Multilinear extension evaluation speedup (arkworks-rs#603)

* Reduce # multiplications in dense ml extension eval

* Reduce # multiplications in sparse ml extension eval

* Eliminate extra vector allocation in precompute_eq

* Add entry to changelog on mul reduction in eval functions

CHANGELOG.md

@@ -2,6 +2,8 @@
 
 ## Pending
 
+- (`ark-poly`) Reduce the number of field multiplications performed by `SparseMultilinearExtension::evaluate` and `DenseMultilinearExtension::evaluate`
+
 ### Breaking changes
 
 ### Features

poly/src/evaluations/multivariate/multilinear/dense.rs

@@ -119,7 +119,9 @@ impl<F: Field> MultilinearExtension<F> for DenseMultilinearExtension<F> {
         for i in 1..dim + 1 {
             let r = partial_point[i - 1];
             for b in 0..(1 << (nv - i)) {
-                poly[b] = poly[b << 1] * (F::one() - r) + poly[(b << 1) + 1] * r;
+                let left = poly[b << 1];
+                let right = poly[(b << 1) + 1];
+                poly[b] = left + r * (right - left);
             }
         }
         Self::from_evaluations_slice(nv - dim, &poly[..(1 << (nv - dim))])

poly/src/evaluations/multivariate/multilinear/sparse.rs

@@ -102,15 +102,14 @@ impl<F: Field> SparseMultilinearExtension<F> {
 /// utility: precompute f(x) = eq(g,x)
 fn precompute_eq<F: Field>(g: &[F]) -> Vec<F> {
     let dim = g.len();
-    let mut dp = Vec::with_capacity(1 << dim);
-    dp.resize(1 << dim, F::zero());
+    let mut dp = vec![F::zero(); 1 << dim];
     dp[0] = F::one() - g[0];
     dp[1] = g[0];
     for i in 1..dim {
-        let dp_prev = dp[0..(1 << i)].to_vec();
         for b in 0..(1 << i) {
-            dp[b] = dp_prev[b] * (F::one() - g[i]);
-            dp[b + (1 << i)] = dp_prev[b] * g[i];
+            let prev = dp[b];
+            dp[b + (1 << i)] = prev * g[i];
+            dp[b] = prev - dp[b + (1 << i)];
         }
     }
     dp