Kahan summation on Q4_1

ggml.c

@@ -1704,6 +1704,9 @@ inline static void ggml_vec_dot_q4_1(const int n, float * restrict s, const void
     // Accumulator for constant offsets
     __m128 acc_offset = _mm_setzero_ps(); //0.0f;
 
+    __m256 acc_err = _mm256_setzero_ps();
+    __m128 acc_offset_err = _mm_setzero_ps();
+
     // Main loop
     for (int i = 0; i < nb; ++i) {
         const float * m0 = (const float *) (pm0 + i*bs);
@@ -1756,17 +1759,30 @@ inline static void ggml_vec_dot_q4_1(const int n, float * restrict s, const void
         __m256i sumsi = _mm256_or_si256( xsumi, _mm256_slli_si256( ysumi, 4 ) );
         __m256  sums  = _mm256_cvtepi32_ps( sumsi );
 
-        // Apply the scales, and accumulate
-        // acc += d0*m1*x + d1*m0*y
-        acc = _mm256_fmadd_ps( cross_scales, sums, acc );
-
         // Convert int32_t to float
         __m256 p = _mm256_cvtepi32_ps( i32 );
-        // acc += d0*d1*x*y
-        acc = _mm256_fmadd_ps( scale_01, p, acc );
+
+        // Apply the scales, and accumulate
+        // Use Kahan error compensation
+        // acc += d0*m1*x + d1*m0*y + d0*d1*x*y
+        __m256 delta = _mm256_mul_ps( scale_01, p );
+        delta = _mm256_fmadd_ps( cross_scales, sums, delta );
+        delta = _mm256_sub_ps( delta, acc_err );
+
+        __m256 acc_next = _mm256_add_ps( acc, delta );
+        acc_err = _mm256_sub_ps( _mm256_sub_ps( acc_next, acc ), delta );
+
+        acc = acc_next;
+
+        __m128 offs_delta = _mm_mul_ss( _mm256_castps256_ps128( m0v ), _mm256_castps256_ps128( m1v ) );
+        offs_delta = _mm_sub_ss( offs_delta, acc_offset_err );
+
+        __m128 offs_next = _mm_add_ss( acc_offset, offs_delta );
+        acc_offset_err = _mm_sub_ss( _mm_sub_ss( offs_next, acc_offset ), offs_delta );
+        acc_offset = offs_next;
 
         // acc_offset += m0*m1 (avoid reloading from RAM)
-        acc_offset = _mm_fmadd_ss( _mm256_castps256_ps128( m0v ), _mm256_castps256_ps128( m1v ), acc_offset );
+        //acc_offset = _mm_fmadd_ss( _mm256_castps256_ps128( m0v ), _mm256_castps256_ps128( m1v ), acc_offset );
     }
 
     // Return horizontal sum of the acc vector