time-to-botec

snanvariancepn.c (5868B)

---

 /**
 * @license Apache-2.0
 *
 * Copyright (c) 2020 The Stdlib Authors.
 *
 * Licensed under the Apache License, Version 2.0 (the "License");
 * you may not use this file except in compliance with the License.
 * You may obtain a copy of the License at
 *
 *    http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */
 
 #include "stdlib/stats/base/snanvariancepn.h"
 #include <stdint.h>
 
 /**
 * Computes the sum of single-precision floating-point strided array elements, ignoring `NaN` values and using pairwise summation.
 *
 * ## Method
 *
 * -   This implementation uses pairwise summation, which accrues rounding error `O(log2 N)` instead of `O(N)`. The recursion depth is also `O(log2 N)`.
 *
 * ## References
 *
 * -   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).
 *
 * @private
 * @param N       number of indexed elements
 * @param W       two-element output array
 * @param X       input array
 * @param stride  stride length
 * @return        output value
 */
 static void snansumpw( const int64_t N, double *W, const float *X, const int64_t stride ) {
 	int64_t ix;
 	float *xp1;
 	float *xp2;
 	float sum;
 	int64_t M;
 	int64_t n;
 	int64_t i;
 	float s0;
 	float s1;
 	float s2;
 	float s3;
 	float s4;
 	float s5;
 	float s6;
 	float s7;
 	float v;
 
 	if ( stride < 0 ) {
 		ix = (1-N) * stride;
 	} else {
 		ix = 0;
 	}
 	if ( N < 8 ) {
 		// Use simple summation...
 		sum = 0.0f;
 		n = 0;
 		for ( i = 0; i < N; i++ ) {
 			v = X[ ix ];
 			if ( v == v ) {
 				sum += X[ ix ];
 				n += 1;
 			}
 			ix += stride;
 		}
 		W[ 0 ] += (double)sum;
 		W[ 1 ] += (double)n;
 		return;
 	}
 	// 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`.)
 	if ( N <= 128 ) {
 		// Sum a block with 8 accumulators (by loop unrolling, we lower the effective blocksize to 16)...
 		s0 = 0.0f;
 		s1 = 0.0f;
 		s2 = 0.0f;
 		s3 = 0.0f;
 		s4 = 0.0f;
 		s5 = 0.0f;
 		s6 = 0.0f;
 		s7 = 0.0f;
 		n = 0;
 
 		M = N % 8;
 		for ( i = 0; i < N-M; i += 8 ) {
 			v = X[ ix ];
 			if ( v == v ) {
 				s0 += v;
 				n += 1;
 			}
 			ix += stride;
 			v = X[ ix ];
 			if ( v == v ) {
 				s1 += v;
 				n += 1;
 			}
 			ix += stride;
 			v = X[ ix ];
 			if ( v == v ) {
 				s2 += v;
 				n += 1;
 			}
 			ix += stride;
 			v = X[ ix ];
 			if ( v == v ) {
 				s3 += v;
 				n += 1;
 			}
 			ix += stride;
 			v = X[ ix ];
 			if ( v == v ) {
 				s4 += v;
 				n += 1;
 			}
 			ix += stride;
 			v = X[ ix ];
 			if ( v == v ) {
 				s5 += v;
 				n += 1;
 			}
 			ix += stride;
 			v = X[ ix ];
 			if ( v == v ) {
 				s6 += v;
 				n += 1;
 			}
 			ix += stride;
 			v = X[ ix ];
 			if ( v == v ) {
 				s7 += v;
 				n += 1;
 			}
 			ix += stride;
 		}
 		// Pairwise sum the accumulators:
 		sum = ((s0+s1) + (s2+s3)) + ((s4+s5) + (s6+s7));
 
 		// Clean-up loop...
 		for (; i < N; i++ ) {
 			v = X[ ix ];
 			if ( v == v ) {
 				sum += X[ ix ];
 				n += 1;
 			}
 			ix += stride;
 		}
 		W[ 0 ] += (double)sum;
 		W[ 1 ] += (double)n;
 		return;
 	}
 	// Recurse by dividing by two, but avoiding non-multiples of unroll factor...
 	n = N / 2;
 	n -= n % 8;
 	if ( stride < 0 ) {
 		xp1 = (float *)X + ( (n-N)*stride );
 		xp2 = (float *)X;
 	} else {
 		xp1 = (float *)X;
 		xp2 = (float *)X + ( n*stride );
 	}
 	snansumpw( n, W, xp1, stride );
 	snansumpw( N-n, W, xp2, stride );
 }
 
 /**
 * Computes the variance of a single-precision floating-point strided array ignoring `NaN` values and using a two-pass algorithm.
 *
 * ## Method
 *
 * -   This implementation uses a two-pass approach, as suggested by Neely (1966).
 *
 * ## References
 *
 * -   Neely, Peter M. 1966. "Comparison of Several Algorithms for Computation of Means, Standard Deviations and Correlation Coefficients." _Communications of the ACM_ 9 (7). Association for Computing Machinery: 496–99. doi:[10.1145/365719.365958](https://doi.org/10.1145/365719.365958).
 * -   Schubert, Erich, and Michael Gertz. 2018. "Numerically Stable Parallel Computation of (Co-)Variance." In _Proceedings of the 30th International Conference on Scientific and Statistical Database Management_. New York, NY, USA: Association for Computing Machinery. doi:[10.1145/3221269.3223036](https://doi.org/10.1145/3221269.3223036).
 *
 * @param N           number of indexed elements
 * @param correction  degrees of freedom adjustment
 * @param X           input array
 * @param stride      stride length
 * @return            output value
 */
 float stdlib_strided_snanvariancepn( const int64_t N, const float correction, const float *X, const int64_t stride ) {
 	double W[] = { 0.0, 0.0 };
 	int64_t ix;
 	int64_t i;
 	double nc;
 	double dM;
 	double n;
 	float M2;
 	float mu;
 	float M;
 	float d;
 	float v;
 
 	if ( N <= 0 ) {
 		return 0.0f / 0.0f; // NaN
 	}
 	if ( N == 1 || stride == 0 ) {
 		v = X[ 0 ];
 		if ( v == v && (double)N-(double)correction > 0.0 ) {
 			return 0.0f;
 		}
 		return 0.0f / 0.0f; // NaN
 	}
 	// Compute an estimate for the mean...
 	snansumpw( N, W, X, stride );
 	n = W[ 1 ];
 	nc = n - (double)correction;
 	if ( nc <= 0.0 ) {
 		return 0.0f / 0.0f; // NaN
 	}
 	if ( stride < 0 ) {
 		ix = (1-N) * stride;
 	} else {
 		ix = 0;
 	}
 	mu = (float)( W[ 0 ] / n );
 
 	// Compute the variance...
 	M2 = 0.0f;
 	M = 0.0f;
 	for ( i = 0; i < N; i++ ) {
 		v = X[ ix ];
 		if ( v == v ) {
 			d = v - mu;
 			M2 += d * d;
 			M += d;
 			n += 1;
 		}
 		ix += stride;
 	}
 	dM = (double)M;
 	return (float)((double)M2/nc) - ( (float)(dM/n) * (float)(dM/nc) );
 }