[rust-lightning] / lightning / src / routing / scoring.rs

// This file is Copyright its original authors, visible in version control
// history.
//
// This file is licensed under the Apache License, Version 2.0 <LICENSE-APACHE
// or http://www.apache.org/licenses/LICENSE-2.0> or the MIT license
// <LICENSE-MIT or http://opensource.org/licenses/MIT>, at your option.
// You may not use this file except in accordance with one or both of these
// licenses.

//! Utilities for scoring payment channels.
//!
//! [`ProbabilisticScorer`] may be given to [`find_route`] to score payment channels during path
//! finding when a custom [`Score`] implementation is not needed.
//!
//! # Example
//!
//! ```
//! # extern crate bitcoin;
//! #
//! # use lightning::routing::network_graph::NetworkGraph;
//! # use lightning::routing::router::{RouteParameters, find_route};
//! # use lightning::routing::scoring::{ProbabilisticScorer, ProbabilisticScoringParameters};
//! # use lightning::chain::keysinterface::{KeysManager, KeysInterface};
//! # use lightning::util::logger::{Logger, Record};
//! # use bitcoin::secp256k1::PublicKey;
//! #
//! # struct FakeLogger {};
//! # impl Logger for FakeLogger {
//! #     fn log(&self, record: &Record) { unimplemented!() }
//! # }
//! # fn find_scored_route(payer: PublicKey, route_params: RouteParameters, network_graph: NetworkGraph) {
//! # let logger = FakeLogger {};
//! #
//! // Use the default channel penalties.
//! let params = ProbabilisticScoringParameters::default();
//! let scorer = ProbabilisticScorer::new(params, &network_graph, &logger);
//!
//! // Or use custom channel penalties.
//! let params = ProbabilisticScoringParameters {
//!     liquidity_penalty_multiplier_msat: 2 * 1000,
//!     ..ProbabilisticScoringParameters::default()
//! };
//! let scorer = ProbabilisticScorer::new(params, &network_graph, &logger);
//! # let random_seed_bytes = [42u8; 32];
//!
//! let route = find_route(&payer, &route_params, &network_graph, None, &logger, &scorer, &random_seed_bytes);
//! # }
//! ```
//!
//! # Note
//!
//! Persisting when built with feature `no-std` and restoring without it, or vice versa, uses
//! different types and thus is undefined.
//!
//! [`find_route`]: crate::routing::router::find_route

use ln::msgs::DecodeError;
use routing::network_graph::{EffectiveCapacity, NetworkGraph, NodeId};
use routing::router::RouteHop;
use util::ser::{Readable, ReadableArgs, Writeable, Writer};
use util::logger::Logger;
use util::time::Time;

use prelude::*;
use core::fmt;
use core::cell::{RefCell, RefMut};
use core::ops::{Deref, DerefMut};
use core::time::Duration;
use io::{self, Read};
use sync::{Mutex, MutexGuard};

/// We define Score ever-so-slightly differently based on whether we are being built for C bindings
/// or not. For users, `LockableScore` must somehow be writeable to disk. For Rust users, this is
/// no problem - you move a `Score` that implements `Writeable` into a `Mutex`, lock it, and now
/// you have the original, concrete, `Score` type, which presumably implements `Writeable`.
///
/// For C users, once you've moved the `Score` into a `LockableScore` all you have after locking it
/// is an opaque trait object with an opaque pointer with no type info. Users could take the unsafe
/// approach of blindly casting that opaque pointer to a concrete type and calling `Writeable` from
/// there, but other languages downstream of the C bindings (e.g. Java) can't even do that.
/// Instead, we really want `Score` and `LockableScore` to implement `Writeable` directly, which we
/// do here by defining `Score` differently for `cfg(c_bindings)`.
macro_rules! define_score { ($($supertrait: path)*) => {
/// An interface used to score payment channels for path finding.
///
///     Scoring is in terms of fees willing to be paid in order to avoid routing through a channel.
pub trait Score $(: $supertrait)* {
        /// Returns the fee in msats willing to be paid to avoid routing `send_amt_msat` through the
        /// given channel in the direction from `source` to `target`.
        ///
        /// The channel's capacity (less any other MPP parts that are also being considered for use in
        /// the same payment) is given by `capacity_msat`. It may be determined from various sources
        /// such as a chain data, network gossip, or invoice hints. For invoice hints, a capacity near
        /// [`u64::max_value`] is given to indicate sufficient capacity for the invoice's full amount.
        /// Thus, implementations should be overflow-safe.
        fn channel_penalty_msat(
                &self, short_channel_id: u64, source: &NodeId, target: &NodeId, usage: ChannelUsage
        ) -> u64;

        /// Handles updating channel penalties after failing to route through a channel.
        fn payment_path_failed(&mut self, path: &[&RouteHop], short_channel_id: u64);

        /// Handles updating channel penalties after successfully routing along a path.
        fn payment_path_successful(&mut self, path: &[&RouteHop]);
}

impl<S: Score, T: DerefMut<Target=S> $(+ $supertrait)*> Score for T {
        fn channel_penalty_msat(
                &self, short_channel_id: u64, source: &NodeId, target: &NodeId, usage: ChannelUsage
        ) -> u64 {
                self.deref().channel_penalty_msat(short_channel_id, source, target, usage)
        }

        fn payment_path_failed(&mut self, path: &[&RouteHop], short_channel_id: u64) {
                self.deref_mut().payment_path_failed(path, short_channel_id)
        }

        fn payment_path_successful(&mut self, path: &[&RouteHop]) {
                self.deref_mut().payment_path_successful(path)
        }
}
} }

#[cfg(c_bindings)]
define_score!(Writeable);
#[cfg(not(c_bindings))]
define_score!();

/// A scorer that is accessed under a lock.
///
/// Needed so that calls to [`Score::channel_penalty_msat`] in [`find_route`] can be made while
/// having shared ownership of a scorer but without requiring internal locking in [`Score`]
/// implementations. Internal locking would be detrimental to route finding performance and could
/// result in [`Score::channel_penalty_msat`] returning a different value for the same channel.
///
/// [`find_route`]: crate::routing::router::find_route
pub trait LockableScore<'a> {
        /// The locked [`Score`] type.
        type Locked: 'a + Score;

        /// Returns the locked scorer.
        fn lock(&'a self) -> Self::Locked;
}

/// Refers to a scorer that is accessible under lock and also writeable to disk
///
/// We need this trait to be able to pass in a scorer to `lightning-background-processor` that will enable us to
/// use the Persister to persist it.
pub trait WriteableScore<'a>: LockableScore<'a> + Writeable {}

impl<'a, T> WriteableScore<'a> for T where T: LockableScore<'a> + Writeable {}

/// (C-not exported)
impl<'a, T: 'a + Score> LockableScore<'a> for Mutex<T> {
        type Locked = MutexGuard<'a, T>;

        fn lock(&'a self) -> MutexGuard<'a, T> {
                Mutex::lock(self).unwrap()
        }
}

impl<'a, T: 'a + Score> LockableScore<'a> for RefCell<T> {
        type Locked = RefMut<'a, T>;

        fn lock(&'a self) -> RefMut<'a, T> {
                self.borrow_mut()
        }
}

#[cfg(c_bindings)]
/// A concrete implementation of [`LockableScore`] which supports multi-threading.
pub struct MultiThreadedLockableScore<S: Score> {
        score: Mutex<S>,
}
#[cfg(c_bindings)]
/// (C-not exported)
impl<'a, T: Score + 'a> LockableScore<'a> for MultiThreadedLockableScore<T> {
        type Locked = MutexGuard<'a, T>;

        fn lock(&'a self) -> MutexGuard<'a, T> {
                Mutex::lock(&self.score).unwrap()
        }
}

#[cfg(c_bindings)]
impl<T: Score> MultiThreadedLockableScore<T> {
        /// Creates a new [`MultiThreadedLockableScore`] given an underlying [`Score`].
        pub fn new(score: T) -> Self {
                MultiThreadedLockableScore { score: Mutex::new(score) }
        }
}

#[cfg(c_bindings)]
/// (C-not exported)
impl<'a, T: Writeable> Writeable for RefMut<'a, T> {
        fn write<W: Writer>(&self, writer: &mut W) -> Result<(), io::Error> {
                T::write(&**self, writer)
        }
}

#[cfg(c_bindings)]
/// (C-not exported)
impl<'a, S: Writeable> Writeable for MutexGuard<'a, S> {
        fn write<W: Writer>(&self, writer: &mut W) -> Result<(), io::Error> {
                S::write(&**self, writer)
        }
}

/// Proposed use of a channel passed as a parameter to [`Score::channel_penalty_msat`].
#[derive(Clone, Copy)]
pub struct ChannelUsage {
        /// The amount to send through the channel, denominated in millisatoshis.
        pub amount_msat: u64,

        /// Total amount, denominated in millisatoshis, already allocated to send through the channel
        /// as part of a multi-path payment.
        pub inflight_htlc_msat: u64,

        /// The effective capacity of the channel.
        pub effective_capacity: EffectiveCapacity,
}

#[derive(Clone)]
/// [`Score`] implementation that uses a fixed penalty.
pub struct FixedPenaltyScorer {
        penalty_msat: u64,
}

impl FixedPenaltyScorer {
        /// Creates a new scorer using `penalty_msat`.
        pub fn with_penalty(penalty_msat: u64) -> Self {
                Self { penalty_msat }
        }
}

impl Score for FixedPenaltyScorer {
        fn channel_penalty_msat(&self, _: u64, _: &NodeId, _: &NodeId, _: ChannelUsage) -> u64 {
                self.penalty_msat
        }

        fn payment_path_failed(&mut self, _path: &[&RouteHop], _short_channel_id: u64) {}

        fn payment_path_successful(&mut self, _path: &[&RouteHop]) {}
}

impl Writeable for FixedPenaltyScorer {
        #[inline]
        fn write<W: Writer>(&self, w: &mut W) -> Result<(), io::Error> {
                write_tlv_fields!(w, {});
                Ok(())
        }
}

impl ReadableArgs<u64> for FixedPenaltyScorer {
        #[inline]
        fn read<R: Read>(r: &mut R, penalty_msat: u64) -> Result<Self, DecodeError> {
                read_tlv_fields!(r, {});
                Ok(Self { penalty_msat })
        }
}

#[cfg(not(feature = "no-std"))]
type ConfiguredTime = std::time::Instant;
#[cfg(feature = "no-std")]
use util::time::Eternity;
#[cfg(feature = "no-std")]
type ConfiguredTime = Eternity;

/// [`Score`] implementation using channel success probability distributions.
///
/// Based on *Optimally Reliable & Cheap Payment Flows on the Lightning Network* by Rene Pickhardt
/// and Stefan Richter [[1]]. Given the uncertainty of channel liquidity balances, probability
/// distributions are defined based on knowledge learned from successful and unsuccessful attempts.
/// Then the negative `log10` of the success probability is used to determine the cost of routing a
/// specific HTLC amount through a channel.
///
/// Knowledge about channel liquidity balances takes the form of upper and lower bounds on the
/// possible liquidity. Certainty of the bounds is decreased over time using a decay function. See
/// [`ProbabilisticScoringParameters`] for details.
///
/// Since the scorer aims to learn the current channel liquidity balances, it works best for nodes
/// with high payment volume or that actively probe the [`NetworkGraph`]. Nodes with low payment
/// volume are more likely to experience failed payment paths, which would need to be retried.
///
/// # Note
///
/// Mixing the `no-std` feature between serialization and deserialization results in undefined
/// behavior.
///
/// [1]: https://arxiv.org/abs/2107.05322
pub type ProbabilisticScorer<G, L> = ProbabilisticScorerUsingTime::<G, L, ConfiguredTime>;

/// Probabilistic [`Score`] implementation.
///
/// (C-not exported) generally all users should use the [`ProbabilisticScorer`] type alias.
pub struct ProbabilisticScorerUsingTime<G: Deref<Target = NetworkGraph>, L: Deref, T: Time> where L::Target: Logger {
        params: ProbabilisticScoringParameters,
        network_graph: G,
        logger: L,
        // TODO: Remove entries of closed channels.
        channel_liquidities: HashMap<u64, ChannelLiquidity<T>>,
}

/// Parameters for configuring [`ProbabilisticScorer`].
///
/// Used to configure base, liquidity, and amount penalties, the sum of which comprises the channel
/// penalty (i.e., the amount in msats willing to be paid to avoid routing through the channel).
#[derive(Clone, Copy)]
pub struct ProbabilisticScoringParameters {
        /// A fixed penalty in msats to apply to each channel.
        ///
        /// Default value: 500 msat
        pub base_penalty_msat: u64,

        /// A multiplier used in conjunction with the negative `log10` of the channel's success
        /// probability for a payment to determine the liquidity penalty.
        ///
        /// The penalty is based in part on the knowledge learned from prior successful and unsuccessful
        /// payments. This knowledge is decayed over time based on [`liquidity_offset_half_life`]. The
        /// penalty is effectively limited to `2 * liquidity_penalty_multiplier_msat` (corresponding to
        /// lower bounding the success probability to `0.01`) when the amount falls within the
        /// uncertainty bounds of the channel liquidity balance. Amounts above the upper bound will
        /// result in a `u64::max_value` penalty, however.
        ///
        /// Default value: 40,000 msat
        ///
        /// [`liquidity_offset_half_life`]: Self::liquidity_offset_half_life
        pub liquidity_penalty_multiplier_msat: u64,

        /// The time required to elapse before any knowledge learned about channel liquidity balances is
        /// cut in half.
        ///
        /// The bounds are defined in terms of offsets and are initially zero. Increasing the offsets
        /// gives tighter bounds on the channel liquidity balance. Thus, halving the offsets decreases
        /// the certainty of the channel liquidity balance.
        ///
        /// Default value: 1 hour
        ///
        /// # Note
        ///
        /// When built with the `no-std` feature, time will never elapse. Therefore, the channel
        /// liquidity knowledge will never decay except when the bounds cross.
        pub liquidity_offset_half_life: Duration,

        /// A multiplier used in conjunction with a payment amount and the negative `log10` of the
        /// channel's success probability for the payment to determine the amount penalty.
        ///
        /// The purpose of the amount penalty is to avoid having fees dominate the channel cost (i.e.,
        /// fees plus penalty) for large payments. The penalty is computed as the product of this
        /// multiplier and `2^20`ths of the payment amount, weighted by the negative `log10` of the
        /// success probability.
        ///
        /// `-log10(success_probability) * amount_penalty_multiplier_msat * amount_msat / 2^20`
        ///
        /// In practice, this means for 0.1 success probability (`-log10(0.1) == 1`) each `2^20`th of
        /// the amount will result in a penalty of the multiplier. And, as the success probability
        /// decreases, the negative `log10` weighting will increase dramatically. For higher success
        /// probabilities, the multiplier will have a decreasing effect as the negative `log10` will
        /// fall below `1`.
        ///
        /// Default value: 256 msat
        pub amount_penalty_multiplier_msat: u64,
}

/// Accounting for channel liquidity balance uncertainty.
///
/// Direction is defined in terms of [`NodeId`] partial ordering, where the source node is the
/// first node in the ordering of the channel's counterparties. Thus, swapping the two liquidity
/// offset fields gives the opposite direction.
struct ChannelLiquidity<T: Time> {
        /// Lower channel liquidity bound in terms of an offset from zero.
        min_liquidity_offset_msat: u64,

        /// Upper channel liquidity bound in terms of an offset from the effective capacity.
        max_liquidity_offset_msat: u64,

        /// Time when the liquidity bounds were last modified.
        last_updated: T,
}

/// A snapshot of [`ChannelLiquidity`] in one direction assuming a certain channel capacity and
/// decayed with a given half life.
struct DirectedChannelLiquidity<L: Deref<Target = u64>, T: Time, U: Deref<Target = T>> {
        min_liquidity_offset_msat: L,
        max_liquidity_offset_msat: L,
        capacity_msat: u64,
        last_updated: U,
        now: T,
        half_life: Duration,
}

impl<G: Deref<Target = NetworkGraph>, L: Deref, T: Time> ProbabilisticScorerUsingTime<G, L, T> where L::Target: Logger {
        /// Creates a new scorer using the given scoring parameters for sending payments from a node
        /// through a network graph.
        pub fn new(params: ProbabilisticScoringParameters, network_graph: G, logger: L) -> Self {
                Self {
                        params,
                        network_graph,
                        logger,
                        channel_liquidities: HashMap::new(),
                }
        }

        #[cfg(test)]
        fn with_channel(mut self, short_channel_id: u64, liquidity: ChannelLiquidity<T>) -> Self {
                assert!(self.channel_liquidities.insert(short_channel_id, liquidity).is_none());
                self
        }

        /// Dump the contents of this scorer into the configured logger.
        ///
        /// Note that this writes roughly one line per channel for which we have a liquidity estimate,
        /// which may be a substantial amount of log output.
        pub fn debug_log_liquidity_stats(&self) {
                let graph = self.network_graph.read_only();
                for (scid, liq) in self.channel_liquidities.iter() {
                        if let Some(chan_debug) = graph.channels().get(scid) {
                                let log_direction = |source, target| {
                                        if let Some((directed_info, _)) = chan_debug.as_directed_to(target) {
                                                let amt = directed_info.effective_capacity().as_msat();
                                                let dir_liq = liq.as_directed(source, target, amt, self.params.liquidity_offset_half_life);
                                                log_debug!(self.logger, "Liquidity from {:?} to {:?} via {} is in the range ({}, {})",
                                                        source, target, scid, dir_liq.min_liquidity_msat(), dir_liq.max_liquidity_msat());
                                        } else {
                                                log_debug!(self.logger, "No amount known for SCID {} from {:?} to {:?}", scid, source, target);
                                        }
                                };

                                log_direction(&chan_debug.node_one, &chan_debug.node_two);
                                log_direction(&chan_debug.node_two, &chan_debug.node_one);
                        } else {
                                log_debug!(self.logger, "No network graph entry for SCID {}", scid);
                        }
                }
        }
}

impl ProbabilisticScoringParameters {
        #[cfg(test)]
        fn zero_penalty() -> Self {
                Self {
                        base_penalty_msat: 0,
                        liquidity_penalty_multiplier_msat: 0,
                        liquidity_offset_half_life: Duration::from_secs(3600),
                        amount_penalty_multiplier_msat: 0,
                }
        }
}

impl Default for ProbabilisticScoringParameters {
        fn default() -> Self {
                Self {
                        base_penalty_msat: 500,
                        liquidity_penalty_multiplier_msat: 40_000,
                        liquidity_offset_half_life: Duration::from_secs(3600),
                        amount_penalty_multiplier_msat: 256,
                }
        }
}

impl<T: Time> ChannelLiquidity<T> {
        #[inline]
        fn new() -> Self {
                Self {
                        min_liquidity_offset_msat: 0,
                        max_liquidity_offset_msat: 0,
                        last_updated: T::now(),
                }
        }

        /// Returns a view of the channel liquidity directed from `source` to `target` assuming
        /// `capacity_msat`.
        fn as_directed(
                &self, source: &NodeId, target: &NodeId, capacity_msat: u64, half_life: Duration
        ) -> DirectedChannelLiquidity<&u64, T, &T> {
                let (min_liquidity_offset_msat, max_liquidity_offset_msat) = if source < target {
                        (&self.min_liquidity_offset_msat, &self.max_liquidity_offset_msat)
                } else {
                        (&self.max_liquidity_offset_msat, &self.min_liquidity_offset_msat)
                };

                DirectedChannelLiquidity {
                        min_liquidity_offset_msat,
                        max_liquidity_offset_msat,
                        capacity_msat,
                        last_updated: &self.last_updated,
                        now: T::now(),
                        half_life,
                }
        }

        /// Returns a mutable view of the channel liquidity directed from `source` to `target` assuming
        /// `capacity_msat`.
        fn as_directed_mut(
                &mut self, source: &NodeId, target: &NodeId, capacity_msat: u64, half_life: Duration
        ) -> DirectedChannelLiquidity<&mut u64, T, &mut T> {
                let (min_liquidity_offset_msat, max_liquidity_offset_msat) = if source < target {
                        (&mut self.min_liquidity_offset_msat, &mut self.max_liquidity_offset_msat)
                } else {
                        (&mut self.max_liquidity_offset_msat, &mut self.min_liquidity_offset_msat)
                };

                DirectedChannelLiquidity {
                        min_liquidity_offset_msat,
                        max_liquidity_offset_msat,
                        capacity_msat,
                        last_updated: &mut self.last_updated,
                        now: T::now(),
                        half_life,
                }
        }
}

/// Bounds `-log10` to avoid excessive liquidity penalties for payments with low success
/// probabilities.
const NEGATIVE_LOG10_UPPER_BOUND: u64 = 2;

/// The rough cutoff at which our precision falls off and we should stop bothering to try to log a
/// ratio, as X in 1/X.
const PRECISION_LOWER_BOUND_DENOMINATOR: u64 = approx::LOWER_BITS_BOUND;

/// The divisor used when computing the amount penalty.
const AMOUNT_PENALTY_DIVISOR: u64 = 1 << 20;

impl<L: Deref<Target = u64>, T: Time, U: Deref<Target = T>> DirectedChannelLiquidity<L, T, U> {
        /// Returns a penalty for routing the given HTLC `amount_msat` through the channel in this
        /// direction.
        fn penalty_msat(&self, amount_msat: u64, params: ProbabilisticScoringParameters) -> u64 {
                let max_liquidity_msat = self.max_liquidity_msat();
                let min_liquidity_msat = core::cmp::min(self.min_liquidity_msat(), max_liquidity_msat);
                if amount_msat <= min_liquidity_msat {
                        0
                } else if amount_msat >= max_liquidity_msat {
                        if amount_msat > max_liquidity_msat {
                                u64::max_value()
                        } else if max_liquidity_msat != self.capacity_msat {
                                // Avoid using the failed channel on retry.
                                u64::max_value()
                        } else {
                                // Equivalent to hitting the else clause below with the amount equal to the
                                // effective capacity and without any certainty on the liquidity upper bound.
                                let negative_log10_times_2048 = NEGATIVE_LOG10_UPPER_BOUND * 2048;
                                self.combined_penalty_msat(amount_msat, negative_log10_times_2048, params)
                        }
                } else {
                        let numerator = (max_liquidity_msat - amount_msat).saturating_add(1);
                        let denominator = (max_liquidity_msat - min_liquidity_msat).saturating_add(1);
                        if amount_msat - min_liquidity_msat < denominator / PRECISION_LOWER_BOUND_DENOMINATOR {
                                // If the failure probability is < 1.5625% (as 1 - numerator/denominator < 1/64),
                                // don't bother trying to use the log approximation as it gets too noisy to be
                                // particularly helpful, instead just round down to 0 and return the base penalty.
                                params.base_penalty_msat
                        } else {
                                let negative_log10_times_2048 =
                                        approx::negative_log10_times_2048(numerator, denominator);
                                self.combined_penalty_msat(amount_msat, negative_log10_times_2048, params)
                        }
                }
        }

        /// Computes the liquidity and amount penalties and adds them to the base penalty.
        #[inline(always)]
        fn combined_penalty_msat(
                &self, amount_msat: u64, negative_log10_times_2048: u64,
                params: ProbabilisticScoringParameters
        ) -> u64 {
                let liquidity_penalty_msat = {
                        // Upper bound the liquidity penalty to ensure some channel is selected.
                        let multiplier_msat = params.liquidity_penalty_multiplier_msat;
                        let max_penalty_msat = multiplier_msat.saturating_mul(NEGATIVE_LOG10_UPPER_BOUND);
                        (negative_log10_times_2048.saturating_mul(multiplier_msat) / 2048).min(max_penalty_msat)
                };
                let amount_penalty_msat = negative_log10_times_2048
                        .saturating_mul(params.amount_penalty_multiplier_msat)
                        .saturating_mul(amount_msat) / 2048 / AMOUNT_PENALTY_DIVISOR;

                params.base_penalty_msat
                        .saturating_add(liquidity_penalty_msat)
                        .saturating_add(amount_penalty_msat)
        }

        /// Returns the lower bound of the channel liquidity balance in this direction.
        fn min_liquidity_msat(&self) -> u64 {
                self.decayed_offset_msat(*self.min_liquidity_offset_msat)
        }

        /// Returns the upper bound of the channel liquidity balance in this direction.
        fn max_liquidity_msat(&self) -> u64 {
                self.capacity_msat
                        .checked_sub(self.decayed_offset_msat(*self.max_liquidity_offset_msat))
                        .unwrap_or(0)
        }

        fn decayed_offset_msat(&self, offset_msat: u64) -> u64 {
                self.now.duration_since(*self.last_updated).as_secs()
                        .checked_div(self.half_life.as_secs())
                        .and_then(|decays| offset_msat.checked_shr(decays as u32))
                        .unwrap_or(0)
        }
}

impl<L: DerefMut<Target = u64>, T: Time, U: DerefMut<Target = T>> DirectedChannelLiquidity<L, T, U> {
        /// Adjusts the channel liquidity balance bounds when failing to route `amount_msat`.
        fn failed_at_channel<Log: Deref>(&mut self, amount_msat: u64, chan_descr: fmt::Arguments, logger: &Log) where Log::Target: Logger {
                if amount_msat < self.max_liquidity_msat() {
                        log_debug!(logger, "Setting max liquidity of {} to {}", chan_descr, amount_msat);
                        self.set_max_liquidity_msat(amount_msat);
                } else {
                        log_trace!(logger, "Max liquidity of {} already more than {}", chan_descr, amount_msat);
                }
        }

        /// Adjusts the channel liquidity balance bounds when failing to route `amount_msat` downstream.
        fn failed_downstream<Log: Deref>(&mut self, amount_msat: u64, chan_descr: fmt::Arguments, logger: &Log) where Log::Target: Logger {
                if amount_msat > self.min_liquidity_msat() {
                        log_debug!(logger, "Setting min liquidity of {} to {}", chan_descr, amount_msat);
                        self.set_min_liquidity_msat(amount_msat);
                } else {
                        log_trace!(logger, "Min liquidity of {} already less than {}", chan_descr, amount_msat);
                }
        }

        /// Adjusts the channel liquidity balance bounds when successfully routing `amount_msat`.
        fn successful<Log: Deref>(&mut self, amount_msat: u64, chan_descr: fmt::Arguments, logger: &Log) where Log::Target: Logger {
                let max_liquidity_msat = self.max_liquidity_msat().checked_sub(amount_msat).unwrap_or(0);
                log_debug!(logger, "Subtracting {} from max liquidity of {} (setting it to {})", amount_msat, chan_descr, max_liquidity_msat);
                self.set_max_liquidity_msat(max_liquidity_msat);
        }

        /// Adjusts the lower bound of the channel liquidity balance in this direction.
        fn set_min_liquidity_msat(&mut self, amount_msat: u64) {
                *self.min_liquidity_offset_msat = amount_msat;
                *self.max_liquidity_offset_msat = if amount_msat > self.max_liquidity_msat() {
                        0
                } else {
                        self.decayed_offset_msat(*self.max_liquidity_offset_msat)
                };
                *self.last_updated = self.now;
        }

        /// Adjusts the upper bound of the channel liquidity balance in this direction.
        fn set_max_liquidity_msat(&mut self, amount_msat: u64) {
                *self.max_liquidity_offset_msat = self.capacity_msat.checked_sub(amount_msat).unwrap_or(0);
                *self.min_liquidity_offset_msat = if amount_msat < self.min_liquidity_msat() {
                        0
                } else {
                        self.decayed_offset_msat(*self.min_liquidity_offset_msat)
                };
                *self.last_updated = self.now;
        }
}

impl<G: Deref<Target = NetworkGraph>, L: Deref, T: Time> Score for ProbabilisticScorerUsingTime<G, L, T> where L::Target: Logger {
        fn channel_penalty_msat(
                &self, short_channel_id: u64, source: &NodeId, target: &NodeId, usage: ChannelUsage
        ) -> u64 {
                if let EffectiveCapacity::ExactLiquidity { liquidity_msat } = usage.effective_capacity {
                        if usage.amount_msat > liquidity_msat {
                                return u64::max_value();
                        } else {
                                return self.params.base_penalty_msat;
                        };
                }

                let liquidity_offset_half_life = self.params.liquidity_offset_half_life;
                let amount_msat = usage.amount_msat;
                let capacity_msat = usage.effective_capacity.as_msat()
                        .saturating_sub(usage.inflight_htlc_msat);
                self.channel_liquidities
                        .get(&short_channel_id)
                        .unwrap_or(&ChannelLiquidity::new())
                        .as_directed(source, target, capacity_msat, liquidity_offset_half_life)
                        .penalty_msat(amount_msat, self.params)
        }

        fn payment_path_failed(&mut self, path: &[&RouteHop], short_channel_id: u64) {
                let amount_msat = path.split_last().map(|(hop, _)| hop.fee_msat).unwrap_or(0);
                let liquidity_offset_half_life = self.params.liquidity_offset_half_life;
                log_trace!(self.logger, "Scoring path through to SCID {} as having failed at {} msat", short_channel_id, amount_msat);
                let network_graph = self.network_graph.read_only();
                for (hop_idx, hop) in path.iter().enumerate() {
                        let target = NodeId::from_pubkey(&hop.pubkey);
                        let channel_directed_from_source = network_graph.channels()
                                .get(&hop.short_channel_id)
                                .and_then(|channel| channel.as_directed_to(&target));

                        if hop.short_channel_id == short_channel_id && hop_idx == 0 {
                                log_warn!(self.logger, "Payment failed at the first hop - we do not attempt to learn channel info in such cases as we can directly observe local state.\n\tBecause we know the local state, we should generally not see failures here - this may be an indication that your channel peer on channel {} is broken and you may wish to close the channel.", hop.short_channel_id);
                        }

                        // Only score announced channels.
                        if let Some((channel, source)) = channel_directed_from_source {
                                let capacity_msat = channel.effective_capacity().as_msat();
                                if hop.short_channel_id == short_channel_id {
                                        self.channel_liquidities
                                                .entry(hop.short_channel_id)
                                                .or_insert_with(ChannelLiquidity::new)
                                                .as_directed_mut(source, &target, capacity_msat, liquidity_offset_half_life)
                                                .failed_at_channel(amount_msat, format_args!("SCID {}, towards {:?}", hop.short_channel_id, target), &self.logger);
                                        break;
                                }

                                self.channel_liquidities
                                        .entry(hop.short_channel_id)
                                        .or_insert_with(ChannelLiquidity::new)
                                        .as_directed_mut(source, &target, capacity_msat, liquidity_offset_half_life)
                                        .failed_downstream(amount_msat, format_args!("SCID {}, towards {:?}", hop.short_channel_id, target), &self.logger);
                        } else {
                                log_debug!(self.logger, "Not able to penalize channel with SCID {} as we do not have graph info for it (likely a route-hint last-hop).",
                                        hop.short_channel_id);
                        }
                }
        }

        fn payment_path_successful(&mut self, path: &[&RouteHop]) {
                let amount_msat = path.split_last().map(|(hop, _)| hop.fee_msat).unwrap_or(0);
                let liquidity_offset_half_life = self.params.liquidity_offset_half_life;
                log_trace!(self.logger, "Scoring path through SCID {} as having succeeded at {} msat.",
                        path.split_last().map(|(hop, _)| hop.short_channel_id).unwrap_or(0), amount_msat);
                let network_graph = self.network_graph.read_only();
                for hop in path {
                        let target = NodeId::from_pubkey(&hop.pubkey);
                        let channel_directed_from_source = network_graph.channels()
                                .get(&hop.short_channel_id)
                                .and_then(|channel| channel.as_directed_to(&target));

                        // Only score announced channels.
                        if let Some((channel, source)) = channel_directed_from_source {
                                let capacity_msat = channel.effective_capacity().as_msat();
                                self.channel_liquidities
                                        .entry(hop.short_channel_id)
                                        .or_insert_with(ChannelLiquidity::new)
                                        .as_directed_mut(source, &target, capacity_msat, liquidity_offset_half_life)
                                        .successful(amount_msat, format_args!("SCID {}, towards {:?}", hop.short_channel_id, target), &self.logger);
                        } else {
                                log_debug!(self.logger, "Not able to learn for channel with SCID {} as we do not have graph info for it (likely a route-hint last-hop).",
                                        hop.short_channel_id);
                        }
                }
        }
}

mod approx {
        const BITS: u32 = 64;
        const HIGHEST_BIT: u32 = BITS - 1;
        const LOWER_BITS: u32 = 6;
        pub(super) const LOWER_BITS_BOUND: u64 = 1 << LOWER_BITS;
        const LOWER_BITMASK: u64 = (1 << LOWER_BITS) - 1;

        /// Look-up table for `log10(x) * 2048` where row `i` is used for each `x` having `i` as the
        /// most significant bit. The next 4 bits of `x`, if applicable, are used for the second index.
        const LOG10_TIMES_2048: [[u16; (LOWER_BITS_BOUND) as usize]; BITS as usize] = [
                [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
                        0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
                        0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
                        0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
                [617, 617, 617, 617, 617, 617, 617, 617, 617, 617, 617, 617, 617, 617, 617, 617,
                        617, 617, 617, 617, 617, 617, 617, 617, 617, 617, 617, 617, 617, 617, 617, 617,
                        977, 977, 977, 977, 977, 977, 977, 977, 977, 977, 977, 977, 977, 977, 977, 977,
                        977, 977, 977, 977, 977, 977, 977, 977, 977, 977, 977, 977, 977, 977, 977, 977],
                [1233, 1233, 1233, 1233, 1233, 1233, 1233, 1233, 1233, 1233, 1233, 1233, 1233, 1233, 1233, 1233,
                        1431, 1431, 1431, 1431, 1431, 1431, 1431, 1431, 1431, 1431, 1431, 1431, 1431, 1431, 1431, 1431,
                        1594, 1594, 1594, 1594, 1594, 1594, 1594, 1594, 1594, 1594, 1594, 1594, 1594, 1594, 1594, 1594,
                        1731, 1731, 1731, 1731, 1731, 1731, 1731, 1731, 1731, 1731, 1731, 1731, 1731, 1731, 1731, 1731],
                [1850, 1850, 1850, 1850, 1850, 1850, 1850, 1850, 1954, 1954, 1954, 1954, 1954, 1954, 1954, 1954,
                        2048, 2048, 2048, 2048, 2048, 2048, 2048, 2048, 2133, 2133, 2133, 2133, 2133, 2133, 2133, 2133,
                        2210, 2210, 2210, 2210, 2210, 2210, 2210, 2210, 2281, 2281, 2281, 2281, 2281, 2281, 2281, 2281,
                        2347, 2347, 2347, 2347, 2347, 2347, 2347, 2347, 2409, 2409, 2409, 2409, 2409, 2409, 2409, 2409],
                [2466, 2466, 2466, 2466, 2520, 2520, 2520, 2520, 2571, 2571, 2571, 2571, 2619, 2619, 2619, 2619,
                        2665, 2665, 2665, 2665, 2708, 2708, 2708, 2708, 2749, 2749, 2749, 2749, 2789, 2789, 2789, 2789,
                        2827, 2827, 2827, 2827, 2863, 2863, 2863, 2863, 2898, 2898, 2898, 2898, 2931, 2931, 2931, 2931,
                        2964, 2964, 2964, 2964, 2995, 2995, 2995, 2995, 3025, 3025, 3025, 3025, 3054, 3054, 3054, 3054],
                [3083, 3083, 3110, 3110, 3136, 3136, 3162, 3162, 3187, 3187, 3212, 3212, 3235, 3235, 3259, 3259,
                        3281, 3281, 3303, 3303, 3324, 3324, 3345, 3345, 3366, 3366, 3386, 3386, 3405, 3405, 3424, 3424,
                        3443, 3443, 3462, 3462, 3479, 3479, 3497, 3497, 3514, 3514, 3531, 3531, 3548, 3548, 3564, 3564,
                        3580, 3580, 3596, 3596, 3612, 3612, 3627, 3627, 3642, 3642, 3656, 3656, 3671, 3671, 3685, 3685],
                [3699, 3713, 3726, 3740, 3753, 3766, 3779, 3791, 3804, 3816, 3828, 3840, 3852, 3864, 3875, 3886,
                        3898, 3909, 3919, 3930, 3941, 3951, 3962, 3972, 3982, 3992, 4002, 4012, 4022, 4031, 4041, 4050,
                        4060, 4069, 4078, 4087, 4096, 4105, 4114, 4122, 4131, 4139, 4148, 4156, 4164, 4173, 4181, 4189,
                        4197, 4205, 4213, 4220, 4228, 4236, 4243, 4251, 4258, 4266, 4273, 4280, 4287, 4294, 4302, 4309],
                [4316, 4329, 4343, 4356, 4369, 4382, 4395, 4408, 4420, 4433, 4445, 4457, 4468, 4480, 4492, 4503,
                        4514, 4525, 4536, 4547, 4557, 4568, 4578, 4589, 4599, 4609, 4619, 4629, 4638, 4648, 4657, 4667,
                        4676, 4685, 4695, 4704, 4713, 4721, 4730, 4739, 4747, 4756, 4764, 4773, 4781, 4789, 4797, 4805,
                        4813, 4821, 4829, 4837, 4845, 4852, 4860, 4867, 4875, 4882, 4889, 4897, 4904, 4911, 4918, 4925],
                [4932, 4946, 4959, 4973, 4986, 4999, 5012, 5024, 5037, 5049, 5061, 5073, 5085, 5097, 5108, 5119,
                        5131, 5142, 5153, 5163, 5174, 5184, 5195, 5205, 5215, 5225, 5235, 5245, 5255, 5264, 5274, 5283,
                        5293, 5302, 5311, 5320, 5329, 5338, 5347, 5355, 5364, 5372, 5381, 5389, 5397, 5406, 5414, 5422,
                        5430, 5438, 5446, 5453, 5461, 5469, 5476, 5484, 5491, 5499, 5506, 5513, 5520, 5527, 5535, 5542],
                [5549, 5562, 5576, 5589, 5603, 5615, 5628, 5641, 5653, 5666, 5678, 5690, 5701, 5713, 5725, 5736,
                        5747, 5758, 5769, 5780, 5790, 5801, 5811, 5822, 5832, 5842, 5852, 5862, 5871, 5881, 5890, 5900,
                        5909, 5918, 5928, 5937, 5946, 5954, 5963, 5972, 5980, 5989, 5997, 6006, 6014, 6022, 6030, 6038,
                        6046, 6054, 6062, 6070, 6078, 6085, 6093, 6100, 6108, 6115, 6122, 6130, 6137, 6144, 6151, 6158],
                [6165, 6179, 6192, 6206, 6219, 6232, 6245, 6257, 6270, 6282, 6294, 6306, 6318, 6330, 6341, 6352,
                        6364, 6375, 6386, 6396, 6407, 6417, 6428, 6438, 6448, 6458, 6468, 6478, 6488, 6497, 6507, 6516,
                        6526, 6535, 6544, 6553, 6562, 6571, 6580, 6588, 6597, 6605, 6614, 6622, 6630, 6639, 6647, 6655,
                        6663, 6671, 6679, 6686, 6694, 6702, 6709, 6717, 6724, 6732, 6739, 6746, 6753, 6761, 6768, 6775],
                [6782, 6795, 6809, 6822, 6836, 6849, 6861, 6874, 6886, 6899, 6911, 6923, 6934, 6946, 6958, 6969,
                        6980, 6991, 7002, 7013, 7023, 7034, 7044, 7055, 7065, 7075, 7085, 7095, 7104, 7114, 7124, 7133,
                        7142, 7151, 7161, 7170, 7179, 7187, 7196, 7205, 7213, 7222, 7230, 7239, 7247, 7255, 7263, 7271,
                        7279, 7287, 7295, 7303, 7311, 7318, 7326, 7333, 7341, 7348, 7355, 7363, 7370, 7377, 7384, 7391],
                [7398, 7412, 7425, 7439, 7452, 7465, 7478, 7490, 7503, 7515, 7527, 7539, 7551, 7563, 7574, 7585,
                        7597, 7608, 7619, 7629, 7640, 7651, 7661, 7671, 7681, 7691, 7701, 7711, 7721, 7731, 7740, 7749,
                        7759, 7768, 7777, 7786, 7795, 7804, 7813, 7821, 7830, 7838, 7847, 7855, 7864, 7872, 7880, 7888,
                        7896, 7904, 7912, 7919, 7927, 7935, 7942, 7950, 7957, 7965, 7972, 7979, 7986, 7994, 8001, 8008],
                [8015, 8028, 8042, 8055, 8069, 8082, 8094, 8107, 8119, 8132, 8144, 8156, 8167, 8179, 8191, 8202,
                        8213, 8224, 8235, 8246, 8256, 8267, 8277, 8288, 8298, 8308, 8318, 8328, 8337, 8347, 8357, 8366,
                        8375, 8384, 8394, 8403, 8412, 8420, 8429, 8438, 8446, 8455, 8463, 8472, 8480, 8488, 8496, 8504,
                        8512, 8520, 8528, 8536, 8544, 8551, 8559, 8566, 8574, 8581, 8588, 8596, 8603, 8610, 8617, 8624],
                [8631, 8645, 8659, 8672, 8685, 8698, 8711, 8723, 8736, 8748, 8760, 8772, 8784, 8796, 8807, 8818,
                        8830, 8841, 8852, 8862, 8873, 8884, 8894, 8904, 8914, 8924, 8934, 8944, 8954, 8964, 8973, 8982,
                        8992, 9001, 9010, 9019, 9028, 9037, 9046, 9054, 9063, 9071, 9080, 9088, 9097, 9105, 9113, 9121,
                        9129, 9137, 9145, 9152, 9160, 9168, 9175, 9183, 9190, 9198, 9205, 9212, 9219, 9227, 9234, 9241],
                [9248, 9261, 9275, 9288, 9302, 9315, 9327, 9340, 9352, 9365, 9377, 9389, 9400, 9412, 9424, 9435,
                        9446, 9457, 9468, 9479, 9490, 9500, 9510, 9521, 9531, 9541, 9551, 9561, 9570, 9580, 9590, 9599,
                        9608, 9617, 9627, 9636, 9645, 9653, 9662, 9671, 9679, 9688, 9696, 9705, 9713, 9721, 9729, 9737,
                        9745, 9753, 9761, 9769, 9777, 9784, 9792, 9799, 9807, 9814, 9821, 9829, 9836, 9843, 9850, 9857],
                [9864, 9878, 9892, 9905, 9918, 9931, 9944, 9956, 9969, 9981, 9993, 10005, 10017, 10029, 10040, 10051,
                        10063, 10074, 10085, 10095, 10106, 10117, 10127, 10137, 10147, 10157, 10167, 10177, 10187, 10197, 10206, 10215,
                        10225, 10234, 10243, 10252, 10261, 10270, 10279, 10287, 10296, 10304, 10313, 10321, 10330, 10338, 10346, 10354,
                        10362, 10370, 10378, 10385, 10393, 10401, 10408, 10416, 10423, 10431, 10438, 10445, 10452, 10460, 10467, 10474],
                [10481, 10494, 10508, 10521, 10535, 10548, 10560, 10573, 10585, 10598, 10610, 10622, 10634, 10645, 10657, 10668,
                        10679, 10690, 10701, 10712, 10723, 10733, 10743, 10754, 10764, 10774, 10784, 10794, 10803, 10813, 10823, 10832,
                        10841, 10851, 10860, 10869, 10878, 10886, 10895, 10904, 10912, 10921, 10929, 10938, 10946, 10954, 10962, 10970,
                        10978, 10986, 10994, 11002, 11010, 11017, 11025, 11032, 11040, 11047, 11054, 11062, 11069, 11076, 11083, 11090],
                [11097, 11111, 11125, 11138, 11151, 11164, 11177, 11189, 11202, 11214, 11226, 11238, 11250, 11262, 11273, 11284,
                        11296, 11307, 11318, 11328, 11339, 11350, 11360, 11370, 11380, 11390, 11400, 11410, 11420, 11430, 11439, 11448,
                        11458, 11467, 11476, 11485, 11494, 11503, 11512, 11520, 11529, 11538, 11546, 11554, 11563, 11571, 11579, 11587,
                        11595, 11603, 11611, 11618, 11626, 11634, 11641, 11649, 11656, 11664, 11671, 11678, 11685, 11693, 11700, 11707],
                [11714, 11727, 11741, 11754, 11768, 11781, 11793, 11806, 11818, 11831, 11843, 11855, 11867, 11878, 11890, 11901,
                        11912, 11923, 11934, 11945, 11956, 11966, 11976, 11987, 11997, 12007, 12017, 12027, 12036, 12046, 12056, 12065,
                        12074, 12084, 12093, 12102, 12111, 12119, 12128, 12137, 12146, 12154, 12162, 12171, 12179, 12187, 12195, 12203,
                        12211, 12219, 12227, 12235, 12243, 12250, 12258, 12265, 12273, 12280, 12287, 12295, 12302, 12309, 12316, 12323],
                [12330, 12344, 12358, 12371, 12384, 12397, 12410, 12423, 12435, 12447, 12459, 12471, 12483, 12495, 12506, 12517,
                        12529, 12540, 12551, 12561, 12572, 12583, 12593, 12603, 12613, 12623, 12633, 12643, 12653, 12663, 12672, 12682,
                        12691, 12700, 12709, 12718, 12727, 12736, 12745, 12753, 12762, 12771, 12779, 12787, 12796, 12804, 12812, 12820,
                        12828, 12836, 12844, 12851, 12859, 12867, 12874, 12882, 12889, 12897, 12904, 12911, 12918, 12926, 12933, 12940],
                [12947, 12960, 12974, 12987, 13001, 13014, 13026, 13039, 13051, 13064, 13076, 13088, 13100, 13111, 13123, 13134,
                        13145, 13156, 13167, 13178, 13189, 13199, 13209, 13220, 13230, 13240, 13250, 13260, 13269, 13279, 13289, 13298,
                        13307, 13317, 13326, 13335, 13344, 13352, 13361, 13370, 13379, 13387, 13395, 13404, 13412, 13420, 13428, 13436,
                        13444, 13452, 13460, 13468, 13476, 13483, 13491, 13498, 13506, 13513, 13521, 13528, 13535, 13542, 13549, 13556],
                [13563, 13577, 13591, 13604, 13617, 13630, 13643, 13656, 13668, 13680, 13692, 13704, 13716, 13728, 13739, 13750,
                        13762, 13773, 13784, 13794, 13805, 13816, 13826, 13836, 13846, 13857, 13866, 13876, 13886, 13896, 13905, 13915,
                        13924, 13933, 13942, 13951, 13960, 13969, 13978, 13986, 13995, 14004, 14012, 14020, 14029, 14037, 14045, 14053,
                        14061, 14069, 14077, 14084, 14092, 14100, 14107, 14115, 14122, 14130, 14137, 14144, 14151, 14159, 14166, 14173],
                [14180, 14194, 14207, 14220, 14234, 14247, 14259, 14272, 14284, 14297, 14309, 14321, 14333, 14344, 14356, 14367,
                        14378, 14389, 14400, 14411, 14422, 14432, 14443, 14453, 14463, 14473, 14483, 14493, 14502, 14512, 14522, 14531,
                        14540, 14550, 14559, 14568, 14577, 14586, 14594, 14603, 14612, 14620, 14628, 14637, 14645, 14653, 14661, 14669,
                        14677, 14685, 14693, 14701, 14709, 14716, 14724, 14731, 14739, 14746, 14754, 14761, 14768, 14775, 14782, 14789],
                [14796, 14810, 14824, 14837, 14850, 14863, 14876, 14889, 14901, 14913, 14925, 14937, 14949, 14961, 14972, 14984,
                        14995, 15006, 15017, 15027, 15038, 15049, 15059, 15069, 15079, 15090, 15099, 15109, 15119, 15129, 15138, 15148,
                        15157, 15166, 15175, 15184, 15193, 15202, 15211, 15219, 15228, 15237, 15245, 15253, 15262, 15270, 15278, 15286,
                        15294, 15302, 15310, 15317, 15325, 15333, 15340, 15348, 15355, 15363, 15370, 15377, 15384, 15392, 15399, 15406],
                [15413, 15427, 15440, 15453, 15467, 15480, 15492, 15505, 15517, 15530, 15542, 15554, 15566, 15577, 15589, 15600,
                        15611, 15622, 15633, 15644, 15655, 15665, 15676, 15686, 15696, 15706, 15716, 15726, 15736, 15745, 15755, 15764,
                        15773, 15783, 15792, 15801, 15810, 15819, 15827, 15836, 15845, 15853, 15862, 15870, 15878, 15886, 15894, 15903,
                        15910, 15918, 15926, 15934, 15942, 15949, 15957, 15964, 15972, 15979, 15987, 15994, 16001, 16008, 16015, 16022],
                [16029, 16043, 16057, 16070, 16083, 16096, 16109, 16122, 16134, 16146, 16158, 16170, 16182, 16194, 16205, 16217,
                        16228, 16239, 16250, 16260, 16271, 16282, 16292, 16302, 16312, 16323, 16332, 16342, 16352, 16362, 16371, 16381,
                        16390, 16399, 16408, 16417, 16426, 16435, 16444, 16452, 16461, 16470, 16478, 16486, 16495, 16503, 16511, 16519,
                        16527, 16535, 16543, 16550, 16558, 16566, 16573, 16581, 16588, 16596, 16603, 16610, 16618, 16625, 16632, 16639],
                [16646, 16660, 16673, 16686, 16700, 16713, 16725, 16738, 16751, 16763, 16775, 16787, 16799, 16810, 16822, 16833,
                        16844, 16855, 16866, 16877, 16888, 16898, 16909, 16919, 16929, 16939, 16949, 16959, 16969, 16978, 16988, 16997,
                        17006, 17016, 17025, 17034, 17043, 17052, 17060, 17069, 17078, 17086, 17095, 17103, 17111, 17119, 17127, 17136,
                        17143, 17151, 17159, 17167, 17175, 17182, 17190, 17197, 17205, 17212, 17220, 17227, 17234, 17241, 17248, 17255],
                [17262, 17276, 17290, 17303, 17316, 17329, 17342, 17355, 17367, 17379, 17391, 17403, 17415, 17427, 17438, 17450,
                        17461, 17472, 17483, 17493, 17504, 17515, 17525, 17535, 17546, 17556, 17565, 17575, 17585, 17595, 17604, 17614,
                        17623, 17632, 17641, 17650, 17659, 17668, 17677, 17685, 17694, 17703, 17711, 17719, 17728, 17736, 17744, 17752,
                        17760, 17768, 17776, 17784, 17791, 17799, 17806, 17814, 17821, 17829, 17836, 17843, 17851, 17858, 17865, 17872],
                [17879, 17893, 17906, 17920, 17933, 17946, 17958, 17971, 17984, 17996, 18008, 18020, 18032, 18043, 18055, 18066,
                        18077, 18088, 18099, 18110, 18121, 18131, 18142, 18152, 18162, 18172, 18182, 18192, 18202, 18211, 18221, 18230,
                        18239, 18249, 18258, 18267, 18276, 18285, 18293, 18302, 18311, 18319, 18328, 18336, 18344, 18352, 18360, 18369,
                        18377, 18384, 18392, 18400, 18408, 18415, 18423, 18430, 18438, 18445, 18453, 18460, 18467, 18474, 18481, 18488],
                [18495, 18509, 18523, 18536, 18549, 18562, 18575, 18588, 18600, 18612, 18624, 18636, 18648, 18660, 18671, 18683,
                        18694, 18705, 18716, 18726, 18737, 18748, 18758, 18768, 18779, 18789, 18799, 18808, 18818, 18828, 18837, 18847,
                        18856, 18865, 18874, 18883, 18892, 18901, 18910, 18919, 18927, 18936, 18944, 18952, 18961, 18969, 18977, 18985,
                        18993, 19001, 19009, 19017, 19024, 19032, 19039, 19047, 19054, 19062, 19069, 19076, 19084, 19091, 19098, 19105],
                [19112, 19126, 19139, 19153, 19166, 19179, 19191, 19204, 19217, 19229, 19241, 19253, 19265, 19276, 19288, 19299,
                        19310, 19321, 19332, 19343, 19354, 19364, 19375, 19385, 19395, 19405, 19415, 19425, 19435, 19444, 19454, 19463,
                        19472, 19482, 19491, 19500, 19509, 19518, 19526, 19535, 19544, 19552, 19561, 19569, 19577, 19585, 19594, 19602,
                        19610, 19617, 19625, 19633, 19641, 19648, 19656, 19663, 19671, 19678, 19686, 19693, 19700, 19707, 19714, 19721],
                [19728, 19742, 19756, 19769, 19782, 19795, 19808, 19821, 19833, 19845, 19857, 19869, 19881, 19893, 19904, 19916,
                        19927, 19938, 19949, 19960, 19970, 19981, 19991, 20001, 20012, 20022, 20032, 20041, 20051, 20061, 20070, 20080,
                        20089, 20098, 20107, 20116, 20125, 20134, 20143, 20152, 20160, 20169, 20177, 20185, 20194, 20202, 20210, 20218,
                        20226, 20234, 20242, 20250, 20257, 20265, 20272, 20280, 20287, 20295, 20302, 20309, 20317, 20324, 20331, 20338],
                [20345, 20359, 20372, 20386, 20399, 20412, 20425, 20437, 20450, 20462, 20474, 20486, 20498, 20509, 20521, 20532,
                        20543, 20554, 20565, 20576, 20587, 20597, 20608, 20618, 20628, 20638, 20648, 20658, 20668, 20677, 20687, 20696,
                        20705, 20715, 20724, 20733, 20742, 20751, 20759, 20768, 20777, 20785, 20794, 20802, 20810, 20818, 20827, 20835,
                        20843, 20850, 20858, 20866, 20874, 20881, 20889, 20896, 20904, 20911, 20919, 20926, 20933, 20940, 20947, 20954],
                [20961, 20975, 20989, 21002, 21015, 21028, 21041, 21054, 21066, 21078, 21090, 21102, 21114, 21126, 21137, 21149,
                        21160, 21171, 21182, 21193, 21203, 21214, 21224, 21234, 21245, 21255, 21265, 21274, 21284, 21294, 21303, 21313,
                        21322, 21331, 21340, 21349, 21358, 21367, 21376, 21385, 21393, 21402, 21410, 21418, 21427, 21435, 21443, 21451,
                        21459, 21467, 21475, 21483, 21490, 21498, 21505, 21513, 21520, 21528, 21535, 21542, 21550, 21557, 21564, 21571],
                [21578, 21592, 21605, 21619, 21632, 21645, 21658, 21670, 21683, 21695, 21707, 21719, 21731, 21742, 21754, 21765,
                        21776, 21787, 21798, 21809, 21820, 21830, 21841, 21851, 21861, 21871, 21881, 21891, 21901, 21910, 21920, 21929,
                        21938, 21948, 21957, 21966, 21975, 21984, 21992, 22001, 22010, 22018, 22027, 22035, 22043, 22051, 22060, 22068,
                        22076, 22083, 22091, 22099, 22107, 22114, 22122, 22129, 22137, 22144, 22152, 22159, 22166, 22173, 22180, 22187],
                [22194, 22208, 22222, 22235, 22248, 22261, 22274, 22287, 22299, 22311, 22323, 22335, 22347, 22359, 22370, 22382,
                        22393, 22404, 22415, 22426, 22436, 22447, 22457, 22467, 22478, 22488, 22498, 22507, 22517, 22527, 22536, 22546,
                        22555, 22564, 22573, 22582, 22591, 22600, 22609, 22618, 22626, 22635, 22643, 22651, 22660, 22668, 22676, 22684,
                        22692, 22700, 22708, 22716, 22723, 22731, 22738, 22746, 22753, 22761, 22768, 22775, 22783, 22790, 22797, 22804],
                [22811, 22825, 22838, 22852, 22865, 22878, 22891, 22903, 22916, 22928, 22940, 22952, 22964, 22975, 22987, 22998,
                        23009, 23020, 23031, 23042, 23053, 23063, 23074, 23084, 23094, 23104, 23114, 23124, 23134, 23143, 23153, 23162,
                        23171, 23181, 23190, 23199, 23208, 23217, 23225, 23234, 23243, 23251, 23260, 23268, 23276, 23284, 23293, 23301,
                        23309, 23316, 23324, 23332, 23340, 23347, 23355, 23363, 23370, 23377, 23385, 23392, 23399, 23406, 23413, 23420],
                [23427, 23441, 23455, 23468, 23481, 23494, 23507, 23520, 23532, 23544, 23556, 23568, 23580, 23592, 23603, 23615,
                        23626, 23637, 23648, 23659, 23669, 23680, 23690, 23700, 23711, 23721, 23731, 23740, 23750, 23760, 23769, 23779,
                        23788, 23797, 23806, 23815, 23824, 23833, 23842, 23851, 23859, 23868, 23876, 23884, 23893, 23901, 23909, 23917,
                        23925, 23933, 23941, 23949, 23956, 23964, 23972, 23979, 23986, 23994, 24001, 24008, 24016, 24023, 24030, 24037],
                [24044, 24058, 24071, 24085, 24098, 24111, 24124, 24136, 24149, 24161, 24173, 24185, 24197, 24208, 24220, 24231,
                        24242, 24253, 24264, 24275, 24286, 24296, 24307, 24317, 24327, 24337, 24347, 24357, 24367, 24376, 24386, 24395,
                        24405, 24414, 24423, 24432, 24441, 24450, 24458, 24467, 24476, 24484, 24493, 24501, 24509, 24517, 24526, 24534,
                        24542, 24550, 24557, 24565, 24573, 24580, 24588, 24596, 24603, 24610, 24618, 24625, 24632, 24639, 24646, 24653],
                [24660, 24674, 24688, 24701, 24714, 24727, 24740, 24753, 24765, 24777, 24790, 24801, 24813, 24825, 24836, 24848,
                        24859, 24870, 24881, 24892, 24902, 24913, 24923, 24933, 24944, 24954, 24964, 24973, 24983, 24993, 25002, 25012,
                        25021, 25030, 25039, 25048, 25057, 25066, 25075, 25084, 25092, 25101, 25109, 25117, 25126, 25134, 25142, 25150,
                        25158, 25166, 25174, 25182, 25189, 25197, 25205, 25212, 25219, 25227, 25234, 25241, 25249, 25256, 25263, 25270],
                [25277, 25291, 25304, 25318, 25331, 25344, 25357, 25369, 25382, 25394, 25406, 25418, 25430, 25441, 25453, 25464,
                        25475, 25486, 25497, 25508, 25519, 25529, 25540, 25550, 25560, 25570, 25580, 25590, 25600, 25609, 25619, 25628,
                        25638, 25647, 25656, 25665, 25674, 25683, 25691, 25700, 25709, 25717, 25726, 25734, 25742, 25750, 25759, 25767,
                        25775, 25783, 25790, 25798, 25806, 25813, 25821, 25829, 25836, 25843, 25851, 25858, 25865, 25872, 25879, 25886],
                [25893, 25907, 25921, 25934, 25947, 25960, 25973, 25986, 25998, 26010, 26023, 26034, 26046, 26058, 26069, 26081,
                        26092, 26103, 26114, 26125, 26135, 26146, 26156, 26166, 26177, 26187, 26197, 26206, 26216, 26226, 26235, 26245,
                        26254, 26263, 26272, 26281, 26290, 26299, 26308, 26317, 26325, 26334, 26342, 26351, 26359, 26367, 26375, 26383,
                        26391, 26399, 26407, 26415, 26422, 26430, 26438, 26445, 26453, 26460, 26467, 26474, 26482, 26489, 26496, 26503],
                [26510, 26524, 26537, 26551, 26564, 26577, 26590, 26602, 26615, 26627, 26639, 26651, 26663, 26674, 26686, 26697,
                        26708, 26719, 26730, 26741, 26752, 26762, 26773, 26783, 26793, 26803, 26813, 26823, 26833, 26842, 26852, 26861,
                        26871, 26880, 26889, 26898, 26907, 26916, 26924, 26933, 26942, 26950, 26959, 26967, 26975, 26983, 26992, 27000,
                        27008, 27016, 27023, 27031, 27039, 27046, 27054, 27062, 27069, 27076, 27084, 27091, 27098, 27105, 27112, 27119],
                [27126, 27140, 27154, 27167, 27180, 27193, 27206, 27219, 27231, 27243, 27256, 27267, 27279, 27291, 27302, 27314,
                        27325, 27336, 27347, 27358, 27368, 27379, 27389, 27399, 27410, 27420, 27430, 27439, 27449, 27459, 27468, 27478,
                        27487, 27496, 27505, 27514, 27523, 27532, 27541, 27550, 27558, 27567, 27575, 27584, 27592, 27600, 27608, 27616,
                        27624, 27632, 27640, 27648, 27655, 27663, 27671, 27678, 27686, 27693, 27700, 27707, 27715, 27722, 27729, 27736],
                [27743, 27757, 27770, 27784, 27797, 27810, 27823, 27835, 27848, 27860, 27872, 27884, 27896, 27907, 27919, 27930,
                        27941, 27952, 27963, 27974, 27985, 27995, 28006, 28016, 28026, 28036, 28046, 28056, 28066, 28075, 28085, 28094,
                        28104, 28113, 28122, 28131, 28140, 28149, 28157, 28166, 28175, 28183, 28192, 28200, 28208, 28217, 28225, 28233,
                        28241, 28249, 28256, 28264, 28272, 28280, 28287, 28295, 28302, 28309, 28317, 28324, 28331, 28338, 28345, 28352],
                [28359, 28373, 28387, 28400, 28413, 28426, 28439, 28452, 28464, 28476, 28489, 28501, 28512, 28524, 28535, 28547,
                        28558, 28569, 28580, 28591, 28601, 28612, 28622, 28633, 28643, 28653, 28663, 28672, 28682, 28692, 28701, 28711,
                        28720, 28729, 28738, 28747, 28756, 28765, 28774, 28783, 28791, 28800, 28808, 28817, 28825, 28833, 28841, 28849,
                        28857, 28865, 28873, 28881, 28888, 28896, 28904, 28911, 28919, 28926, 28933, 28941, 28948, 28955, 28962, 28969],
                [28976, 28990, 29003, 29017, 29030, 29043, 29056, 29068, 29081, 29093, 29105, 29117, 29129, 29140, 29152, 29163,
                        29174, 29185, 29196, 29207, 29218, 29228, 29239, 29249, 29259, 29269, 29279, 29289, 29299, 29308, 29318, 29327,
                        29337, 29346, 29355, 29364, 29373, 29382, 29390, 29399, 29408, 29416, 29425, 29433, 29441, 29450, 29458, 29466,
                        29474, 29482, 29489, 29497, 29505, 29513, 29520, 29528, 29535, 29542, 29550, 29557, 29564, 29571, 29578, 29585],
                [29592, 29606, 29620, 29633, 29646, 29659, 29672, 29685, 29697, 29709, 29722, 29734, 29745, 29757, 29768, 29780,
                        29791, 29802, 29813, 29824, 29834, 29845, 29855, 29866, 29876, 29886, 29896, 29906, 29915, 29925, 29934, 29944,
                        29953, 29962, 29971, 29980, 29989, 29998, 30007, 30016, 30024, 30033, 30041, 30050, 30058, 30066, 30074, 30082,
                        30090, 30098, 30106, 30114, 30121, 30129, 30137, 30144, 30152, 30159, 30166, 30174, 30181, 30188, 30195, 30202],
                [30209, 30223, 30236, 30250, 30263, 30276, 30289, 30301, 30314, 30326, 30338, 30350, 30362, 30373, 30385, 30396,
                        30407, 30418, 30429, 30440, 30451, 30461, 30472, 30482, 30492, 30502, 30512, 30522, 30532, 30541, 30551, 30560,
                        30570, 30579, 30588, 30597, 30606, 30615, 30624, 30632, 30641, 30649, 30658, 30666, 30674, 30683, 30691, 30699,
                        30707, 30715, 30722, 30730, 30738, 30746, 30753, 30761, 30768, 30775, 30783, 30790, 30797, 30804, 30811, 30818],
                [30825, 30839, 30853, 30866, 30879, 30892, 30905, 30918, 30930, 30943, 30955, 30967, 30978, 30990, 31001, 31013,
                        31024, 31035, 31046, 31057, 31067, 31078, 31088, 31099, 31109, 31119, 31129, 31139, 31148, 31158, 31167, 31177,
                        31186, 31195, 31204, 31213, 31222, 31231, 31240, 31249, 31257, 31266, 31274, 31283, 31291, 31299, 31307, 31315,
                        31323, 31331, 31339, 31347, 31354, 31362, 31370, 31377, 31385, 31392, 31399, 31407, 31414, 31421, 31428, 31435],
                [31442, 31456, 31469, 31483, 31496, 31509, 31522, 31534, 31547, 31559, 31571, 31583, 31595, 31606, 31618, 31629,
                        31640, 31652, 31662, 31673, 31684, 31694, 31705, 31715, 31725, 31735, 31745, 31755, 31765, 31774, 31784, 31793,
                        31803, 31812, 31821, 31830, 31839, 31848, 31857, 31865, 31874, 31882, 31891, 31899, 31907, 31916, 31924, 31932,
                        31940, 31948, 31955, 31963, 31971, 31979, 31986, 31994, 32001, 32008, 32016, 32023, 32030, 32037, 32044, 32052],
                [32058, 32072, 32086, 32099, 32112, 32125, 32138, 32151, 32163, 32176, 32188, 32200, 32211, 32223, 32234, 32246,
                        32257, 32268, 32279, 32290, 32300, 32311, 32321, 32332, 32342, 32352, 32362, 32372, 32381, 32391, 32400, 32410,
                        32419, 32428, 32437, 32446, 32455, 32464, 32473, 32482, 32490, 32499, 32507, 32516, 32524, 32532, 32540, 32548,
                        32556, 32564, 32572, 32580, 32587, 32595, 32603, 32610, 32618, 32625, 32632, 32640, 32647, 32654, 32661, 32668],
                [32675, 32689, 32702, 32716, 32729, 32742, 32755, 32767, 32780, 32792, 32804, 32816, 32828, 32839, 32851, 32862,
                        32873, 32885, 32895, 32906, 32917, 32927, 32938, 32948, 32958, 32968, 32978, 32988, 32998, 33007, 33017, 33026,
                        33036, 33045, 33054, 33063, 33072, 33081, 33090, 33098, 33107, 33115, 33124, 33132, 33140, 33149, 33157, 33165,
                        33173, 33181, 33188, 33196, 33204, 33212, 33219, 33227, 33234, 33241, 33249, 33256, 33263, 33270, 33278, 33285],
                [33292, 33305, 33319, 33332, 33345, 33358, 33371, 33384, 33396, 33409, 33421, 33433, 33444, 33456, 33467, 33479,
                        33490, 33501, 33512, 33523, 33533, 33544, 33554, 33565, 33575, 33585, 33595, 33605, 33614, 33624, 33633, 33643,
                        33652, 33661, 33670, 33680, 33688, 33697, 33706, 33715, 33723, 33732, 33740, 33749, 33757, 33765, 33773, 33781,
                        33789, 33797, 33805, 33813, 33820, 33828, 33836, 33843, 33851, 33858, 33865, 33873, 33880, 33887, 33894, 33901],
                [33908, 33922, 33935, 33949, 33962, 33975, 33988, 34000, 34013, 34025, 34037, 34049, 34061, 34072, 34084, 34095,
                        34106, 34118, 34128, 34139, 34150, 34160, 34171, 34181, 34191, 34201, 34211, 34221, 34231, 34240, 34250, 34259,
                        34269, 34278, 34287, 34296, 34305, 34314, 34323, 34331, 34340, 34348, 34357, 34365, 34373, 34382, 34390, 34398,
                        34406, 34414, 34422, 34429, 34437, 34445, 34452, 34460, 34467, 34475, 34482, 34489, 34496, 34503, 34511, 34518],
                [34525, 34538, 34552, 34565, 34578, 34591, 34604, 34617, 34629, 34642, 34654, 34666, 34677, 34689, 34700, 34712,
                        34723, 34734, 34745, 34756, 34766, 34777, 34787, 34798, 34808, 34818, 34828, 34838, 34847, 34857, 34866, 34876,
                        34885, 34894, 34904, 34913, 34921, 34930, 34939, 34948, 34956, 34965, 34973, 34982, 34990, 34998, 35006, 35014,
                        35022, 35030, 35038, 35046, 35053, 35061, 35069, 35076, 35084, 35091, 35098, 35106, 35113, 35120, 35127, 35134],
                [35141, 35155, 35168, 35182, 35195, 35208, 35221, 35233, 35246, 35258, 35270, 35282, 35294, 35306, 35317, 35328,
                        35340, 35351, 35361, 35372, 35383, 35393, 35404, 35414, 35424, 35434, 35444, 35454, 35464, 35473, 35483, 35492,
                        35502, 35511, 35520, 35529, 35538, 35547, 35556, 35564, 35573, 35581, 35590, 35598, 35606, 35615, 35623, 35631,
                        35639, 35647, 35655, 35662, 35670, 35678, 35685, 35693, 35700, 35708, 35715, 35722, 35729, 35736, 35744, 35751],
                [35758, 35771, 35785, 35798, 35811, 35824, 35837, 35850, 35862, 35875, 35887, 35899, 35910, 35922, 35934, 35945,
                        35956, 35967, 35978, 35989, 35999, 36010, 36020, 36031, 36041, 36051, 36061, 36071, 36080, 36090, 36099, 36109,
                        36118, 36127, 36137, 36146, 36154, 36163, 36172, 36181, 36189, 36198, 36206, 36215, 36223, 36231, 36239, 36247,
                        36255, 36263, 36271, 36279, 36287, 36294, 36302, 36309, 36317, 36324, 36331, 36339, 36346, 36353, 36360, 36367],
                [36374, 36388, 36401, 36415, 36428, 36441, 36454, 36466, 36479, 36491, 36503, 36515, 36527, 36539, 36550, 36561,
                        36573, 36584, 36594, 36605, 36616, 36626, 36637, 36647, 36657, 36667, 36677, 36687, 36697, 36706, 36716, 36725,
                        36735, 36744, 36753, 36762, 36771, 36780, 36789, 36797, 36806, 36814, 36823, 36831, 36839, 36848, 36856, 36864,
                        36872, 36880, 36888, 36895, 36903, 36911, 36918, 36926, 36933, 36941, 36948, 36955, 36962, 36969, 36977, 36984],
                [36991, 37004, 37018, 37031, 37044, 37057, 37070, 37083, 37095, 37108, 37120, 37132, 37143, 37155, 37167, 37178,
                        37189, 37200, 37211, 37222, 37232, 37243, 37253, 37264, 37274, 37284, 37294, 37304, 37313, 37323, 37332, 37342,
                        37351, 37360, 37370, 37379, 37388, 37396, 37405, 37414, 37422, 37431, 37439, 37448, 37456, 37464, 37472, 37480,
                        37488, 37496, 37504, 37512, 37520, 37527, 37535, 37542, 37550, 37557, 37564, 37572, 37579, 37586, 37593, 37600],
                [37607, 37621, 37634, 37648, 37661, 37674, 37687, 37699, 37712, 37724, 37736, 37748, 37760, 37772, 37783, 37794,
                        37806, 37817, 37828, 37838, 37849, 37859, 37870, 37880, 37890, 37900, 37910, 37920, 37930, 37939, 37949, 37958,
                        37968, 37977, 37986, 37995, 38004, 38013, 38022, 38030, 38039, 38047, 38056, 38064, 38072, 38081, 38089, 38097,
                        38105, 38113, 38121, 38128, 38136, 38144, 38151, 38159, 38166, 38174, 38181, 38188, 38195, 38202, 38210, 38217],
                [38224, 38237, 38251, 38264, 38278, 38290, 38303, 38316, 38328, 38341, 38353, 38365, 38376, 38388, 38400, 38411,
                        38422, 38433, 38444, 38455, 38465, 38476, 38486, 38497, 38507, 38517, 38527, 38537, 38546, 38556, 38565, 38575,
                        38584, 38593, 38603, 38612, 38621, 38629, 38638, 38647, 38655, 38664, 38672, 38681, 38689, 38697, 38705, 38713,
                        38721, 38729, 38737, 38745, 38753, 38760, 38768, 38775, 38783, 38790, 38797, 38805, 38812, 38819, 38826, 38833],
                [38840, 38854, 38867, 38881, 38894, 38907, 38920, 38932, 38945, 38957, 38969, 38981, 38993, 39005, 39016, 39027,
                        39039, 39050, 39061, 39071, 39082, 39092, 39103, 39113, 39123, 39133, 39143, 39153, 39163, 39172, 39182, 39191,
                        39201, 39210, 39219, 39228, 39237, 39246, 39255, 39263, 39272, 39280, 39289, 39297, 39305, 39314, 39322, 39330,
                        39338, 39346, 39354, 39361, 39369, 39377, 39384, 39392, 39399, 39407, 39414, 39421, 39428, 39436, 39443, 39450],
        ];

        /// Approximate `log10(numerator / denominator) * 2048` using a look-up table.
        #[inline]
        pub fn negative_log10_times_2048(numerator: u64, denominator: u64) -> u64 {
                // Multiply the -1 through to avoid needing to use signed numbers.
                (log10_times_2048(denominator) - log10_times_2048(numerator)) as u64
        }

        #[inline]
        fn log10_times_2048(x: u64) -> u16 {
                debug_assert_ne!(x, 0);
                let most_significant_bit = HIGHEST_BIT - x.leading_zeros();
                let lower_bits = (x >> most_significant_bit.saturating_sub(LOWER_BITS)) & LOWER_BITMASK;
                LOG10_TIMES_2048[most_significant_bit as usize][lower_bits as usize]
        }

        #[cfg(test)]
        mod tests {
                use super::*;

                #[test]
                fn prints_negative_log10_times_2048_lookup_table() {
                        for msb in 0..BITS {
                                for i in 0..LOWER_BITS_BOUND {
                                        let x = ((LOWER_BITS_BOUND + i) << (HIGHEST_BIT - LOWER_BITS)) >> (HIGHEST_BIT - msb);
                                        let log10_times_2048 = ((x as f64).log10() * 2048.0).round() as u16;
                                        assert_eq!(log10_times_2048, LOG10_TIMES_2048[msb as usize][i as usize]);

                                        if i % LOWER_BITS_BOUND == 0 {
                                                print!("\t\t[{}, ", log10_times_2048);
                                        } else if i % LOWER_BITS_BOUND == LOWER_BITS_BOUND - 1 {
                                                println!("{}],", log10_times_2048);
                                        } else if i % (LOWER_BITS_BOUND/4) == LOWER_BITS_BOUND/4 - 1 {
                                                print!("{},\n\t\t\t", log10_times_2048);
                                        } else {
                                                print!("{}, ", log10_times_2048);
                                        }
                                }
                        }
                }
        }
}

impl<G: Deref<Target = NetworkGraph>, L: Deref, T: Time> Writeable for ProbabilisticScorerUsingTime<G, L, T> where L::Target: Logger {
        #[inline]
        fn write<W: Writer>(&self, w: &mut W) -> Result<(), io::Error> {
                write_tlv_fields!(w, {
                        (0, self.channel_liquidities, required)
                });
                Ok(())
        }
}

impl<G: Deref<Target = NetworkGraph>, L: Deref, T: Time>
ReadableArgs<(ProbabilisticScoringParameters, G, L)> for ProbabilisticScorerUsingTime<G, L, T> where L::Target: Logger {
        #[inline]
        fn read<R: Read>(
                r: &mut R, args: (ProbabilisticScoringParameters, G, L)
        ) -> Result<Self, DecodeError> {
                let (params, network_graph, logger) = args;
                let mut channel_liquidities = HashMap::new();
                read_tlv_fields!(r, {
                        (0, channel_liquidities, required)
                });
                Ok(Self {
                        params,
                        network_graph,
                        logger,
                        channel_liquidities,
                })
        }
}

impl<T: Time> Writeable for ChannelLiquidity<T> {
        #[inline]
        fn write<W: Writer>(&self, w: &mut W) -> Result<(), io::Error> {
                let duration_since_epoch = T::duration_since_epoch() - self.last_updated.elapsed();
                write_tlv_fields!(w, {
                        (0, self.min_liquidity_offset_msat, required),
                        (2, self.max_liquidity_offset_msat, required),
                        (4, duration_since_epoch, required),
                });
                Ok(())
        }
}

impl<T: Time> Readable for ChannelLiquidity<T> {
        #[inline]
        fn read<R: Read>(r: &mut R) -> Result<Self, DecodeError> {
                let mut min_liquidity_offset_msat = 0;
                let mut max_liquidity_offset_msat = 0;
                let mut duration_since_epoch = Duration::from_secs(0);
                read_tlv_fields!(r, {
                        (0, min_liquidity_offset_msat, required),
                        (2, max_liquidity_offset_msat, required),
                        (4, duration_since_epoch, required),
                });
                Ok(Self {
                        min_liquidity_offset_msat,
                        max_liquidity_offset_msat,
                        last_updated: T::now() - (T::duration_since_epoch() - duration_since_epoch),
                })
        }
}

#[cfg(test)]
mod tests {
        use super::{ChannelLiquidity, ProbabilisticScoringParameters, ProbabilisticScorerUsingTime};
        use util::time::Time;
        use util::time::tests::SinceEpoch;

        use ln::features::{ChannelFeatures, NodeFeatures};
        use ln::msgs::{ChannelAnnouncement, ChannelUpdate, OptionalField, UnsignedChannelAnnouncement, UnsignedChannelUpdate};
        use routing::scoring::{ChannelUsage, Score};
        use routing::network_graph::{EffectiveCapacity, NetworkGraph, NodeId};
        use routing::router::RouteHop;
        use util::ser::{ReadableArgs, Writeable};
        use util::test_utils::TestLogger;

        use bitcoin::blockdata::constants::genesis_block;
        use bitcoin::hashes::Hash;
        use bitcoin::hashes::sha256d::Hash as Sha256dHash;
        use bitcoin::network::constants::Network;
        use bitcoin::secp256k1::{PublicKey, Secp256k1, SecretKey};
        use core::time::Duration;
        use io;

        fn source_privkey() -> SecretKey {
                SecretKey::from_slice(&[42; 32]).unwrap()
        }

        fn target_privkey() -> SecretKey {
                SecretKey::from_slice(&[43; 32]).unwrap()
        }

        fn source_pubkey() -> PublicKey {
                let secp_ctx = Secp256k1::new();
                PublicKey::from_secret_key(&secp_ctx, &source_privkey())
        }

        fn target_pubkey() -> PublicKey {
                let secp_ctx = Secp256k1::new();
                PublicKey::from_secret_key(&secp_ctx, &target_privkey())
        }

        fn source_node_id() -> NodeId {
                NodeId::from_pubkey(&source_pubkey())
        }

        fn target_node_id() -> NodeId {
                NodeId::from_pubkey(&target_pubkey())
        }

        // `ProbabilisticScorer` tests

        /// A probabilistic scorer for testing with time that can be manually advanced.
        type ProbabilisticScorer<'a> = ProbabilisticScorerUsingTime::<&'a NetworkGraph, &'a TestLogger, SinceEpoch>;

        fn sender_privkey() -> SecretKey {
                SecretKey::from_slice(&[41; 32]).unwrap()
        }

        fn recipient_privkey() -> SecretKey {
                SecretKey::from_slice(&[45; 32]).unwrap()
        }

        fn sender_pubkey() -> PublicKey {
                let secp_ctx = Secp256k1::new();
                PublicKey::from_secret_key(&secp_ctx, &sender_privkey())
        }

        fn recipient_pubkey() -> PublicKey {
                let secp_ctx = Secp256k1::new();
                PublicKey::from_secret_key(&secp_ctx, &recipient_privkey())
        }

        fn sender_node_id() -> NodeId {
                NodeId::from_pubkey(&sender_pubkey())
        }

        fn recipient_node_id() -> NodeId {
                NodeId::from_pubkey(&recipient_pubkey())
        }

        fn network_graph() -> NetworkGraph {
                let genesis_hash = genesis_block(Network::Testnet).header.block_hash();
                let mut network_graph = NetworkGraph::new(genesis_hash);
                add_channel(&mut network_graph, 42, source_privkey(), target_privkey());
                add_channel(&mut network_graph, 43, target_privkey(), recipient_privkey());

                network_graph
        }

        fn add_channel(
                network_graph: &mut NetworkGraph, short_channel_id: u64, node_1_key: SecretKey,
                node_2_key: SecretKey
        ) {
                let genesis_hash = genesis_block(Network::Testnet).header.block_hash();
                let node_1_secret = &SecretKey::from_slice(&[39; 32]).unwrap();
                let node_2_secret = &SecretKey::from_slice(&[40; 32]).unwrap();
                let secp_ctx = Secp256k1::new();
                let unsigned_announcement = UnsignedChannelAnnouncement {
                        features: ChannelFeatures::known(),
                        chain_hash: genesis_hash,
                        short_channel_id,
                        node_id_1: PublicKey::from_secret_key(&secp_ctx, &node_1_key),
                        node_id_2: PublicKey::from_secret_key(&secp_ctx, &node_2_key),
                        bitcoin_key_1: PublicKey::from_secret_key(&secp_ctx, &node_1_secret),
                        bitcoin_key_2: PublicKey::from_secret_key(&secp_ctx, &node_2_secret),
                        excess_data: Vec::new(),
                };
                let msghash = hash_to_message!(&Sha256dHash::hash(&unsigned_announcement.encode()[..])[..]);
                let signed_announcement = ChannelAnnouncement {
                        node_signature_1: secp_ctx.sign_ecdsa(&msghash, &node_1_key),
                        node_signature_2: secp_ctx.sign_ecdsa(&msghash, &node_2_key),
                        bitcoin_signature_1: secp_ctx.sign_ecdsa(&msghash, &node_1_secret),
                        bitcoin_signature_2: secp_ctx.sign_ecdsa(&msghash, &node_2_secret),
                        contents: unsigned_announcement,
                };
                let chain_source: Option<&::util::test_utils::TestChainSource> = None;
                network_graph.update_channel_from_announcement(
                        &signed_announcement, &chain_source, &secp_ctx).unwrap();
                update_channel(network_graph, short_channel_id, node_1_key, 0);
                update_channel(network_graph, short_channel_id, node_2_key, 1);
        }

        fn update_channel(
                network_graph: &mut NetworkGraph, short_channel_id: u64, node_key: SecretKey, flags: u8
        ) {
                let genesis_hash = genesis_block(Network::Testnet).header.block_hash();
                let secp_ctx = Secp256k1::new();
                let unsigned_update = UnsignedChannelUpdate {
                        chain_hash: genesis_hash,
                        short_channel_id,
                        timestamp: 100,
                        flags,
                        cltv_expiry_delta: 18,
                        htlc_minimum_msat: 0,
                        htlc_maximum_msat: OptionalField::Present(1_000),
                        fee_base_msat: 1,
                        fee_proportional_millionths: 0,
                        excess_data: Vec::new(),
                };
                let msghash = hash_to_message!(&Sha256dHash::hash(&unsigned_update.encode()[..])[..]);
                let signed_update = ChannelUpdate {
                        signature: secp_ctx.sign_ecdsa(&msghash, &node_key),
                        contents: unsigned_update,
                };
                network_graph.update_channel(&signed_update, &secp_ctx).unwrap();
        }

        fn payment_path_for_amount(amount_msat: u64) -> Vec<RouteHop> {
                vec![
                        RouteHop {
                                pubkey: source_pubkey(),
                                node_features: NodeFeatures::known(),
                                short_channel_id: 41,
                                channel_features: ChannelFeatures::known(),
                                fee_msat: 1,
                                cltv_expiry_delta: 18,
                        },
                        RouteHop {
                                pubkey: target_pubkey(),
                                node_features: NodeFeatures::known(),
                                short_channel_id: 42,
                                channel_features: ChannelFeatures::known(),
                                fee_msat: 2,
                                cltv_expiry_delta: 18,
                        },
                        RouteHop {
                                pubkey: recipient_pubkey(),
                                node_features: NodeFeatures::known(),
                                short_channel_id: 43,
                                channel_features: ChannelFeatures::known(),
                                fee_msat: amount_msat,
                                cltv_expiry_delta: 18,
                        },
                ]
        }

        #[test]
        fn liquidity_bounds_directed_from_lowest_node_id() {
                let logger = TestLogger::new();
                let last_updated = SinceEpoch::now();
                let network_graph = network_graph();
                let params = ProbabilisticScoringParameters::default();
                let mut scorer = ProbabilisticScorer::new(params, &network_graph, &logger)
                        .with_channel(42,
                                ChannelLiquidity {
                                        min_liquidity_offset_msat: 700, max_liquidity_offset_msat: 100, last_updated
                                })
                        .with_channel(43,
                                ChannelLiquidity {
                                        min_liquidity_offset_msat: 700, max_liquidity_offset_msat: 100, last_updated
                                });
                let source = source_node_id();
                let target = target_node_id();
                let recipient = recipient_node_id();
                assert!(source > target);
                assert!(target < recipient);

                // Update minimum liquidity.

                let liquidity_offset_half_life = scorer.params.liquidity_offset_half_life;
                let liquidity = scorer.channel_liquidities.get(&42).unwrap()
                        .as_directed(&source, &target, 1_000, liquidity_offset_half_life);
                assert_eq!(liquidity.min_liquidity_msat(), 100);
                assert_eq!(liquidity.max_liquidity_msat(), 300);

                let liquidity = scorer.channel_liquidities.get(&42).unwrap()
                        .as_directed(&target, &source, 1_000, liquidity_offset_half_life);
                assert_eq!(liquidity.min_liquidity_msat(), 700);
                assert_eq!(liquidity.max_liquidity_msat(), 900);

                scorer.channel_liquidities.get_mut(&42).unwrap()
                        .as_directed_mut(&source, &target, 1_000, liquidity_offset_half_life)
                        .set_min_liquidity_msat(200);

                let liquidity = scorer.channel_liquidities.get(&42).unwrap()
                        .as_directed(&source, &target, 1_000, liquidity_offset_half_life);
                assert_eq!(liquidity.min_liquidity_msat(), 200);
                assert_eq!(liquidity.max_liquidity_msat(), 300);

                let liquidity = scorer.channel_liquidities.get(&42).unwrap()
                        .as_directed(&target, &source, 1_000, liquidity_offset_half_life);
                assert_eq!(liquidity.min_liquidity_msat(), 700);
                assert_eq!(liquidity.max_liquidity_msat(), 800);

                // Update maximum liquidity.

                let liquidity = scorer.channel_liquidities.get(&43).unwrap()
                        .as_directed(&target, &recipient, 1_000, liquidity_offset_half_life);
                assert_eq!(liquidity.min_liquidity_msat(), 700);
                assert_eq!(liquidity.max_liquidity_msat(), 900);

                let liquidity = scorer.channel_liquidities.get(&43).unwrap()
                        .as_directed(&recipient, &target, 1_000, liquidity_offset_half_life);
                assert_eq!(liquidity.min_liquidity_msat(), 100);
                assert_eq!(liquidity.max_liquidity_msat(), 300);

                scorer.channel_liquidities.get_mut(&43).unwrap()
                        .as_directed_mut(&target, &recipient, 1_000, liquidity_offset_half_life)
                        .set_max_liquidity_msat(200);

                let liquidity = scorer.channel_liquidities.get(&43).unwrap()
                        .as_directed(&target, &recipient, 1_000, liquidity_offset_half_life);
                assert_eq!(liquidity.min_liquidity_msat(), 0);
                assert_eq!(liquidity.max_liquidity_msat(), 200);

                let liquidity = scorer.channel_liquidities.get(&43).unwrap()
                        .as_directed(&recipient, &target, 1_000, liquidity_offset_half_life);
                assert_eq!(liquidity.min_liquidity_msat(), 800);
                assert_eq!(liquidity.max_liquidity_msat(), 1000);
        }

        #[test]
        fn resets_liquidity_upper_bound_when_crossed_by_lower_bound() {
                let logger = TestLogger::new();
                let last_updated = SinceEpoch::now();
                let network_graph = network_graph();
                let params = ProbabilisticScoringParameters::default();
                let mut scorer = ProbabilisticScorer::new(params, &network_graph, &logger)
                        .with_channel(42,
                                ChannelLiquidity {
                                        min_liquidity_offset_msat: 200, max_liquidity_offset_msat: 400, last_updated
                                });
                let source = source_node_id();
                let target = target_node_id();
                assert!(source > target);

                // Check initial bounds.
                let liquidity_offset_half_life = scorer.params.liquidity_offset_half_life;
                let liquidity = scorer.channel_liquidities.get(&42).unwrap()
                        .as_directed(&source, &target, 1_000, liquidity_offset_half_life);
                assert_eq!(liquidity.min_liquidity_msat(), 400);
                assert_eq!(liquidity.max_liquidity_msat(), 800);

                let liquidity = scorer.channel_liquidities.get(&42).unwrap()
                        .as_directed(&target, &source, 1_000, liquidity_offset_half_life);
                assert_eq!(liquidity.min_liquidity_msat(), 200);
                assert_eq!(liquidity.max_liquidity_msat(), 600);

                // Reset from source to target.
                scorer.channel_liquidities.get_mut(&42).unwrap()
                        .as_directed_mut(&source, &target, 1_000, liquidity_offset_half_life)
                        .set_min_liquidity_msat(900);

                let liquidity = scorer.channel_liquidities.get(&42).unwrap()
                        .as_directed(&source, &target, 1_000, liquidity_offset_half_life);
                assert_eq!(liquidity.min_liquidity_msat(), 900);
                assert_eq!(liquidity.max_liquidity_msat(), 1_000);

                let liquidity = scorer.channel_liquidities.get(&42).unwrap()
                        .as_directed(&target, &source, 1_000, liquidity_offset_half_life);
                assert_eq!(liquidity.min_liquidity_msat(), 0);
                assert_eq!(liquidity.max_liquidity_msat(), 100);

                // Reset from target to source.
                scorer.channel_liquidities.get_mut(&42).unwrap()
                        .as_directed_mut(&target, &source, 1_000, liquidity_offset_half_life)
                        .set_min_liquidity_msat(400);

                let liquidity = scorer.channel_liquidities.get(&42).unwrap()
                        .as_directed(&source, &target, 1_000, liquidity_offset_half_life);
                assert_eq!(liquidity.min_liquidity_msat(), 0);
                assert_eq!(liquidity.max_liquidity_msat(), 600);

                let liquidity = scorer.channel_liquidities.get(&42).unwrap()
                        .as_directed(&target, &source, 1_000, liquidity_offset_half_life);
                assert_eq!(liquidity.min_liquidity_msat(), 400);
                assert_eq!(liquidity.max_liquidity_msat(), 1_000);
        }

        #[test]
        fn resets_liquidity_lower_bound_when_crossed_by_upper_bound() {
                let logger = TestLogger::new();
                let last_updated = SinceEpoch::now();
                let network_graph = network_graph();
                let params = ProbabilisticScoringParameters::default();
                let mut scorer = ProbabilisticScorer::new(params, &network_graph, &logger)
                        .with_channel(42,
                                ChannelLiquidity {
                                        min_liquidity_offset_msat: 200, max_liquidity_offset_msat: 400, last_updated
                                });
                let source = source_node_id();
                let target = target_node_id();
                assert!(source > target);

                // Check initial bounds.
                let liquidity_offset_half_life = scorer.params.liquidity_offset_half_life;
                let liquidity = scorer.channel_liquidities.get(&42).unwrap()
                        .as_directed(&source, &target, 1_000, liquidity_offset_half_life);
                assert_eq!(liquidity.min_liquidity_msat(), 400);
                assert_eq!(liquidity.max_liquidity_msat(), 800);

                let liquidity = scorer.channel_liquidities.get(&42).unwrap()
                        .as_directed(&target, &source, 1_000, liquidity_offset_half_life);
                assert_eq!(liquidity.min_liquidity_msat(), 200);
                assert_eq!(liquidity.max_liquidity_msat(), 600);

                // Reset from source to target.
                scorer.channel_liquidities.get_mut(&42).unwrap()
                        .as_directed_mut(&source, &target, 1_000, liquidity_offset_half_life)
                        .set_max_liquidity_msat(300);

                let liquidity = scorer.channel_liquidities.get(&42).unwrap()
                        .as_directed(&source, &target, 1_000, liquidity_offset_half_life);
                assert_eq!(liquidity.min_liquidity_msat(), 0);
                assert_eq!(liquidity.max_liquidity_msat(), 300);

                let liquidity = scorer.channel_liquidities.get(&42).unwrap()
                        .as_directed(&target, &source, 1_000, liquidity_offset_half_life);
                assert_eq!(liquidity.min_liquidity_msat(), 700);
                assert_eq!(liquidity.max_liquidity_msat(), 1_000);

                // Reset from target to source.
                scorer.channel_liquidities.get_mut(&42).unwrap()
                        .as_directed_mut(&target, &source, 1_000, liquidity_offset_half_life)
                        .set_max_liquidity_msat(600);

                let liquidity = scorer.channel_liquidities.get(&42).unwrap()
                        .as_directed(&source, &target, 1_000, liquidity_offset_half_life);
                assert_eq!(liquidity.min_liquidity_msat(), 400);
                assert_eq!(liquidity.max_liquidity_msat(), 1_000);

                let liquidity = scorer.channel_liquidities.get(&42).unwrap()
                        .as_directed(&target, &source, 1_000, liquidity_offset_half_life);
                assert_eq!(liquidity.min_liquidity_msat(), 0);
                assert_eq!(liquidity.max_liquidity_msat(), 600);
        }

        #[test]
        fn increased_penalty_nearing_liquidity_upper_bound() {
                let logger = TestLogger::new();
                let network_graph = network_graph();
                let params = ProbabilisticScoringParameters {
                        liquidity_penalty_multiplier_msat: 1_000,
                        ..ProbabilisticScoringParameters::zero_penalty()
                };
                let scorer = ProbabilisticScorer::new(params, &network_graph, &logger);
                let source = source_node_id();
                let target = target_node_id();

                let usage = ChannelUsage {
                        amount_msat: 1_024,
                        inflight_htlc_msat: 0,
                        effective_capacity: EffectiveCapacity::Total { capacity_msat: 1_024_000 },
                };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 0);
                let usage = ChannelUsage { amount_msat: 10_240, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 0);
                let usage = ChannelUsage { amount_msat: 102_400, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 47);
                let usage = ChannelUsage { amount_msat: 1_024_000, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 2_000);

                let usage = ChannelUsage {
                        amount_msat: 128,
                        inflight_htlc_msat: 0,
                        effective_capacity: EffectiveCapacity::Total { capacity_msat: 1_024 },
                };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 58);
                let usage = ChannelUsage { amount_msat: 256, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 125);
                let usage = ChannelUsage { amount_msat: 374, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 198);
                let usage = ChannelUsage { amount_msat: 512, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 300);
                let usage = ChannelUsage { amount_msat: 640, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 425);
                let usage = ChannelUsage { amount_msat: 768, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 602);
                let usage = ChannelUsage { amount_msat: 896, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 902);
        }

        #[test]
        fn constant_penalty_outside_liquidity_bounds() {
                let logger = TestLogger::new();
                let last_updated = SinceEpoch::now();
                let network_graph = network_graph();
                let params = ProbabilisticScoringParameters {
                        liquidity_penalty_multiplier_msat: 1_000,
                        ..ProbabilisticScoringParameters::zero_penalty()
                };
                let scorer = ProbabilisticScorer::new(params, &network_graph, &logger)
                        .with_channel(42,
                                ChannelLiquidity {
                                        min_liquidity_offset_msat: 40, max_liquidity_offset_msat: 40, last_updated
                                });
                let source = source_node_id();
                let target = target_node_id();

                let usage = ChannelUsage {
                        amount_msat: 39,
                        inflight_htlc_msat: 0,
                        effective_capacity: EffectiveCapacity::Total { capacity_msat: 100 },
                };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 0);
                let usage = ChannelUsage { amount_msat: 50, ..usage };
                assert_ne!(scorer.channel_penalty_msat(42, &source, &target, usage), 0);
                assert_ne!(scorer.channel_penalty_msat(42, &source, &target, usage), u64::max_value());
                let usage = ChannelUsage { amount_msat: 61, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), u64::max_value());
        }

        #[test]
        fn does_not_further_penalize_own_channel() {
                let logger = TestLogger::new();
                let network_graph = network_graph();
                let params = ProbabilisticScoringParameters {
                        liquidity_penalty_multiplier_msat: 1_000,
                        ..ProbabilisticScoringParameters::zero_penalty()
                };
                let mut scorer = ProbabilisticScorer::new(params, &network_graph, &logger);
                let sender = sender_node_id();
                let source = source_node_id();
                let usage = ChannelUsage {
                        amount_msat: 500,
                        inflight_htlc_msat: 0,
                        effective_capacity: EffectiveCapacity::Total { capacity_msat: 1_000 },
                };
                let failed_path = payment_path_for_amount(500);
                let successful_path = payment_path_for_amount(200);

                assert_eq!(scorer.channel_penalty_msat(41, &sender, &source, usage), 301);

                scorer.payment_path_failed(&failed_path.iter().collect::<Vec<_>>(), 41);
                assert_eq!(scorer.channel_penalty_msat(41, &sender, &source, usage), 301);

                scorer.payment_path_successful(&successful_path.iter().collect::<Vec<_>>());
                assert_eq!(scorer.channel_penalty_msat(41, &sender, &source, usage), 301);
        }

        #[test]
        fn sets_liquidity_lower_bound_on_downstream_failure() {
                let logger = TestLogger::new();
                let network_graph = network_graph();
                let params = ProbabilisticScoringParameters {
                        liquidity_penalty_multiplier_msat: 1_000,
                        ..ProbabilisticScoringParameters::zero_penalty()
                };
                let mut scorer = ProbabilisticScorer::new(params, &network_graph, &logger);
                let source = source_node_id();
                let target = target_node_id();
                let path = payment_path_for_amount(500);

                let usage = ChannelUsage {
                        amount_msat: 250,
                        inflight_htlc_msat: 0,
                        effective_capacity: EffectiveCapacity::Total { capacity_msat: 1_000 },
                };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 128);
                let usage = ChannelUsage { amount_msat: 500, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 301);
                let usage = ChannelUsage { amount_msat: 750, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 602);

                scorer.payment_path_failed(&path.iter().collect::<Vec<_>>(), 43);

                let usage = ChannelUsage { amount_msat: 250, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 0);
                let usage = ChannelUsage { amount_msat: 500, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 0);
                let usage = ChannelUsage { amount_msat: 750, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 300);
        }

        #[test]
        fn sets_liquidity_upper_bound_on_failure() {
                let logger = TestLogger::new();
                let network_graph = network_graph();
                let params = ProbabilisticScoringParameters {
                        liquidity_penalty_multiplier_msat: 1_000,
                        ..ProbabilisticScoringParameters::zero_penalty()
                };
                let mut scorer = ProbabilisticScorer::new(params, &network_graph, &logger);
                let source = source_node_id();
                let target = target_node_id();
                let path = payment_path_for_amount(500);

                let usage = ChannelUsage {
                        amount_msat: 250,
                        inflight_htlc_msat: 0,
                        effective_capacity: EffectiveCapacity::Total { capacity_msat: 1_000 },
                };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 128);
                let usage = ChannelUsage { amount_msat: 500, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 301);
                let usage = ChannelUsage { amount_msat: 750, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 602);

                scorer.payment_path_failed(&path.iter().collect::<Vec<_>>(), 42);

                let usage = ChannelUsage { amount_msat: 250, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 300);
                let usage = ChannelUsage { amount_msat: 500, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), u64::max_value());
                let usage = ChannelUsage { amount_msat: 750, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), u64::max_value());
        }

        #[test]
        fn reduces_liquidity_upper_bound_along_path_on_success() {
                let logger = TestLogger::new();
                let network_graph = network_graph();
                let params = ProbabilisticScoringParameters {
                        liquidity_penalty_multiplier_msat: 1_000,
                        ..ProbabilisticScoringParameters::zero_penalty()
                };
                let mut scorer = ProbabilisticScorer::new(params, &network_graph, &logger);
                let sender = sender_node_id();
                let source = source_node_id();
                let target = target_node_id();
                let recipient = recipient_node_id();
                let usage = ChannelUsage {
                        amount_msat: 250,
                        inflight_htlc_msat: 0,
                        effective_capacity: EffectiveCapacity::Total { capacity_msat: 1_000 },
                };
                let path = payment_path_for_amount(500);

                assert_eq!(scorer.channel_penalty_msat(41, &sender, &source, usage), 128);
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 128);
                assert_eq!(scorer.channel_penalty_msat(43, &target, &recipient, usage), 128);

                scorer.payment_path_successful(&path.iter().collect::<Vec<_>>());

                assert_eq!(scorer.channel_penalty_msat(41, &sender, &source, usage), 128);
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 300);
                assert_eq!(scorer.channel_penalty_msat(43, &target, &recipient, usage), 300);
        }

        #[test]
        fn decays_liquidity_bounds_over_time() {
                let logger = TestLogger::new();
                let network_graph = network_graph();
                let params = ProbabilisticScoringParameters {
                        liquidity_penalty_multiplier_msat: 1_000,
                        liquidity_offset_half_life: Duration::from_secs(10),
                        ..ProbabilisticScoringParameters::zero_penalty()
                };
                let mut scorer = ProbabilisticScorer::new(params, &network_graph, &logger);
                let source = source_node_id();
                let target = target_node_id();

                let usage = ChannelUsage {
                        amount_msat: 0,
                        inflight_htlc_msat: 0,
                        effective_capacity: EffectiveCapacity::Total { capacity_msat: 1_024 },
                };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 0);
                let usage = ChannelUsage { amount_msat: 1_024, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 2_000);

                scorer.payment_path_failed(&payment_path_for_amount(768).iter().collect::<Vec<_>>(), 42);
                scorer.payment_path_failed(&payment_path_for_amount(128).iter().collect::<Vec<_>>(), 43);

                let usage = ChannelUsage { amount_msat: 128, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 0);
                let usage = ChannelUsage { amount_msat: 256, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 93);
                let usage = ChannelUsage { amount_msat: 768, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 1_479);
                let usage = ChannelUsage { amount_msat: 896, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), u64::max_value());

                SinceEpoch::advance(Duration::from_secs(9));
                let usage = ChannelUsage { amount_msat: 128, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 0);
                let usage = ChannelUsage { amount_msat: 256, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 93);
                let usage = ChannelUsage { amount_msat: 768, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 1_479);
                let usage = ChannelUsage { amount_msat: 896, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), u64::max_value());

                SinceEpoch::advance(Duration::from_secs(1));
                let usage = ChannelUsage { amount_msat: 64, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 0);
                let usage = ChannelUsage { amount_msat: 128, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 34);
                let usage = ChannelUsage { amount_msat: 896, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 1_970);
                let usage = ChannelUsage { amount_msat: 960, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), u64::max_value());

                // Fully decay liquidity lower bound.
                SinceEpoch::advance(Duration::from_secs(10 * 7));
                let usage = ChannelUsage { amount_msat: 0, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 0);
                let usage = ChannelUsage { amount_msat: 1, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 0);
                let usage = ChannelUsage { amount_msat: 1_023, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 2_000);
                let usage = ChannelUsage { amount_msat: 1_024, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 2_000);

                // Fully decay liquidity upper bound.
                SinceEpoch::advance(Duration::from_secs(10));
                let usage = ChannelUsage { amount_msat: 0, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 0);
                let usage = ChannelUsage { amount_msat: 1_024, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 2_000);

                SinceEpoch::advance(Duration::from_secs(10));
                let usage = ChannelUsage { amount_msat: 0, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 0);
                let usage = ChannelUsage { amount_msat: 1_024, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 2_000);
        }

        #[test]
        fn decays_liquidity_bounds_without_shift_overflow() {
                let logger = TestLogger::new();
                let network_graph = network_graph();
                let params = ProbabilisticScoringParameters {
                        liquidity_penalty_multiplier_msat: 1_000,
                        liquidity_offset_half_life: Duration::from_secs(10),
                        ..ProbabilisticScoringParameters::zero_penalty()
                };
                let mut scorer = ProbabilisticScorer::new(params, &network_graph, &logger);
                let source = source_node_id();
                let target = target_node_id();
                let usage = ChannelUsage {
                        amount_msat: 256,
                        inflight_htlc_msat: 0,
                        effective_capacity: EffectiveCapacity::Total { capacity_msat: 1_024 },
                };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 125);

                scorer.payment_path_failed(&payment_path_for_amount(512).iter().collect::<Vec<_>>(), 42);
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 281);

                // An unchecked right shift 64 bits or more in DirectedChannelLiquidity::decayed_offset_msat
                // would cause an overflow.
                SinceEpoch::advance(Duration::from_secs(10 * 64));
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 125);

                SinceEpoch::advance(Duration::from_secs(10));
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 125);
        }

        #[test]
        fn restricts_liquidity_bounds_after_decay() {
                let logger = TestLogger::new();
                let network_graph = network_graph();
                let params = ProbabilisticScoringParameters {
                        liquidity_penalty_multiplier_msat: 1_000,
                        liquidity_offset_half_life: Duration::from_secs(10),
                        ..ProbabilisticScoringParameters::zero_penalty()
                };
                let mut scorer = ProbabilisticScorer::new(params, &network_graph, &logger);
                let source = source_node_id();
                let target = target_node_id();
                let usage = ChannelUsage {
                        amount_msat: 512,
                        inflight_htlc_msat: 0,
                        effective_capacity: EffectiveCapacity::Total { capacity_msat: 1_024 },
                };

                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 300);

                // More knowledge gives higher confidence (256, 768), meaning a lower penalty.
                scorer.payment_path_failed(&payment_path_for_amount(768).iter().collect::<Vec<_>>(), 42);
                scorer.payment_path_failed(&payment_path_for_amount(256).iter().collect::<Vec<_>>(), 43);
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 281);

                // Decaying knowledge gives less confidence (128, 896), meaning a higher penalty.
                SinceEpoch::advance(Duration::from_secs(10));
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 291);

                // Reducing the upper bound gives more confidence (128, 832) that the payment amount (512)
                // is closer to the upper bound, meaning a higher penalty.
                scorer.payment_path_successful(&payment_path_for_amount(64).iter().collect::<Vec<_>>());
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 331);

                // Increasing the lower bound gives more confidence (256, 832) that the payment amount (512)
                // is closer to the lower bound, meaning a lower penalty.
                scorer.payment_path_failed(&payment_path_for_amount(256).iter().collect::<Vec<_>>(), 43);
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 245);

                // Further decaying affects the lower bound more than the upper bound (128, 928).
                SinceEpoch::advance(Duration::from_secs(10));
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 280);
        }

        #[test]
        fn restores_persisted_liquidity_bounds() {
                let logger = TestLogger::new();
                let network_graph = network_graph();
                let params = ProbabilisticScoringParameters {
                        liquidity_penalty_multiplier_msat: 1_000,
                        liquidity_offset_half_life: Duration::from_secs(10),
                        ..ProbabilisticScoringParameters::zero_penalty()
                };
                let mut scorer = ProbabilisticScorer::new(params, &network_graph, &logger);
                let source = source_node_id();
                let target = target_node_id();
                let usage = ChannelUsage {
                        amount_msat: 500,
                        inflight_htlc_msat: 0,
                        effective_capacity: EffectiveCapacity::Total { capacity_msat: 1_000 },
                };

                scorer.payment_path_failed(&payment_path_for_amount(500).iter().collect::<Vec<_>>(), 42);
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), u64::max_value());

                SinceEpoch::advance(Duration::from_secs(10));
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 473);

                scorer.payment_path_failed(&payment_path_for_amount(250).iter().collect::<Vec<_>>(), 43);
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 300);

                let mut serialized_scorer = Vec::new();
                scorer.write(&mut serialized_scorer).unwrap();

                let mut serialized_scorer = io::Cursor::new(&serialized_scorer);
                let deserialized_scorer =
                        <ProbabilisticScorer>::read(&mut serialized_scorer, (params, &network_graph, &logger)).unwrap();
                assert_eq!(deserialized_scorer.channel_penalty_msat(42, &source, &target, usage), 300);
        }

        #[test]
        fn decays_persisted_liquidity_bounds() {
                let logger = TestLogger::new();
                let network_graph = network_graph();
                let params = ProbabilisticScoringParameters {
                        liquidity_penalty_multiplier_msat: 1_000,
                        liquidity_offset_half_life: Duration::from_secs(10),
                        ..ProbabilisticScoringParameters::zero_penalty()
                };
                let mut scorer = ProbabilisticScorer::new(params, &network_graph, &logger);
                let source = source_node_id();
                let target = target_node_id();
                let usage = ChannelUsage {
                        amount_msat: 500,
                        inflight_htlc_msat: 0,
                        effective_capacity: EffectiveCapacity::Total { capacity_msat: 1_000 },
                };

                scorer.payment_path_failed(&payment_path_for_amount(500).iter().collect::<Vec<_>>(), 42);
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), u64::max_value());

                let mut serialized_scorer = Vec::new();
                scorer.write(&mut serialized_scorer).unwrap();

                SinceEpoch::advance(Duration::from_secs(10));

                let mut serialized_scorer = io::Cursor::new(&serialized_scorer);
                let deserialized_scorer =
                        <ProbabilisticScorer>::read(&mut serialized_scorer, (params, &network_graph, &logger)).unwrap();
                assert_eq!(deserialized_scorer.channel_penalty_msat(42, &source, &target, usage), 473);

                scorer.payment_path_failed(&payment_path_for_amount(250).iter().collect::<Vec<_>>(), 43);
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 300);

                SinceEpoch::advance(Duration::from_secs(10));
                assert_eq!(deserialized_scorer.channel_penalty_msat(42, &source, &target, usage), 365);
        }

        #[test]
        fn scores_realistic_payments() {
                // Shows the scores of "realistic" sends of 100k sats over channels of 1-10m sats (with a
                // 50k sat reserve).
                let logger = TestLogger::new();
                let network_graph = network_graph();
                let params = ProbabilisticScoringParameters::default();
                let scorer = ProbabilisticScorer::new(params, &network_graph, &logger);
                let source = source_node_id();
                let target = target_node_id();

                let usage = ChannelUsage {
                        amount_msat: 100_000_000,
                        inflight_htlc_msat: 0,
                        effective_capacity: EffectiveCapacity::Total { capacity_msat: 950_000_000 },
                };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 3613);
                let usage = ChannelUsage {
                        effective_capacity: EffectiveCapacity::Total { capacity_msat: 1_950_000_000 }, ..usage
                };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 1977);
                let usage = ChannelUsage {
                        effective_capacity: EffectiveCapacity::Total { capacity_msat: 2_950_000_000 }, ..usage
                };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 1474);
                let usage = ChannelUsage {
                        effective_capacity: EffectiveCapacity::Total { capacity_msat: 3_950_000_000 }, ..usage
                };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 1223);
                let usage = ChannelUsage {
                        effective_capacity: EffectiveCapacity::Total { capacity_msat: 4_950_000_000 }, ..usage
                };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 877);
                let usage = ChannelUsage {
                        effective_capacity: EffectiveCapacity::Total { capacity_msat: 5_950_000_000 }, ..usage
                };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 845);
                let usage = ChannelUsage {
                        effective_capacity: EffectiveCapacity::Total { capacity_msat: 6_950_000_000 }, ..usage
                };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 500);
                let usage = ChannelUsage {
                        effective_capacity: EffectiveCapacity::Total { capacity_msat: 7_450_000_000 }, ..usage
                };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 500);
                let usage = ChannelUsage {
                        effective_capacity: EffectiveCapacity::Total { capacity_msat: 7_950_000_000 }, ..usage
                };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 500);
                let usage = ChannelUsage {
                        effective_capacity: EffectiveCapacity::Total { capacity_msat: 8_950_000_000 }, ..usage
                };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 500);
                let usage = ChannelUsage {
                        effective_capacity: EffectiveCapacity::Total { capacity_msat: 9_950_000_000 }, ..usage
                };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 500);
        }

        #[test]
        fn adds_base_penalty_to_liquidity_penalty() {
                let logger = TestLogger::new();
                let network_graph = network_graph();
                let source = source_node_id();
                let target = target_node_id();
                let usage = ChannelUsage {
                        amount_msat: 128,
                        inflight_htlc_msat: 0,
                        effective_capacity: EffectiveCapacity::Total { capacity_msat: 1_024 },
                };

                let params = ProbabilisticScoringParameters {
                        liquidity_penalty_multiplier_msat: 1_000,
                        ..ProbabilisticScoringParameters::zero_penalty()
                };
                let scorer = ProbabilisticScorer::new(params, &network_graph, &logger);
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 58);

                let params = ProbabilisticScoringParameters {
                        base_penalty_msat: 500, liquidity_penalty_multiplier_msat: 1_000, ..Default::default()
                };
                let scorer = ProbabilisticScorer::new(params, &network_graph, &logger);
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 558);
        }

        #[test]
        fn adds_amount_penalty_to_liquidity_penalty() {
                let logger = TestLogger::new();
                let network_graph = network_graph();
                let source = source_node_id();
                let target = target_node_id();
                let usage = ChannelUsage {
                        amount_msat: 512_000,
                        inflight_htlc_msat: 0,
                        effective_capacity: EffectiveCapacity::Total { capacity_msat: 1_024_000 },
                };

                let params = ProbabilisticScoringParameters {
                        liquidity_penalty_multiplier_msat: 1_000,
                        amount_penalty_multiplier_msat: 0,
                        ..ProbabilisticScoringParameters::zero_penalty()
                };
                let scorer = ProbabilisticScorer::new(params, &network_graph, &logger);
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 300);

                let params = ProbabilisticScoringParameters {
                        liquidity_penalty_multiplier_msat: 1_000,
                        amount_penalty_multiplier_msat: 256,
                        ..ProbabilisticScoringParameters::zero_penalty()
                };
                let scorer = ProbabilisticScorer::new(params, &network_graph, &logger);
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 337);
        }

        #[test]
        fn calculates_log10_without_overflowing_u64_max_value() {
                let logger = TestLogger::new();
                let network_graph = network_graph();
                let source = source_node_id();
                let target = target_node_id();
                let usage = ChannelUsage {
                        amount_msat: u64::max_value(),
                        inflight_htlc_msat: 0,
                        effective_capacity: EffectiveCapacity::Infinite,
                };

                let params = ProbabilisticScoringParameters {
                        liquidity_penalty_multiplier_msat: 40_000,
                        ..ProbabilisticScoringParameters::zero_penalty()
                };
                let scorer = ProbabilisticScorer::new(params, &network_graph, &logger);
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), 80_000);
        }

        #[test]
        fn accounts_for_inflight_htlc_usage() {
                let network_graph = network_graph();
                let logger = TestLogger::new();
                let params = ProbabilisticScoringParameters::default();
                let scorer = ProbabilisticScorer::new(params, &network_graph, &logger);
                let source = source_node_id();
                let target = target_node_id();

                let usage = ChannelUsage {
                        amount_msat: 750,
                        inflight_htlc_msat: 0,
                        effective_capacity: EffectiveCapacity::Total { capacity_msat: 1_000 },
                };
                assert_ne!(scorer.channel_penalty_msat(42, &source, &target, usage), u64::max_value());

                let usage = ChannelUsage { inflight_htlc_msat: 251, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), u64::max_value());
        }

        #[test]
        fn removes_uncertainity_when_exact_liquidity_known() {
                let network_graph = network_graph();
                let logger = TestLogger::new();
                let params = ProbabilisticScoringParameters::default();
                let scorer = ProbabilisticScorer::new(params, &network_graph, &logger);
                let source = source_node_id();
                let target = target_node_id();

                let base_penalty_msat = params.base_penalty_msat;
                let usage = ChannelUsage {
                        amount_msat: 750,
                        inflight_htlc_msat: 0,
                        effective_capacity: EffectiveCapacity::ExactLiquidity { liquidity_msat: 1_000 },
                };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), base_penalty_msat);

                let usage = ChannelUsage { amount_msat: 1_000, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), base_penalty_msat);

                let usage = ChannelUsage { amount_msat: 1_001, ..usage };
                assert_eq!(scorer.channel_penalty_msat(42, &source, &target, usage), u64::max_value());
        }
}