libcruft-util/maths.hpp

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/*
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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
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* 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.
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*
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* Copyright 2010-2014 Danny Robson <danny@nerdcruft.net>
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*/
#ifndef __MATHS_HPP
#define __MATHS_HPP
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#include "debug.hpp"
#include "types/traits.hpp"
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#include <cmath>
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#include <cstdint>
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#include <limits>
#include <type_traits>
#include <utility>
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template <typename T>
T
abs (T value)
{ return value > 0 ? value : -value; }
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namespace util {
template <typename T> T abs (T t) { return ::abs<T> (t); }
}
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///////////////////////////////////////////////////////////////////////////////
// exponentials
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namespace util {
template <typename T>
constexpr T
pow2 [[gnu::pure]] (T value)
{ return value * value; }
}
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template <typename T> constexpr T pow2 [[gnu::pure]] (T value) { return util::pow2 (value); }
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//-----------------------------------------------------------------------------
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template <typename T>
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constexpr T
pow [[gnu::pure]] (T x, unsigned y);
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namespace util {
template <typename T>
constexpr T pow (T x, unsigned y) { return ::pow (x, y); }
}
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//-----------------------------------------------------------------------------
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template <typename T>
bool
is_pow2 [[gnu::pure]] (T value);
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//-----------------------------------------------------------------------------
// Logarithms
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template <typename T>
T
log2 [[gnu::pure]] (T val);
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template <typename T>
T
log2up [[gnu::pure]] (T val);
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//-----------------------------------------------------------------------------
// Roots
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template <typename T>
double
rootsquare [[gnu::pure]] (T a, T b);
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//-----------------------------------------------------------------------------
// Rounding
template <typename T, typename U>
typename std::common_type<T, U>::type
align [[gnu::pure]] (T value, U size);
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template <typename T>
T
round_pow2 [[gnu::pure]] (T value);
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template <typename T, typename U>
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constexpr T
divup [[gnu::pure]] (const T a, const U b)
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{ return (a + b - 1) / b; }
//-----------------------------------------------------------------------------
// Classification
template <typename T>
bool
is_integer [[gnu::pure]] (const T& value);
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//-----------------------------------------------------------------------------
// Properties
template <typename T>
unsigned
digits [[gnu::pure]] (const T& value);
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//-----------------------------------------------------------------------------
// 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 ();
}
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//-----------------------------------------------------------------------------
constexpr int sign (int);
constexpr float sign (float);
constexpr double sign (double);
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//-----------------------------------------------------------------------------
template <typename T>
const T&
identity (const T& t)
{
return t;
}
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//-----------------------------------------------------------------------------
// Comparisons
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template <typename T>
bool
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almost_equal [[gnu::pure]] (const T &a, const T &b)
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{ return a == b; }
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template <>
bool
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almost_equal [[gnu::pure]] (const float &a, const float &b);
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template <>
bool
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almost_equal [[gnu::pure]] (const double &a, const double &b);
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template <typename Ta, typename Tb>
typename std::enable_if<
std::is_arithmetic<Ta>::value && std::is_arithmetic<Tb>::value,
bool
>::type
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almost_equal [[gnu::pure]] (Ta a, Tb b) {
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return almost_equal <decltype(a + b)> (static_cast<decltype(a + b)>(a),
static_cast<decltype(a + b)>(b));
}
template <typename Ta, typename Tb>
typename std::enable_if<
!std::is_arithmetic<Ta>::value || !std::is_arithmetic<Tb>::value,
bool
>::type
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almost_equal [[gnu::pure]] (const Ta &a, const Tb &b)
{ return a == b; }
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// Useful for explictly ignore equality warnings
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wfloat-equal"
template <typename T, typename U>
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bool
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exactly_equal [[gnu::pure]] (const T &a, const U &b)
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{ return a == b; }
#pragma GCC diagnostic pop
template <typename T>
bool
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almost_zero [[gnu::pure]] (T a)
{ return almost_equal (a, 0); }
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template <typename T>
bool
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exactly_zero [[gnu::pure]] (T a)
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{ return exactly_equal (a, static_cast<T> (0)); }
//-----------------------------------------------------------------------------
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// angles, trig
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template <typename T>
constexpr T PI = T(3.141592653589793238462643);
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template <typename T>
constexpr T E = T(2.71828182845904523536028747135266250);
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template <typename T>
constexpr T
to_degrees [[gnu::pure]] (T radians)
{
static_assert (std::is_floating_point<T>::value, "undefined for integral types");
return radians * 180 / PI<T>;
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}
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template <typename T>
constexpr T
to_radians [[gnu::pure]] (T degrees)
{
static_assert (std::is_floating_point<T>::value, "undefined for integral types");
return degrees / 180 * PI<T>;
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}
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//! Normalised sinc function
template <typename T>
constexpr T
sincn [[gnu::pure]] (T x)
{
return almost_zero (x) ? 1 : std::sin (PI<T> * x) / (PI<T> * x);
}
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//! Unnormalised sinc function
template <typename T>
constexpr T
sincu [[gnu::pure]] (T x)
{
return almost_zero (x) ? 1 : std::sin (x) / x;
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}
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//-----------------------------------------------------------------------------
constexpr uintmax_t
factorial [[gnu::pure]] (unsigned i)
{
return i <= 1 ? 0 : i * factorial (i - 1);
}
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/// stirlings approximation of factorials
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constexpr uintmax_t
stirling [[gnu::pure]] (unsigned n)
{
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return static_cast<uintmax_t> (
std::sqrt (2 * PI<float> * n) * std::pow (n / E<float>, n)
);
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}
constexpr uintmax_t
combination [[gnu::pure]] (unsigned n, unsigned k)
{
return factorial (n) / (factorial (k) / (factorial (n - k)));
}
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//-----------------------------------------------------------------------------
// 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;
}
//-----------------------------------------------------------------------------
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/// Variadic minimum
namespace util {
template <typename T>
constexpr T
min [[gnu::pure]] (const T a)
{ return a; }
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template <typename T, typename U, typename ...Args>
constexpr typename std::enable_if<
std::is_unsigned<typename std::decay<T>::type>::value == std::is_unsigned<typename std::decay<U>::type>::value &&
std::is_integral<typename std::decay<T>::type>::value == std::is_integral<typename std::decay<U>::type>::value,
typename std::common_type<T,U>::type
>::type
min [[gnu::pure]] (const T a, const U b, Args ...args)
{
return min (a < b ? a : b, std::forward<Args> (args)...);
}
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//-----------------------------------------------------------------------------
/// Variadic maximum
template <typename T>
constexpr T
max [[gnu::pure]] (const T a)
{ return a; }
template <typename T, typename U, typename ...Args>
constexpr typename std::enable_if<
std::is_unsigned<typename std::decay<T>::type>::value == std::is_unsigned<typename std::decay<U>::type>::value &&
std::is_integral<typename std::decay<T>::type>::value == std::is_integral<typename std::decay<U>::type>::value,
typename std::common_type<T,U>::type
>::type
max [[gnu::pure]] (const T a, const U b, Args ...args)
{
return max (a > b ? a : b, std::forward<Args> (args)...);
}
}
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//-----------------------------------------------------------------------------
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// Limiting functions
// min/max clamping
template <typename T, typename U, typename V>
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constexpr T
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limit [[gnu::pure]] (const T val, const U lo, const V hi)
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{
lo <= hi ? (void)0 : panic ();
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return val > hi ? hi:
val < lo ? lo:
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val;
}
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// clamped cubic hermite interpolation
template <typename T>
T
smoothstep [[gnu::pure]] (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);
}
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#include "types/string.hpp"
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///////////////////////////////////////////////////////////////////////////////
// renormalisation of unit floating point and/or normalised integers
// int -> float
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template <typename T, typename U>
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constexpr
typename std::enable_if<
!std::is_floating_point<T>::value && std::is_floating_point<U>::value, U
>::type
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renormalise [[gnu::pure]] (T t)
{
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return t / static_cast<U> (std::numeric_limits<T>::max ());
}
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// 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 [[gnu::pure]] (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);
}
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// 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 [[gnu::pure]] (T t)
{
return static_cast<U> (t);
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}
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// 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 [[gnu::pure]] (T t)
{
static_assert (sizeof (T) > sizeof (U),
"assumes right shift is sufficient");
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// 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 [[gnu::pure]] (T t)
{
static_assert (sizeof (T) < sizeof (U),
"assumes bit creation is required to fill space");
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// 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;
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return out;
}
template <typename T, typename U>
constexpr
typename std::enable_if<
std::is_same<T,U>::value, U
>::type
renormalise [[gnu::pure]] (T t)
{ return t; }
#include "maths.ipp"
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#endif // __MATHS_HPP