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/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>();
+}