libcruft-util/maths.hpp

/*
 * This Source Code Form is subject to the terms of the Mozilla Public
 * License, v. 2.0. If a copy of the MPL was not distributed with this
 * file, You can obtain one at http://mozilla.org/MPL/2.0/.
 *
 * Copyright 2010-2018 Danny Robson <danny@nerdcruft.net>
 */

#pragma once

// DO NOT INCLUDE debug.hpp
// it triggers a circular dependency; debug -> format -> maths -> debug
// instead, just use cassert

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

#include <algorithm>
#include <cassert>
#include <cmath>
#include <cstdint>
#include <limits>
#include <numeric>
#include <type_traits>


///////////////////////////////////////////////////////////////////////////////
// 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 cruft {
    ///////////////////////////////////////////////////////////////////////////
    template <concepts::arithmetic T>
    constexpr T
    abs [[gnu::const]] (T t)
    {
        return t > 0 ? t : -t;
    }


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


    ///////////////////////////////////////////////////////////////////////////
    // Comparisons
    ///
    /// check that a query value is within a specified relative percentage of
    /// a ground truth value.
    ///
    /// eg: relatively_equal(355/113.f,M_PI,1e-2f);
    template <typename ValueT, typename PercentageT>
    auto
    relatively_equal (ValueT truth, ValueT test, PercentageT percentage)
    {
        // we want to do 1 - b / a, but a might be zero. if we have FE_INVALID
        // enabled then we'll pretty quickly throw an exception.
        //
        // instead we use |a - b | / (1 + |truth|). note that it's not as
        // accurate when the test values aren't close to 1.
        return abs (truth - test) / (1 + abs (truth)) < percentage;
    }

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


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


    //-----------------------------------------------------------------------------
    template <typename ValueA, typename ValueB>
    constexpr auto
    almost_equal (const ValueA &a, const ValueB &b)
    {
        if constexpr (std::is_floating_point_v<ValueA> && std::is_floating_point_v<ValueB>) {
            using common_t = std::common_type_t<ValueA,ValueB>;
            return almost_equal (common_t {a}, common_t{b});
        } else if constexpr (std::is_integral_v<ValueA> &&
                             std::is_integral_v<ValueB> &&
                             std::is_signed_v<ValueA> == std::is_signed_v<ValueB>)
        {
            using common_t = std::common_type_t<ValueA,ValueB>;
            return common_t {a} == common_t {b};
        } else {
            return equal (a, b);
        }
    }


    //-----------------------------------------------------------------------------
    template <typename T>
    constexpr bool
    almost_zero (T t)
    {
        if constexpr (std::is_integral_v<T>) {
            return t == 0;
        } else {
            return almost_equal (t, T{0});
        }
    }


    //-------------------------------------------------------------------------
    template <typename T>
    constexpr bool
    exactly_zero (T t)
    {
        if constexpr (std::is_integral_v<T>) {
            return equal (t, T{0});
        } else {
            return equal (t, T{0});
        }
    }


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

    // reciprocal sqrt, provided so that we can search for usages and replace
    // at a later point with a more efficient implementation.
    inline float
    rsqrt (float val)
    {
        return 1 / std::sqrt (val);
    }

    template <
        typename BaseT,
        concepts::integral ExponentT
    >
    constexpr BaseT
    pow [[gnu::const]] (BaseT base, ExponentT exponent)
    {
        assert (exponent >= 0);

        if (exponent == 1)
            return base;
        if (exponent == 0)
            return BaseT{1};
        return base * pow (base, exponent - 1);
    }


    //-------------------------------------------------------------------------
    template <typename ValueT>
    constexpr auto
    pow2 [[gnu::const]] (ValueT const &val)
    { return val * val; }


    //-------------------------------------------------------------------------
    template <concepts::integral T>
    constexpr bool
    is_pow2 [[gnu::const]] (T value)
    {
        return value && !(value & (value - 1));
    }


    ///////////////////////////////////////////////////////////////////////////////
    /// Calculate the base-2 logarithm of an integer, truncating to the next
    /// lowest integer.
    ///
    /// `val` must be strictly greater than zero, otherwise the results are
    /// undefined.
    template <concepts::integral T>
    constexpr T
    log2 (T val)
    {
        assert (val > 0);

        T tally = 0;
        while (val >>= 1)
            ++tally;
        return tally;
    }


    ///------------------------------------------------------------------------
    /// Calculates the base-2 logarithm of an integer, rounding up to the next
    /// highest integer.
    template <typename T>
    T
    log2up  (T val)
    {

        return log2 ((val << 1) - 1);
    }


    /// Naively calculates the integer log of `val` in `base`, rounded down.
    ///
    /// We deliberately restrict this to consteval to limit unexpected issues
    /// with runtime performance given the simplistic construction.
    ///
    /// It's useful for sizing temporary arrays.
    template <concepts::integral T>
    consteval T
    ilog (T val, T base)
    {
        T tally = 0;
        while (val /= base)
            ++tally;
        return tally;
    }


    ///////////////////////////////////////////////////////////////////////////////
    /// round T up to the nearest multiple of U
    template <concepts::integral T, concepts::integral U>
    inline
    std::common_type_t<T, U>
    round_up (T value, U size)
    {
        // we perform this as two steps to avoid unnecessarily incrementing when
        // remainder is zero.
        if (value % size)
            value += size - value % size;
        return value;
    }


    ///----------------------------------------------------------------------------
    /// round T up to the nearest power-of-2
    template <concepts::integral T>
    constexpr auto
    round_pow2  (T value)
    {
        --value;

        for (unsigned i = 1; i < sizeof (T) * 8; i <<= 1) {
            value |= value >> i;
        }

        ++value;
        return value;
    }


    ///----------------------------------------------------------------------------
    /// round T up to the nearest multiple of U and return the quotient.
    template <
        concepts::integral T,
        concepts::integral U
    >
    constexpr auto
    divup  (T const a, U const b)
    {
        return (a + b - 1) / b;
    }


    ///////////////////////////////////////////////////////////////////////////////
    // Properties
    template <concepts::integral T>
    constexpr bool
    is_integer (T)
    {
        return true;
    }


    template <concepts::floating_point T>
    constexpr bool
    is_integer (T t)
    {
        T i = 0;
        return equal (std::modf (t, &i), T{0});
    }


    //-------------------------------------------------------------------------
    template <concepts::integral NumericT>
    constexpr auto
    digits10 (NumericT v) noexcept
    {
        // cascading conditionals are faster, but it's super annoying to write
        // out for arbitrarily sized types so we use this base case unti
        // there's actually a performance reason to use another algorithm.
        int tally = 0;
        do {
            v /= 10;
            ++tally;
        } while (v);

        return tally;

        /*
        return (v >= 1000000000) ? 10 :
               (v >=  100000000) ?  9 :
               (v >=   10000000) ?  8 :
               (v >=    1000000) ?  7 :
               (v >=     100000) ?  6 :
               (v >=      10000) ?  5 :
               (v >=       1000) ?  4 :
               (v >=        100) ?  3 :
               (v >=         10) ?  2 :
                                    1;
        */
    }


    template <concepts::integral ValueT, concepts::integral BaseT>
    constexpr int
    digits (ValueT value, BaseT base) noexcept
    {
        assert (base > 0);

        if (value < 0)
            value *= -1;

        int tally = 1;
        while (value /= base)
            ++tally;

        return tally;
    }


    ///----------------------------------------------------------------------------
    /// return positive or negative unit value corresponding to the input.
    template <concepts::signed_integral T>
    constexpr 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 <concepts::floating_point T>
    constexpr T
    sign (T t)
    {
        return std::signbit (t) ? -1 : 1;
    }

    //-------------------------------------------------------------------------
    template <typename T>
    constexpr
    bool
    samesign (T a, T b)
    {
        return (a >= 0 && b >= 0) || (a <= 0 && b <= 0);
    }


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


    ///////////////////////////////////////////////////////////////////////////
    // Modulus/etc

    // namespaced wrapper for `man 3 fmod`
    template <concepts::floating_point T>
    constexpr T
    mod (T x, T y)
    {
        return std::fmod (x, y);
    }


    template <concepts::integral T>
    constexpr T
    mod (T x, T y)
    {
        return x % y;
    }


    template <concepts::floating_point ValueT>
    ValueT
    frac (ValueT val)
    {
        return val - static_cast<long> (val);
    }


    ///////////////////////////////////////////////////////////////////////////////
    // angles, trig
    namespace detail {
        template <typename T>
        struct pi;

        template <> struct pi<float> { static constexpr float value = 3.141592653589793238462643f; };
        template <> struct pi<double> { static constexpr double value = 3.141592653589793238462643; };
    };

    template <typename T>
    constexpr auto pi = detail::pi<T>::value;


    //-----------------------------------------------------------------------------
    template <typename T>
    constexpr T E = static_cast<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;
    }


    //-------------------------------------------------------------------------
    // thin wrappers around std trig identities.
    //
    // we have these because it's a little easier to qualify templates when
    // passing function objects as compared to explicitly disambiguating raw
    // functions (ie, with casts or typedefs).
    template <typename ValueT> ValueT cos (ValueT theta) { return ::std::cos (theta); }
    template <typename ValueT> ValueT sin (ValueT theta) { return ::std::sin (theta); }
    template <typename ValueT> ValueT tan (ValueT theta) { return ::std::tan (theta); }

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

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


    //-----------------------------------------------------------------------------
    /// stirlings approximation of factorials
    inline uintmax_t
    stirling  (unsigned n)
    {
        using real_t = double;

        return static_cast<uintmax_t> (
            std::sqrt (2 * pi<real_t> * n) * std::pow (n / E<real_t>, 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 <typename InputT>
    requires
        concepts::legacy_input_iterator<InputT> &&
        concepts::floating_point<typename std::iterator_traits<InputT>::value_type>
    typename std::iterator_traits<InputT>::value_type
    sum (InputT first, InputT last)
    {
        using T = typename std::iterator_traits<InputT>::value_type;

        T sum = 0;
        T c = 0;

        for (auto cursor = first; cursor != last; ++cursor) {
            // Infinities are handled poorly in this implementation. We tend
            // to produce NaNs because of the subtraction where we compute
            // `c'. For the time being just panic in this scenario.
            assert(!std::isinf (*cursor));

            T y = *cursor - c;
            T t = sum + y;

            c = (t - sum) - y;
            sum = t;
        }

        return sum;
    }


    //-------------------------------------------------------------------------
    template <typename InputT>
    requires
        concepts::legacy_input_iterator<InputT> &&
        concepts::integral<typename std::iterator_traits<InputT>::value_type>
    typename std::iterator_traits<InputT>::value_type
    sum (InputT first, InputT last)
    {
        using T = typename std::iterator_traits<InputT>::value_type;
        return std::accumulate (first, last, T{0});
    }


    ///////////////////////////////////////////////////////////////////////////
    /// Variadic minimum
    template <typename T, typename U, typename ...Args>
    constexpr
    std::common_type_t<T,U>
    min  (const T a, const U b, Args ...args)
    {
        if constexpr (sizeof... (args) > 0) {
            return min (a < b ? a : b, std::forward<Args> (args)...);
        } else {
            return a < b ? a : b;
        }
    }


    ///------------------------------------------------------------------------
    /// Unary maximum provided to simplify application of max to template
    /// parameter packs.
    ///
    /// eg, `max (sizeof (T)...)` will otherwise fail with a single type.
    template <concepts::integral ValueT>
    constexpr decltype(auto)
    max (ValueT &&val)
    {
        return std::forward<ValueT> (val);
    }


    /// Variadic maximum
    template <typename T, typename U, typename ...Args>
    constexpr
    std::common_type_t<T,U>
    max  (const T a, const U b, Args ...args)
    {
        if constexpr (sizeof... (args) > 0) {
            return max (a > b ? a : b, std::forward<Args> (args)...);
        } else {
            return a > b ? a : b;
        }
    }


    //-------------------------------------------------------------------------
    template <concepts::container ContainerT>
    typename ContainerT::value_type const&
    max (ContainerT const &vals)
    {
        return *std::max_element (vals.begin (), vals.end ());
    }


    template <concepts::container ValueT>
    typename ValueT::value_type &
    max (ValueT &&) = delete;


    //-------------------------------------------------------------------------
    template <concepts::container ContainerT>
    typename ContainerT::value_type const&
    min (ContainerT const &vals)
    {
        return *std::min_element (vals.begin (), vals.end ());
    }


    template <concepts::container ContainerT>
    typename ContainerT::value_type&
    min (ContainerT &&) = delete;


    ///------------------------------------------------------------------------
    /// Returns an ordered pair where the elements come from the parameters.
    template <typename ValueT>
    std::pair<ValueT, ValueT>
    maxmin (ValueT a, ValueT b)
    {
        if (a >= b)
            return { a, b };
        else
            return { b, a };
    }


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

    // min/max clamping
    template <
        concepts::scalar T,
        concepts::scalar U,
        concepts::scalar V
    >
    constexpr std::common_type_t<T,U,V>
    clamp (T const val, U const lo, V const hi)
    {
        assert (lo <= hi);

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


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


    //-------------------------------------------------------------------------
    template <
        concepts::numeric U,
        concepts::numeric T
    >
    constexpr U
    mix (U const a, U const b, T const t)
    {
        // give some tolerance for floating point rounding
        assert (t >= -0.00001f);
        assert (t <=  1.00001f);

        return a * (1 - t) + b * t;
    }


    ///////////////////////////////////////////////////////////////////////////
    /// convert between different representations of normalised quantities.
    ///
    /// * floating point values must be within [0, 1] (otherwise undefined)
    /// * signed values are handled by converting to unsigned representations
    /// * may introduce small biases when expanding values so that low order
    ///   bits have some meaning (particularly when dealing with UINTMAX)

    // uint -> float
    template <
        concepts::unsigned_integral T,
        concepts::floating_point U
    >
    constexpr U
    renormalise (T t)
    {
        return t / static_cast<U> (std::numeric_limits<T>::max ());
    }


    //-------------------------------------------------------------------------
    // float -> uint
    template <
        concepts::floating_point T,
        concepts::unsigned_integral U
    >
    constexpr U
    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 = clamp (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, avoids identity conversion as we don't want to create
    // ambiguous overloads
    template <typename T, typename U>
    requires
        concepts::floating_point<T> &&
        concepts::floating_point<U> &&
        (!std::is_same_v<T, U>)
    constexpr U
    renormalise (T t)
    {
        return static_cast<U> (t);
    }


    //-------------------------------------------------------------------------
    // hi_uint -> lo_uint
    template <typename T, typename U>
    requires
        concepts::unsigned_integral<T> &&
        concepts::unsigned_integral<U> &&
        (sizeof (T) > sizeof (U))
    constexpr U
    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_uint -> hi_uint
    template <
        typename SrcT,
        typename DstT
    >
    requires
        concepts::unsigned_integral<SrcT> &&
        concepts::unsigned_integral<DstT> &&
        (sizeof (SrcT) < sizeof (DstT))
    constexpr DstT
    renormalise (SrcT src)
    {
        // we can make some simplifying assumptions for the shift distances if
        // we assume the integers are powers of two. this is probably going to
        // be the case for every conceivable input type, but we don't want to
        // get caught out if we extend this routine to more general types
        // (eg, OpenGL DEPTH24).
        static_assert (is_pow2 (sizeof (SrcT)));
        static_assert (is_pow2 (sizeof (DstT)));

        static_assert (sizeof (SrcT) < sizeof (DstT),
                       "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 it allows us to set all output bits high
        // when we receive the maximum allowable input value.
        static_assert (sizeof (DstT) % sizeof (SrcT) == 0,
                       "assumes integer multiple of sizes");


        // clang#xxxx: ideally we wouldn't use a multiplication here, but we
        // trigger a segfault in clang-5.0 when using ld.gold+lto;
        // 'X86 DAG->DAG Instruction Selection'
        //
        // create a mask of locations we'd like copies of the src bit pattern.
        //
        // this replicates repeatedly or'ing and shifting dst with itself.
        DstT dst { 1 };
        for (unsigned i = sizeof (SrcT) * 8; i < sizeof (DstT) * 8; i *= 2)
            dst |= dst << i;
        return dst * src;
    }


    //-------------------------------------------------------------------------
    // identity transformation. must precede the signed cases, as they may rely
    // on this as a side effect of casts.
    template <typename T, typename U>
    requires (std::is_same_v<T, U>)
    constexpr U
    renormalise (T t)
    { return t; }


    //-------------------------------------------------------------------------
    // anything-to-sint
    template <typename T, typename U>
    requires
        concepts::signed_integral<U> &&
        (!std::is_same<T,U>::value)
    constexpr U
    renormalise (T t)
    {
        using uint_t = typename std::make_unsigned<U>::type;

        return static_cast<U> (
            ::cruft::renormalise<T,uint_t> (t) + std::numeric_limits<U>::min ()
        );
    };


    //-------------------------------------------------------------------------
    // sint-to-anything
    template <typename T, typename U>
    requires
        concepts::signed_integral<T> &&
        (!std::is_same<T,U>::value)
    constexpr U
    renormalise (T sint)
    {
        using uint_t = typename std::make_unsigned<T>::type;

        return ::cruft::renormalise<uint_t,U> (
            static_cast<uint_t> (sint) - std::numeric_limits<T>::min ()
        );
    };
}