libcruft-util/maths.hpp

/*
 * 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.
 *
 * Copyright 2010-2014 Danny Robson <danny@nerdcruft.net>
 */

#ifndef __MATHS_HPP
#define __MATHS_HPP

#include "./debug.hpp"
#include "./types/traits.hpp"
#include "./float.hpp"

#include <cmath>
#include <cstdint>
#include <limits>
#include <type_traits>
#include <utility>


///////////////////////////////////////////////////////////////////////////////
// NOTE: You may be tempted to add all sorts of performance enhancing
// attributes (like gnu::const or gnu::pure). DO NOT DO THIS WITHOUT EXTENSIVE
// TESTING. Just about everything will break in some way with these attributes.
//
// In particular: it is safest to apply these only to leaf functions


///////////////////////////////////////////////////////////////////////////////
namespace util {
    template <typename T>
    T
    abs [[gnu::const]] (T t)
    {
        return t > 0 ? t : -t;
    }

    ///////////////////////////////////////////////////////////////////////////
    // exponentials

    template <typename T>
    constexpr T
    pow2  [[gnu::const]] (T value)
    {
        return value * value;
    }


    ///////////////////////////////////////////////////////////////////////////
    template <typename T>
    constexpr T
    pow [[gnu::const]] (T x, unsigned y)
    {
        return y == 0 ? T{1} : x * pow (x, y - 1);
    }


    //-------------------------------------------------------------------------
    template <typename T>
    bool
    is_pow2  (T value);


    //-----------------------------------------------------------------------------
    // Logarithms
    template <typename T>
    T
    log2  (T val);


    //-------------------------------------------------------------------------
    template <typename T>
    T
    log2up  (T val);


    ///////////////////////////////////////////////////////////////////////////////
    // Rounding
    template <typename T, typename U>
    inline
    typename std::common_type<
        std::enable_if_t<std::is_integral<T>::value,T>,
        std::enable_if_t<std::is_integral<U>::value,U>
    >::type
    round_to (T value, U size)
    {
        if (value % size == 0)
            return value;

        return value + (size - value % size);
    }


    //-----------------------------------------------------------------------------
    template <typename T>
    std::enable_if_t<
        std::is_integral<T>::value, T
    >
    round_pow2  (T value);


    //-----------------------------------------------------------------------------
    template <typename T, typename U>
    constexpr std::enable_if_t<
        std::is_integral<T>::value &&
        std::is_integral<U>::value,
        T
    >
    divup  (const T a, const U b)
    {
        return (a + b - 1) / b;
    }


    ///////////////////////////////////////////////////////////////////////////////
    // Properties
    template <typename T>
    bool
    is_integer (const T& value);


    //-----------------------------------------------------------------------------
    template <typename T>
    unsigned
    digits (const T& value);


    ///----------------------------------------------------------------------------
    /// return positive or negative unit value corresponding to the input.
    template <typename T>
    constexpr std::enable_if_t<
        std::is_signed<T>::value && std::is_integral<T>::value, T
    >
    sign (T t)
    {
        return t < 0 ? -1 : 1;
    }

    ///------------------------------------------------------------------------
    /// return positive or negative unit value corresponding to the input.
    /// guaranteed to give correct results for signed zeroes, use another
    /// method if extreme speed is important.
    template <typename T>
    constexpr std::enable_if_t<
        std::is_floating_point<T>::value, T
    >
    sign (T t)
    {
        return std::signbit (t) ? -1 : 1;
    }


    ///////////////////////////////////////////////////////////////////////////////
    // factorisation
    template <typename T>
    constexpr T
    gcd (T a, T b)
    {
        if (a == b) return a;

        if (a > b) return gcd (a - b, b);
        if (b > a) return gcd (a, b - a);

        unreachable ();
    }


    ///////////////////////////////////////////////////////////////////////////////
    // Comparisons
    inline bool
    almost_equal (const float &a, const float &b)
    {
        return ieee_single::almost_equal (a, b);
    }


    //-----------------------------------------------------------------------------
    inline bool
    almost_equal (const double &a, const double &b)
    {
        return ieee_double::almost_equal (a, b);
    }


    //-----------------------------------------------------------------------------
    template <typename A, typename B>
    typename std::enable_if_t<
        std::is_floating_point<A>::value &&
        std::is_floating_point<B>::value,
        bool
    >
    almost_equal (const A &a, const B &b)
    {
        using common_t = std::common_type_t<A,B>;
        return almost_equal<common_t> (static_cast<common_t> (a),
                                       static_cast<common_t> (b));
    }


    //-----------------------------------------------------------------------------
    template <typename A, typename B>
    typename std::enable_if_t<
        std::is_integral<A>::value &&
        std::is_integral<B>::value &&
        std::is_signed<A>::value == std::is_signed<B>::value,
        bool
    >
    almost_equal (const A &a, const B &b) {
        using common_t = std::common_type_t<A,B>;
        return static_cast<common_t> (a) == static_cast<common_t> (b);
    }


    //-----------------------------------------------------------------------------
    template <typename Ta, typename Tb>
    typename std::enable_if<
        !std::is_arithmetic<Ta>::value ||
        !std::is_arithmetic<Tb>::value,
        bool
    >::type
    almost_equal (const Ta &a, const Tb &b)
        { return a == b; }


    //-----------------------------------------------------------------------------
    // Useful for explictly ignore equality warnings
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wfloat-equal"
    template <typename T, typename U>
    bool
    exactly_equal  (const T &a, const U &b)
        { return a == b; }
#pragma GCC diagnostic pop


    //-----------------------------------------------------------------------------
    template <typename T>
    bool
    almost_zero  (T a)
        { return almost_equal (a, T{0}); }


    //-----------------------------------------------------------------------------
    template <typename T>
    bool
    exactly_zero (T a)
        { return exactly_equal (a, T{0}); }


    //-----------------------------------------------------------------------------
    template <typename T>
    const T&
    identity (const T& t)
    {
        return t;
    }


    ///////////////////////////////////////////////////////////////////////////////
    // angles, trig
    template <typename T>
    constexpr T PI = T(3.141592653589793238462643);


    //-----------------------------------------------------------------------------
    template <typename T>
    constexpr T E = T(2.71828182845904523536028747135266250);


    //-----------------------------------------------------------------------------
    template <typename T>
    constexpr T
    to_degrees  (T radians)
    {
        static_assert (std::is_floating_point<T>::value, "undefined for integral types");
        return radians * 180 / PI<T>;
    }


    //-----------------------------------------------------------------------------
    template <typename T>
    constexpr T
    to_radians  (T degrees)
    {
        static_assert (std::is_floating_point<T>::value, "undefined for integral types");
        return degrees / 180 * PI<T>;
    }


    //-----------------------------------------------------------------------------
    //! Normalised sinc function
    template <typename T>
    constexpr T
    sincn  (T x)
    {
        return almost_zero (x) ? 1 : std::sin (PI<T> * x) / (PI<T> * x);
    }


    //-----------------------------------------------------------------------------
    //! Unnormalised sinc function
    template <typename T>
    constexpr T
    sincu  (T x)
    {
        return almost_zero (x) ? 1 : std::sin (x) / x;
    }


    ///////////////////////////////////////////////////////////////////////////////
    // combinatorics

    constexpr uintmax_t
    factorial  (unsigned i)
    {
        return i <= 1 ? 0 : i * factorial (i - 1);
    }


    //-----------------------------------------------------------------------------
    /// stirlings approximation of factorials
    constexpr uintmax_t
    stirling  (unsigned n)
    {
        return static_cast<uintmax_t> (
            std::sqrt (2 * PI<float> * n) * std::pow (n / E<float>, n)
        );
    }


    //-----------------------------------------------------------------------------
    constexpr uintmax_t
    combination  (unsigned n, unsigned k)
    {
        return factorial (n) / (factorial (k) / (factorial (n - k)));
    }


    ///////////////////////////////////////////////////////////////////////////////
    // kahan summation for long floating point sequences

    template <class InputIt>
    typename std::iterator_traits<InputIt>::value_type
    fsum (InputIt first, InputIt last)
    {
        using T = typename std::iterator_traits<InputIt>::value_type;
        static_assert (std::is_floating_point<T>::value,
                       "fsum only works for floating point types");

        T sum = 0;
        T c = 0;

        for (auto cursor = first; cursor != last; ++cursor) {
            T y = *cursor - c;
            T t = sum + y;
            c = (t - sum) - y;
            sum = t;
        }

        return sum;
    }


    ///////////////////////////////////////////////////////////////////////////
    /// Variadic minimum
    template <typename T>
    constexpr T
    min  (const T a)
        { return a; }


    //-------------------------------------------------------------------------
    template <typename T, typename U, typename ...Args>
    constexpr std::enable_if_t<
        std::is_unsigned<std::decay_t<T>>::value == std::is_unsigned<std::decay_t<U>>::value &&
        std::is_integral<std::decay_t<T>>::value == std::is_integral<std::decay_t<U>>::value,
        std::common_type_t<T,U>
    >
    min  (const T a, const U b, Args ...args)
    {
        return min (a < b ? a : b, std::forward<Args> (args)...);
    }


    //-------------------------------------------------------------------------
    /// Variadic maximum
    template <typename T>
    constexpr T
    max  (const T a)
        { return a; }


    //-------------------------------------------------------------------------
    template <typename T, typename U, typename ...Args>
    constexpr std::enable_if_t<
        std::is_unsigned<std::decay_t<T>>::value == std::is_unsigned<std::decay_t<U>>::value &&
        std::is_integral<std::decay_t<T>>::value == std::is_integral<std::decay_t<U>>::value,
        std::common_type_t<T,U>
    >
    max  (const T a, const U b, Args ...args)
    {
        return max (a > b ? a : b, std::forward<Args> (args)...);
    }


    ///////////////////////////////////////////////////////////////////////////
    // Limiting functions

    // min/max clamping
    template <typename T, typename U, typename V>
    constexpr T
    limit (const T val, const U lo, const V hi)
    {
        lo <= hi ? (void)0 : panic ();

        return val > hi ? hi:
               val < lo ? lo:
               val;
    }


    //-------------------------------------------------------------------------
    // clamped cubic hermite interpolation
    template <typename T>
    T
    smoothstep  (T a, T b, T x)
    {
        CHECK_LE(a, b);
        x = limit ((x - a) / (b - a), T{0}, T{1});
        return x * x * (3 - 2 * x);
    }



    ///////////////////////////////////////////////////////////////////////////
    // renormalisation of unit floating point and/or normalised integers

    // int -> float
    template <typename T, typename U>
    constexpr
    typename std::enable_if<
        !std::is_floating_point<T>::value && std::is_floating_point<U>::value, U
    >::type
    renormalise (T t)
    {
        return t / static_cast<U> (std::numeric_limits<T>::max ());
    }


    //-------------------------------------------------------------------------
    // float -> int
    template <typename T, typename U>
    constexpr
    typename std::enable_if<
        std::is_floating_point<T>::value && !std::is_floating_point<U>::value, U
    >::type
    renormalise (T t)
    {
        // Ideally std::ldexp would be involved but it complicates handing
        // integers with greater precision than our floating point type. Also it
        // would prohibit constexpr and involve errno.

        size_t usable    = std::numeric_limits<T>::digits;
        size_t available = sizeof (U) * 8;
        size_t shift     = std::max (available, usable) - usable;

        t = limit (t, 0, 1);

        // construct an integer of the float's mantissa size, multiply it by our
        // parameter, then shift it back into the full range of the integer type.
        U in  = std::numeric_limits<U>::max () >> shift;
        U mid = static_cast<U> (t * in);
        U out = mid << shift;

        // use the top bits of the output to fill the bottom bits which through
        // shifting would otherwise be zero. this gives us the full extent of the
        // integer range, while varying predictably through the entire output
        // space.
        return out | out >> (available - shift);
    }


    //-------------------------------------------------------------------------
    // float -> float, avoid identity conversion as we don't want to create
    // ambiguous overloads
    template <typename T, typename U>
    constexpr
    typename std::enable_if<
        std::is_floating_point<T>::value &&
        std::is_floating_point<U>::value &&
        !std::is_same<T,U>::value, U
    >::type
    renormalise (T t)
    {
        return static_cast<U> (t);
    }


    //-------------------------------------------------------------------------
    // hi_int -> lo_int
    template <typename T, typename U>
    constexpr
    typename std::enable_if<
        std::is_integral<T>::value &&
        std::is_integral<U>::value &&
        (sizeof (T) > sizeof (U)), U
    >::type
    renormalise (T t)
    {
        static_assert (sizeof (T) > sizeof (U),
                       "assumes right shift is sufficient");

        // we have excess bits ,just shift and return
        constexpr auto shift = 8 * (sizeof (T) - sizeof (U));
        return t >> shift;
    }


    //-------------------------------------------------------------------------
    // lo_int -> hi_int
    template <typename T, typename U>
    constexpr
    typename std::enable_if<
        std::is_integral<T>::value &&
        std::is_integral<U>::value &&
        sizeof (T) < sizeof (U), U
    >::type
    renormalise (T t)
    {
        static_assert (sizeof (T) < sizeof (U),
                       "assumes bit creation is required to fill space");

        // we need to create bits. fill the output integer with copies of ourself.
        // this is approximately correct in the general case (introducing a small
        // linear positive bias), but allows us to fill the output space in the
        // case of input maximum.

        static_assert (sizeof (U) % sizeof (T) == 0,
                       "assumes integer multiple of sizes");

        U out = 0;

        for (size_t i = 0; i < sizeof (U) / sizeof (T); ++i)
            out |= U (t) << sizeof (T) * 8 * i;

        return out;
    }


    //-------------------------------------------------------------------------
    template <typename T, typename U>
    constexpr
    typename std::enable_if<
        std::is_same<T,U>::value, U
    >::type
    renormalise (T t)
    { return t; }
}


#endif // __MATHS_HPP