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/*
* 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 2010-2018 Danny Robson <danny@nerdcruft.net>
*/
#ifndef __MATHS_HPP
#define __MATHS_HPP
// DO NOT INCLUDE debug.hpp
// it triggers a circular dependency; debug -> format -> maths -> debug
// instead, just use cassert
#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 util {
///////////////////////////////////////////////////////////////////////////
template <typename T>
std::enable_if_t<std::is_arithmetic_v<T>, 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 Ta, typename Tb>
constexpr
typename std::enable_if_t<
std::is_arithmetic<Ta>::value &&
std::is_arithmetic<Tb>::value,
bool
>
equal (const Ta &a, const Tb &b)
{
// use double not to force conversion for types with explicit bool
// operators on the result of equality
return !!(a == b);
}
//-------------------------------------------------------------------------
template <typename Ta, typename Tb>
inline
typename std::enable_if_t<
!std::is_arithmetic<Ta>::value ||
!std::is_arithmetic<Tb>::value,
bool
>
equal (const Ta &a, const Tb &b)
{
// use double not to force conversion for types with explicit bool
// operators on the result of equality
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 static_cast<common_t> (a) == static_cast<common_t> (b);
} else {
return equal (a, b);
}
}
//-----------------------------------------------------------------------------
template <typename T>
constexpr
std::enable_if_t<
std::is_integral<T>::value, bool
>
almost_zero (T t)
{
return t == 0;
}
//-------------------------------------------------------------------------
template <typename T>
std::enable_if_t<
!std::is_integral<T>::value, bool
>
almost_zero (T a)
{
return almost_equal (a, T{0});
}
//-------------------------------------------------------------------------
template <typename T>
constexpr
typename std::enable_if_t<
std::is_integral<T>::value, bool
>
exactly_zero (T t)
{
return equal (t, T{0});
}
template <typename T>
constexpr
typename std::enable_if_t<
!std::is_integral<T>::value, bool
>
exactly_zero (T t)
{
return equal (t, T{0});
}
//-------------------------------------------------------------------------
template <typename T>
constexpr
typename std::enable_if_t<
std::is_floating_point<T>::value, bool
>
is_nan (T t)
{
return std::isnan (t);
}
//-------------------------------------------------------------------------
template <typename T>
constexpr
typename std::enable_if_t<
!std::is_floating_point<T>::value, bool
>
is_nan (const T&)
{
return false;
}
///////////////////////////////////////////////////////////////////////////
// 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,
typename ExponentT,
typename = std::enable_if_t<
std::is_integral_v<ExponentT>,
void
>
>
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 T>
constexpr
std::enable_if_t<std::is_integral<T>::value, bool>
is_pow2 [[gnu::const]] (T value)
{
return value && !(value & (value - 1));
}
//-----------------------------------------------------------------------------
// Logarithms
template <typename T>
constexpr
std::enable_if_t<std::is_integral_v<T>, T>
log2 (T val)
{
T tally = 0;
while (val >>= 1)
++tally;
return tally;
}
//-------------------------------------------------------------------------
template <typename T>
T
log2up (T val);
///////////////////////////////////////////////////////////////////////////////
/// round T up to the nearest multiple of U
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_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 <typename T>
constexpr
std::enable_if_t<
std::is_integral<T>::value, T
>
round_pow2 (T value)
{
using return_type = std::enable_if_t<std::is_integral<T>::value, T>;
--value;
for (unsigned i = 1; i < sizeof (T) * 8; i <<= 1) {
value |= value >> i;
}
++value;
return (return_type)value;
}
///----------------------------------------------------------------------------
/// round T up to the nearest multiple of U and return the quotient.
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>
constexpr
std::enable_if_t<std::is_integral<T>::value, bool>
is_integer (T)
{
return true;
}
template <typename T>
constexpr
std::enable_if_t<std::is_floating_point<T>::value, bool>
is_integer (T t)
{
T i = 0;
return equal (std::modf (t, &i), T{0});
}
//-------------------------------------------------------------------------
template <
typename NumericT,
typename = std::enable_if_t<
std::is_integral_v<NumericT>,
void
>
>
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 <typename ValueT, typename BaseT>
constexpr
std::enable_if_t<
std::is_integral_v<ValueT> && std::is_integral_v<BaseT>,
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 <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;
}
//-------------------------------------------------------------------------
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 <typename T>
constexpr
std::enable_if_t<
std::is_floating_point<T>::value, T
>
mod (T x, T y)
{
return std::fmod (x, y);
}
template <typename T>
constexpr
std::enable_if_t<
std::is_integral<T>::value, T
>
mod (T x, T y)
{
return x % y;
}
///////////////////////////////////////////////////////////////////////////////
// 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 <class InputT>
std::enable_if_t<
std::is_floating_point<
typename std::iterator_traits<InputT>::value_type
>::value,
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 <class InputT>
std::enable_if_t<
std::is_integral<
typename std::iterator_traits<InputT>::value_type
>::value,
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
// disable the single parameter version for non-arithmetic types so that
// min for coord types is unambiguous. allow pointers too because it
// doesn't add ambiguity and it simplifies some memory juggling.
template <
typename T,
typename = std::enable_if_t<
std::is_arithmetic_v<T> || std::is_pointer_v<T>, void
>
>
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 std::enable_if_t<std::is_arithmetic_v<T> || std::is_pointer_v<T>, 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
std::enable_if_t<
std::is_scalar_v<T> && std::is_scalar_v<U> && std::is_scalar_v<V>,
std::common_type_t<T,U,V>
>
limit (const T val, const U lo, const V 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 = limit ((x - a) / (b - a), T{0}, T{1});
return x * x * (3 - 2 * x);
}
//-------------------------------------------------------------------------
template <typename U, typename T>
constexpr
std::enable_if_t<std::is_floating_point<T>::value, U>
mix (U a, U b, T 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 <typename T, typename U>
constexpr
typename std::enable_if<
std::is_unsigned<T>::value && std::is_floating_point<U>::value, U
>::type
renormalise (T t)
{
return t / static_cast<U> (std::numeric_limits<T>::max ());
}
//-------------------------------------------------------------------------
// float -> uint
template <typename T, typename U>
constexpr
typename std::enable_if<
std::is_floating_point<T>::value && std::is_unsigned<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, avoids 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_uint -> lo_uint
template <typename T, typename U>
constexpr
typename std::enable_if<
std::is_unsigned<T>::value &&
std::is_unsigned<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_uint -> hi_uint
template <
typename SrcT,
typename DstT,
typename = std::enable_if_t<
std::is_unsigned<SrcT>::value &&
std::is_unsigned<DstT>::value &&
sizeof (SrcT) < sizeof (DstT), 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>
constexpr
typename std::enable_if<
std::is_same<T,U>::value, U
>::type
renormalise (T t)
{ return t; }
//-------------------------------------------------------------------------
// anything-to-sint
template <typename T, typename U>
constexpr
typename std::enable_if<
std::is_signed<U>::value &&
std::is_integral<U>::value &&
!std::is_same<T,U>::value,
U
>::type
renormalise (T t)
{
using uint_t = typename std::make_unsigned<U>::type;
return static_cast<U> (
::util::renormalise<T,uint_t> (t) + std::numeric_limits<U>::min ()
);
};
//-------------------------------------------------------------------------
// sint-to-anything
template <typename T, typename U>
constexpr
typename std::enable_if<
std::is_signed<T>::value &&
std::is_integral<T>::value &&
!std::is_same<T,U>::value,
U
>::type
renormalise (T sint)
{
using uint_t = typename std::make_unsigned<T>::type;
return ::util::renormalise<uint_t,U> (
static_cast<uint_t> (sint) - std::numeric_limits<T>::min ()
);
};
}
#endif // __MATHS_HPP
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