fp_utils.h

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// Copyright 2010-2021 Google LLC
// 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.
 
// Utility functions on IEEE floating-point numbers.
// Implemented on float, double, and long double.
//
// Also a placeholder for tools controlling and checking FPU rounding modes.
//
// IMPORTANT NOTICE: you need to compile your binary with -frounding-math if
// you want to use rounding modes.
 
#ifndef OR_TOOLS_UTIL_FP_UTILS_H_
#define OR_TOOLS_UTIL_FP_UTILS_H_
 
#if defined(_MSC_VER)
#pragma fenv_access(on)  // NOLINT
#else
#include <fenv.h>  // NOLINT
#endif
 
#ifdef __SSE__
#include <xmmintrin.h>
#endif
 
#include <stdlib.h>
 
#include <algorithm>
#include <cmath>
#include <limits>
#include <vector>
 
#include "ortools/base/integral_types.h"
#include "ortools/base/logging.h"
 
#if defined(_MSC_VER)
static inline double isnan(double value) { return _isnan(value); }
static inline double round(double value) { return floor(value + 0.5); }
#elif defined(__APPLE__) || __GNUC__ >= 5
using std::isnan;
#endif
 
namespace operations_research {
 
// ScopedFloatingPointEnv is used to easily enable Floating-point exceptions.
// The initial state is automatically restored when the object is deleted.
//
// Note(user): For some reason, this causes an FPE exception to be triggered for
// unknown reasons when compiled in 32 bits. Because of this, we do not turn
// on FPE exception if __x86_64__ is not defined.
//
// TODO(user): Make it work on 32 bits.
// TODO(user): Make it work on msvc, currently calls to _controlfp crash.
 
class ScopedFloatingPointEnv {
 public:
  ScopedFloatingPointEnv() {
#if defined(_MSC_VER)
    // saved_control_ = _controlfp(0, 0);
#elif (defined(__GNUC__) || defined(__llvm__)) && defined(__x86_64__)
    CHECK_EQ(0, fegetenv(&saved_fenv_));
#endif
  }
 
  ~ScopedFloatingPointEnv() {
#if defined(_MSC_VER)
    // CHECK_EQ(saved_control_, _controlfp(saved_control_, 0xFFFFFFFF));
#elif defined(__x86_64__) && defined(__GLIBC__)
    CHECK_EQ(0, fesetenv(&saved_fenv_));
#endif
  }
 
  void EnableExceptions(int excepts) {
#if defined(_MSC_VER)
    // _controlfp(static_cast<unsigned int>(excepts), _MCW_EM);
#elif (defined(__GNUC__) || defined(__llvm__)) && defined(__x86_64__) && \
    !defined(__ANDROID__)
    CHECK_EQ(0, fegetenv(&fenv_));
    excepts &= FE_ALL_EXCEPT;
#if defined(__APPLE__)
    fenv_.__control &= ~excepts;
#elif defined(__FreeBSD__)
    fenv_.__x87.__control &= ~excepts;
#else  // Linux
    fenv_.__control_word &= ~excepts;
#endif
    fenv_.__mxcsr &= ~(excepts << 7);
    CHECK_EQ(0, fesetenv(&fenv_));
#endif
  }
 
 private:
#if defined(_MSC_VER)
  // unsigned int saved_control_;
#elif (defined(__GNUC__) || defined(__llvm__)) && defined(__x86_64__)
  fenv_t fenv_;
  mutable fenv_t saved_fenv_;
#endif
};
 
template <typename FloatType>
inline bool IsPositiveOrNegativeInfinity(FloatType x) {
  return x == std::numeric_limits<FloatType>::infinity() ||
         x == -std::numeric_limits<FloatType>::infinity();
}
 
// Tests whether x and y are close to one another using absolute and relative
// tolerances.
// Returns true if |x - y| <= a (with a being the absolute_tolerance).
// The above case is useful for values that are close to zero.
// Returns true if |x - y| <= max(|x|, |y|) * r. (with r being the relative
//                                                tolerance.)
// The cases for infinities are treated separately to avoid generating NaNs.
template <typename FloatType>
bool AreWithinAbsoluteOrRelativeTolerances(FloatType x, FloatType y,
                                           FloatType relative_tolerance,
                                           FloatType absolute_tolerance) {
  DCHECK_LE(0.0, relative_tolerance);
  DCHECK_LE(0.0, absolute_tolerance);
  DCHECK_GT(1.0, relative_tolerance);
  if (IsPositiveOrNegativeInfinity(x) || IsPositiveOrNegativeInfinity(y)) {
    return x == y;
  }
  const FloatType difference = fabs(x - y);
  if (difference <= absolute_tolerance) {
    return true;
  }
  const FloatType largest_magnitude = std::max(fabs(x), fabs(y));
  return difference <= largest_magnitude * relative_tolerance;
}
 
// Tests whether x and y are close to one another using an absolute tolerance.
// Returns true if |x - y| <= a (with a being the absolute_tolerance).
// The cases for infinities are treated separately to avoid generating NaNs.
template <typename FloatType>
bool AreWithinAbsoluteTolerance(FloatType x, FloatType y,
                                FloatType absolute_tolerance) {
  DCHECK_LE(0.0, absolute_tolerance);
  if (IsPositiveOrNegativeInfinity(x) || IsPositiveOrNegativeInfinity(y)) {
    return x == y;
  }
  return fabs(x - y) <= absolute_tolerance;
}
 
// Returns true if x is less than y or slighlty greater than y with the given
// absolute or relative tolerance.
template <typename FloatType>
bool IsSmallerWithinTolerance(FloatType x, FloatType y, FloatType tolerance) {
  if (IsPositiveOrNegativeInfinity(y)) return x <= y;
  return x <= y + tolerance * std::max(1.0, std::min(std::abs(x), std::abs(y)));
}
 
// Returns true if x is within tolerance of any integer.  Always returns
// false for x equal to +/- infinity.
template <typename FloatType>
inline bool IsIntegerWithinTolerance(FloatType x, FloatType tolerance) {
  DCHECK_LE(0.0, tolerance);
  if (IsPositiveOrNegativeInfinity(x)) return false;
  return std::abs(x - std::round(x)) <= tolerance;
}
 
// Handy alternatives to EXPECT_NEAR(), using relative and absolute tolerance
// instead of relative tolerance only, and with a proper support for infinity.
// TODO(user): investigate moving this to ortools/base/ or some other place.
#define EXPECT_COMPARABLE(expected, obtained, epsilon)                    \
  EXPECT_TRUE(operations_research::AreWithinAbsoluteOrRelativeTolerances( \
      expected, obtained, epsilon, epsilon))                              \
      << obtained << " != expected value " << expected                    \
      << " within epsilon = " << epsilon;
 
#define EXPECT_NOTCOMPARABLE(expected, obtained, epsilon)                  \
  EXPECT_FALSE(operations_research::AreWithinAbsoluteOrRelativeTolerances( \
      expected, obtained, epsilon, epsilon))                               \
      << obtained << " == expected value " << expected                     \
      << " within epsilon = " << epsilon;
 
// Given an array of doubles, this computes a positive scaling factor such that
// the scaled doubles can then be rounded to integers with little or no loss of
// precision, and so that the L1 norm of these integers is <= max_sum. More
// precisely, the following formulas will hold (x[i] is input[i], for brevity):
// - For all i, |round(factor * x[i]) / factor  - x[i]| <= error * |x[i]|
// - The sum over i of |round(factor * x[i])| <= max_sum.
//
// The algorithm tries to minimize "error" (which is the relative error for one
// coefficient). Note however than in really broken cases, the error might be
// infinity and the factor zero.
//
// Note on the algorithm:
// - It only uses factors of the form 2^n (i.e. ldexp(1.0, n)) for simplicity.
// - The error will be zero in many practical instances. For example, if x
//   contains only integers with low magnitude; or if x contains doubles whose
//   exponents cover a small range.
// - It chooses the factor as high as possible under the given constraints, as
//   a result the numbers produced may be large. To balance this, we recommend
//   to divide the scaled integers by their gcd() which will result in no loss
//   of precision and will help in many practical cases.
//
// TODO(user): incorporate the gcd computation here? The issue is that I am
// not sure if I just do factor /= gcd that round(x * factor) will be the same.
void GetBestScalingOfDoublesToInt64(const std::vector<double>& input,
                                    int64_t max_absolute_sum,
                                    double* scaling_factor,
                                    double* max_relative_coeff_error);
 
// Returns the scaling factor like above with the extra conditions:
//  -  The sum over i of min(0, round(factor * x[i])) >= -max_sum.
//  -  The sum over i of max(0, round(factor * x[i])) <= max_sum.
// For any possible values of the x[i] such that x[i] is in [lb[i], ub[i]].
double GetBestScalingOfDoublesToInt64(const std::vector<double>& input,
                                      const std::vector<double>& lb,
                                      const std::vector<double>& ub,
                                      int64_t max_absolute_sum);
// This computes:
//
// The max_relative_coeff_error, which is the maximum over all coeff of
// |round(factor * x[i]) / (factor * x[i])  - 1|.
//
// The max_scaled_sum_error which is a bound on the maximum difference between
// the exact scaled sum and the rounded one. One needs to divide this by
// scaling_factor to have the maximum absolute error on the original sum.
void ComputeScalingErrors(const std::vector<double>& input,
                          const std::vector<double>& lb,
                          const std::vector<double>& ub,
                          const double scaling_factor,
                          double* max_relative_coeff_error,
                          double* max_scaled_sum_error);
 
// Returns the Greatest Common Divisor of the numbers
// round(fabs(x[i] * scaling_factor)). The numbers 0 are ignored and if they are
// all zero then the result is 1. Note that round(fabs()) is the same as
// fabs(round()) since the numbers are rounded away from zero.
int64_t ComputeGcdOfRoundedDoubles(const std::vector<double>& x,
                                   double scaling_factor);
 
// Returns alpha * x + (1 - alpha) * y.
template <typename FloatType>
inline FloatType Interpolate(FloatType x, FloatType y, FloatType alpha) {
  return alpha * x + (1 - alpha) * y;
}
 
// This is a fast implementation of the C99 function ilogb for normalized
// doubles with the caveat that it returns -1023 for zero, and 1024 for infinity
// an NaNs.
int fast_ilogb(const double value);
 
// This is a fast implementation of the C99 function scalbn, with the caveat
// that it works on normalized numbers and if the result underflows, overflows,
// or is applied to a NaN or an +-infinity, the result is undefined behavior.
// Note that the version of the function that takes a reference, modifies the
// given value.
double fast_scalbn(const double value, const int exponent);
void fast_scalbn_inplace(double& mutable_value, const int exponent);
 
}  // namespace operations_research
 
#endif  // OR_TOOLS_UTIL_FP_UTILS_H_