Extract floating point rounding routines. (#8771)

src/common/deterministic.cuh

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+/**
+ * Copyright 2020-2023 by XGBoost Contributors
+ */
+#ifndef XGBOOST_COMMON_DETERMINISTIC_CUH_
+#define XGBOOST_COMMON_DETERMINISTIC_CUH_
+
+#include <cmath>
+#include <limits>          // std::numeric_limits
+
+#include "xgboost/base.h"  // XGBOOST_DEVICE
+
+namespace xgboost {
+namespace common {
+// Following 2 functions are slightly modified version of fbcuda.
+
+/**
+ * \brief Constructs a rounding factor used to truncate elements in a sum such that the
+ *        sum of the truncated elements is the same no matter what the order of the sum
+ *        is.
+ *
+ * Algorithm 5: Reproducible Sequential Sum in 'Fast Reproducible Floating-Point
+ * Summation' by Demmel and Nguyen.
+ */
+template <typename T>
+XGBOOST_DEVICE T CreateRoundingFactor(T max_abs, int n) {
+  T delta = max_abs / (static_cast<T>(1.0) -
+                       static_cast<T>(2.0) * static_cast<T>(n) * std::numeric_limits<T>::epsilon());
+
+  // Calculate ceil(log_2(delta)).
+  // frexpf() calculates exp and returns `x` such that
+  // delta = x * 2^exp, where `x` in (-1.0, -0.5] U [0.5, 1).
+  // Because |x| < 1, exp is exactly ceil(log_2(delta)).
+  int exp;
+  std::frexp(delta, &exp);
+
+  // return M = 2 ^ ceil(log_2(delta))
+  return std::ldexp(static_cast<T>(1.0), exp);
+}
+
+template <typename T>
+XGBOOST_DEVICE T TruncateWithRounding(T const rounding_factor, T const x) {
+  return (rounding_factor + x) - rounding_factor;
+}
+}  // namespace common
+}  // namespace xgboost
+#endif  // XGBOOST_COMMON_DETERMINISTIC_CUH_