dsapxsumpw.c (3712B)
1 /** 2 * @license Apache-2.0 3 * 4 * Copyright (c) 2020 The Stdlib Authors. 5 * 6 * Licensed under the Apache License, Version 2.0 (the "License"); 7 * you may not use this file except in compliance with the License. 8 * You may obtain a copy of the License at 9 * 10 * http://www.apache.org/licenses/LICENSE-2.0 11 * 12 * Unless required by applicable law or agreed to in writing, software 13 * distributed under the License is distributed on an "AS IS" BASIS, 14 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 15 * See the License for the specific language governing permissions and 16 * limitations under the License. 17 */ 18 19 #include "stdlib/blas/ext/base/dsapxsumpw.h" 20 #include <stdint.h> 21 22 /** 23 * Adds a constant to each single-precision floating-point strided array element and computes the sum using pairwise summation with extended accumulation and returning an extended precision result. 24 * 25 * ## Method 26 * 27 * - This implementation uses pairwise summation, which accrues rounding error `O(log2 N)` instead of `O(N)`. The recursion depth is also `O(log2 N)`. 28 * 29 * ## References 30 * 31 * - Higham, Nicholas J. 1993. "The Accuracy of Floating Point Summation." _SIAM Journal on Scientific Computing_ 14 (4): 783–99. doi:[10.1137/0914050](https://doi.org/10.1137/0914050). 32 * 33 * @param N number of indexed elements 34 * @param alpha constant 35 * @param X input array 36 * @param stride stride length 37 * @return output value 38 */ 39 double stdlib_strided_dsapxsumpw( const int64_t N, const float alpha, const float *X, const int64_t stride ) { 40 float *xp1; 41 float *xp2; 42 double sum; 43 int64_t ix; 44 int64_t M; 45 int64_t n; 46 int64_t i; 47 double s0; 48 double s1; 49 double s2; 50 double s3; 51 double s4; 52 double s5; 53 double s6; 54 double s7; 55 double a; 56 57 if ( N <= 0 ) { 58 return 0.0; 59 } 60 a = (double)alpha; 61 if ( N == 1 || stride == 0 ) { 62 return a + (double)X[ 0 ]; 63 } 64 if ( stride < 0 ) { 65 ix = (1-N) * stride; 66 } else { 67 ix = 0; 68 } 69 if ( N < 8 ) { 70 // Use simple summation... 71 sum = 0.0; 72 for ( i = 0; i < N; i++ ) { 73 sum += a + (double)X[ ix ]; 74 ix += stride; 75 } 76 return sum; 77 } 78 // Blocksize for pairwise summation: 128 (NOTE: decreasing the blocksize decreases rounding error as more pairs are summed, but also decreases performance. Because the inner loop is unrolled eight times, the blocksize is effectively `16`.) 79 if ( N <= 128 ) { 80 // Sum a block with 8 accumulators (by loop unrolling, we lower the effective blocksize to 16)... 81 s0 = a + (double)X[ ix ]; 82 s1 = a + (double)X[ ix+stride ]; 83 s2 = a + (double)X[ ix+(2*stride) ]; 84 s3 = a + (double)X[ ix+(3*stride) ]; 85 s4 = a + (double)X[ ix+(4*stride) ]; 86 s5 = a + (double)X[ ix+(5*stride) ]; 87 s6 = a + (double)X[ ix+(6*stride) ]; 88 s7 = a + (double)X[ ix+(7*stride) ]; 89 ix += 8 * stride; 90 91 M = N % 8; 92 for ( i = 8; i < N-M; i += 8 ) { 93 s0 += a + (double)X[ ix ]; 94 s1 += a + (double)X[ ix+stride ]; 95 s2 += a + (double)X[ ix+(2*stride) ]; 96 s3 += a + (double)X[ ix+(3*stride) ]; 97 s4 += a + (double)X[ ix+(4*stride) ]; 98 s5 += a + (double)X[ ix+(5*stride) ]; 99 s6 += a + (double)X[ ix+(6*stride) ]; 100 s7 += a + (double)X[ ix+(7*stride) ]; 101 ix += 8 * stride; 102 } 103 // Pairwise sum the accumulators: 104 sum = ((s0+s1) + (s2+s3)) + ((s4+s5) + (s6+s7)); 105 106 // Clean-up loop... 107 for (; i < N; i++ ) { 108 sum += a + (double)X[ ix ]; 109 ix += stride; 110 } 111 return sum; 112 } 113 // Recurse by dividing by two, but avoiding non-multiples of unroll factor... 114 n = N / 2; 115 n -= n % 8; 116 if ( stride < 0 ) { 117 xp1 = (float *)X + ( (n-N)*stride ); 118 xp2 = (float *)X; 119 } else { 120 xp1 = (float *)X; 121 xp2 = (float *)X + ( n*stride ); 122 } 123 return stdlib_strided_dsapxsumpw( n, alpha, xp1, stride ) + stdlib_strided_dsapxsumpw( N-n, alpha, xp2, stride ); 124 }