cstd.hh

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#ifndef CSTD_HH
#define CSTD_HH
 
#include <cassert>
#include <cmath>
#include <cstddef>
#include <functional>
#include <string_view>
#include <utility>
 
namespace cstd {
 
// Inspired by this post:
//    http://tristanbrindle.com/posts/a-more-useful-compile-time-quicksort
 
// Constexpr reimplementations of standard algorithms or data-structures.
//
// Everything implemented in 'cstd' will very likely become part of standard
// C++ in the future. So then we can remove our (re)implementation and change
// the callers from 'cstd::xxx()' to 'std::xxx()'.
 
//
// Various constexpr reimplementation of STL algorithms.
//
 
template<typename Iter1, typename Iter2>
constexpr void iter_swap(Iter1 a, Iter2 b)
{
        auto temp = std::move(*a);
        *a = std::move(*b);
        *b = std::move(temp);
}
 
template<typename InputIt, typename UnaryPredicate>
[[nodiscard]] constexpr InputIt find_if_not(InputIt first, InputIt last, UnaryPredicate p)
{
        for (; first != last; ++first) {
                if (!p(*first)) {
                        return first;
                }
        }
        return last;
}
 
template<typename ForwardIt, typename UnaryPredicate>
[[nodiscard]] constexpr ForwardIt partition(ForwardIt first, ForwardIt last, UnaryPredicate p)
{
        first = cstd::find_if_not(first, last, p);
        if (first == last) return first;
 
        for (ForwardIt i = first + 1; i != last; ++i) {
                if (p(*i)) {
                        cstd::iter_swap(i, first);
                        ++first;
                }
        }
        return first;
}
 
// Textbook implementation of quick sort. Not optimized, but the intention is
// that it only gets executed during compile-time.
template<typename RAIt, typename Compare = std::less<>>
constexpr void sort(RAIt first, RAIt last, Compare cmp = Compare{})
{
        auto const N = last - first;
        if (N <= 1) return;
        auto const pivot = *(first + N / 2);
        auto const middle1 = cstd::partition(
                first, last, [&](const auto& elem) { return cmp(elem, pivot); });
        auto const middle2 = cstd::partition(
                middle1, last, [&](const auto& elem) { return !cmp(pivot, elem); });
        cstd::sort(first, middle1, cmp);
        cstd::sort(middle2, last, cmp);
}
 
template<typename RandomAccessRange, typename Compare = std::less<>>
constexpr void sort(RandomAccessRange&& range, Compare cmp = Compare{})
{
        cstd::sort(std::begin(range), std::end(range), cmp);
}
 
template<typename InputIt1, typename InputIt2>
[[nodiscard]] constexpr bool lexicographical_compare(InputIt1 first1, InputIt1 last1,
                                                     InputIt2 first2, InputIt2 last2)
{
        for (; (first1 != last1) && (first2 != last2); ++first1, ++first2) {
                if (*first1 < *first2) return true;
                if (*first2 < *first1) return false;
        }
        return (first1 == last1) && (first2 != last2);
}
 
 
// Reimplementation of various mathematical functions. You must specify an
// iteration count, this controls how accurate the result will be.
#if (defined(__GNUC__) && !defined(__clang__))
 
// Gcc has constexpr versions of most mathematical functions (this is a
// non-standard extension). Prefer those over our implementations.
template<int>      [[nodiscard]] constexpr double sin  (double x) { return std::sin  (x); }
template<int>      [[nodiscard]] constexpr double cos  (double x) { return std::cos  (x); }
template<int, int> [[nodiscard]] constexpr double log  (double x) { return std::log  (x); }
template<int, int> [[nodiscard]] constexpr double log2 (double x) { return    ::log  (x) / ::log(2); } // should be std::log2(x) but this doesn't seem to compile in g++-4.8/g++-4.9 (bug?)
template<int, int> [[nodiscard]] constexpr double log10(double x) { return std::log10(x); }
template<int>      [[nodiscard]] constexpr double exp  (double x) { return std::exp  (x); }
template<int>      [[nodiscard]] constexpr double exp2 (double x) { return    ::exp2 (x); } // see log2, but apparently no need to use exp(log(2) * x) here?!
template<int, int> [[nodiscard]] constexpr double pow(double x, double y) { return std::pow(x, y); }
[[nodiscard]] inline constexpr double round(double x) { return ::round(x); } // should be std::round(), see above
[[nodiscard]] inline constexpr float  round(float  x) { return ::round(x); }
 
#else
 
[[nodiscard]] constexpr double upow(double x, unsigned u)
{
        double y = 1.0;
        while (u) {
                if (u & 1) y *= x;
                x *= x;
                u >>= 1;
        }
        return y;
}
 
[[nodiscard]] constexpr double ipow(double x, int i)
{
        return (i >= 0) ? upow(x, i) : upow(x, -i);
}
 
template<int ITERATIONS>
[[nodiscard]] constexpr double exp(double x)
{
        // Split x into integral and fractional part:
        //   exp(x) = exp(i + f) = exp(i) * exp(f)
        //   with: i an int     (undefined if out of range)
        //         -1 < f < 1
        int i = int(x);
        double f = x - i;
 
        // Approximate exp(f) with Taylor series.
        double y = 1.0;
        double t = f;
        double n = 1.0;
        for (int k = 2; k < (2 + ITERATIONS); ++k) {
                y += t / n;
                t *= f;
                n *= k;
        }
 
        // Approximate exp(i) by squaring.
        int p = (i >= 0) ? i : -i; // abs(i);
        double s = upow(M_E, p);
 
        // Combine the results.
        if (i >= 0) {
                return y * s;
        } else {
                return y / s;
        }
}
 
[[nodiscard]] constexpr double simple_fmod(double x, double y)
{
        assert(y > 0.0);
        return x - int(x / y) * y;
}
 
template<int ITERATIONS>
[[nodiscard]] constexpr double sin_iter(double x)
{
        double x2 = x * x;
        double y = 0.0;
        double t = x;
        double n = 1.0;
        for (int k = 1; k < (1 + 4 * ITERATIONS); ) {
                y += t / n;
                t *= x2;
                n *= ++k;
                n *= ++k;
 
                y -= t / n;
                t *= x2;
                n *= ++k;
                n *= ++k;
        }
        return y;
}
 
template<int ITERATIONS>
[[nodiscard]] constexpr double cos_iter(double x)
{
        double x2 = x * x;
        double y = 1.0;
        double t = x2;
        double n = 2.0;
        for (int k = 2; k < (2 + 4 * ITERATIONS); ) {
                y -= t / n;
                t *= x2;
                n *= ++k;
                n *= ++k;
 
                y += t / n;
                t *= x2;
                n *= ++k;
                n *= ++k;
        }
        return y;
}
 
template<int ITERATIONS>
[[nodiscard]] constexpr double sin(double x)
{
        double sign = 1.0;
 
        // reduce to [0, +inf)
        if (x < 0.0) {
                sign = -1.0;
                x = -x;
        }
 
        // reduce to [0, 2pi)
        x = simple_fmod(x, 2 * M_PI);
 
        // reduce to [0, pi]
        if (x > M_PI) {
                sign = -sign;
                x -= M_PI;
        }
 
        // reduce to [0, pi/2]
        if (x > M_PI/2) {
                x = M_PI - x;
        }
 
        // reduce to [0, pi/4]
        if (x > M_PI/4) {
                x = M_PI/2 - x;
                return sign * cos_iter<ITERATIONS>(x);
        } else {
                return sign * sin_iter<ITERATIONS>(x);
        }
}
 
template<int ITERATIONS>
[[nodiscard]] constexpr double cos(double x)
{
        double sign = 1.0;
 
        // reduce to [0, +inf)
        if (x < 0.0) {
                x = -x;
        }
 
        // reduce to [0, 2pi)
        x = simple_fmod(x, 2 * M_PI);
 
        // reduce to [0, pi]
        if (x > M_PI) {
                x = 2.0 * M_PI - x;
        }
 
        // reduce to [0, pi/2]
        if (x > M_PI/2) {
                sign = -sign;
                x = M_PI - x;
        }
 
        // reduce to [0, pi/4]
        if (x > M_PI/4) {
                x = M_PI/2 - x;
                return sign * sin_iter<ITERATIONS>(x);
        } else {
                return sign * cos_iter<ITERATIONS>(x);
        }
}
 
 
// https://en.wikipedia.org/wiki/Natural_logarithm#High_precision
template<int E_ITERATIONS, int L_ITERATIONS>
[[nodiscard]] constexpr double log(double x)
{
        int a = 0;
        while (x <= 0.25) {
                x *= M_E;
                ++a;
        }
        double y = 0.0;
        for (int i = 0; i < L_ITERATIONS; ++i) {
                auto ey = cstd::exp<E_ITERATIONS>(y);
                y = y + 2.0 * (x - ey) / (x + ey);
        }
        return y - a;
}
 
template<int E_ITERATIONS, int L_ITERATIONS>
[[nodiscard]] constexpr double log2(double x)
{
        return cstd::log<E_ITERATIONS, L_ITERATIONS>(x) / M_LN2;
}
 
template<int E_ITERATIONS, int L_ITERATIONS>
[[nodiscard]] constexpr double log10(double x)
{
        return cstd::log<E_ITERATIONS, L_ITERATIONS>(x) / M_LN10;
}
 
template<int E_ITERATIONS, int L_ITERATIONS>
[[nodiscard]] constexpr double pow(double x, double y)
{
        return cstd::exp<E_ITERATIONS>(cstd::log<E_ITERATIONS, L_ITERATIONS>(x) * y);
}
 
template<int ITERATIONS>
[[nodiscard]] constexpr double exp2(double x)
{
        return cstd::exp<ITERATIONS>(M_LN2 * x);
}
 
[[nodiscard]] constexpr double round(double x)
{
        return (x >= 0) ?  int( x + 0.5)
                        : -int(-x + 0.5);
}
 
[[nodiscard]] constexpr float round(float x)
{
        return (x >= 0) ?  int( x + 0.5f)
                        : -int(-x + 0.5f);
}
 
#endif
 
} // namespace cstd
 
#endif