876 lines
25 KiB
C++
876 lines
25 KiB
C++
/*
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* This Source Code Form is subject to the terms of the Mozilla Public
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* License, v. 2.0. If a copy of the MPL was not distributed with this
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* 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
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// it triggers a circular dependency; debug -> format -> maths -> debug
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// instead, just use cassert
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#include "concepts/traits.hpp"
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#include "concepts/named.hpp"
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#include "concepts.hpp"
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#include "types/traits.hpp"
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#include "float.hpp"
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#include <algorithm>
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#include <cassert>
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#include <cmath>
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#include <cstdint>
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#include <limits>
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#include <numeric>
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#include <type_traits>
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///////////////////////////////////////////////////////////////////////////////
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// NOTE: You may be tempted to add all sorts of performance enhancing
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// attributes (like gnu::const or gnu::pure). DO NOT DO THIS WITHOUT EXTENSIVE
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// TESTING. Just about everything will break in some way with these attributes.
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//
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// In particular: it is safest to apply these only to leaf functions
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///////////////////////////////////////////////////////////////////////////////
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namespace cruft {
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///////////////////////////////////////////////////////////////////////////
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template <concepts::traits::arithmetic T>
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constexpr T
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abs [[gnu::const]] (T t)
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{
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return t > 0 ? t : -t;
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}
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//-----------------------------------------------------------------------------
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// Useful for explictly ignore equality warnings
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#pragma GCC diagnostic push
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#pragma GCC diagnostic ignored "-Wfloat-equal"
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template <typename ValueA, typename ValueB>
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constexpr auto
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equal (ValueA const &a, ValueB const &b)
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{
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return a == b;
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}
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#pragma GCC diagnostic pop
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///////////////////////////////////////////////////////////////////////////
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// Comparisons
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///
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/// check that a query value is within a specified relative percentage of
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/// a ground truth value.
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///
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/// eg: relatively_equal(355/113.f,M_PI,1e-2f);
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template <typename ValueT, typename PercentageT>
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auto
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relatively_equal (ValueT truth, ValueT test, PercentageT percentage)
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{
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// we want to do 1 - b / a, but a might be zero. if we have FE_INVALID
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// enabled then we'll pretty quickly throw an exception.
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//
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// instead we use |a - b | / (1 + |truth|). note that it's not as
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// accurate when the test values aren't close to 1.
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return abs (truth - test) / (1 + abs (truth)) < percentage;
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}
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//-------------------------------------------------------------------------
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inline bool
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almost_equal (float a, float b)
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{
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return ieee_single::almost_equal (a, b);
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}
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//-----------------------------------------------------------------------------
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inline bool
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almost_equal (double a, double b)
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{
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return ieee_double::almost_equal (a, b);
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}
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//-----------------------------------------------------------------------------
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template <typename ValueA, typename ValueB>
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constexpr auto
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almost_equal (const ValueA &a, const ValueB &b)
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{
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if constexpr (std::is_floating_point_v<ValueA> && std::is_floating_point_v<ValueB>) {
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using common_t = std::common_type_t<ValueA,ValueB>;
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return almost_equal (common_t {a}, common_t{b});
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} else if constexpr (std::is_integral_v<ValueA> &&
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std::is_integral_v<ValueB> &&
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std::is_signed_v<ValueA> == std::is_signed_v<ValueB>)
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{
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using common_t = std::common_type_t<ValueA,ValueB>;
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return common_t {a} == common_t {b};
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} else {
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return equal (a, b);
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}
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}
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//-----------------------------------------------------------------------------
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template <typename T>
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constexpr bool
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almost_zero (T t)
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{
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if constexpr (std::is_integral_v<T>) {
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return t == 0;
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} else {
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return almost_equal (t, T{0});
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}
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}
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//-------------------------------------------------------------------------
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template <typename T>
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constexpr bool
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exactly_zero (T t)
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{
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if constexpr (std::is_integral_v<T>) {
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return equal (t, T{0});
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} else {
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return equal (t, T{0});
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}
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}
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///////////////////////////////////////////////////////////////////////////
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// exponentials
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// reciprocal sqrt, provided so that we can search for usages and replace
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// at a later point with a more efficient implementation.
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inline float
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rsqrt (float val)
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{
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return 1 / std::sqrt (val);
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}
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template <
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typename BaseT,
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std::integral ExponentT
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>
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constexpr BaseT
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pow [[gnu::const]] (BaseT base, ExponentT exponent)
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{
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assert (exponent >= 0);
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if (exponent == 1)
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return base;
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if (exponent == 0)
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return BaseT{1};
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return base * pow (base, exponent - 1);
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}
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//-------------------------------------------------------------------------
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template <typename ValueT>
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constexpr auto
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pow2 [[gnu::const]] (ValueT const &val)
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{ return val * val; }
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//-------------------------------------------------------------------------
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template <std::integral T>
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constexpr bool
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is_pow2 [[gnu::const]] (T value)
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{
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return value && !(value & (value - 1));
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}
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///////////////////////////////////////////////////////////////////////////////
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/// Calculate the base-2 logarithm of an integer, truncating to the next
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/// lowest integer.
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///
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/// `val` must be strictly greater than zero, otherwise the results are
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/// undefined.
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template <std::integral T>
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constexpr T
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log2 (T val)
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{
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assert (val > 0);
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T tally = 0;
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while (val >>= 1)
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++tally;
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return tally;
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}
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///------------------------------------------------------------------------
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/// Calculates the base-2 logarithm of an integer, rounding up to the next
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/// highest integer.
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template <std::integral T>
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constexpr T
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log2up (T val)
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{
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return log2 ((val << 1) - 1);
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}
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/// Naively calculates the integer log of `val` in `base`, rounded down.
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///
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/// We deliberately restrict this to consteval to limit unexpected issues
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/// with runtime performance given the simplistic construction.
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///
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/// It's useful for sizing temporary arrays.
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template <std::integral T>
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consteval T
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ilog (T val, T base)
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{
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T tally = 0;
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while (val /= base)
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++tally;
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return tally;
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}
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///////////////////////////////////////////////////////////////////////////////
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/// round T up to the nearest multiple of U
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template <std::integral T, std::integral U>
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inline
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std::common_type_t<T, U>
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round_up (T value, U size)
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{
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// we perform this as two steps to avoid unnecessarily incrementing when
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// remainder is zero.
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if (value % size)
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value += size - value % size;
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return value;
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}
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///----------------------------------------------------------------------------
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/// round T up to the nearest power-of-2
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template <std::integral T>
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constexpr auto
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round_pow2 (T value)
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{
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--value;
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for (unsigned i = 1; i < sizeof (T) * 8; i <<= 1) {
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value |= value >> i;
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}
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++value;
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return value;
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}
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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 <
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std::integral T,
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std::integral U
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>
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constexpr auto
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divup (T const a, U const b)
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{
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return (a + b - 1) / b;
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}
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///////////////////////////////////////////////////////////////////////////////
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// Properties
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template <std::integral T>
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constexpr bool
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is_integer (T)
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{
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return true;
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}
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template <std::floating_point T>
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constexpr bool
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is_integer (T t)
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{
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T i = 0;
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return equal (std::modf (t, &i), T{0});
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}
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//-------------------------------------------------------------------------
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template <std::integral NumericT>
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constexpr auto
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digits10 (NumericT v) noexcept
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{
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// cascading conditionals are faster, but it's super annoying to write
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// out for arbitrarily sized types so we use this base case unti
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// there's actually a performance reason to use another algorithm.
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int tally = 0;
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do {
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v /= 10;
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++tally;
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} while (v);
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return tally;
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/*
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return (v >= 1000000000) ? 10 :
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(v >= 100000000) ? 9 :
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(v >= 10000000) ? 8 :
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(v >= 1000000) ? 7 :
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(v >= 100000) ? 6 :
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(v >= 10000) ? 5 :
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(v >= 1000) ? 4 :
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(v >= 100) ? 3 :
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(v >= 10) ? 2 :
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1;
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*/
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}
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template <std::integral ValueT, std::integral BaseT>
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constexpr int
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digits (ValueT value, BaseT base) noexcept
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{
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assert (base > 0);
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if (value < 0)
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value *= -1;
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int tally = 1;
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while (value /= base)
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++tally;
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return tally;
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}
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///----------------------------------------------------------------------------
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/// return positive or negative unit value corresponding to the input.
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template <std::signed_integral T>
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constexpr T
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sign (T t)
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{
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return t < 0 ? -1 : 1;
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}
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///------------------------------------------------------------------------
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/// return positive or negative unit value corresponding to the input.
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/// guaranteed to give correct results for signed zeroes, use another
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/// method if extreme speed is important.
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template <std::floating_point T>
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constexpr T
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sign (T t)
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{
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return std::signbit (t) ? -1 : 1;
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}
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//-------------------------------------------------------------------------
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template <typename A, typename B>
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constexpr
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bool
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samesign (A &&a, B &&b)
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{
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return (a >= 0 && b >= 0) || (a <= 0 && b <= 0);
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}
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///////////////////////////////////////////////////////////////////////////////
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// factorisation
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template <typename T>
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const T&
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identity (const T& t)
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{
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return t;
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}
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///////////////////////////////////////////////////////////////////////////
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// Modulus/etc
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// namespaced wrapper for `man 3 fmod`
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template <std::floating_point T>
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constexpr T
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mod (T x, T y)
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{
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return std::fmod (x, y);
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}
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template <std::integral T>
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constexpr T
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mod (T x, T y)
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{
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return x % y;
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}
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template <std::floating_point ValueT>
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ValueT
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frac (ValueT val)
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{
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return val - static_cast<long> (val);
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}
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///////////////////////////////////////////////////////////////////////////////
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// angles, trig
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namespace detail {
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template <typename T>
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struct pi;
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template <> struct pi<float> { static constexpr float value = 3.141592653589793238462643f; };
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template <> struct pi<double> { static constexpr double value = 3.141592653589793238462643; };
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};
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template <typename T>
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constexpr auto pi = detail::pi<T>::value;
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//-----------------------------------------------------------------------------
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template <typename T>
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constexpr T E = static_cast<T> (2.71828182845904523536028747135266250);
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//-----------------------------------------------------------------------------
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template <typename T>
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constexpr T
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to_degrees (T radians)
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{
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static_assert (std::is_floating_point<T>::value, "undefined for integral types");
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return radians * 180 / pi<T>;
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}
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//-----------------------------------------------------------------------------
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template <typename T>
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constexpr T
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to_radians (T degrees)
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{
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static_assert (std::is_floating_point<T>::value, "undefined for integral types");
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return degrees / 180 * pi<T>;
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}
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//-----------------------------------------------------------------------------
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//! Normalised sinc function
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template <typename T>
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constexpr T
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sincn (T x)
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{
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return almost_zero (x) ? 1 : std::sin (pi<T> * x) / (pi<T> * x);
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}
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//-----------------------------------------------------------------------------
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//! Unnormalised sinc function
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template <typename T>
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constexpr T
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sincu (T x)
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{
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return almost_zero (x) ? 1 : std::sin (x) / x;
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}
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//-------------------------------------------------------------------------
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// thin wrappers around std trig identities.
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//
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// we have these because it's a little easier to qualify templates when
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// passing function objects as compared to explicitly disambiguating raw
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// functions (ie, with casts or typedefs).
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template <typename ValueT> ValueT cos (ValueT theta) { return ::std::cos (theta); }
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template <typename ValueT> ValueT sin (ValueT theta) { return ::std::sin (theta); }
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template <typename ValueT> ValueT tan (ValueT theta) { return ::std::tan (theta); }
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///////////////////////////////////////////////////////////////////////////////
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// combinatorics
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constexpr uintmax_t
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factorial (unsigned i)
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{
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return i <= 1 ? 0 : i * factorial (i - 1);
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}
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//-----------------------------------------------------------------------------
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/// stirlings approximation of factorials
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inline uintmax_t
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stirling (unsigned n)
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{
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using real_t = double;
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return static_cast<uintmax_t> (
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std::sqrt (2 * pi<real_t> * n) * std::pow (n / E<real_t>, n)
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);
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}
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//-----------------------------------------------------------------------------
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constexpr uintmax_t
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combination (unsigned n, unsigned k)
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{
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return factorial (n) / (factorial (k) / (factorial (n - k)));
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}
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///////////////////////////////////////////////////////////////////////////////
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// kahan summation for long floating point sequences
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template <typename InputT>
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requires
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concepts::named::legacy_input_iterator<InputT> &&
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std::floating_point<typename std::iterator_traits<InputT>::value_type>
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typename std::iterator_traits<InputT>::value_type
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sum (InputT first, InputT last)
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{
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using T = typename std::iterator_traits<InputT>::value_type;
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T sum = 0;
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T c = 0;
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for (auto cursor = first; cursor != last; ++cursor) {
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// Infinities are handled poorly in this implementation. We tend
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// to produce NaNs because of the subtraction where we compute
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// `c'. For the time being just panic in this scenario.
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assert(!std::isinf (*cursor));
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T y = *cursor - c;
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T t = sum + y;
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c = (t - sum) - y;
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sum = t;
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}
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return sum;
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}
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template <concepts::named::container ContainerT>
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typename ContainerT::value_type
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sum (ContainerT const &src)
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{
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return sum (src.begin (), src.end ());
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}
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//-------------------------------------------------------------------------
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template <typename InputT>
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requires
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concepts::named::legacy_input_iterator<InputT> &&
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std::integral<typename std::iterator_traits<InputT>::value_type>
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typename std::iterator_traits<InputT>::value_type
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sum (InputT first, InputT last)
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{
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using T = typename std::iterator_traits<InputT>::value_type;
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return std::accumulate (first, last, T{0});
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}
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|
|
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///////////////////////////////////////////////////////////////////////////
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/// Variadic minimum
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template <typename T, typename U, typename ...Args>
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constexpr
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std::common_type_t<T,U>
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min (const T a, const U b, Args ...args)
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{
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if constexpr (sizeof... (args) > 0) {
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return min (a < b ? a : b, std::forward<Args> (args)...);
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} else {
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return a < b ? a : b;
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}
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}
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///------------------------------------------------------------------------
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/// Unary maximum provided to simplify application of max to template
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/// parameter packs.
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///
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/// eg, `max (sizeof (T)...)` will otherwise fail with a single type.
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template <std::integral ValueT>
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constexpr decltype(auto)
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max (ValueT &&val)
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{
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return std::forward<ValueT> (val);
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}
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|
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/// Variadic maximum
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template <typename T, typename U, typename ...Args>
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constexpr
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std::common_type_t<T,U>
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max (const T a, const U b, Args ...args)
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{
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if constexpr (sizeof... (args) > 0) {
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return max (a > b ? a : b, std::forward<Args> (args)...);
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} else {
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return a > b ? a : b;
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}
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}
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|
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//-------------------------------------------------------------------------
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template <concepts::named::container ContainerT>
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typename ContainerT::value_type const&
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max (ContainerT const &vals)
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{
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return *std::max_element (vals.begin (), vals.end ());
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}
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|
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template <concepts::named::container ValueT>
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typename ValueT::value_type &
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max (ValueT &&) = delete;
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//-------------------------------------------------------------------------
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template <concepts::named::container ContainerT>
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typename ContainerT::value_type const&
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min (ContainerT const &vals)
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{
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return *std::min_element (vals.begin (), vals.end ());
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}
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template <concepts::named::container ContainerT>
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typename ContainerT::value_type&
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min (ContainerT &&) = delete;
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///------------------------------------------------------------------------
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/// Returns an ordered pair where the elements come from the parameters.
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template <typename ValueT>
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std::pair<ValueT, ValueT>
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maxmin (ValueT a, ValueT b)
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{
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if (a >= b)
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return { a, b };
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else
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return { b, a };
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}
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///////////////////////////////////////////////////////////////////////////
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// Limiting functions
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// min/max clamping
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template <
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concepts::traits::scalar T,
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concepts::traits::scalar U,
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concepts::traits::scalar V
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>
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constexpr std::common_type_t<T,U,V>
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clamp (T const val, U const lo, V const hi)
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{
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assert (lo <= hi);
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return val > hi ? hi:
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val < lo ? lo:
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val;
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}
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|
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//-------------------------------------------------------------------------
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// clamped cubic hermite interpolation
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template <typename T>
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constexpr
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T
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smoothstep (T a, T b, T x)
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{
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assert (a <= b);
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x = clamp ((x - a) / (b - a), T{0}, T{1});
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return x * x * (3 - 2 * x);
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}
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|
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//-------------------------------------------------------------------------
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template <
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concepts::numeric U,
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concepts::numeric T
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>
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constexpr U
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mix (U const a, U const b, T const t)
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{
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// give some tolerance for floating point rounding
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assert (t >= -0.00001f);
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assert (t <= 1.00001f);
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return a * (1 - t) + b * t;
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}
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|
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///////////////////////////////////////////////////////////////////////////
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/// convert between different representations of normalised quantities.
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///
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/// * floating point values must be within [0, 1] (otherwise undefined)
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/// * signed values are handled by converting to unsigned representations
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/// * may introduce small biases when expanding values so that low order
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/// bits have some meaning (particularly when dealing with UINTMAX)
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// uint -> float
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template <
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std::unsigned_integral T,
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std::floating_point U
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>
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constexpr U
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renormalise (T t)
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|
{
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return t / static_cast<U> (std::numeric_limits<T>::max ());
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}
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|
|
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//-------------------------------------------------------------------------
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// float -> uint
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template <
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std::floating_point T,
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std::unsigned_integral U
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>
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|
constexpr U
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renormalise (T t)
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|
{
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|
// Ideally std::ldexp would be involved but it complicates handing
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// integers with greater precision than our floating point type. Also it
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// would prohibit constexpr and involve errno.
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size_t usable = std::numeric_limits<T>::digits;
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size_t available = sizeof (U) * 8;
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size_t shift = std::max (available, usable) - usable;
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|
t = clamp (t, 0, 1);
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|
|
// construct an integer of the float's mantissa size, multiply it by our
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// parameter, then shift it back into the full range of the integer type.
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U in = std::numeric_limits<U>::max () >> shift;
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U mid = static_cast<U> (t * in);
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|
U out = mid << shift;
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|
|
|
// 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
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|
// ambiguous overloads
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|
template <typename T, typename U>
|
|
requires
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|
std::floating_point<T> &&
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|
std::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
|
|
std::unsigned_integral<T> &&
|
|
std::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
|
|
std::unsigned_integral<SrcT> &&
|
|
std::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
|
|
std::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
|
|
std::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 ()
|
|
);
|
|
};
|
|
}
|