Compute yearly inflation on-chain allowing to change x_ideal according to number of slots. (#8332)

* new crate

* Update frame/staking/reward-fn/src/lib.rs

Co-authored-by: Gavin Wood <gavin@parity.io>

* fix doc

Co-authored-by: Gavin Wood <gavin@parity.io>

substrate/Cargo.lock

@@ -5372,6 +5372,14 @@ dependencies = [
  "syn",
 ]
 
+[[package]]
+name = "pallet-staking-reward-fn"
+version = "3.0.0"
+dependencies = [
+ "log",
+ "sp-arithmetic",
+]
+
 [[package]]
 name = "pallet-sudo"
 version = "3.0.0"

substrate/Cargo.toml

@@ -107,6 +107,7 @@ members = [
 	"frame/staking",
 	"frame/staking/fuzzer",
 	"frame/staking/reward-curve",
+	"frame/staking/reward-fn",
 	"frame/sudo",
 	"frame/support",
 	"frame/support/procedural",

substrate/frame/staking/reward-fn/Cargo.toml

@@ -0,0 +1,25 @@
+[package]
+name = "pallet-staking-reward-fn"
+version = "3.0.0"
+authors = ["Parity Technologies <admin@parity.io>"]
+edition = "2018"
+license = "Apache-2.0"
+homepage = "https://substrate.dev"
+repository = "https://github.com/paritytech/substrate/"
+description = "Reward function for FRAME staking pallet"
+
+[package.metadata.docs.rs]
+targets = ["x86_64-unknown-linux-gnu"]
+
+[lib]
+
+[dependencies]
+sp-arithmetic = { version = "3.0.0", default-features = false, path = "../../../primitives/arithmetic" }
+log = { version = "0.4.14", default-features = false }
+
+[features]
+default = ["std"]
+std = [
+	"sp-arithmetic/std",
+	"log/std",
+]

substrate/frame/staking/reward-fn/src/lib.rs

@@ -0,0 +1,235 @@
+// This file is part of Substrate.
+
+// Copyright (C) 2021 Parity Technologies (UK) Ltd.
+// SPDX-License-Identifier: Apache-2.0
+
+// Licensed under the Apache License, Version 2.0 (the "License");
+// you may not use this file except in compliance with the License.
+// You may obtain a copy of the License at
+//
+// 	http://www.apache.org/licenses/LICENSE-2.0
+//
+// Unless required by applicable law or agreed to in writing, software
+// distributed under the License is distributed on an "AS IS" BASIS,
+// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+// See the License for the specific language governing permissions and
+// limitations under the License.
+
+//! Useful function for inflation for nominated proof of stake.
+
+use sp_arithmetic::{Perquintill, PerThing, biguint::BigUint, traits::{Zero, SaturatedConversion}};
+use core::convert::TryFrom;
+
+/// Compute yearly inflation using function
+///
+/// ```ignore
+/// I(x) = for x between 0 and x_ideal: x / x_ideal,
+/// for x between x_ideal and 1: 2^((x_ideal - x) / d)
+/// ```
+///
+/// where:
+/// * x is the stake rate, i.e. fraction of total issued tokens that actively staked behind
+///   validators.
+/// * d is the falloff or `decay_rate`
+/// * x_ideal: the ideal stake rate.
+///
+/// The result is meant to be scaled with minimum inflation and maximum inflation.
+///
+/// (as detailed
+/// [here](https://research.web3.foundation/en/latest/polkadot/economics/1-token-economics.html#inflation-model-with-parachains))
+///
+/// Arguments are:
+/// * `stake`:
+///   The fraction of total issued tokens that actively staked behind
+///   validators. Known as `x` in the literature.
+///   Must be between 0 and 1.
+/// * `ideal_stake`:
+///   The fraction of total issued tokens that should be actively staked behind
+///   validators. Known as `x_ideal` in the literature.
+///   Must be between 0 and 1.
+/// * `falloff`:
+///   Known as `decay_rate` in the literature. A co-efficient dictating the strength of
+///   the global incentivization to get the `ideal_stake`. A higher number results in less typical
+///   inflation at the cost of greater volatility for validators.
+///   Must be more than 0.01.
+pub fn compute_inflation<P: PerThing>(
+	stake: P,
+	ideal_stake: P,
+	falloff: P,
+) -> P {
+	if stake < ideal_stake {
+		// ideal_stake is more than 0 because it is strictly more than stake
+		return stake / ideal_stake
+	}
+
+	if falloff < P::from_percent(1.into()) {
+		log::error!("Invalid inflation computation: falloff less than 1% is not supported");
+		return PerThing::zero()
+	}
+
+	let accuracy = {
+		let mut a = BigUint::from(Into::<u128>::into(P::ACCURACY));
+		a.lstrip();
+		a
+	};
+	
+	let mut falloff = BigUint::from(falloff.deconstruct().into());
+	falloff.lstrip();
+
+	let ln2 = {
+		let ln2 = P::from_rational(LN2.deconstruct().into(), Perquintill::ACCURACY.into());
+		BigUint::from(ln2.deconstruct().into())
+	};
+
+	// falloff is stripped above.
+	let ln2_div_d = div_by_stripped(ln2.mul(&accuracy), &falloff);
+
+	let inpos_param = INPoSParam {
+		x_ideal: BigUint::from(ideal_stake.deconstruct().into()),
+		x: BigUint::from(stake.deconstruct().into()),
+		accuracy,
+		ln2_div_d,
+	};
+
+	let res = compute_taylor_serie_part(&inpos_param);
+
+	match u128::try_from(res.clone()) {
+		Ok(res) if res <= Into::<u128>::into(P::ACCURACY) => {
+			P::from_parts(res.saturated_into())
+		},
+		// If result is beyond bounds there is nothing we can do
+		_ => {
+			log::error!("Invalid inflation computation: unexpected result {:?}", res);
+			P::zero()
+		},
+	}
+}
+
+
+/// Internal struct holding parameter info alongside other cached value.
+///
+/// All expressed in part from `accuracy`
+struct INPoSParam {
+	ln2_div_d: BigUint,
+	x_ideal: BigUint,
+	x: BigUint,
+	/// Must be stripped and have no leading zeros.
+	accuracy: BigUint,
+}
+
+/// `ln(2)` expressed in as perquintillionth.
+const LN2: Perquintill = Perquintill::from_parts(0_693_147_180_559_945_309);
+
+/// Compute `2^((x_ideal - x) / d)` using taylor serie.
+///
+/// x must be strictly more than x_ideal.
+///
+/// result is expressed with accuracy `INPoSParam.accuracy`
+fn compute_taylor_serie_part(p: &INPoSParam) -> BigUint {
+	// The last computed taylor term.
+	let mut last_taylor_term = p.accuracy.clone();
+
+	// Whereas taylor sum is positive.
+	let mut taylor_sum_positive = true;
+
+	// The sum of all taylor term.
+	let mut taylor_sum = last_taylor_term.clone();
+
+	for k in 1..300 {
+		last_taylor_term = compute_taylor_term(k, &last_taylor_term, p);
+
+		if last_taylor_term.is_zero() {
+			break
+		}
+
+		let last_taylor_term_positive = k % 2 == 0;
+
+		if taylor_sum_positive == last_taylor_term_positive {
+			taylor_sum = taylor_sum.add(&last_taylor_term);
+		} else {
+			if taylor_sum >= last_taylor_term {
+				taylor_sum = taylor_sum.sub(&last_taylor_term)
+					// NOTE: Should never happen as checked above
+					.unwrap_or_else(|e| e);
+			} else {
+				taylor_sum_positive = !taylor_sum_positive;
+				taylor_sum = last_taylor_term.clone().sub(&taylor_sum)
+					// NOTE: Should never happen as checked above
+					.unwrap_or_else(|e| e);
+			}
+		}
+	}
+
+	if !taylor_sum_positive {
+		return BigUint::zero()
+	}
+
+	taylor_sum.lstrip();
+	taylor_sum
+}
+
+/// Return the absolute value of k-th taylor term of `2^((x_ideal - x))/d` i.e.
+/// `((x - x_ideal) * ln(2) / d)^k / k!`
+///
+/// x must be strictly more x_ideal.
+///
+/// We compute the term from the last term using this formula:
+///
+/// `((x - x_ideal) * ln(2) / d)^k / k! == previous_term * (x - x_ideal) * ln(2) / d / k`
+///
+/// `previous_taylor_term` and result are expressed with accuracy `INPoSParam.accuracy`
+fn compute_taylor_term(k: u32, previous_taylor_term: &BigUint, p: &INPoSParam) -> BigUint {
+	let x_minus_x_ideal = p.x.clone().sub(&p.x_ideal)
+		// NOTE: Should never happen, as x must be more than x_ideal
+		.unwrap_or_else(|_| BigUint::zero());
+
+	let res = previous_taylor_term.clone()
+		.mul(&x_minus_x_ideal)
+		.mul(&p.ln2_div_d)
+		.div_unit(k);
+
+	// p.accuracy is stripped by definition.
+	let res = div_by_stripped(res, &p.accuracy);
+	let mut res = div_by_stripped(res, &p.accuracy);
+
+	res.lstrip();
+	res
+}
+
+/// Compute a div b.
+///
+/// requires `b` to be stripped and have no leading zeros.
+fn div_by_stripped(mut a: BigUint, b: &BigUint) -> BigUint {
+	a.lstrip();
+
+	if b.len() == 0 {
+		log::error!("Computation error: Invalid division");
+		return BigUint::zero()
+	}
+
+	if b.len() == 1 {
+		return a.div_unit(b.checked_get(0).unwrap_or(1))
+	}
+
+	if b.len() > a.len() {
+		return BigUint::zero()
+	}
+
+	if b.len() == a.len() {
+		// 100_000^2 is more than 2^32-1, thus `new_a` has more limbs than `b`.
+		let mut new_a = a.mul(&BigUint::from(100_000u64.pow(2)));
+		new_a.lstrip();
+
+		debug_assert!(new_a.len() > b.len());
+		return new_a
+			.div(b, false)
+			.map(|res| res.0)
+			.unwrap_or_else(|| BigUint::zero())
+			.div_unit(100_000)
+			.div_unit(100_000)
+	}
+	
+	a.div(b, false)
+		.map(|res| res.0)
+		.unwrap_or_else(|| BigUint::zero())
+}

substrate/frame/staking/reward-fn/tests/test.rs

@@ -0,0 +1,101 @@
+// This file is part of Substrate.
+
+// Copyright (C) 2021 Parity Technologies (UK) Ltd.
+// SPDX-License-Identifier: Apache-2.0
+
+// Licensed under the Apache License, Version 2.0 (the "License");
+// you may not use this file except in compliance with the License.
+// You may obtain a copy of the License at
+//
+// 	http://www.apache.org/licenses/LICENSE-2.0
+//
+// Unless required by applicable law or agreed to in writing, software
+// distributed under the License is distributed on an "AS IS" BASIS,
+// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+// See the License for the specific language governing permissions and
+// limitations under the License.
+
+use sp_arithmetic::{PerThing, Perbill, PerU16, Percent, Perquintill};
+
+/// This test the precision and panics if error too big error.
+///
+/// error is asserted to be less or equal to 8/accuracy or 8*f64::EPSILON
+fn test_precision<P: PerThing>(stake: P, ideal_stake: P, falloff: P) {
+	let accuracy_f64 = Into::<u128>::into(P::ACCURACY) as f64;
+	let res = pallet_staking_reward_fn::compute_inflation(stake, ideal_stake, falloff);
+	let res = Into::<u128>::into(res.deconstruct()) as f64 / accuracy_f64;
+
+	let expect = float_i_npos(stake, ideal_stake, falloff);
+
+	let error = (res - expect).abs();
+
+	if error > 8f64 / accuracy_f64 && error > 8.0 * f64::EPSILON {
+		panic!(
+			"stake: {:?}, ideal_stake: {:?}, falloff: {:?}, res: {}, expect: {}",
+			stake, ideal_stake, falloff, res , expect
+		);
+	}
+}
+
+/// compute the inflation using floats
+fn float_i_npos<P: PerThing>(stake: P, ideal_stake: P, falloff: P) -> f64 {
+	let accuracy_f64 = Into::<u128>::into(P::ACCURACY) as f64;
+
+	let ideal_stake = Into::<u128>::into(ideal_stake.deconstruct()) as f64 / accuracy_f64;
+	let stake = Into::<u128>::into(stake.deconstruct()) as f64 / accuracy_f64;
+	let falloff = Into::<u128>::into(falloff.deconstruct()) as f64 / accuracy_f64;
+
+	let x_ideal = ideal_stake;
+	let x = stake;
+	let d = falloff;
+
+	if x < x_ideal {
+		x / x_ideal
+	} else {
+		2_f64.powf((x_ideal - x) / d)
+	}
+}
+
+#[test]
+fn test_precision_for_minimum_falloff() {
+	fn test_falloff_precision_for_minimum_falloff<P: PerThing>() {
+		for stake in 0..1_000 {
+			let stake = P::from_rational(stake, 1_000);
+			let ideal_stake = P::zero();
+			let falloff = P::from_rational(1, 100);
+			test_precision(stake, ideal_stake, falloff);
+		}
+	}
+
+	test_falloff_precision_for_minimum_falloff::<Perquintill>();
+
+	test_falloff_precision_for_minimum_falloff::<PerU16>();
+
+	test_falloff_precision_for_minimum_falloff::<Perbill>();
+
+	test_falloff_precision_for_minimum_falloff::<Percent>();
+}
+
+#[test]
+fn compute_inflation_works() {
+	fn compute_inflation_works<P: PerThing>() {
+		for stake in 0..100 {
+			for ideal_stake in 0..10 {
+				for falloff in 1..10 {
+					let stake = P::from_rational(stake, 100);
+					let ideal_stake = P::from_rational(ideal_stake, 10);
+					let falloff = P::from_rational(falloff, 100);
+					test_precision(stake, ideal_stake, falloff);
+				}
+			}
+		}
+	}
+
+	compute_inflation_works::<Perquintill>();
+
+	compute_inflation_works::<PerU16>();
+
+	compute_inflation_works::<Perbill>();
+
+	compute_inflation_works::<Percent>();
+}