std::to_string(expectedMantissa) +
+                    ", got: " + std::to_string(normalized.first) + " @ " + std::to_string(line));
+            BEAST_EXPECTS(
+                normalized.second == expectedExponent,
+                "Number " + to_string(n) + " scaled to " + std::to_string(rangeMax) +
+                    ". Expected exponent:" + std::to_string(expectedExponent) +
+                    ", got: " + std::to_string(normalized.second) + " @ " + std::to_string(line));
+        };
+
+        std::int64_t constexpr iRangeMin = 100;
+        std::int64_t constexpr iRangeMax = 999;
+
+        std::uint64_t constexpr uRangeMin = 100;
+        std::uint64_t constexpr uRangeMax = 999;
+
+        constexpr static MantissaRange largeRange{MantissaRange::large};
+
+        std::int64_t constexpr iBigMin = largeRange.min;
+        std::int64_t constexpr iBigMax = largeRange.max;
+
+        auto const testSuite = [&](Number const& n,
+                                   auto const expectedSmallMantissa,
+                                   auto const expectedSmallExponent,
+                                   auto const expectedLargeMantissa,
+                                   auto const expectedLargeExponent,
+                                   auto const line) {
+            test(n, iRangeMin, iRangeMax, expectedSmallMantissa, expectedSmallExponent, line);
+            test(n, iBigMin, iBigMax, expectedLargeMantissa, expectedLargeExponent, line);
+
+            // Only test non-negative. testing a negative number with an
+            // unsigned range will assert, and asserts can't be tested.
+            if (n.signum() >= 0)
+            {
+                test(n, uRangeMin, uRangeMax, expectedSmallMantissa, expectedSmallExponent, line);
+                test(n, largeRange.min, largeRange.max, expectedLargeMantissa, expectedLargeExponent, line);
+            }
+        };
+
+        {
+            // zero
+            Number const n{0};
+
+            testSuite(n, 0, std::numeric_limits<int>::lowest(), 0, std::numeric_limits<int>::lowest(), __LINE__);
+        }
+        {
+            // Small positive number
+            Number const n{2};
+
+            testSuite(n, 200, -2, 2'000'000'000'000'000'000, -18, __LINE__);
+        }
+        {
+            // Negative number
+            Number const n{-2};
+
+            testSuite(n, -200, -2, -2'000'000'000'000'000'000, -18, __LINE__);
+        }
+        {
+            // Biggest valid mantissa
+            Number const n{Number::largestMantissa, 0, Number::normalized{}};
+
+            if (scale == MantissaRange::small)
+                // With the small mantissa range, the value rounds up. Because
+                // it rounds up, when scaling up to the full int64 range, it
+                // can't go over the max, so it is one digit smaller than the
+                // full value.
+                testSuite(n, 922, 16, 922'337'203'685'477'600, 1, __LINE__);
+            else
+                testSuite(n, 922, 16, Number::largestMantissa, 0, __LINE__);
+        }
+        {
+            // Biggest valid mantissa + 1
+            Number const n{Number::largestMantissa + 1, 0, Number::normalized{}};
+
+            if (scale == MantissaRange::small)
+                // With the small mantissa range, the value rounds up. Because
+                // it rounds up, when scaling up to the full int64 range, it
+                // can't go over the max, so it is one digit smaller than the
+                // full value.
+                testSuite(n, 922, 16, 922'337'203'685'477'600, 1, __LINE__);
+            else
+                testSuite(n, 922, 16, 922'337'203'685'477'581, 1, __LINE__);
+        }
+        {
+            // Biggest valid mantissa + 2
+            Number const n{Number::largestMantissa + 2, 0, Number::normalized{}};
+
+            if (scale == MantissaRange::small)
+                // With the small mantissa range, the value rounds up. Because
+                // it rounds up, when scaling up to the full int64 range, it
+                // can't go over the max, so it is one digit smaller than the
+                // full value.
+                testSuite(n, 922, 16, 922'337'203'685'477'600, 1, __LINE__);
+            else
+                testSuite(n, 922, 16, 922'337'203'685'477'581, 1, __LINE__);
+        }
+        {
+            // Biggest valid mantissa + 3
+            Number const n{Number::largestMantissa + 3, 0, Number::normalized{}};
+
+            if (scale == MantissaRange::small)
+                // With the small mantissa range, the value rounds up. Because
+                // it rounds up, when scaling up to the full int64 range, it
+                // can't go over the max, so it is one digit smaller than the
+                // full value.
+                testSuite(n, 922, 16, 922'337'203'685'477'600, 1, __LINE__);
+            else
+                testSuite(n, 922, 16, 922'337'203'685'477'581, 1, __LINE__);
+        }
+        {
+            // int64 min
+            Number const n{std::numeric_limits<std::int64_t>::min(), 0};
+
+            if (scale == MantissaRange::small)
+                testSuite(n, -922, 16, -922'337'203'685'477'600, 1, __LINE__);
+            else
+                testSuite(n, -922, 16, -922'337'203'685'477'581, 1, __LINE__);
+        }
+        {
+            // int64 min + 1
+            Number const n{std::numeric_limits<std::int64_t>::min() + 1, 0};
+
+            if (scale == MantissaRange::small)
+                testSuite(n, -922, 16, -922'337'203'685'477'600, 1, __LINE__);
+            else
+                testSuite(n, -922, 16, -9'223'372'036'854'775'807, 0, __LINE__);
+        }
+        {
+            // int64 min - 1
+            // Need to cast to uint, even though we're dealing with a negative
+            // number to avoid overflow and UB
+            Number const n{
+                true,
+                -static_cast<std::uint64_t>(std::numeric_limits<std::int64_t>::min()) + 1,
+                0,
+                Number::normalized{}};
+
+            if (scale == MantissaRange::small)
+                testSuite(n, -922, 16, -922'337'203'685'477'600, 1, __LINE__);
+            else
+                testSuite(n, -922, 16, -922'337'203'685'477'581, 1, __LINE__);
         }
     }
 
@@ -1361,6 +1698,7 @@ public:
             test_truncate();
             testRounding();
             testInt64();
+            testNormalizeToRange();
         }
     }
 };