Fixing more tests with integral rework

src/data/projEntry.tsx

@@ -4,7 +4,11 @@ import { createResource, trackBest, trackOOMPS, trackTotal } from "features/reso
 import type { GenericTree } from "features/trees/tree";
 import { branchedResetPropagation, createTree } from "features/trees/tree";
 import { globalBus } from "game/events";
-import Formula, { calculateCost, calculateMaxAffordable } from "game/formulas/formulas";
+import Formula, {
+    calculateCost,
+    calculateMaxAffordable,
+    findNonInvertible
+} from "game/formulas/formulas";
 import type { BaseLayer, GenericLayer } from "game/layers";
 import { createLayer } from "game/layers";
 import type { Player } from "game/player";
@@ -18,6 +22,7 @@ import prestige from "./layers/prestige";
 window.Formula = Formula;
 window.calculateMaxAffordable = calculateMaxAffordable;
 window.calculateCost = calculateCost;
+window.findNonInvertible = findNonInvertible;
 window.unref = unref;
 window.ref = ref;
 window.createResource = createResource;

src/game/formulas/formulas.ts

@@ -1,7 +1,8 @@
 import { Resource } from "features/resources/resource";
 import Decimal, { DecimalSource } from "util/bignum";
 import { Computable, convertComputable, ProcessedComputable } from "util/computed";
-import { computed, ComputedRef, Ref, ref, unref } from "vue";
+import { computed, ComputedRef, ref, unref } from "vue";
+import * as ops from "./operations";
 import type {
     EvaluateFunction,
     FormulaOptions,
@@ -18,7 +19,6 @@ import type {
     SubstitutionFunction,
     SubstitutionStack
 } from "./types";
-import * as ops from "./operations";
 
 export function hasVariable(value: FormulaSource): value is InvertibleFormula {
     return value instanceof Formula && value.hasVariable();
@@ -32,13 +32,6 @@ function integrateVariable(this: GenericFormula) {
     return Formula.pow(this, 2).div(2);
 }
 
-function integrateVariableInner(this: GenericFormula, variable?: DecimalSource) {
-    if (variable == null && this.innermostVariable == null) {
-        throw new Error("Cannot integrate non-existent variable");
-    }
-    return variable ?? unref(this.innermostVariable);
-}
-
 /**
  * A class that can be used for cost/goal functions. It can be evaluated similar to a cost function, but also provides extra features for supported formulas. For example, a lot of math functions can be inverted.
  * Typically, the use of these extra features is to support cost/goal functions that have multiple levels purchased/completed at once efficiently.
@@ -53,7 +46,7 @@ export default class Formula<T extends [FormulaSource] | FormulaSource[]> {
     private readonly internalIntegrate: IntegrateFunction<T> | undefined;
     private readonly internalIntegrateInner: IntegrateFunction<T> | undefined;
     private readonly applySubstitution: SubstitutionFunction<T> | undefined;
-    private readonly internalHasVariable: boolean;
+    private readonly internalVariables: number;
 
     public readonly innermostVariable: ProcessedComputable<DecimalSource> | undefined;
 
@@ -69,7 +62,7 @@ export default class Formula<T extends [FormulaSource] | FormulaSource[]> {
             readonlyProperties = this.setupFormula(options);
         }
         this.inputs = readonlyProperties.inputs;
-        this.internalHasVariable = readonlyProperties.internalHasVariable;
+        this.internalVariables = readonlyProperties.internalVariables;
         this.innermostVariable = readonlyProperties.innermostVariable;
         this.internalEvaluate = readonlyProperties.internalEvaluate;
         this.internalInvert = readonlyProperties.internalInvert;
@@ -85,10 +78,9 @@ export default class Formula<T extends [FormulaSource] | FormulaSource[]> {
     }): InternalFormulaProperties<T> {
         return {
             inputs: [variable] as T,
-            internalHasVariable: true,
+            internalVariables: 1,
             innermostVariable: variable,
-            internalIntegrate: integrateVariable as unknown as IntegrateFunction<T>,
+            internalIntegrate: integrateVariable,
-            internalIntegrateInner: integrateVariableInner as unknown as IntegrateFunction<T>,
             applySubstitution: ops.passthrough as unknown as SubstitutionFunction<T>
         };
     }
@@ -99,68 +91,49 @@ export default class Formula<T extends [FormulaSource] | FormulaSource[]> {
         }
         return {
             inputs: inputs as T,
-            internalHasVariable: false
+            internalVariables: 0
         };
     }
 
     private setupFormula(options: GeneralFormulaOptions<T>): InternalFormulaProperties<T> {
-        const {
+        const { inputs, evaluate, invert, integrate, integrateInner, applySubstitution } = options;
-            inputs,
+        const numVariables = inputs.reduce<number>(
-            evaluate,
+            (acc, input) => acc + (input instanceof Formula ? input.internalVariables : 0),
-            invert,
+            0
-            integrate,
+        );
-            integrateInner,
-            applySubstitution,
-            hasVariable
-        } = options;
-        if (invert == null && hasVariable) {
-            throw new Error(
-                "A formula cannot be marked as having a variable if it is not invertible"
-            );
-        }
-
-        const numVariables = inputs.filter(
-            input => input instanceof Formula && input.hasVariable()
-        ).length;
         const variable = inputs.find(input => input instanceof Formula && input.hasVariable()) as
             | GenericFormula
             | undefined;
 
-        const internalHasVariable =
+        const innermostVariable = numVariables === 1 ? variable?.innermostVariable : undefined;
-            numVariables === 1 || (numVariables === 0 && hasVariable === true);
-        const innermostVariable = internalHasVariable ? variable?.innermostVariable : undefined;
-        const internalInvert = internalHasVariable && variable?.isInvertible() ? invert : undefined;
 
         return {
             inputs,
             internalEvaluate: evaluate,
-            internalInvert,
+            internalInvert: invert,
             internalIntegrate: integrate,
             internalIntegrateInner: integrateInner,
             applySubstitution,
             innermostVariable,
-            internalHasVariable
+            internalVariables: numVariables
         };
     }
 
     private calculateConstantOfIntegration() {
-        // Calculate C based on the knowledge that at 1 purchase, the total sum would be the cost of that one purchase
+        // Calculate C based on the knowledge that at x=1, the integral should be the average between f(0) and f(1)
         const integral = this.getIntegralFormula().evaluate(1);
-        const actualCost = this.evaluate(0);
+        const actualCost = Decimal.add(this.evaluate(0), this.evaluate(1)).div(2);
         return Decimal.sub(actualCost, integral);
     }
 
     /** Type predicate that this formula can be inverted. */
     isInvertible(): this is InvertibleFormula {
-        return (
+        return this.hasVariable() && (this.internalInvert != null || this.internalEvaluate == null);
-            this.internalHasVariable &&
-            (this.internalInvert != null || this.internalEvaluate == null)
-        );
     }
 
     /** Type predicate that this formula can be integrated. */
     isIntegrable(): this is IntegrableFormula {
-        return this.internalHasVariable && this.internalIntegrate != null;
+        return this.hasVariable() && this.internalIntegrate != null;
     }
 
     /** Type predicate that this formula has an integral function that can be inverted. */
@@ -173,7 +146,7 @@ export default class Formula<T extends [FormulaSource] | FormulaSource[]> {
 
     /** Whether or not this formula has a singular variable inside it, which can be accessed via {@link innermostVariable}. */
     hasVariable(): boolean {
-        return this.internalHasVariable;
+        return this.internalVariables === 1;
     }
 
     /**
@@ -188,7 +161,7 @@ export default class Formula<T extends [FormulaSource] | FormulaSource[]> {
                     unrefFormulaSource(input, variable)
                 ) as GuardedFormulasToDecimals<T>)
             ) ??
-            (this.internalHasVariable ? variable : null) ??
+            (this.hasVariable() ? variable : null) ??
             unrefFormulaSource(this.inputs[0])
         );
     }
@@ -199,9 +172,9 @@ export default class Formula<T extends [FormulaSource] | FormulaSource[]> {
      * @see {@link isInvertible}
      */
     invert(value: DecimalSource): DecimalSource {
-        if (this.internalInvert) {
+        if (this.internalInvert && this.hasVariable()) {
             return this.internalInvert.call(this, value, ...this.inputs);
-        } else if (this.inputs.length === 1 && this.internalHasVariable) {
+        } else if (this.inputs.length === 1 && this.hasVariable()) {
             return value;
         }
         throw new Error("Cannot invert non-invertible formula");
@@ -228,7 +201,7 @@ export default class Formula<T extends [FormulaSource] | FormulaSource[]> {
      * @see {@link isIntegralInvertible}
      */
     invertIntegral(value: DecimalSource): DecimalSource {
-        if (this.integralFormula?.isInvertible()) {
+        if (!this.isIntegrable() || !this.getIntegralFormula().isInvertible()) {
             throw new Error("Cannot invert integral of formula without invertible integral");
         }
         return this.getIntegralFormula().invert(value);
@@ -236,24 +209,12 @@ export default class Formula<T extends [FormulaSource] | FormulaSource[]> {
 
     /**
      * Get a formula that will evaluate to the integral of this formula. May also be invertible.
-     * @param variable The variable that will be used to evaluate this integral at a given x value
+     * @param stack For nested formulas, a stack of operations that occur outside the complex operation.
-     * @param stack For nested formulas, a stack of operations that occur outside the complex operation
      */
-    getIntegralFormula(
+    getIntegralFormula(stack?: SubstitutionStack): GenericFormula {
-        variable?: ProcessedComputable<DecimalSource>,
+        if (this.integralFormula != null) {
-        stack?: SubstitutionStack
-    ): GenericFormula {
-        if (variable == null && this.integralFormula != null) {
             return this.integralFormula;
         }
-        let formula;
-        const variablePresent = variable != null;
-        if (variable == null) {
-            variable = this.innermostVariable;
-            if (variable == null) {
-                throw new Error("Cannot integrate formula without variable");
-            }
-        }
         if (stack == null) {
             // "Outer" part of the formula
             if (this.applySubstitution == null) {
@@ -262,21 +223,24 @@ export default class Formula<T extends [FormulaSource] | FormulaSource[]> {
                 if (this.internalIntegrate == null) {
                     throw new Error("Cannot integrate formula with non-integrable operation");
                 }
-                let value = this.internalIntegrate.call(this, variable, stack, ...this.inputs);
+                let value = this.internalIntegrate.call(this, stack, ...this.inputs);
                 stack.forEach(func => (value = func(value)));
-                formula = value;
+                this.integralFormula = value;
             } else {
                 // Continue digging into the formula
                 if (this.internalIntegrate) {
-                    formula = this.internalIntegrate.call(
+                    this.integralFormula = this.internalIntegrate.call(
                         this,
-                        variable,
                         undefined,
                         ...this.inputs
                     );
-                } else if (this.inputs.length === 1 && this.internalHasVariable) {
+                } else if (
+                    this.inputs.length === 1 &&
+                    this.internalEvaluate == null &&
+                    this.hasVariable()
+                ) {
                     // eslint-disable-next-line @typescript-eslint/no-this-alias
-                    formula = this;
+                    this.integralFormula = this;
                 } else {
                     throw new Error("Cannot integrate formula without variable");
                 }
@@ -291,20 +255,25 @@ export default class Formula<T extends [FormulaSource] | FormulaSource[]> {
                 this.applySubstitution!.call(this, variable, ...this.inputs)
             );
             if (this.internalIntegrateInner) {
-                formula = this.internalIntegrateInner.call(this, variable, stack, ...this.inputs);
+                this.integralFormula = this.internalIntegrateInner.call(
+                    this,
+                    stack,
+                    ...this.inputs
+                );
             } else if (this.internalIntegrate) {
-                formula = this.internalIntegrate.call(this, variable, stack, ...this.inputs);
+                this.integralFormula = this.internalIntegrate.call(this, stack, ...this.inputs);
-            } else if (this.inputs.length === 1 && this.internalHasVariable) {
+            } else if (
+                this.inputs.length === 1 &&
+                this.internalEvaluate == null &&
+                this.hasVariable()
+            ) {
                 // eslint-disable-next-line @typescript-eslint/no-this-alias
-                formula = this;
+                this.integralFormula = this;
             } else {
                 throw new Error("Cannot integrate formula without variable");
             }
         }
-        if (!variablePresent) {
+        return this.integralFormula;
-            this.integralFormula = formula;
-        }
-        return formula;
     }
 
     /**
@@ -324,7 +293,7 @@ export default class Formula<T extends [FormulaSource] | FormulaSource[]> {
             this.internalEvaluate === other.internalEvaluate &&
             this.internalInvert === other.internalInvert &&
             this.internalIntegrate === other.internalIntegrate &&
-            this.internalHasVariable === other.internalHasVariable
+            this.internalVariables === other.internalVariables
         );
     }
 
@@ -556,17 +525,8 @@ export default class Formula<T extends [FormulaSource] | FormulaSource[]> {
     public static reciprocal = Formula.recip;
     public static reciprocate = Formula.recip;
 
-    public static max(value: FormulaSource, other: FormulaSource): GenericFormula {
+    // TODO these functions should ostensibly be integrable, and the integrals should be invertible
-        return new Formula({
+    public static max = ops.createPassthroughBinaryFormula(Decimal.max);
-            inputs: [value, other],
-            evaluate: Decimal.max,
-            invert: ops.passthrough as (
-                value: DecimalSource,
-                ...inputs: [FormulaSource, FormulaSource]
-            ) => DecimalSource
-        });
-    }
-
     public static min = ops.createPassthroughBinaryFormula(Decimal.min);
     public static minabs = ops.createPassthroughBinaryFormula(Decimal.minabs);
     public static maxabs = ops.createPassthroughBinaryFormula(Decimal.maxabs);
@@ -1356,6 +1316,24 @@ export default class Formula<T extends [FormulaSource] | FormulaSource[]> {
     }
 }
 
+/**
+ * Utility for recursively searching through a formula for the cause of non-invertibility.
+ * @param formula The formula to search for a non-invertible formula within
+ */
+export function findNonInvertible(formula: GenericFormula): GenericFormula | null {
+    // eslint-disable-next-line @typescript-eslint/ban-ts-comment
+    // @ts-ignore
+    if (formula.internalInvert == null && formula.internalEvaluate != null) {
+        return formula;
+    }
+    for (const input of formula.inputs) {
+        if (hasVariable(input)) {
+            return findNonInvertible(input);
+        }
+    }
+    return null;
+}
+
 /**
  * Utility for calculating the maximum amount of purchases possible with a given formula and resource. If {@ref spendResources} is changed to false, the calculation will be much faster with higher numbers.
  * @param formula The formula to use for calculating buy max from

src/game/formulas/operations.ts

@@ -1,8 +1,9 @@
 import Decimal, { DecimalSource } from "util/bignum";
-import { Ref } from "vue";
 import Formula, { hasVariable, unrefFormulaSource } from "./formulas";
 import { FormulaSource, GenericFormula, InvertFunction, SubstitutionStack } from "./types";
 
+const ln10 = Decimal.ln(10);
+
 export function passthrough<T extends GenericFormula | DecimalSource>(value: T): T {
     return value;
 }
@@ -14,13 +15,9 @@ export function invertNeg(value: DecimalSource, lhs: FormulaSource) {
     throw new Error("Could not invert due to no input being a variable");
 }
 
-export function integrateNeg(
+export function integrateNeg(stack: SubstitutionStack, lhs: FormulaSource) {
-    variable: Ref<DecimalSource>,
-    stack: SubstitutionStack,
-    lhs: FormulaSource
-) {
     if (hasVariable(lhs)) {
-        return Formula.neg(lhs.getIntegralFormula(variable, stack));
+        return Formula.neg(lhs.getIntegralFormula(stack));
     }
     throw new Error("Could not integrate due to no input being a variable");
 }
@@ -38,33 +35,27 @@ export function invertAdd(value: DecimalSource, lhs: FormulaSource, rhs: Formula
     throw new Error("Could not invert due to no input being a variable");
 }
 
-export function integrateAdd(
+export function integrateAdd(stack: SubstitutionStack, lhs: FormulaSource, rhs: FormulaSource) {
-    variable: Ref<DecimalSource>,
-    stack: SubstitutionStack,
-    lhs: FormulaSource,
-    rhs: FormulaSource
-) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        const x = lhs.getIntegralFormula(stack);
-        return Formula.times(rhs, variable ?? lhs.innermostVariable ?? 0).add(x);
+        return Formula.times(rhs, lhs.innermostVariable ?? 0).add(x);
     } else if (hasVariable(rhs)) {
-        const x = rhs.getIntegralFormula(variable, stack);
+        const x = rhs.getIntegralFormula(stack);
-        return Formula.times(lhs, variable ?? rhs.innermostVariable ?? 0).add(x);
+        return Formula.times(lhs, rhs.innermostVariable ?? 0).add(x);
     }
     throw new Error("Could not integrate due to no input being a variable");
 }
 
 export function integrateInnerAdd(
-    variable: Ref<DecimalSource>,
     stack: SubstitutionStack,
     lhs: FormulaSource,
     rhs: FormulaSource
 ) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        const x = lhs.getIntegralFormula(stack);
         return Formula.add(x, rhs);
     } else if (hasVariable(rhs)) {
-        const x = rhs.getIntegralFormula(variable, stack);
+        const x = rhs.getIntegralFormula(stack);
         return Formula.add(x, lhs);
     }
     throw new Error("Could not integrate due to no input being a variable");
@@ -79,33 +70,27 @@ export function invertSub(value: DecimalSource, lhs: FormulaSource, rhs: Formula
     throw new Error("Could not invert due to no input being a variable");
 }
 
-export function integrateSub(
+export function integrateSub(stack: SubstitutionStack, lhs: FormulaSource, rhs: FormulaSource) {
-    variable: Ref<DecimalSource>,
-    stack: SubstitutionStack,
-    lhs: FormulaSource,
-    rhs: FormulaSource
-) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        const x = lhs.getIntegralFormula(stack);
-        return Formula.sub(x, Formula.times(rhs, variable ?? lhs.innermostVariable ?? 0));
+        return Formula.sub(x, Formula.times(rhs, lhs.innermostVariable ?? 0));
     } else if (hasVariable(rhs)) {
-        const x = rhs.getIntegralFormula(variable, stack);
+        const x = rhs.getIntegralFormula(stack);
-        return Formula.times(lhs, variable ?? rhs.innermostVariable ?? 0).sub(x);
+        return Formula.times(lhs, rhs.innermostVariable ?? 0).sub(x);
     }
     throw new Error("Could not integrate due to no input being a variable");
 }
 
 export function integrateInnerSub(
-    variable: Ref<DecimalSource>,
     stack: SubstitutionStack,
     lhs: FormulaSource,
     rhs: FormulaSource
 ) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        const x = lhs.getIntegralFormula(stack);
         return Formula.sub(x, rhs);
     } else if (hasVariable(rhs)) {
-        const x = rhs.getIntegralFormula(variable, stack);
+        const x = rhs.getIntegralFormula(stack);
         return Formula.sub(x, lhs);
     }
     throw new Error("Could not integrate due to no input being a variable");
@@ -120,17 +105,12 @@ export function invertMul(value: DecimalSource, lhs: FormulaSource, rhs: Formula
     throw new Error("Could not invert due to no input being a variable");
 }
 
-export function integrateMul(
+export function integrateMul(stack: SubstitutionStack, lhs: FormulaSource, rhs: FormulaSource) {
-    variable: Ref<DecimalSource>,
-    stack: SubstitutionStack,
-    lhs: FormulaSource,
-    rhs: FormulaSource
-) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        const x = lhs.getIntegralFormula(stack);
         return Formula.times(x, rhs);
     } else if (hasVariable(rhs)) {
-        const x = rhs.getIntegralFormula(variable, stack);
+        const x = rhs.getIntegralFormula(stack);
         return Formula.times(x, lhs);
     }
     throw new Error("Could not integrate due to no input being a variable");
@@ -158,17 +138,12 @@ export function invertDiv(value: DecimalSource, lhs: FormulaSource, rhs: Formula
     throw new Error("Could not invert due to no input being a variable");
 }
 
-export function integrateDiv(
+export function integrateDiv(stack: SubstitutionStack, lhs: FormulaSource, rhs: FormulaSource) {
-    variable: Ref<DecimalSource>,
-    stack: SubstitutionStack,
-    lhs: FormulaSource,
-    rhs: FormulaSource
-) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        const x = lhs.getIntegralFormula(stack);
         return Formula.div(x, rhs);
     } else if (hasVariable(rhs)) {
-        const x = rhs.getIntegralFormula(variable, stack);
+        const x = rhs.getIntegralFormula(stack);
         return Formula.div(lhs, x);
     }
     throw new Error("Could not integrate due to no input being a variable");
@@ -194,13 +169,9 @@ export function invertRecip(value: DecimalSource, lhs: FormulaSource) {
     throw new Error("Could not invert due to no input being a variable");
 }
 
-export function integrateRecip(
+export function integrateRecip(stack: SubstitutionStack, lhs: FormulaSource) {
-    variable: Ref<DecimalSource>,
-    stack: SubstitutionStack,
-    lhs: FormulaSource
-) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        const x = lhs.getIntegralFormula(stack);
         return Formula.ln(x);
     }
     throw new Error("Could not integrate due to no input being a variable");
@@ -213,14 +184,25 @@ export function invertLog10(value: DecimalSource, lhs: FormulaSource) {
     throw new Error("Could not invert due to no input being a variable");
 }
 
-export function integrateLog10(
+function internalIntegrateLog10(lhs: DecimalSource) {
-    variable: Ref<DecimalSource>,
+    return Decimal.ln(lhs).sub(1).times(lhs).div(ln10);
-    stack: SubstitutionStack,
+}
-    lhs: FormulaSource
+
-) {
+function internalInvertIntegralLog10(value: DecimalSource, lhs: FormulaSource) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        const numerator = ln10.times(value);
-        return Formula.ln(x).sub(1).times(x).div(Formula.ln(10));
+        return lhs.invert(numerator.div(numerator.div(Math.E).lambertw()));
+    }
+    throw new Error("Could not invert due to no input being a variable");
+}
+
+export function integrateLog10(stack: SubstitutionStack, lhs: FormulaSource) {
+    if (hasVariable(lhs)) {
+        return new Formula({
+            inputs: [lhs.getIntegralFormula(stack)],
+            evaluate: internalIntegrateLog10,
+            invert: internalInvertIntegralLog10
+        });
     }
     throw new Error("Could not integrate due to no input being a variable");
 }
@@ -234,15 +216,25 @@ export function invertLog(value: DecimalSource, lhs: FormulaSource, rhs: Formula
     throw new Error("Could not invert due to no input being a variable");
 }
 
-export function integrateLog(
+function internalIntegrateLog(lhs: DecimalSource, rhs: DecimalSource) {
-    variable: Ref<DecimalSource>,
+    return Decimal.ln(lhs).sub(1).times(lhs).div(Decimal.ln(rhs));
-    stack: SubstitutionStack,
+}
-    lhs: FormulaSource,
+
-    rhs: FormulaSource
+function internalInvertIntegralLog(value: DecimalSource, lhs: FormulaSource, rhs: FormulaSource) {
-) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        const numerator = Decimal.ln(unrefFormulaSource(rhs)).times(value);
-        return Formula.ln(x).sub(1).times(x).div(Formula.ln(rhs));
+        return lhs.invert(numerator.div(numerator.div(Math.E).lambertw()));
+    }
+    throw new Error("Could not invert due to no input being a variable");
+}
+
+export function integrateLog(stack: SubstitutionStack, lhs: FormulaSource, rhs: FormulaSource) {
+    if (hasVariable(lhs)) {
+        return new Formula({
+            inputs: [lhs.getIntegralFormula(stack), rhs],
+            evaluate: internalIntegrateLog,
+            invert: internalInvertIntegralLog
+        });
     }
     throw new Error("Could not integrate due to no input being a variable");
 }
@@ -254,14 +246,25 @@ export function invertLog2(value: DecimalSource, lhs: FormulaSource) {
     throw new Error("Could not invert due to no input being a variable");
 }
 
-export function integrateLog2(
+function internalIntegrateLog2(lhs: DecimalSource) {
-    variable: Ref<DecimalSource>,
+    return Decimal.ln(lhs).sub(1).times(lhs).div(Decimal.ln(2));
-    stack: SubstitutionStack,
+}
-    lhs: FormulaSource
+
-) {
+function internalInvertIntegralLog2(value: DecimalSource, lhs: FormulaSource) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        const numerator = Decimal.ln(2).times(value);
-        return Formula.ln(x).sub(1).times(x).div(Formula.ln(2));
+        return lhs.invert(numerator.div(numerator.div(Math.E).lambertw()));
+    }
+    throw new Error("Could not invert due to no input being a variable");
+}
+
+export function integrateLog2(stack: SubstitutionStack, lhs: FormulaSource) {
+    if (hasVariable(lhs)) {
+        return new Formula({
+            inputs: [lhs.getIntegralFormula(stack)],
+            evaluate: internalIntegrateLog2,
+            invert: internalInvertIntegralLog2
+        });
     }
     throw new Error("Could not integrate due to no input being a variable");
 }
@@ -273,14 +276,24 @@ export function invertLn(value: DecimalSource, lhs: FormulaSource) {
     throw new Error("Could not invert due to no input being a variable");
 }
 
-export function integrateLn(
+function internalIntegrateLn(lhs: DecimalSource) {
-    variable: Ref<DecimalSource>,
+    return Decimal.ln(lhs).sub(1).times(lhs);
-    stack: SubstitutionStack,
+}
-    lhs: FormulaSource
+
-) {
+function internalInvertIntegralLn(value: DecimalSource, lhs: FormulaSource) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        return lhs.invert(Decimal.div(value, Decimal.div(value, Math.E).lambertw()));
-        return Formula.ln(x).sub(1).times(x);
+    }
+    throw new Error("Could not invert due to no input being a variable");
+}
+
+export function integrateLn(stack: SubstitutionStack, lhs: FormulaSource) {
+    if (hasVariable(lhs)) {
+        return new Formula({
+            inputs: [lhs.getIntegralFormula(stack)],
+            evaluate: internalIntegrateLn,
+            invert: internalInvertIntegralLn
+        });
     }
     throw new Error("Could not integrate due to no input being a variable");
 }
@@ -294,18 +307,13 @@ export function invertPow(value: DecimalSource, lhs: FormulaSource, rhs: Formula
     throw new Error("Could not invert due to no input being a variable");
 }
 
-export function integratePow(
+export function integratePow(stack: SubstitutionStack, lhs: FormulaSource, rhs: FormulaSource) {
-    variable: Ref<DecimalSource>,
-    stack: SubstitutionStack,
-    lhs: FormulaSource,
-    rhs: FormulaSource
-) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        const x = lhs.getIntegralFormula(stack);
         const pow = Formula.add(rhs, 1);
         return Formula.pow(x, pow).div(pow);
     } else if (hasVariable(rhs)) {
-        const x = rhs.getIntegralFormula(variable, stack);
+        const x = rhs.getIntegralFormula(stack);
         return Formula.pow(lhs, x).div(Formula.ln(lhs));
     }
     throw new Error("Could not integrate due to no input being a variable");
@@ -318,14 +326,24 @@ export function invertPow10(value: DecimalSource, lhs: FormulaSource) {
     throw new Error("Could not invert due to no input being a variable");
 }
 
-export function integratePow10(
+function internalIntegratePow10(lhs: DecimalSource) {
-    variable: Ref<DecimalSource>,
+    return Decimal.pow10(lhs).div(Decimal.ln(lhs));
-    stack: SubstitutionStack,
+}
-    lhs: FormulaSource
+
-) {
+function internalInvertIntegralPow10(value: DecimalSource, lhs: FormulaSource) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        return lhs.invert(ln10.times(value).ln().div(ln10));
-        return Formula.ln(x).sub(1).times(x).div(Decimal.ln(10));
+    }
+    throw new Error("Could not invert due to no input being a variable");
+}
+
+export function integratePow10(stack: SubstitutionStack, lhs: FormulaSource) {
+    if (hasVariable(lhs)) {
+        return new Formula({
+            inputs: [lhs.getIntegralFormula(stack)],
+            evaluate: internalIntegratePow10,
+            invert: internalInvertIntegralPow10
+        });
     }
     throw new Error("Could not integrate due to no input being a variable");
 }
@@ -339,17 +357,12 @@ export function invertPowBase(value: DecimalSource, lhs: FormulaSource, rhs: For
     throw new Error("Could not invert due to no input being a variable");
 }
 
-export function integratePowBase(
+export function integratePowBase(stack: SubstitutionStack, lhs: FormulaSource, rhs: FormulaSource) {
-    variable: Ref<DecimalSource>,
-    stack: SubstitutionStack,
-    lhs: FormulaSource,
-    rhs: FormulaSource
-) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        const x = lhs.getIntegralFormula(stack);
         return Formula.pow(rhs, x).div(Formula.ln(rhs));
     } else if (hasVariable(rhs)) {
-        const x = rhs.getIntegralFormula(variable, stack);
+        const x = rhs.getIntegralFormula(stack);
         const denominator = Formula.add(lhs, 1);
         return Formula.pow(x, denominator).div(denominator);
     }
@@ -365,14 +378,9 @@ export function invertRoot(value: DecimalSource, lhs: FormulaSource, rhs: Formul
     throw new Error("Could not invert due to no input being a variable");
 }
 
-export function integrateRoot(
+export function integrateRoot(stack: SubstitutionStack, lhs: FormulaSource, rhs: FormulaSource) {
-    variable: Ref<DecimalSource>,
-    stack: SubstitutionStack,
-    lhs: FormulaSource,
-    rhs: FormulaSource
-) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        const x = lhs.getIntegralFormula(stack);
         return Formula.pow(x, Formula.recip(rhs).add(1)).times(rhs).div(Formula.add(rhs, 1));
     }
     throw new Error("Could not integrate due to no input being a variable");
@@ -385,13 +393,9 @@ export function invertExp(value: DecimalSource, lhs: FormulaSource) {
     throw new Error("Could not invert due to no input being a variable");
 }
 
-export function integrateExp(
+export function integrateExp(stack: SubstitutionStack, lhs: FormulaSource) {
-    variable: Ref<DecimalSource>,
-    stack: SubstitutionStack,
-    lhs: FormulaSource
-) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        const x = lhs.getIntegralFormula(stack);
         return Formula.exp(x);
     }
     throw new Error("Could not integrate due to no input being a variable");
@@ -520,13 +524,9 @@ export function invertSin(value: DecimalSource, lhs: FormulaSource) {
     throw new Error("Could not invert due to no input being a variable");
 }
 
-export function integrateSin(
+export function integrateSin(stack: SubstitutionStack, lhs: FormulaSource) {
-    variable: Ref<DecimalSource>,
-    stack: SubstitutionStack,
-    lhs: FormulaSource
-) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        const x = lhs.getIntegralFormula(stack);
         return Formula.cos(x).neg();
     }
     throw new Error("Could not integrate due to no input being a variable");
@@ -539,13 +539,9 @@ export function invertCos(value: DecimalSource, lhs: FormulaSource) {
     throw new Error("Could not invert due to no input being a variable");
 }
 
-export function integrateCos(
+export function integrateCos(stack: SubstitutionStack, lhs: FormulaSource) {
-    variable: Ref<DecimalSource>,
-    stack: SubstitutionStack,
-    lhs: FormulaSource
-) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        const x = lhs.getIntegralFormula(stack);
         return Formula.sin(x);
     }
     throw new Error("Could not integrate due to no input being a variable");
@@ -558,13 +554,9 @@ export function invertTan(value: DecimalSource, lhs: FormulaSource) {
     throw new Error("Could not invert due to no input being a variable");
 }
 
-export function integrateTan(
+export function integrateTan(stack: SubstitutionStack, lhs: FormulaSource) {
-    variable: Ref<DecimalSource>,
-    stack: SubstitutionStack,
-    lhs: FormulaSource
-) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        const x = lhs.getIntegralFormula(stack);
         return Formula.cos(x).ln().neg();
     }
     throw new Error("Could not integrate due to no input being a variable");
@@ -577,13 +569,9 @@ export function invertAsin(value: DecimalSource, lhs: FormulaSource) {
     throw new Error("Could not invert due to no input being a variable");
 }
 
-export function integrateAsin(
+export function integrateAsin(stack: SubstitutionStack, lhs: FormulaSource) {
-    variable: Ref<DecimalSource>,
-    stack: SubstitutionStack,
-    lhs: FormulaSource
-) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        const x = lhs.getIntegralFormula(stack);
         return Formula.asin(x)
             .times(x)
             .add(Formula.sqrt(Formula.sub(1, Formula.pow(x, 2))));
@@ -598,13 +586,9 @@ export function invertAcos(value: DecimalSource, lhs: FormulaSource) {
     throw new Error("Could not invert due to no input being a variable");
 }
 
-export function integrateAcos(
+export function integrateAcos(stack: SubstitutionStack, lhs: FormulaSource) {
-    variable: Ref<DecimalSource>,
-    stack: SubstitutionStack,
-    lhs: FormulaSource
-) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        const x = lhs.getIntegralFormula(stack);
         return Formula.acos(x)
             .times(x)
             .sub(Formula.sqrt(Formula.sub(1, Formula.pow(x, 2))));
@@ -619,13 +603,9 @@ export function invertAtan(value: DecimalSource, lhs: FormulaSource) {
     throw new Error("Could not invert due to no input being a variable");
 }
 
-export function integrateAtan(
+export function integrateAtan(stack: SubstitutionStack, lhs: FormulaSource) {
-    variable: Ref<DecimalSource>,
-    stack: SubstitutionStack,
-    lhs: FormulaSource
-) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        const x = lhs.getIntegralFormula(stack);
         return Formula.atan(x)
             .times(x)
             .sub(Formula.ln(Formula.pow(x, 2).add(1)).div(2));
@@ -640,13 +620,9 @@ export function invertSinh(value: DecimalSource, lhs: FormulaSource) {
     throw new Error("Could not invert due to no input being a variable");
 }
 
-export function integrateSinh(
+export function integrateSinh(stack: SubstitutionStack, lhs: FormulaSource) {
-    variable: Ref<DecimalSource>,
-    stack: SubstitutionStack,
-    lhs: FormulaSource
-) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        const x = lhs.getIntegralFormula(stack);
         return Formula.cosh(x);
     }
     throw new Error("Could not integrate due to no input being a variable");
@@ -659,13 +635,9 @@ export function invertCosh(value: DecimalSource, lhs: FormulaSource) {
     throw new Error("Could not invert due to no input being a variable");
 }
 
-export function integrateCosh(
+export function integrateCosh(stack: SubstitutionStack, lhs: FormulaSource) {
-    variable: Ref<DecimalSource>,
-    stack: SubstitutionStack,
-    lhs: FormulaSource
-) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        const x = lhs.getIntegralFormula(stack);
         return Formula.sinh(x);
     }
     throw new Error("Could not integrate due to no input being a variable");
@@ -678,13 +650,9 @@ export function invertTanh(value: DecimalSource, lhs: FormulaSource) {
     throw new Error("Could not invert due to no input being a variable");
 }
 
-export function integrateTanh(
+export function integrateTanh(stack: SubstitutionStack, lhs: FormulaSource) {
-    variable: Ref<DecimalSource>,
-    stack: SubstitutionStack,
-    lhs: FormulaSource
-) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        const x = lhs.getIntegralFormula(stack);
         return Formula.cosh(x).ln();
     }
     throw new Error("Could not integrate due to no input being a variable");
@@ -697,13 +665,9 @@ export function invertAsinh(value: DecimalSource, lhs: FormulaSource) {
     throw new Error("Could not invert due to no input being a variable");
 }
 
-export function integrateAsinh(
+export function integrateAsinh(stack: SubstitutionStack, lhs: FormulaSource) {
-    variable: Ref<DecimalSource>,
-    stack: SubstitutionStack,
-    lhs: FormulaSource
-) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        const x = lhs.getIntegralFormula(stack);
         return Formula.asinh(x).times(x).sub(Formula.pow(x, 2).add(1).sqrt());
     }
     throw new Error("Could not integrate due to no input being a variable");
@@ -716,13 +680,9 @@ export function invertAcosh(value: DecimalSource, lhs: FormulaSource) {
     throw new Error("Could not invert due to no input being a variable");
 }
 
-export function integrateAcosh(
+export function integrateAcosh(stack: SubstitutionStack, lhs: FormulaSource) {
-    variable: Ref<DecimalSource>,
-    stack: SubstitutionStack,
-    lhs: FormulaSource
-) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        const x = lhs.getIntegralFormula(stack);
         return Formula.acosh(x)
             .times(x)
             .sub(Formula.add(x, 1).sqrt().times(Formula.sub(x, 1).sqrt()));
@@ -737,13 +697,9 @@ export function invertAtanh(value: DecimalSource, lhs: FormulaSource) {
     throw new Error("Could not invert due to no input being a variable");
 }
 
-export function integrateAtanh(
+export function integrateAtanh(stack: SubstitutionStack, lhs: FormulaSource) {
-    variable: Ref<DecimalSource>,
-    stack: SubstitutionStack,
-    lhs: FormulaSource
-) {
     if (hasVariable(lhs)) {
-        const x = lhs.getIntegralFormula(variable, stack);
+        const x = lhs.getIntegralFormula(stack);
         return Formula.atanh(x)
             .times(x)
             .add(Formula.sub(1, Formula.pow(x, 2)).ln().div(2));

src/game/formulas/types.d.ts

@@ -22,7 +22,6 @@ type EvaluateFunction<T> = (
 type InvertFunction<T> = (this: Formula<T>, value: DecimalSource, ...inputs: T) => DecimalSource;
 type IntegrateFunction<T> = (
     this: Formula<T>,
-    variable: Ref<DecimalSource>,
     stack: SubstitutionStack | undefined,
     ...inputs: T
 ) => GenericFormula;
@@ -43,7 +42,6 @@ type GeneralFormulaOptions<T extends [FormulaSource] | FormulaSource[]> = {
     integrate?: IntegrateFunction<T>;
     integrateInner?: IntegrateFunction<T>;
     applySubstitution?: SubstitutionFunction<T>;
-    hasVariable?: boolean;
 };
 type FormulaOptions<T extends [FormulaSource] | FormulaSource[]> =
     | VariableFormulaOptions
@@ -52,7 +50,7 @@ type FormulaOptions<T extends [FormulaSource] | FormulaSource[]> =
 
 type InternalFormulaProperties<T extends [FormulaSource] | FormulaSource[]> = {
     inputs: T;
-    internalHasVariable: boolean;
+    internalVariables: number;
     internalEvaluate?: EvaluateFunction<T>;
     internalInvert?: InvertFunction<T>;
     internalIntegrate?: IntegrateFunction<T>;

tests/game/formulas.test.ts

@@ -96,21 +96,21 @@ const invertibleIntegralZeroPramFunctionNames = [
     "sqr",
     "sqrt",
     "cube",
-    "cbrt"
+    "cbrt",
-] as const;
-const nonInvertibleIntegralZeroPramFunctionNames = [
-    ...nonIntegrableZeroParamFunctionNames,
     "neg",
     "exp",
     "sin",
     "cos",
     "tan",
+    "sinh",
+    "cosh",
+    "tanh"
+] as const;
+const nonInvertibleIntegralZeroPramFunctionNames = [
+    ...nonIntegrableZeroParamFunctionNames,
     "asin",
     "acos",
     "atan",
-    "sinh",
-    "cosh",
-    "tanh",
     "asinh",
     "acosh",
     "atanh"
@@ -493,8 +493,8 @@ describe("Integrating", () => {
 
     test("variable.evaluateIntegral() calculates correctly", () =>
         expect(variable.evaluateIntegral()).compare_tolerance(Decimal.pow(10, 2).div(2)));
-    test("evaluateIntegral(variable) overrides variable value", () =>
+    test("variable.evaluateIntegral(variable) overrides variable value", () =>
-        expect(variable.add(10).evaluateIntegral(20)).compare_tolerance(400));
+        expect(variable.evaluateIntegral(20)).compare_tolerance(Decimal.pow(20, 2).div(2)));
 
     describe("Integrable functions marked as such", () => {
         function checkFormula(formula: GenericFormula) {
@@ -668,32 +668,13 @@ describe("Inverting integrals", () => {
 
     test("Inverting integral of nested formulas", () => {
         const formula = Formula.add(variable, constant).times(constant).pow(2).times(30);
-        expect(formula.invertIntegral(formula.evaluateIntegral())).compare_tolerance(10);
+        expect(formula.invertIntegral(formula.evaluateIntegral())).compare_tolerance(10, 0.01);
     });
 
     test("Inverting integral of nested complex formulas", () => {
         const formula = Formula.pow(1.05, variable).times(100).pow(0.5);
         expect(() => formula.invertIntegral(100)).toThrow();
     });
-
-    describe("Inverting integral pass-throughs", () => {
-        test("max", () =>
-            expect(Formula.max(variable, constant).invertIntegral(10)).compare_tolerance(10));
-        test("min", () =>
-            expect(Formula.min(variable, constant).invertIntegral(10)).compare_tolerance(10));
-        test("minabs", () =>
-            expect(Formula.minabs(variable, constant).invertIntegral(10)).compare_tolerance(10));
-        test("maxabs", () =>
-            expect(Formula.maxabs(variable, constant).invertIntegral(10)).compare_tolerance(10));
-        test("clampMax", () =>
-            expect(Formula.clampMax(variable, constant).invertIntegral(10)).compare_tolerance(10));
-        test("clampMin", () =>
-            expect(Formula.clampMin(variable, constant).invertIntegral(10)).compare_tolerance(10));
-        test("clamp", () =>
-            expect(
-                Formula.clamp(variable, constant, constant).invertIntegral(10)
-            ).compare_tolerance(10));
-    });
 });
 
 describe("Step-wise", () => {
@@ -914,8 +895,7 @@ describe("Custom Formulas", () => {
                 new Formula({
                     inputs: [],
                     evaluate: () => 6,
-                    invert: value => value,
+                    invert: value => value
-                    hasVariable: true
                 }).invert(10)
             ).toThrow());
         test("One input inverts correctly", () =>
@@ -923,8 +903,7 @@ describe("Custom Formulas", () => {
                 new Formula({
                     inputs: [variable],
                     evaluate: () => 10,
-                    invert: (value, v1) => v1.evaluate(),
+                    invert: (value, v1) => v1.evaluate()
-                    hasVariable: true
                 }).invert(10)
             ).compare_tolerance(1));
         test("Two inputs inverts correctly", () =>
@@ -932,37 +911,36 @@ describe("Custom Formulas", () => {
                 new Formula({
                     inputs: [variable, 2],
                     evaluate: () => 10,
-                    invert: (value, v1, v2) => v2,
+                    invert: (value, v1, v2) => v2
-                    hasVariable: true
                 }).invert(10)
             ).compare_tolerance(2));
     });
 
     describe("Formula with integrate", () => {
-        test("Zero input integrates correctly", () =>
+        test("Zero input cannot integrate", () =>
-            expect(
+            expect(() =>
                 new Formula({
                     inputs: [],
-                    evaluate: () => 10,
+                    evaluate: () => 0,
-                    integrate: variable => variable
+                    integrate: stack => variable
                 }).evaluateIntegral()
-            ).compare_tolerance(20));
+            ).toThrow());
         test("One input integrates correctly", () =>
             expect(
                 new Formula({
                     inputs: [variable],
-                    evaluate: () => 10,
+                    evaluate: v1 => Decimal.add(v1, 19.5),
-                    integrate: (variable, stack, v1) => Formula.add(variable, v1)
+                    integrate: (stack, v1) => Formula.add(v1, 10)
                 }).evaluateIntegral()
             ).compare_tolerance(20));
         test("Two inputs integrates correctly", () =>
             expect(
                 new Formula({
-                    inputs: [variable, 2],
+                    inputs: [variable, 10],
-                    evaluate: (v1, v2) => 10,
+                    evaluate: v1 => Decimal.add(v1, 19.5),
-                    integrate: (variable, v1, v2) => variable
+                    integrate: (stack, v1, v2) => Formula.add(v1, v2)
                 }).evaluateIntegral()
-            ).compare_tolerance(3));
+            ).compare_tolerance(20));
     });
 
     describe("Formula with invertIntegral", () => {
@@ -970,29 +948,26 @@ describe("Custom Formulas", () => {
             expect(() =>
                 new Formula({
                     inputs: [],
-                    evaluate: () => 10,
+                    evaluate: () => 0,
-                    integrate: variable => variable,
+                    integrate: stack => variable
-                    hasVariable: true
+                }).invertIntegral(20)
-                }).invertIntegral(8)
             ).toThrow());
         test("One input inverts integral correctly", () =>
             expect(
                 new Formula({
                     inputs: [variable],
-                    evaluate: () => 10,
+                    evaluate: v1 => Decimal.add(v1, 19.5),
-                    integrate: (variable, stack, v1) => variable,
+                    integrate: (stack, v1) => Formula.add(v1, 10)
-                    hasVariable: true
+                }).invertIntegral(20)
-                }).invertIntegral(8)
+            ).compare_tolerance(10));
-            ).compare_tolerance(1));
         test("Two inputs inverts integral correctly", () =>
             expect(
                 new Formula({
-                    inputs: [variable, 2],
+                    inputs: [variable, 10],
-                    evaluate: (v1, v2) => 10,
+                    evaluate: v1 => Decimal.add(v1, 19.5),
-                    integrate: (variable, v1, v2) => variable,
+                    integrate: (stack, v1, v2) => Formula.add(v1, v2)
-                    hasVariable: true
+                }).invertIntegral(20)
-                }).invertIntegral(8)
+            ).compare_tolerance(10));
-            ).compare_tolerance(1));
     });
 
     describe.todo("Formula as input");

tests/utils.ts

@@ -2,7 +2,7 @@ import Decimal, { DecimalSource, format } from "util/bignum";
 import { expect } from "vitest";
 
 interface CustomMatchers<R = unknown> {
-    compare_tolerance(expected: DecimalSource): R;
+    compare_tolerance(expected: DecimalSource, tolerance?: number): R;
 }
 
 declare global {
@@ -16,7 +16,7 @@ declare global {
 }
 
 expect.extend({
-    compare_tolerance(received: DecimalSource, expected: DecimalSource) {
+    compare_tolerance(received: DecimalSource, expected: DecimalSource, tolerance?: number) {
         const { isNot } = this;
         let pass = false;
         if (!Decimal.isFinite(expected)) {
@@ -24,7 +24,7 @@ expect.extend({
         } else if (Decimal.isNaN(expected)) {
             pass = Decimal.isNaN(received);
         } else {
-            pass = Decimal.eq_tolerance(received, expected);
+            pass = Decimal.eq_tolerance(received, expected, tolerance);
         }
         return {
             // do not alter your "pass" based on isNot. Vitest does it for you