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Make grow formulas faster and more accurate. (#1044)
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@ -24,8 +24,8 @@ export const CONSTANTS: {
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NeuroFluxGovernorLevelMult: number;
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NumNetscriptPorts: number;
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HomeComputerMaxRam: number;
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ServerBaseGrowthRate: number;
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ServerMaxGrowthRate: number;
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ServerBaseGrowthIncr: number;
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ServerMaxGrowthLog: number;
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ServerFortifyAmount: number;
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ServerWeakenAmount: number;
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PurchasedServerLimit: number;
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@ -125,8 +125,8 @@ export const CONSTANTS: {
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// Server-related constants
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HomeComputerMaxRam: 1073741824, // 2 ^ 30
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ServerBaseGrowthRate: 1.03, // Unadjusted Growth rate
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ServerMaxGrowthRate: 1.0035, // Maximum possible growth rate (max rate accounting for server security)
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ServerBaseGrowthIncr: 0.03, // Unadjusted growth increment (growth rate is this * adjustment + 1)
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ServerMaxGrowthLog: 0.00349388925425578, // Maximum possible growth rate accounting for server security, precomputed as log1p(.0035)
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ServerFortifyAmount: 0.002, // Amount by which server's security increases when its hacked/grown
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ServerWeakenAmount: 0.05, // Amount by which server's security decreases when weakened
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@ -1,9 +1,8 @@
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import { GetServer, createUniqueRandomIp, ipExists } from "./AllServers";
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import { Server, IConstructorParams } from "./Server";
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import { BaseServer } from "./BaseServer";
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import { calculateServerGrowth } from "./formulas/grow";
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import { calculateServerGrowth, calculateServerGrowthLog } from "./formulas/grow";
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import { currentNodeMults } from "../BitNode/BitNodeMultipliers";
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import { CONSTANTS } from "../Constants";
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import { Player } from "@player";
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import { CompletedProgramName, LiteratureName } from "@enums";
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@ -52,24 +51,7 @@ export function safelyCreateUniqueServer(params: IConstructorParams): Server {
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*/
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export function numCycleForGrowth(server: IServer, growth: number, cores = 1): number {
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if (!server.serverGrowth) return Infinity;
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const hackDifficulty = server.hackDifficulty ?? 100;
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let ajdGrowthRate = 1 + (CONSTANTS.ServerBaseGrowthRate - 1) / hackDifficulty;
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if (ajdGrowthRate > CONSTANTS.ServerMaxGrowthRate) {
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ajdGrowthRate = CONSTANTS.ServerMaxGrowthRate;
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}
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const serverGrowthPercentage = server.serverGrowth / 100;
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const coreBonus = getCoreBonus(cores);
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const cycles =
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Math.log(growth) /
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(Math.log(ajdGrowthRate) *
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Player.mults.hacking_grow *
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serverGrowthPercentage *
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currentNodeMults.ServerGrowthRate *
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coreBonus);
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return cycles;
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return Math.log(growth) / calculateServerGrowthLog(server, 1, Player, cores);
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}
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/**
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@ -91,130 +73,106 @@ export function numCycleForGrowthCorrected(
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): number {
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if (!server.serverGrowth) return Infinity;
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const moneyMax = server.moneyMax ?? 1;
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const hackDifficulty = server.hackDifficulty ?? 100;
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if (startMoney < 0) startMoney = 0; // servers "can't" have less than 0 dollars on them
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if (targetMoney > moneyMax) targetMoney = moneyMax; // can't grow a server to more than its moneyMax
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if (targetMoney <= startMoney) return 0; // no growth --> no threads
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// exponential base adjusted by security
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const adjGrowthRate = 1 + (CONSTANTS.ServerBaseGrowthRate - 1) / hackDifficulty;
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const exponentialBase = Math.min(adjGrowthRate, CONSTANTS.ServerMaxGrowthRate); // cap growth rate
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// total of all grow thread multipliers
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const serverGrowthPercentage = server.serverGrowth / 100.0;
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const coreMultiplier = getCoreBonus(cores);
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const threadMultiplier =
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serverGrowthPercentage * person.mults.hacking_grow * coreMultiplier * currentNodeMults.ServerGrowthRate;
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const k = calculateServerGrowthLog(server, 1, person, cores);
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/* To understand what is done below we need to do some math. I hope the explanation is clear enough.
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* First of, the names will be shortened for ease of manipulation:
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* n:= targetMoney (n for new), o:= startMoney (o for old), b:= exponentialBase, t:= threadMultiplier, c:= cycles/threads
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* c is what we are trying to compute.
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* n:= targetMoney (n for new), o:= startMoney (o for old), k:= calculateServerGrowthLog, x:= threads
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* x is what we are trying to compute.
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*
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* After growing, the money on a server is n = (o + c) * b^(c*t)
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* c appears in an exponent and outside it, this is usually solved using the productLog/lambert's W special function
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* this function will be noted W in the following
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* The idea behind lambert's W function is W(x)*exp(W(x)) = x, or in other words, solving for y, y*exp(y) = x, as a function of x
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* This function is provided in some advanced math library but we will compute it ourself here.
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* After growing, the money on a server is n = (o + x) * exp(k*x)
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* x appears in an exponent and outside it, this is usually solved using the productLog/lambert's W special function,
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* but it turns out that due to floating-point range issues this approach is *useless* to us, so it will be ignored.
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*
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* Let's get back to solving the equation. It cannot be rewrote using W immediately because the base of the exponentiation is b
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* b^(c*t) = exp(ln(b)*c*t) (this is how a^b is defined on reals, it matches the definition on integers)
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* so n = (o + c) * exp(ln(b)*c*t) , W still cannot be used directly. We want to eliminate the other terms in 'o + c' and 'ln(b)*c*t'.
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*
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* A change of variable will do. The idea is to add an equation introducing a new variable (w here) in the form c = f(w) (for some f)
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* With this equation we will eliminate all references to c, then solve for w and plug the result in the new equation to get c.
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* The change of variable performed here should get rid of the unwanted terms mentioned above, c = w/(ln(b)*t) - o should help.
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* This change of variable is allowed because whatever the value of c is, there is a value of w such that this equation holds:
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* w = (c + o)*ln(b)*t (see how we used the terms we wanted to eliminate in order to build this variable change)
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*
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* We get n = (o + w/(ln(b)*t) - o) * exp(ln(b)*(w/(ln(b)*t) - o)*t) [ = w/(ln(b)*t) * exp(w - ln(b)*o*t) ]
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* The change of variable exposed exp(w - o*ln(b)*t), we can rewrite that with exp(a - b) = exp(a)/exp(b) to isolate 'w*exp(w)'
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* n = w/(ln(b)*t) * exp(w)/exp(ln(b)*o*t) [ = w*exp(w) / (ln(b) * t * b^(o*t)) ]
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* Almost there, we just need to cancel the denominator on the right side of the equation:
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* n * ln(b) * t * b^(o*t) = w*exp(w), Thus w = W(n * ln(b) * t * b^(o*t))
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* Finally we invert the variable change: c = W(n * ln(b) * t * b^(o*t))/(ln(b)*t) - o
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*
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* There is still an issue left: b^(o*t) doesn't fit inside a double precision float
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* because the typical amount of money on servers is around 10^6~10^9
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* We need to get an approximation of W without computing the power when o is huge
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* Thankfully an approximation giving ~30% error uses log immediately so we will use
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* W(n * ln(b) * t * b^(o*t)) ~= log(n * ln(b) * t * b^(o*t)) = log(n * ln(b) * t) + log(exp(ln(b) * o * t))
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* = log(n * ln(b) * t) + ln(b) * o * t
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* (thanks to Drak for the grow formula, f4113nb34st and Wolfram Alpha for the rewrite, dwRchyngqxs for the explanation)
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*/
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const x = threadMultiplier * Math.log(exponentialBase);
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const y = startMoney * x + Math.log(targetMoney * x);
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/* Code for the approximation of lambert's W function is adapted from
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* https://git.savannah.gnu.org/cgit/gsl.git/tree/specfunc/lambert.c
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* using the articles [1] https://doi.org/10.1007/BF02124750 (algorithm above)
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* and [2] https://doi.org/10.1145/361952.361970 (initial approximation when x < 2.5)
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*/
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let w;
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if (y < Math.log(2.5)) {
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/* exp(y) can be safely computed without overflow.
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* The relative error on the result is better when exp(y) < 2.5
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* using Padé rational fraction approximation [2](5)
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*/
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const ey = Math.exp(y);
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w = (ey + (4 / 3) * ey * ey) / (1 + (7 / 3) * ey + (5 / 6) * ey * ey);
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} else {
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/* obtain initial approximation from rough asymptotic [1](4.18)
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* w = y [- log y when 0 <= y]
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*/
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w = y;
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if (y > 0) w -= Math.log(y);
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}
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let cycles = w / x - startMoney;
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/* Iterative refinement, the goal is to correct c until |(o + c) * b^(c*t) - n| < 1
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* or the correction on the approximation is less than 1
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* The Newton-Raphson method will be used, this method is a classic to find roots of functions
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* (given f, find c such that f(c) = 0).
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* Instead, we proceed directly to Newton-Raphson iteration. We first rewrite the equation in
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* log-form, since iterating it this way has faster convergence: log(n) = log(o+x) + k*x.
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* Now our goal is to find the zero of f(x) = log((o+x)/n) + k*x.
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* (Due to the shape of the function, there will be a single zero.)
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*
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* The idea of this method is to take the horizontal position at which the horizontal axis
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* intersects with of the tangent of the function's curve as the next approximation.
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* It is equivalent to treating the curve as a line (it is called a first order approximation)
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* If the current approximation is c then the new approximated value is c - f(c)/f'(c)
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* If the current approximation is x then the new approximated value is x - f(x)/f'(x)
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* (where f' is the derivative of f).
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*
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* In our case f(c) = (o + c) * b^(c*t) - n, f'(c) = d((o + c) * b^(c*t) - n)/dc
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* = (ln(b)*t * (c + o) + 1) * b^(c*t)
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* And the update step is c[new] = c[old] - ((o + c) * b^(c*t) - n)/((ln(b)*t * (o + c) + 1) * b^(c*t))
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* In our case f(x) = log((o+x)/n) + k*x, f'(x) = d(log((o+x)/n) + k*x)/dx
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* = 1/(o + x) + k
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* And the update step is x[new] = x - (log((o+x)/n) + k*x)/(1/(o+x) + k)
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* We can simplify this by bringing the first term up into the fraction:
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* = (x * (1/(o+x) + k) - log((o+x)/n) - k*x) / (1/(o+x) + k)
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* = (x/(o+x) - log((o+x)/n)) / (1/(o+x) + k) [multiplying top and bottom by (o+x)]
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* = (x - (o+x)*log((o+x)/n)) / (1 + (o+x)*k)
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*
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* The main question to ask when using this method is "does it converges?"
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* The main question to ask when using this method is "does it converge?"
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* (are the approximations getting better?), if it does then it does quickly.
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* DOES IT CONVERGES? In the present case it does. The reason why doesn't help explaining the algorithm.
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* If you are interested then check out the wikipedia page.
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* Since the derivative is always positive but also strictly decreasing, convergence is guaranteed.
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* This also provides the useful knowledge that any x which starts *greater* than the solution will
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* undershoot across to the left, while values *smaller* than the zero will continue to find
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* closer approximations that are still smaller than the final value.
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*
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* Of great importance for reducing the number of iterations is starting with a good initial
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* guess. We use a very simple starting condition: x_0 = n - o. We *know* this will always overshot
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* the target, usually by a vast amount. But we can run it manually through one Newton iteration
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* to get a better start with nice properties:
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* x_1 = ((n - o) - (n - o + o)*log((n-o+o)/n)) / (1 + (n-o+o)*k)
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* = ((n - o) - n * log(n/n)) / (1 + n*k)
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* = ((n - o) - n * 0) / (1 + n*k)
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* = (n - o) / (1 + n*k)
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* We can do the same procedure with the exponential form of Newton's method, starting from x_0 = 0.
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* This gives x_1 = (n - o) / (1 + o*k), (full derivation omitted) which will be an overestimate.
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* We use a weighted average of the denominators to get the final guess:
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* x = (n - o) / (1 + (1/16*n + 15/16*o)*k)
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* The reason for this particular weighting is subtle; it is exactly representable and holds up
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* well under a wide variety of conditions, making it likely that the we start within 1 thread of
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* correct. It particularly bounds the worst-case to 3 iterations, and gives a very wide swatch
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* where 2 iterations is good enough.
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*
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* The accuracy of the initial guess is good for many inputs - often one iteration
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* is sufficient. This means the overall cost is two logs (counting the one in calculateServerGrowthLog),
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* possibly one exp, 5 divisions, and a handful of basic arithmetic.
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*/
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let bt = exponentialBase ** threadMultiplier;
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if (bt == Infinity) bt = 1e300;
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let corr = Infinity;
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// Two sided error because we do not want to get stuck if the error stays on the wrong side
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const guess = (targetMoney - startMoney) / (1 + (targetMoney * (1 / 16) + startMoney * (15 / 16)) * k);
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let x = guess;
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let diff;
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do {
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// c should be above 0 so Halley's method can't be used, we have to stick to Newton-Raphson
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let bct = bt ** cycles;
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if (bct == Infinity) bct = 1e300;
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const opc = startMoney + cycles;
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let diff = opc * bct - targetMoney;
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if (diff == Infinity) diff = 1e300;
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corr = diff / (opc * x + 1.0) / bct;
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cycles -= corr;
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} while (Math.abs(corr) >= 1);
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/* c is now within +/- 1 of the exact result.
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* We want the ceiling of the exact result, so the floor if the approximation is above,
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* the ceiling if the approximation is in the same unit as the exact result,
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* and the ceiling + 1 if the approximation is below.
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const ox = startMoney + x;
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// Have to use division instead of multiplication by inverse, because
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// if targetMoney is MIN_VALUE then inverting gives Infinity
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const newx = (x - ox * Math.log(ox / targetMoney)) / (1 + ox * k);
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diff = newx - x;
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x = newx;
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} while (diff < -1 || diff > 1);
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/* If we see a diff of 1 or less we know all future diffs will be smaller, and the rate of
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* convergence means the *sum* of the diffs will be less than 1.
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* In most cases, our result here will be ceil(x).
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*/
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const fca = Math.floor(cycles);
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if (targetMoney <= (startMoney + fca) * Math.pow(exponentialBase, fca * threadMultiplier)) {
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return fca;
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const ccycle = Math.ceil(x);
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if (ccycle - x > 0.999999) {
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// Rounding-error path: It's possible that we slightly overshot the integer value due to
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// rounding error, and more specifically precision issues with log and the size difference of
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// startMoney vs. x. See if a smaller integer works. Most of the time, x was not close enough
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// that we need to try.
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const fcycle = ccycle - 1;
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if (targetMoney <= (startMoney + fcycle) * Math.exp(k * fcycle)) {
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return fcycle;
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}
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const cca = Math.ceil(cycles);
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if (targetMoney <= (startMoney + cca) * Math.pow(exponentialBase, cca * threadMultiplier)) {
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return cca;
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}
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return cca + 1;
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if (ccycle >= x + ((diff <= 0 ? -diff : diff) + 0.000001)) {
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// Fast-path: We know the true value is somewhere in the range [x, x + |diff|] but the next
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// greatest integer is past this. Since we have to round up grows anyway, we can return this
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// with no more calculation. We need some slop due to rounding errors - we can't fast-path
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// a value that is too small.
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return ccycle;
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}
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if (targetMoney <= (startMoney + ccycle) * Math.exp(k * ccycle)) {
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return ccycle;
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}
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return ccycle + 1;
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}
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//Applied server growth for a single server. Returns the percentage growth
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@ -226,7 +184,7 @@ export function processSingleServerGrowth(server: Server, threads: number, cores
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}
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const oldMoneyAvailable = server.moneyAvailable;
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server.moneyAvailable += 1 * threads; // It can be grown even if it has no money
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server.moneyAvailable += threads; // It can be grown even if it has no money
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server.moneyAvailable *= serverGrowth;
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// in case of data corruption
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@ -2,24 +2,31 @@ import { CONSTANTS } from "../../Constants";
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import { currentNodeMults } from "../../BitNode/BitNodeMultipliers";
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import { Person as IPerson, Server as IServer } from "@nsdefs";
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export function calculateServerGrowth(server: IServer, threads: number, p: IPerson, cores = 1): number {
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if (!server.serverGrowth) return 0;
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// Returns the log of the growth rate. When passing 1 for threads, this gives a useful constant.
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export function calculateServerGrowthLog(server: IServer, threads: number, p: IPerson, cores = 1): number {
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if (!server.serverGrowth) return -Infinity;
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const hackDifficulty = server.hackDifficulty ?? 100;
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const numServerGrowthCycles = Math.max(Math.floor(threads), 0);
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const numServerGrowthCycles = Math.max(threads, 0);
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//Get adjusted growth rate, which accounts for server security
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const growthRate = CONSTANTS.ServerBaseGrowthRate;
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let adjGrowthRate = 1 + (growthRate - 1) / hackDifficulty;
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if (adjGrowthRate > CONSTANTS.ServerMaxGrowthRate) {
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adjGrowthRate = CONSTANTS.ServerMaxGrowthRate;
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//Get adjusted growth log, which accounts for server security
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//log1p computes log(1+p), it is far more accurate for small values.
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let adjGrowthLog = Math.log1p(CONSTANTS.ServerBaseGrowthIncr / hackDifficulty);
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if (adjGrowthLog >= CONSTANTS.ServerMaxGrowthLog) {
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adjGrowthLog = CONSTANTS.ServerMaxGrowthLog;
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}
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//Calculate adjusted server growth rate based on parameters
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const serverGrowthPercentage = server.serverGrowth / 100;
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const numServerGrowthCyclesAdjusted =
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numServerGrowthCycles * serverGrowthPercentage * currentNodeMults.ServerGrowthRate;
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const serverGrowthPercentageAdjusted = serverGrowthPercentage * currentNodeMults.ServerGrowthRate;
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//Apply serverGrowth for the calculated number of growth cycles
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const coreBonus = 1 + (cores - 1) / 16;
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return Math.pow(adjGrowthRate, numServerGrowthCyclesAdjusted * p.mults.hacking_grow * coreBonus);
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const coreBonus = 1 + (cores - 1) * (1 / 16);
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// It is critical that numServerGrowthCycles (aka threads) is multiplied last,
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// so that it rounds the same way as numCycleForGrowthCorrected.
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return adjGrowthLog * serverGrowthPercentageAdjusted * p.mults.hacking_grow * coreBonus * numServerGrowthCycles;
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}
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export function calculateServerGrowth(server: IServer, threads: number, p: IPerson, cores = 1): number {
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if (!server.serverGrowth) return 0;
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return Math.exp(calculateServerGrowthLog(server, threads, p, cores));
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}
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76
test/jest/Grow.test.ts
Normal file
76
test/jest/Grow.test.ts
Normal file
@ -0,0 +1,76 @@
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import { PlayerObject } from "../../src/PersonObjects/Player/PlayerObject";
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import { Server } from "../../src/Server/Server";
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import { calculateServerGrowth } from "../../src/Server/formulas/grow";
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import { numCycleForGrowthCorrected } from "../../src/Server/ServerHelpers";
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test("Grow is accurate", () => {
|
||||
// Tests that certain special values work out to exactly what we'd expect,
|
||||
// given the formulas. The growth multiplier maxes out at 1.0035, and the
|
||||
// increment is .03 so with a difficulty of 10 it should be .003.
|
||||
// These tests are *exact* because the whole point is that the math should
|
||||
// get exactly the right value (or as close as is possible with floating-point).
|
||||
const server = new Server({ hostname: "foo", hackDifficulty: 5, serverGrowth: 100 });
|
||||
const player = new PlayerObject();
|
||||
expect(calculateServerGrowth(server, 1, player)).toBe(1.0035);
|
||||
expect(calculateServerGrowth(server, 2, player)).toBe(1.00701225);
|
||||
server.hackDifficulty = 10;
|
||||
expect(calculateServerGrowth(server, 1, player)).toBe(1.003);
|
||||
expect(calculateServerGrowth(server, 2, player)).toBe(1.006009);
|
||||
expect(calculateServerGrowth(server, 3, player)).toBe(1.009027027);
|
||||
expect(calculateServerGrowth(server, 4, player)).toBe(1.012054108081);
|
||||
});
|
||||
|
||||
describe("numCycleForGrowthCorrected reverses calculateServerGrowth", () => {
|
||||
// Test that numCycleForGrowthCorrected() functions properly as the inverse
|
||||
// of calculateServerGrowth().
|
||||
// calculateServerGrowth() works for *any* given number of threads, but is
|
||||
// usually passed an integer. numCycleForGrowthCorrected() *always* returns
|
||||
// an integer, the ceiling of the floating-point value. When reversing an
|
||||
// integer number of threads, it should *always* return the same integer.
|
||||
// Similarly, if we pass the next-highest representable number as a target,
|
||||
// it should *always* return the next largest integer, since the previous
|
||||
// number is no longer sufficient.
|
||||
|
||||
// This is an arbitrary transcedental constant.
|
||||
const multiplier = Math.exp(1.4);
|
||||
const server = new Server({ hostname: "foo", hackDifficulty: 10 * multiplier, serverGrowth: 100 });
|
||||
server.moneyMax = 1e308; // Not available as a constructor param
|
||||
const player = new PlayerObject();
|
||||
const tests = [];
|
||||
while (server.moneyAvailable < 5e49) {
|
||||
tests.push(server.moneyAvailable);
|
||||
server.moneyAvailable = (server.moneyAvailable + 59) * calculateServerGrowth(server, 59, player);
|
||||
}
|
||||
test.each(tests)("startMoney: %f", (money: number) => {
|
||||
// Do fewer threads to save on test time
|
||||
for (let threads = 0; threads < 30; threads++) {
|
||||
const newMoney = (money + threads) * calculateServerGrowth(server, threads, player);
|
||||
const eps = newMoney ? 2 ** (Math.floor(Math.log2(newMoney)) - 52) : Number.MIN_VALUE;
|
||||
expect(numCycleForGrowthCorrected(server, newMoney, money, 1, player)).toBe(threads);
|
||||
const value = numCycleForGrowthCorrected(server, newMoney + eps, money, 1, player);
|
||||
// Write our own check because Jest is a goblin that can't provide context
|
||||
if (value !== threads + 1) {
|
||||
console.log(
|
||||
`newMoney: ${newMoney} eps: ${eps} newMoney+eps: ${newMoney + eps} value: ${value} threads: ${threads}`,
|
||||
);
|
||||
expect(value).toBe(threads + 1);
|
||||
}
|
||||
}
|
||||
});
|
||||
const tests2 = [];
|
||||
for (let t = 0; t <= 900000; t += 2000) {
|
||||
tests2.push([t, t * calculateServerGrowth(server, t, player)]);
|
||||
}
|
||||
test.each(tests2)("threads: %f newMoney: %f", (threads: number, newMoney: number) => {
|
||||
const eps = newMoney ? 2 ** (Math.floor(Math.log2(newMoney)) - 52) : Number.MIN_VALUE;
|
||||
expect(numCycleForGrowthCorrected(server, newMoney, 0, 1, player)).toBe(threads);
|
||||
const value = numCycleForGrowthCorrected(server, newMoney + eps, 0, 1, player);
|
||||
// Write our own check because Jest is a goblin that can't provide context
|
||||
if (value !== threads + 1) {
|
||||
console.log(
|
||||
`newMoney: ${newMoney} eps: ${eps} newMoney+eps: ${newMoney + eps} value: ${value} threads: ${threads}`,
|
||||
);
|
||||
expect(value).toBe(threads + 1);
|
||||
}
|
||||
});
|
||||
});
|
@ -477,7 +477,10 @@ describe("Stock Market Tests", function () {
|
||||
const initValue = initialValues[stock.symbol];
|
||||
expect(initValue.price).not.toEqual(stock.price);
|
||||
if (initValue.otlkMag === stock.otlkMag && initValue.b === stock.b) {
|
||||
throw new Error("expected either price or otlkMag to be different");
|
||||
throw new Error(
|
||||
"expected either price or otlkMag to be different: " +
|
||||
`stock: ${stockName} otlkMag: ${stock.otlkMag} b: ${stock.b}`,
|
||||
);
|
||||
}
|
||||
}
|
||||
});
|
||||
|
Loading…
Reference in New Issue
Block a user