diff --git a/packages/dual-algebra/package.json b/packages/dual-algebra/package.json
index 4692824e4e..b3a56abfff 100644
--- a/packages/dual-algebra/package.json
+++ b/packages/dual-algebra/package.json
@@ -33,6 +33,7 @@
     "devDependencies": {
         "@istanbuljs/nyc-config-typescript": "^1.0.1",
         "@microsoft/api-extractor": "^7.9.11",
+        "@thi.ng/api": "^6.12.3",
         "@types/mocha": "^8.0.3",
         "@types/node": "^14.6.1",
         "mocha": "^8.1.2",
@@ -41,9 +42,6 @@
         "typedoc": "^0.18.0",
         "typescript": "^4.0.2"
     },
-    "dependencies": {
-        "@thi.ng/api": "^6.12.3"
-    },
     "files": [
         "*.js",
         "*.d.ts",
diff --git a/packages/dual-algebra/src/api.ts b/packages/dual-algebra/src/api.ts
index 9290b89d3b..c31ffbefe3 100644
--- a/packages/dual-algebra/src/api.ts
+++ b/packages/dual-algebra/src/api.ts
@@ -1,9 +1,26 @@
-import type { Fn, Fn2, Fn3, Fn4, NumericArray } from "@thi.ng/api";
+import type {
+    Fn,
+    Fn2,
+    FnU2,
+    FnU3,
+    FnU4,
+    FnU5,
+    FnU6,
+    NumericArray,
+} from "@thi.ng/api";
 
 export type Dual = NumericArray;
 
 export type Op1 = Fn<Dual, Dual>;
+
 export type Op1N = Fn2<Dual, number, Dual>;
-export type Op2 = Fn2<Dual, Dual, Dual>;
-export type Op3 = Fn3<Dual, Dual, Dual, Dual>;
-export type Op4 = Fn4<Dual, Dual, Dual, Dual, Dual>;
+
+export type Op2 = FnU2<Dual>;
+
+export type Op3 = FnU3<Dual>;
+
+export type Op4 = FnU4<Dual>;
+
+export type Op5 = FnU5<Dual>;
+
+export type Op6 = FnU6<Dual>;
diff --git a/packages/dual-algebra/src/ops.ts b/packages/dual-algebra/src/ops.ts
index 0fa457127d..4a6bcb6f20 100644
--- a/packages/dual-algebra/src/ops.ts
+++ b/packages/dual-algebra/src/ops.ts
@@ -1,4 +1,4 @@
-import { Op1, Op1N, Op2, Op3, Op4 } from "./api";
+import type { Op1, Op1N, Op2, Op3, Op4 } from "./api";
 
 /**
  * Creates a multivariate dual number.
@@ -8,7 +8,7 @@ import { Op1, Op1N, Op2, Op3, Op4 } from "./api";
  * @param i - variable index (0 < i <= n)
  */
 export const dual = (real: number, n = 1, i = 0) => {
-    const out = new Array(n + 1).fill(0);
+    const out = new Array(n + 1).fill(0, 1);
     out[0] = real;
     i > 0 && (out[i] = 1);
     return out;
@@ -77,7 +77,7 @@ export const mul = defOp<Op2>(
         const ar = a[0];
         const br = b[0];
         const out = [ar * br];
-        for (let i = 1, n = a.length; i < n; i++) {
+        for (let i = a.length; --i >= 1; ) {
             out[i] = ar * b[i] + a[i] * br;
         }
         return out;
@@ -91,7 +91,7 @@ export const div = defOp<Op2>(
         const br = b[0];
         const ibr = 1 / (br * br);
         const out = [ar / br];
-        for (let i = 1, n = a.length; i < n; i++) {
+        for (let i = a.length; --i >= 1; ) {
             out[i] = (a[i] * br - ar * b[i]) * ibr;
         }
         return out;
@@ -103,7 +103,7 @@ export const abs = defOp<Op1>(
     (a) => {
         const s = Math.sign(a[0]);
         const out = [Math.abs(a[0])];
-        for (let i = 1, n = a.length; i < n; i++) {
+        for (let i = a.length; --i >= 1; ) {
             out[i] = s * a[i];
         }
         return out;
@@ -119,7 +119,7 @@ export const sqrt = defOp<Op1>(
         const s = Math.sqrt(a[0]);
         const si = 0.5 / s;
         const out = [s];
-        for (let i = 1, n = a.length; i < n; i++) {
+        for (let i = a.length; --i >= 1; ) {
             out[i] = si * a[i];
         }
         return out;
@@ -134,7 +134,7 @@ export const exp = defOp<Op1>(
     (a) => {
         const ar = Math.exp(a[0]);
         const out = [ar];
-        for (let i = 1, n = a.length; i < n; i++) {
+        for (let i = a.length; --i >= 1; ) {
             out[i] = ar * a[i];
         }
         return out;
@@ -147,7 +147,7 @@ export const log = defOp<Op1>(
         const ar = Math.log(a[0]);
         const iar = 1 / ar;
         const out = [ar];
-        for (let i = 1, n = a.length; i < n; i++) {
+        for (let i = a.length; --i >= 1; ) {
             out[i] = iar * a[i];
         }
         return out;
@@ -159,7 +159,7 @@ export const pow = defOp<Op1N>(
     (a, k) => {
         const f = k * a[0] ** (k - 1);
         const out = [a[0] ** k];
-        for (let i = 1, n = a.length; i < n; i++) {
+        for (let i = a.length; --i >= 1; ) {
             out[i] = f * a[i];
         }
         return out;
@@ -171,7 +171,7 @@ export const sin = defOp<Op1>(
     (a) => {
         const c = Math.cos(a[0]);
         const out = [Math.sin(a[0])];
-        for (let i = 1, n = a.length; i < n; i++) {
+        for (let i = a.length; --i >= 1; ) {
             out[i] = c * a[i];
         }
         return out;
@@ -183,7 +183,7 @@ export const cos = defOp<Op1>(
     (a) => {
         const s = -Math.sin(a[0]);
         const out = [Math.cos(a[0])];
-        for (let i = 1, n = a.length; i < n; i++) {
+        for (let i = a.length; --i >= 1; ) {
             out[i] = s * a[i];
         }
         return out;
@@ -199,7 +199,7 @@ export const tan = defOp<Op1>(
         const c = Math.cos(a[0]);
         const ic = 1 / (c * c);
         const out = [Math.tan(a[0])];
-        for (let i = 1, n = a.length; i < n; i++) {
+        for (let i = a.length; --i >= 1; ) {
             out[i] = ic * a[i];
         }
         return out;
@@ -212,7 +212,7 @@ export const atan = defOp<Op1>(
         const ar = a[0];
         const iar = 1 / (1 + ar * ar);
         const out = [Math.atan(ar)];
-        for (let i = 1, n = a.length; i < n; i++) {
+        for (let i = a.length; --i >= 1; ) {
             out[i] = iar * a[i];
         }
         return out;
diff --git a/packages/dual-algebra/src/poly.ts b/packages/dual-algebra/src/poly.ts
index 4055d87aad..bb6a6ea5ad 100644
--- a/packages/dual-algebra/src/poly.ts
+++ b/packages/dual-algebra/src/poly.ts
@@ -1,8 +1,9 @@
-import { Dual, Op4 } from "./api";
+import type { FnU4, FnU5, FnU6 } from "@thi.ng/api";
+import type { Dual, Op4, Op5 } from "./api";
 import { add, mul } from "./ops";
 
 /**
- * Computes: `ax^2 + bx + c`
+ * Computes: `ax^2 + bx + c`. All args must have same size/arity.
  *
  * @param x
  * @param a
@@ -21,11 +22,11 @@ export const quadratic: Op4 = (x, a, b, c) =>
  * @param b
  * @param c
  */
-export const quadraticS = (x: number, a: number, b: number, c: number) =>
+export const quadraticS: FnU4<number, Dual> = (x, a, b, c) =>
     quadratic([x, 1], [a, 0], [b, 0], [c, 0]);
 
 /**
- * Computes: `ax^3 + bx^2 + cx + d`
+ * Computes: `ax^3 + bx^2 + cx + d`. All args must have same size/arity.
  *
  * @param x
  * @param a
@@ -33,7 +34,7 @@ export const quadraticS = (x: number, a: number, b: number, c: number) =>
  * @param c
  * @param d
  */
-export const cubic = (x: Dual, a: Dual, b: Dual, c: Dual, d: Dual) => {
+export const cubic: Op5 = (x, a, b, c, d) => {
     const x2 = mul(x, x);
     return add(add(add(mul(a, mul(x2, x)), mul(b, x2)), mul(c, x)), d);
 };
@@ -48,11 +49,11 @@ export const cubic = (x: Dual, a: Dual, b: Dual, c: Dual, d: Dual) => {
  * @param c
  * @param d
  */
-export const cubicS = (x: number, a: number, b: number, c: number, d: number) =>
+export const cubicS: FnU5<number, Dual> = (x, a, b, c, d) =>
     cubic([x, 1], [a, 0], [b, 0], [c, 0], [d, 0]);
 
 /**
- * Computes: `ax^4 + bx^3 + cx^2 + dx + e`
+ * Computes: `ax^4 + bx^3 + cx^2 + dx + e`. All args must have same size/arity.
  *
  * @param x
  * @param a
@@ -88,11 +89,5 @@ export const quartic = (
  * @param d
  * @param e
  */
-export const quarticS = (
-    x: number,
-    a: number,
-    b: number,
-    c: number,
-    d: number,
-    e: number
-) => quartic([x, 1], [a, 0], [b, 0], [c, 0], [d, 0], [e, 0]);
+export const quarticS: FnU6<number, Dual> = (x, a, b, c, d, e) =>
+    quartic([x, 1], [a, 0], [b, 0], [c, 0], [d, 0], [e, 0]);