libcruft-util/maths.hpp

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/*
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* 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/.
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*
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* Copyright 2010-2018 Danny Robson <danny@nerdcruft.net>
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*/
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#pragma once
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// DO NOT INCLUDE debug.hpp
// it triggers a circular dependency; debug -> format -> maths -> debug
// instead, just use cassert
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#include "concepts.hpp"
#include "types/traits.hpp"
#include "coord/traits.hpp"
#include "float.hpp"
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#include <algorithm>
#include <cassert>
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#include <cmath>
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#include <cstdint>
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#include <limits>
#include <numeric>
#include <type_traits>
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///////////////////////////////////////////////////////////////////////////////
// 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 {
///////////////////////////////////////////////////////////////////////////
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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)
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{
// 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;
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}
//-------------------------------------------------------------------------
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>
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constexpr bool
almost_zero (T t)
{
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if constexpr (std::is_integral_v<T>) {
return t == 0;
} else {
return almost_equal (t, T{0});
}
}
//-------------------------------------------------------------------------
template <typename T>
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constexpr bool
exactly_zero (T t)
{
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if constexpr (std::is_integral_v<T>) {
return equal (t, T{0});
} else {
return equal (t, T{0});
}
}
///////////////////////////////////////////////////////////////////////////
// exponentials
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// 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,
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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);
}
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//-------------------------------------------------------------------------
template <typename ValueT>
constexpr auto
pow2 [[gnu::const]] (ValueT const &val)
{ return val * val; }
//-------------------------------------------------------------------------
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template <concepts::integral T>
constexpr bool
is_pow2 [[gnu::const]] (T value)
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{
return value && !(value & (value - 1));
}
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///////////////////////////////////////////////////////////////////////////////
/// 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.
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template <concepts::integral T>
constexpr T
log2 (T val)
{
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assert (val > 0);
T tally = 0;
while (val >>= 1)
++tally;
return tally;
}
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///------------------------------------------------------------------------
/// Calculates the base-2 logarithm of an integer, rounding up to the next
/// highest integer.
template <typename T>
T
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log2up (T val)
{
return log2 ((val << 1) - 1);
}
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/// 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
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template <concepts::integral T, concepts::integral U>
inline
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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;
}
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///----------------------------------------------------------------------------
/// round T up to the nearest power-of-2
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template <concepts::integral T>
constexpr auto
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round_pow2 (T value)
{
--value;
for (unsigned i = 1; i < sizeof (T) * 8; i <<= 1) {
value |= value >> i;
}
++value;
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return value;
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}
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///----------------------------------------------------------------------------
/// round T up to the nearest multiple of U and return the quotient.
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template <
concepts::integral T,
concepts::integral U
>
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constexpr auto
divup (T const a, U const b)
{
return (a + b - 1) / b;
}
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///////////////////////////////////////////////////////////////////////////////
// Properties
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template <concepts::integral T>
constexpr bool
is_integer (T)
{
return true;
}
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template <concepts::floating_point T>
constexpr bool
is_integer (T t)
{
T i = 0;
return equal (std::modf (t, &i), T{0});
}
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//-------------------------------------------------------------------------
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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;
*/
}
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template <concepts::integral ValueT, concepts::integral BaseT>
constexpr int
digits (ValueT value, BaseT base) noexcept
{
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assert (base > 0);
if (value < 0)
value *= -1;
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int tally = 1;
while (value /= base)
++tally;
return tally;
}
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///----------------------------------------------------------------------------
/// return positive or negative unit value corresponding to the input.
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template <concepts::signed_integral T>
constexpr T
sign (T t)
{
return t < 0 ? -1 : 1;
}
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///------------------------------------------------------------------------
/// 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.
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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)
{
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return (a >= 0 && b >= 0) || (a <= 0 && b <= 0);
}
///////////////////////////////////////////////////////////////////////////////
// factorisation
template <typename T>
const T&
identity (const T& t)
{
return t;
}
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///////////////////////////////////////////////////////////////////////////
// Modulus/etc
// namespaced wrapper for `man 3 fmod`
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template <concepts::floating_point T>
constexpr T
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mod (T x, T y)
{
return std::fmod (x, y);
}
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template <concepts::integral T>
constexpr T
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mod (T x, T y)
{
return x % y;
}
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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;
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//-----------------------------------------------------------------------------
template <typename T>
constexpr T E = static_cast<T> (2.71828182845904523536028747135266250);
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//-----------------------------------------------------------------------------
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>;
}
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//-----------------------------------------------------------------------------
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>;
}
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//-----------------------------------------------------------------------------
//! Normalised sinc function
template <typename T>
constexpr T
sincn (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 (T x)
{
return almost_zero (x) ? 1 : std::sin (x) / x;
}
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//-------------------------------------------------------------------------
// 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);
}
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//-----------------------------------------------------------------------------
/// 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)
);
}
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//-----------------------------------------------------------------------------
constexpr uintmax_t
combination (unsigned n, unsigned k)
{
return factorial (n) / (factorial (k) / (factorial (n - k)));
}
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///////////////////////////////////////////////////////////////////////////////
// kahan summation for long floating point sequences
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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;
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T sum = 0;
T c = 0;
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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;
}
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return sum;
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}
//-------------------------------------------------------------------------
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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>
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constexpr std::enable_if_t<
!is_coord_v<T> && !is_coord_v<U>,
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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;
}
}
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//-------------------------------------------------------------------------
/// Variadic maximum
template <typename T, typename U, typename ...Args>
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constexpr std::enable_if_t<
!is_coord_v<T> && !is_coord_v<U>,
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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;
}
}
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//-------------------------------------------------------------------------
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template <concepts::container ContainerT>
typename ContainerT::value_type const&
max (ContainerT const &vals)
{
return *std::max_element (vals.begin (), vals.end ());
}
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template <concepts::container ValueT>
typename ValueT::value_type &
max (ValueT &&) = delete;
//-------------------------------------------------------------------------
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template <concepts::container ContainerT>
typename ContainerT::value_type const&
min (ContainerT const &vals)
{
return *std::min_element (vals.begin (), vals.end ());
}
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template <concepts::container ContainerT>
typename ContainerT::value_type&
min (ContainerT &&) = delete;
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///------------------------------------------------------------------------
/// Returns an ordered pair where the elements come from the parameters.
template <typename ValueT>
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std::pair<ValueT, ValueT>
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maxmin (ValueT a, ValueT b)
{
if (a >= b)
return { a, b };
else
return { b, a };
}
///////////////////////////////////////////////////////////////////////////
// Limiting functions
// min/max clamping
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template <
concepts::scalar T,
concepts::scalar U,
concepts::scalar V
>
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constexpr std::common_type_t<T,U,V>
clamp (T const val, U const lo, V const hi)
{
assert (lo <= hi);
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return val > hi ? hi:
val < lo ? lo:
val;
}
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//-------------------------------------------------------------------------
// clamped cubic hermite interpolation
template <typename T>
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constexpr
T
smoothstep (T a, T b, T x)
{
assert (a <= b);
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x = clamp ((x - a) / (b - a), T{0}, T{1});
return x * x * (3 - 2 * x);
}
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//-------------------------------------------------------------------------
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template <
concepts::numeric U,
concepts::numeric T
>
constexpr U
mix (U const a, U const b, T const t)
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{
// give some tolerance for floating point rounding
assert (t >= -0.00001f);
assert (t <= 1.00001f);
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return a * (1 - t) + b * t;
}
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///////////////////////////////////////////////////////////////////////////
/// 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
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template <
concepts::unsigned_integral T,
concepts::floating_point U
>
constexpr U
renormalise (T t)
{
return t / static_cast<U> (std::numeric_limits<T>::max ());
}
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//-------------------------------------------------------------------------
// float -> uint
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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;
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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);
}
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//-------------------------------------------------------------------------
// float -> float, avoids identity conversion as we don't want to create
// ambiguous overloads
template <typename T, typename U>
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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>
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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");
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// we have excess bits ,just shift and return
constexpr auto shift = 8 * (sizeof (T) - sizeof (U));
return t >> shift;
}
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//-------------------------------------------------------------------------
// lo_uint -> hi_uint
template <
typename SrcT,
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typename DstT
>
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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");
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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 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");
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// 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;
}
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//-------------------------------------------------------------------------
// identity transformation. must precede the signed cases, as they may rely
// on this as a side effect of casts.
template <typename T, typename U>
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requires (std::is_same_v<T, U>)
constexpr U
renormalise (T t)
{ return t; }
//-------------------------------------------------------------------------
// anything-to-sint
template <typename T, typename U>
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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>
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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 ()
);
};
}