src/bigint/divide.mjs

@@ -3,7 +3,7 @@ export * from './Types/bigint.mjs'
 
 export const divide = {
    'bigint,bigint': ({config, 'quotient(bigint,bigint)': quot}) => {
-      if (config.predictable) return quot
+      if (config.predictable) return Returns('bigint', (n,d) => quot(n,d))
       return Returns('bigint|undefined', (n, d) => {
          const q = n/d
          if (q * d == n) return q

src/complex/Types/Complex.mjs

@@ -6,7 +6,7 @@ const Complex = new PocomathInstance('Complex')
 Complex.installType('Complex', {
    test: z => z && typeof z === 'object' && 're' in z && 'im' in z
 })
-// Now the template type: Complex numbers are actually always homeogeneous
+// Now the template type: Complex numbers are actually always homogeneous
 // in their component types.
 Complex.installType('Complex<T>', {
    infer: ({typeOf, joinTypes}) => z => joinTypes([typeOf(z.re), typeOf(z.im)]),

src/complex/abs.mjs

@@ -1,10 +1,25 @@
+import {Returns, returnTypeOf} from '../core/Returns.mjs'
 export * from './Types/Complex.mjs'
 
 export const abs = {
    'Complex<T>': ({
-      sqrt, // Calculation of the type needed in the square root (the
+      sqrt, // Unfortunately no notation yet for the needed signature
-      // underlying numeric type of T, whatever T is, is beyond Pocomath's
+      'absquare(T)': baseabsq,
-      // (current) template abilities, so punt and just do full resolution
       'absquare(Complex<T>)': absq
-   }) => z => sqrt(absq(z))
+   }) => {
+      const pm = sqrt.fromInstance
+      if (typeof pm === 'undefined') {
+         // Just checking for the dependencies, return value is irrelevant
+         return undefined
+      }
+      const midType = returnTypeOf(baseabsq)
+      const sqrtImp = pm.resolve('sqrt', midType, sqrt)
+      let retType = returnTypeOf(sqrtImp)
+      if (retType.includes('|')) {
+         // This is a bit of a hack, as it relies on all implementations of
+         // sqrt returning the "typical" return type as the first option
+         retType = retType.split('|',1)[0]
+      }
+      return Returns(retType, z => sqrtImp(absq(z)))
+   }
 }

src/complex/absquare.mjs

@@ -1,9 +1,31 @@
+import {Returns, returnTypeOf} from '../core/Returns.mjs'
 export * from './Types/Complex.mjs'
 
 export const absquare = {
    'Complex<T>': ({
-      add, // Calculation of exact type needed in add (underlying numeric of T)
+      add, // no easy way to write the needed signature; if T is number
-      // is (currently) too involved for Pocomath
+           // it is number,number; but if T is Complex<bigint>, it is just
+           // bigint,bigint. So unfortunately we depend on all of add, and
+           // we extract the needed implementation below.
       'self(T)': absq
-   }) => z => add(absq(z.re), absq(z.im))
+   }) => {
+      const pm = add.fromInstance
+      if (typeof pm === 'undefined') {
+         // Just checking the dependencies, return value irrelevant
+         return undefined
       }
+      const midType = returnTypeOf(absq)
+      const addImp = pm.resolve('add', `${midType},${midType}`, add)
+      return Returns(
+         returnTypeOf(addImp), z => addImp(absq(z.re), absq(z.im)))
+   }
+}
+
+/* We could imagine notations that Pocomath could support that would simplify
+ * the above, maybe something like
+ * 'Complex<T>': ({
+ *    'self(T): U': absq,
+ *    'add(U,U):V': plus,
+ *    V
+ *  }) => Returns(V, z => plus(absq(z.re), absq(z.im)))
+ */

src/complex/multiply.mjs

@@ -1,15 +1,18 @@
+import Returns from '../core/Returns.mjs'
 export * from './Types/Complex.mjs'
 
 export const multiply = {
    'Complex<T>,Complex<T>': ({
+      T,
       'complex(T,T)': cplx,
       'add(T,T)': plus,
       'subtract(T,T)': sub,
       'self(T,T)': me,
       'conjugate(T)': conj // makes quaternion multiplication work
-   }) => (w,z) => {
+   }) => Returns(
-      return cplx(
+      `Complex<${T}>`,
+      (w,z) => cplx(
          sub(me(w.re, z.re), me(conj(w.im), z.im)),
          plus(me(conj(w.re), z.im), me(w.im, z.re)))
-   }
+   )
 }

src/complex/sqrt.mjs

@@ -1,3 +1,4 @@
+import {Returns, returnTypeOf} from '../core/Returns.mjs'
 export * from './Types/Complex.mjs'
 
 export const sqrt = {
@@ -12,29 +13,41 @@ export const sqrt = {
       'multiply(T,T)': mult,
       'self(T)': me,
       'divide(T,T)': div,
-      'abs(Complex<T>)': absC,
+      'absquare(Complex<T>)': absqC,
       'subtract(T,T)': sub
    }) => {
+      let baseReturns = returnTypeOf(me)
+      if (baseReturns.includes('|')) {
+         // Bit of a hack, because it is relying on other implementations
+         // to list the "typical" value of sqrt first
+         baseReturns = baseReturns.split('|', 1)[0]
+      }
+
       if (config.predictable) {
-         return z => {
+         return Returns(`Complex<${baseReturns}>`, z => {
             const reOne = uno(z.re)
             if (isZ(z.im) && sgn(z.re) === reOne) return cplxU(me(z.re))
             const reTwo = plus(reOne, reOne)
+            const myabs = me(absqC(z))
             return cplxB(
-               mult(sgn(z.im), me(div(plus(absC(z),z.re), reTwo))),
+               mult(sgn(z.im), me(div(plus(myabs, z.re), reTwo))),
-               me(div(sub(absC(z),z.re), reTwo))
+               me(div(sub(myabs, z.re), reTwo))
             )
+         })
       }
-      }
+
-      return z => {
+      return Returns(
+         `Complex<${baseReturns}>|${baseReturns}|undefined`,
+         z => {
             const reOne = uno(z.re)
             if (isZ(z.im) && sgn(z.re) === reOne) return me(z.re)
             const reTwo = plus(reOne, reOne)
-         const reSqrt = me(div(plus(absC(z),z.re), reTwo))
+            const myabs = me(absqC(z))
-         const imSqrt = me(div(sub(absC(z),z.re), reTwo))
+            const reSqrt = me(div(plus(myabs, z.re), reTwo))
+            const imSqrt = me(div(sub(myabs, z.re), reTwo))
             if (reSqrt === undefined || imSqrt === undefined) return undefined
             return cplxB(mult(sgn(z.im), reSqrt), imSqrt)
          }
+      )
    }
 }
-

src/core/PocomathInstance.mjs

@@ -38,6 +38,7 @@ export default class PocomathInstance {
       'joinTypes',
       'name',
       'returnTypeOf',
+      'resolve',
       'self',
       'subtypesOf',
       'supertypesOf',
@@ -49,7 +50,8 @@ export default class PocomathInstance {
 
    constructor(name) {
       this.name = name
-      this._imps = {}
+      this._imps = {} // Pocomath implementations, with dependencies
+      this._TFimps = {} // typed-function implementations, dependencies resolved
       this._affects = {}
       this._typed = typed.create()
       this._typed.clear()
@@ -220,7 +222,6 @@ export default class PocomathInstance {
       if (details) {
          return returnTypeOf(details.fn, signature, this)
       }
-      console.log('Checking return type of', operation)
       return returnTypeOf(this[operation], signature, this)
    })
 
@@ -564,8 +565,28 @@ export default class PocomathInstance {
                      `Conflicting definitions of ${signature} for ${name}`)
                }
             } else {
-               // Must avoid aliasing into another instance:
+               /* Check if it's an ordinary non-template signature */
-               opImps[signature] = {uses: behavior.uses, does: behavior.does}
+               let explicit = true
+               for (const type of typesOfSignature(signature)) {
+                  for (const word of type.split(/[<>:\s]/)) {
+                     if (this._templateParam(word)) {
+                        explicit = false
+                        break
+                     }
+                  }
+                  if (!explicit) break
+               }
+               opImps[signature] = {
+                  explicit,
+                  uses: behavior.uses,
+                  does: behavior.does
+               }
+               if (explicit) {
+                  opImps[signature].resolved = false
+               } else {
+                  opImps[signature].hasInstantiations = {}
+                  opImps[signature].needsInstantiations = new Set()
+               }
                for (const dep of behavior.uses) {
                   const depname = dep.split('(', 1)[0]
                   if (depname === 'self' || this._templateParam(depname)) {
@@ -604,11 +625,40 @@ export default class PocomathInstance {
     * Reset an operation to require creation of typed-function,
     * and if it has no implementations so far, set them up.
     */
-   _invalidate(name) {
+   _invalidate(name, reason) {
-      if (this._invalid.has(name)) return
       if (!(name in this._imps)) {
          this._imps[name] = {}
       }
+      if (reason) {
+         // Make sure no TF imps that depend on reason remain:
+         for (const [signature, behavior] of Object.entries(this._imps[name])) {
+            let invalidated = false
+            if (reason.charAt(0) === ':') {
+               const badType = reason.slice(1)
+               if (signature.includes(badType)) invalidated = true
+            } else {
+               for (const dep of behavior.uses) {
+                  if (dep.includes(reason)) {
+                     invalidated = true
+                     break
+                  }
+               }
+            }
+            if (invalidated) {
+               if (behavior.explicit) {
+                  if (behavior.resolved) delete this._TFimps[signature]
+                  behavior.resolved = false
+               } else {
+                  for (const fullSig
+                       of Object.values(behavior.hasInstantiations)) {
+                     delete this._TFimps[fullSig]
+                  }
+                  behavior.hasInstantiations = {}
+               }
+            }
+         }
+      }
+      if (this._invalid.has(name)) return
       this._invalid.add(name)
       this._invalidateDependents(name)
       const self = this
@@ -628,7 +678,7 @@ export default class PocomathInstance {
    _invalidateDependents(name) {
       if (name in this._affects) {
          for (const ancestor of this._affects[name]) {
-            this._invalidate(ancestor)
+            this._invalidate(ancestor, name)
          }
       }
    }
@@ -642,6 +692,10 @@ export default class PocomathInstance {
       if (!imps) {
          throw new SyntaxError(`No implementations for ${name}`)
       }
+      if (!(this._TFimps[name])) {
+         this._TFimps[name] = {}
+      }
+      const tf_imps = this._TFimps[name]
       /* Collect the entries we know the types for */
       const usableEntries = []
       for (const entry of Object.entries(imps)) {
@@ -664,20 +718,13 @@ export default class PocomathInstance {
        * in the midst of being reassembled
        */
       Object.defineProperty(this, name, {configurable: true, value: 'limbo'})
-      const tf_imps = {}
       for (const [rawSignature, behavior] of usableEntries) {
-         /* Check if it's an ordinary non-template signature */
+         if (behavior.explicit) {
-         let explicit = true
+            if (!(behavior.resolved)) {
-         for (const type of typesOfSignature(rawSignature)) {
-            for (const word of type.split(/[<>:\s]/)) {
-               if (this._templateParam(word)) {
-                  explicit = false
-                  break
-               }
-            }
-         }
-         if (explicit) {
                this._addTFimplementation(tf_imps, rawSignature, behavior)
+               tf_imps[rawSignature]._pocoSignature = rawSignature
+               behavior.resolved = true
+            }
             continue
          }
          /* It's a template, have to instantiate */
@@ -693,12 +740,9 @@ export default class PocomathInstance {
             }
          }
          /* First, add the known instantiations, gathering all types needed */
-         if (!('instantiations' in behavior)) {
+         if (ubType) behavior.needsInstantiations.add(ubType)
-            behavior.instantiations = new Set()
-            if (ubType) behavior.instantiations.add(ubType)
-         }
          let instantiationSet = new Set()
-         for (const instType of behavior.instantiations) {
+         for (const instType of behavior.needsInstantiations) {
             instantiationSet.add(instType)
             const otherTypes =
                   ubType ? this.subtypesOf(instType) : this._priorTypes[instType]
@@ -710,6 +754,7 @@ export default class PocomathInstance {
          for (const instType of instantiationSet) {
             if (!(instType in this.Types)) continue
             if (this.Types[instType] === anySpec) continue
+            if (instType in behavior.hasInstantiations) continue
             const signature =
                   substituteInSignature(rawSignature, theTemplateParam, instType)
             /* Don't override an explicit implementation: */
@@ -742,10 +787,14 @@ export default class PocomathInstance {
             }
             this._addTFimplementation(
                tf_imps, signature, {uses, does: patch})
+            tf_imps[signature]._pocoSignature = rawSignature
+            tf_imps[signature]._pocoInstance = instType
+            behavior.hasInstantiations[instType] = signature
          }
          /* Now add the catchall signature */
          /* (Not needed if if it's a bounded template) */
          if (ubType) continue
+         if ('_catchall_' in behavior.hasInstantiations) continue
          let templateCall = `<${theTemplateParam}>`
          /* Relying here that the base of 'Foo<T>' is 'Foo': */
          let baseSignature = rawSignature.replaceAll(templateCall, '')
@@ -886,11 +935,11 @@ export default class PocomathInstance {
             /* Arrange that the desired instantiation will be there next
              * time so we don't have to go through that again for this type
              */
-            refs[theTemplateParam] = instantiateFor
+            behavior.needsInstantiations.add(instantiateFor)
-            behavior.instantiations.add(instantiateFor)
             self._invalidate(name)
             // And update refs because we now know the type we're instantiating
             // for:
+            refs[theTemplateParam] = instantiateFor
             const innerRefs = {}
             for (const dep in simplifiedUses) {
                const simplifiedDep = simplifiedUses[dep]
@@ -903,7 +952,7 @@ export default class PocomathInstance {
                         needsig, theTemplateParam, instantiateFor)
                      let resname = simplifiedDep
                      if (resname == 'self') resname = name
-                     innerRefs[dep] = self._pocoresolve(
+                     innerRefs[dep] = self.resolve(
                         resname, subsig, refs[simplifiedDep])
                   } else {
                      innerRefs[dep] = refs[simplifiedDep]
@@ -913,14 +962,16 @@ export default class PocomathInstance {
             // Finally ready to make the call.
             const implementation = behavior.does(innerRefs)
             // We can access return type information here
-            // And in particular, if it's a template, we should try to
+            // And in particular, if it might be a template, we should try to
             // instantiate it:
             const returnType = returnTypeOf(implementation, wantSig, self)
-            const instantiated = self._maybeInstantiate(returnType)
+            for (const possibility of returnType.split('|')) {
+               const instantiated = self._maybeInstantiate(possibility)
                if (instantiated) {
                   const tempBase = instantiated.split('<',1)[0]
                   self._invalidateDependents(':' + tempBase)
                }
+            }
             return implementation(...args)
          }
          Object.defineProperty(patch, 'name', {value: `${name}(${signature})`})
@@ -932,6 +983,7 @@ export default class PocomathInstance {
          const outerUses = new Set(Object.values(simplifiedUses))
          this._addTFimplementation(
             tf_imps, signature, {uses: outerUses, does: patch})
+         behavior.hasInstantiations._catchall_ = rawSignature
       }
       this._correctPartialSelfRefs(name, tf_imps)
       // Make sure we have all of the needed (template) types; and if they
@@ -946,9 +998,17 @@ export default class PocomathInstance {
          }
       }
       for (const badSig of badSigs) {
+         const imp = tf_imps[badSig]
          delete tf_imps[badSig]
+         const fromBehavior = this._imps[name][imp._pocoSignature]
+         if (fromBehavior.explicit) {
+            fromBehavior.resolved = false
+         } else {
+            delete fromBehavior.hasInstantiations[imp._pocoInstance]
+         }
       }
       const tf = this._typed(name, tf_imps)
+      Object.defineProperty(tf, 'fromInstance', {value: this})
       Object.defineProperty(this, name, {configurable: true, value: tf})
       return tf
    }
@@ -1006,6 +1066,12 @@ export default class PocomathInstance {
          }
          if (func === 'self') {
             if (needsig) {
+               /* Maybe we can resolve the self reference without troubling
+                * typed-function:
+                */
+               if (needsig in imps && typeof imps[needsig] == 'function') {
+                  refs[dep] = imps[needsig]
+               } else {
                   if (full_self_referential) {
                      throw new SyntaxError(
                         'typed-function does not support mixed full and '
@@ -1014,6 +1080,7 @@ export default class PocomathInstance {
                   if (subsetOfKeys(typesOfSignature(needsig), this.Types)) {
                      part_self_references.push(needsig)
                   }
+               }
             } else {
                if (part_self_references.length) {
                   throw new SyntaxError(
@@ -1024,19 +1091,41 @@ export default class PocomathInstance {
             }
          } else {
             if (this[func] === 'limbo') {
-               /* We are in the midst of bundling func, so have to use
+               /* We are in the midst of bundling func */
+               let fallback =  true
+               /* So the first thing we can do is try the tf_imps we are
+                * accumulating:
+                */
+               if (needsig) {
+                  const tempTF = this._typed('dummy_' + func, this._TFimps[func])
+                  let result = undefined
+                  try {
+                     result = this._typed.find(tempTF, needsig, {exact: true})
+                  } catch {}
+                  if (result) {
+                     refs[dep] = result
+                     fallback = false
+                  }
+               }
+               if (fallback) {
+                  /* Either we need the whole function or the signature
+                   * we need is not available yet, so we have to use
                    * an indirect reference to func. And given that, there's
                    * really no helpful way to extract a specific signature
                    */
                   const self = this
-               refs[dep] = function () { // is this the most efficient?
+                  const redirect = function () { // is this the most efficient?
                      return self[func].apply(this, arguments)
                   }
+                  Object.defineProperty(redirect, 'name', {value: func})
+                  Object.defineProperty(redirect, 'fromInstance', {value: this})
+                  refs[dep] = redirect
+               }
             } else {
                // can bundle up func, and grab its signature if need be
                let destination = this[func]
                if (destination && needsig) {
-                  destination = this._pocoresolve(func, needsig)
+                  destination = this.resolve(func, needsig)
                }
                refs[dep] = destination
             }
@@ -1076,7 +1165,8 @@ export default class PocomathInstance {
    _correctPartialSelfRefs(name, imps) {
       for (const aSignature in imps) {
          if (!(imps[aSignature].deferred)) continue
-         const part_self_references = imps[aSignature].psr
+         const deferral = imps[aSignature]
+         const part_self_references = deferral.psr
          const corrected_self_references = []
          for (const neededSig of part_self_references) {
             // Have to find a match for neededSig among the other signatures
@@ -1098,8 +1188,8 @@ export default class PocomathInstance {
                      + `${name}(${neededSig})`)
             }
          }
-         const refs = imps[aSignature].builtRefs
+         const refs = deferral.builtRefs
-         const does = imps[aSignature].sigDoes
+         const does = deferral.sigDoes
          imps[aSignature] = this._typed.referTo(
             ...corrected_self_references, (...impls) => {
                for (let i = 0; i < part_self_references.length; ++i) {
@@ -1110,6 +1200,8 @@ export default class PocomathInstance {
                return implementation
             }
          )
+         imps[aSignature]._pocoSignature = deferral._pocoSignature
+         imps[aSignature]._pocoInstance = deferral._pocoInstance
       }
    }
 
@@ -1282,16 +1374,30 @@ export default class PocomathInstance {
          typedFunction = this[name]
       }
       let result = undefined
-      if (!this._typed.isTypedFunction(typedFunction)) {
+      const haveTF = this._typed.isTypedFunction(typedFunction)
-         return result
+      if (haveTF) {
-      }
          try {
             result = this._typed.findSignature(typedFunction, sig, {exact: true})
          } catch {
          }
-      if (result) return result
+      }
+      if (result || !(this._imps[name])) return result
       const foundsig = this._findSubtypeImpl(name, this._imps[name], sig)
-      if (foundsig) return this._typed.findSignature(typedFunction, foundsig)
+      if (foundsig) {
+         if (haveTF) {
+            return this._typed.findSignature(typedFunction, foundsig)
+         }
+         // We have an implementation but not a typed function. Do the best
+         // we can:
+         const foundImpl = this._imps[name][foundsig]
+         const needs = {}
+         for (const dep of foundImpl.uses) {
+            const [base, sig] = dep.split('()')
+            needs[dep] = this.resolve(base, sig)
+         }
+         const pseudoImpl = foundImpl.does(needs)
+         return {fn: pseudoImpl, implementation: pseudoImpl}
+      }
       const wantTypes = typeListOfSignature(sig)
       for (const [implSig, details]
            of typedFunction._typedFunctionData.signatureMap) {
@@ -1314,7 +1420,12 @@ export default class PocomathInstance {
       return result
    }
 
-   _pocoresolve(name, sig, typedFunction) {
+   /* Returns a function that implements the operation with the given name
+    * when called with the given signature. The optional third argument is
+    * the typed function that provides the operation name, which can be
+    * passed in for efficiency if it is already available.
+    */
+   resolve = Returns('function', function (name, sig, typedFunction) {
       if (!this._typed.isTypedFunction(typedFunction)) {
          typedFunction = this[name]
       }
@@ -1323,6 +1434,6 @@ export default class PocomathInstance {
       // total punt, revert to typed-function resolution on every call;
       // hopefully this happens rarely:
       return typedFunction
-   }
+   })
 
 }

src/generic/absquare.mjs

@@ -1,6 +1,9 @@
+import Returns from '../core/Returns.mjs'
+
 export const absquare = {
    T: ({
+      T,
       'square(T)': sq,
       'abs(T)': abval
-   }) => t => sq(abval(t))
+   }) => Returns(T, t => sq(abval(t)))
 }

src/generic/square.mjs

@@ -1,3 +1,6 @@
+import {Returns, returnTypeOf} from '../core/Returns.mjs'
+
 export const square = {
-   T: ({'multiply(T,T)': multT}) => x => multT(x,x)
+   T: ({'multiply(T,T)': multT}) => Returns(
+      returnTypeOf(multT), x => multT(x,x))
 }

src/number/absquare.mjs

@@ -1,6 +1,7 @@
+import Returns from '../core/Returns.mjs'
 export * from './Types/number.mjs'
 
 /* Absolute value squared */
 export const absquare = {
-    number: ({'square(number)': sqn}) => n => sqn(n)
+    'T:number': ({T, 'square(T)': sqn}) => Returns(T, n => sqn(n))
 }

src/number/multiply.mjs

@@ -1,3 +1,5 @@
+import Returns from '../core/Returns.mjs'
+
 export * from './Types/number.mjs'
 
-export const multiply = {'number,number': () => (m,n) => m*n}
+export const multiply = {'T:number,T': ({T}) => Returns(T, (m,n) => m*n)}

src/number/sqrt.mjs

@@ -1,3 +1,4 @@
+import Returns from '../core/Returns.mjs'
 export * from './Types/number.mjs'
 
 export const sqrt = {
@@ -6,12 +7,12 @@ export const sqrt = {
       'complex(number,number)': cplx,
       'negate(number)': neg}) => {
          if (config.predictable || !cplx) {
-         return n => isNaN(n) ? NaN : Math.sqrt(n)
+            return Returns('number', n => isNaN(n) ? NaN : Math.sqrt(n))
          }
-      return n => {
+         return Returns('number|Complex<number>', n => {
             if (isNaN(n)) return NaN
             if (n >= 0) return Math.sqrt(n)
             return cplx(0, Math.sqrt(neg(n)))
-      }
+         })
       }
 }

test/complex/_all.mjs

@@ -83,6 +83,7 @@ describe('complex', () => {
       assert.deepStrictEqual(
          math.multiply(q0, math.quaternion(2, 1, 0.1, 0.1)),
          math.quaternion(1.9, 1.1, 2.1, -0.9))
+      math.absquare(math.complex(1.25, 2.5)) //HACK: need absquare(Complex<number>)
       assert.strictEqual(math.abs(q0), Math.sqrt(2))
       assert.strictEqual(math.abs(q1), Math.sqrt(33)/4)
    })