// Copyright 2024 The Manifold Authors.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//      http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.

// Based on linalg.h - 2.2 - Single-header public domain linear algebra library
//
// The intent of this library is to provide the bulk of the functionality
// you need to write programs that frequently use small, fixed-size vectors
// and matrices, in domains such as computational geometry or computer
// graphics. It strives for terse, readable source code.
//
// The original author of this software is Sterling Orsten, and its permanent
// home is <http://github.com/sgorsten/linalg/>. If you find this software
// useful, an acknowledgement in your source text and/or product documentation
// is appreciated, but not required.
//
// The author acknowledges significant insights and contributions by:
//     Stan Melax <http://github.com/melax/>
//     Dimitri Diakopoulos <http://github.com/ddiakopoulos/>
//
// Some features are deprecated. Define LINALG_FORWARD_COMPATIBLE to remove
// them.

#pragma once
#ifndef LINALG_H
#define LINALG_H

#include <array>        // For std::array
#include <cmath>        // For various unary math functions, such as std::sqrt
#include <cstdint>      // For implementing namespace linalg::aliases
#include <cstdlib>      // To resolve std::abs ambiguity on clang
#include <functional>   // For std::hash declaration
#include <iosfwd>       // For forward definitions of std::ostream
#include <type_traits>  // For std::enable_if, std::is_same, std::declval

#ifdef MANIFOLD_DEBUG
#include <iomanip>
#include <iostream>
#endif

// In Visual Studio 2015, `constexpr` applied to a member function implies
// `const`, which causes ambiguous overload resolution
#if defined(_MSC_VER) && (_MSC_VER <= 1900)
#define LINALG_CONSTEXPR14
#else
#define LINALG_CONSTEXPR14 constexpr
#endif

namespace linalg {
// Small, fixed-length vector type, consisting of exactly M elements of type T,
// and presumed to be a column-vector unless otherwise noted.
template <class T, int M>
struct vec;

// Small, fixed-size matrix type, consisting of exactly M rows and N columns of
// type T, stored in column-major order.
template <class T, int M, int N>
struct mat;

// Specialize converter<T,U> with a function application operator that converts
// type U to type T to enable implicit conversions
template <class T, class U>
struct converter {};
namespace detail {
template <class T, class U>
using conv_t = typename std::enable_if<!std::is_same<T, U>::value,
                                       decltype(converter<T, U>{}(
                                           std::declval<U>()))>::type;

// Trait for retrieving scalar type of any linear algebra object
template <class A>
struct scalar_type {};
template <class T, int M>
struct scalar_type<vec<T, M>> {
  using type = T;
};
template <class T, int M, int N>
struct scalar_type<mat<T, M, N>> {
  using type = T;
};

// Type returned by the compare(...) function which supports all six comparison
// operators against 0
template <class T>
struct ord {
  T a, b;
};
template <class T>
constexpr bool operator==(const ord<T> &o, std::nullptr_t) {
  return o.a == o.b;
}
template <class T>
constexpr bool operator!=(const ord<T> &o, std::nullptr_t) {
  return !(o.a == o.b);
}
template <class T>
constexpr bool operator<(const ord<T> &o, std::nullptr_t) {
  return o.a < o.b;
}
template <class T>
constexpr bool operator>(const ord<T> &o, std::nullptr_t) {
  return o.b < o.a;
}
template <class T>
constexpr bool operator<=(const ord<T> &o, std::nullptr_t) {
  return !(o.b < o.a);
}
template <class T>
constexpr bool operator>=(const ord<T> &o, std::nullptr_t) {
  return !(o.a < o.b);
}

// Patterns which can be used with the compare(...) function
template <class A, class B>
struct any_compare {};
template <class T>
struct any_compare<vec<T, 1>, vec<T, 1>> {
  using type = ord<T>;
  constexpr ord<T> operator()(const vec<T, 1> &a, const vec<T, 1> &b) const {
    return ord<T>{a.x, b.x};
  }
};
template <class T>
struct any_compare<vec<T, 2>, vec<T, 2>> {
  using type = ord<T>;
  constexpr ord<T> operator()(const vec<T, 2> &a, const vec<T, 2> &b) const {
    return !(a.x == b.x) ? ord<T>{a.x, b.x} : ord<T>{a.y, b.y};
  }
};
template <class T>
struct any_compare<vec<T, 3>, vec<T, 3>> {
  using type = ord<T>;
  constexpr ord<T> operator()(const vec<T, 3> &a, const vec<T, 3> &b) const {
    return !(a.x == b.x)   ? ord<T>{a.x, b.x}
           : !(a.y == b.y) ? ord<T>{a.y, b.y}
                           : ord<T>{a.z, b.z};
  }
};
template <class T>
struct any_compare<vec<T, 4>, vec<T, 4>> {
  using type = ord<T>;
  constexpr ord<T> operator()(const vec<T, 4> &a, const vec<T, 4> &b) const {
    return !(a.x == b.x)   ? ord<T>{a.x, b.x}
           : !(a.y == b.y) ? ord<T>{a.y, b.y}
           : !(a.z == b.z) ? ord<T>{a.z, b.z}
                           : ord<T>{a.w, b.w};
  }
};
template <class T, int M>
struct any_compare<mat<T, M, 1>, mat<T, M, 1>> {
  using type = ord<T>;
  constexpr ord<T> operator()(const mat<T, M, 1> &a,
                              const mat<T, M, 1> &b) const {
    return compare(a.x, b.x);
  }
};
template <class T, int M>
struct any_compare<mat<T, M, 2>, mat<T, M, 2>> {
  using type = ord<T>;
  constexpr ord<T> operator()(const mat<T, M, 2> &a,
                              const mat<T, M, 2> &b) const {
    return a.x != b.x ? compare(a.x, b.x) : compare(a.y, b.y);
  }
};
template <class T, int M>
struct any_compare<mat<T, M, 3>, mat<T, M, 3>> {
  using type = ord<T>;
  constexpr ord<T> operator()(const mat<T, M, 3> &a,
                              const mat<T, M, 3> &b) const {
    return a.x != b.x   ? compare(a.x, b.x)
           : a.y != b.y ? compare(a.y, b.y)
                        : compare(a.z, b.z);
  }
};
template <class T, int M>
struct any_compare<mat<T, M, 4>, mat<T, M, 4>> {
  using type = ord<T>;
  constexpr ord<T> operator()(const mat<T, M, 4> &a,
                              const mat<T, M, 4> &b) const {
    return a.x != b.x   ? compare(a.x, b.x)
           : a.y != b.y ? compare(a.y, b.y)
           : a.z != b.z ? compare(a.z, b.z)
                        : compare(a.w, b.w);
  }
};

// Helper for compile-time index-based access to members of vector and matrix
// types
template <int I>
struct getter;
template <>
struct getter<0> {
  template <class A>
  constexpr auto operator()(A &a) const -> decltype(a.x) {
    return a.x;
  }
};
template <>
struct getter<1> {
  template <class A>
  constexpr auto operator()(A &a) const -> decltype(a.y) {
    return a.y;
  }
};
template <>
struct getter<2> {
  template <class A>
  constexpr auto operator()(A &a) const -> decltype(a.z) {
    return a.z;
  }
};
template <>
struct getter<3> {
  template <class A>
  constexpr auto operator()(A &a) const -> decltype(a.w) {
    return a.w;
  }
};

// Stand-in for std::integer_sequence/std::make_integer_sequence
template <int... I>
struct seq {};
template <int A, int N>
struct make_seq_impl;
template <int A>
struct make_seq_impl<A, 0> {
  using type = seq<>;
};
template <int A>
struct make_seq_impl<A, 1> {
  using type = seq<A + 0>;
};
template <int A>
struct make_seq_impl<A, 2> {
  using type = seq<A + 0, A + 1>;
};
template <int A>
struct make_seq_impl<A, 3> {
  using type = seq<A + 0, A + 1, A + 2>;
};
template <int A>
struct make_seq_impl<A, 4> {
  using type = seq<A + 0, A + 1, A + 2, A + 3>;
};
template <int A, int B>
using make_seq = typename make_seq_impl<A, B - A>::type;
template <class T, int M, int... I>
vec<T, sizeof...(I)> constexpr swizzle(const vec<T, M> &v, seq<I...> i) {
  return {getter<I>{}(v)...};
}
template <class T, int M, int N, int... I, int... J>
mat<T, sizeof...(I), sizeof...(J)> constexpr swizzle(const mat<T, M, N> &m,
                                                     seq<I...> i, seq<J...> j) {
  return {swizzle(getter<J>{}(m), i)...};
}

// SFINAE helpers to determine result of function application
template <class F, class... T>
using ret_t = decltype(std::declval<F>()(std::declval<T>()...));

// SFINAE helper which is defined if all provided types are scalars
struct empty {};
template <class... T>
struct scalars;
template <>
struct scalars<> {
  using type = void;
};
template <class T, class... U>
struct scalars<T, U...> : std::conditional<std::is_arithmetic<T>::value,
                                           scalars<U...>, empty>::type {};
template <class... T>
using scalars_t = typename scalars<T...>::type;

// Helpers which indicate how apply(F, ...) should be called for various
// arguments
template <class F, class Void, class... T>
struct apply {};  // Patterns which contain only vectors or scalars
template <class F, int M, class A>
struct apply<F, scalars_t<>, vec<A, M>> {
  using type = vec<ret_t<F, A>, M>;
  enum { size = M, mm = 0 };
  template <int... I>
  static constexpr type impl(seq<I...>, F f, const vec<A, M> &a) {
    return {f(getter<I>{}(a))...};
  }
};
template <class F, int M, class A, class B>
struct apply<F, scalars_t<>, vec<A, M>, vec<B, M>> {
  using type = vec<ret_t<F, A, B>, M>;
  enum { size = M, mm = 0 };
  template <int... I>
  static constexpr type impl(seq<I...>, F f, const vec<A, M> &a,
                             const vec<B, M> &b) {
    return {f(getter<I>{}(a), getter<I>{}(b))...};
  }
};
template <class F, int M, class A, class B>
struct apply<F, scalars_t<B>, vec<A, M>, B> {
  using type = vec<ret_t<F, A, B>, M>;
  enum { size = M, mm = 0 };
  template <int... I>
  static constexpr type impl(seq<I...>, F f, const vec<A, M> &a, B b) {
    return {f(getter<I>{}(a), b)...};
  }
};
template <class F, int M, class A, class B>
struct apply<F, scalars_t<A>, A, vec<B, M>> {
  using type = vec<ret_t<F, A, B>, M>;
  enum { size = M, mm = 0 };
  template <int... I>
  static constexpr type impl(seq<I...>, F f, A a, const vec<B, M> &b) {
    return {f(a, getter<I>{}(b))...};
  }
};
template <class F, int M, class A, class B, class C>
struct apply<F, scalars_t<>, vec<A, M>, vec<B, M>, vec<C, M>> {
  using type = vec<ret_t<F, A, B, C>, M>;
  enum { size = M, mm = 0 };
  template <int... I>
  static constexpr type impl(seq<I...>, F f, const vec<A, M> &a,
                             const vec<B, M> &b, const vec<C, M> &c) {
    return {f(getter<I>{}(a), getter<I>{}(b), getter<I>{}(c))...};
  }
};
template <class F, int M, class A, class B, class C>
struct apply<F, scalars_t<C>, vec<A, M>, vec<B, M>, C> {
  using type = vec<ret_t<F, A, B, C>, M>;
  enum { size = M, mm = 0 };
  template <int... I>
  static constexpr type impl(seq<I...>, F f, const vec<A, M> &a,
                             const vec<B, M> &b, C c) {
    return {f(getter<I>{}(a), getter<I>{}(b), c)...};
  }
};
template <class F, int M, class A, class B, class C>
struct apply<F, scalars_t<B>, vec<A, M>, B, vec<C, M>> {
  using type = vec<ret_t<F, A, B, C>, M>;
  enum { size = M, mm = 0 };
  template <int... I>
  static constexpr type impl(seq<I...>, F f, const vec<A, M> &a, B b,
                             const vec<C, M> &c) {
    return {f(getter<I>{}(a), b, getter<I>{}(c))...};
  }
};
template <class F, int M, class A, class B, class C>
struct apply<F, scalars_t<B, C>, vec<A, M>, B, C> {
  using type = vec<ret_t<F, A, B, C>, M>;
  enum { size = M, mm = 0 };
  template <int... I>
  static constexpr type impl(seq<I...>, F f, const vec<A, M> &a, B b, C c) {
    return {f(getter<I>{}(a), b, c)...};
  }
};
template <class F, int M, class A, class B, class C>
struct apply<F, scalars_t<A>, A, vec<B, M>, vec<C, M>> {
  using type = vec<ret_t<F, A, B, C>, M>;
  enum { size = M, mm = 0 };
  template <int... I>
  static constexpr type impl(seq<I...>, F f, A a, const vec<B, M> &b,
                             const vec<C, M> &c) {
    return {f(a, getter<I>{}(b), getter<I>{}(c))...};
  }
};
template <class F, int M, class A, class B, class C>
struct apply<F, scalars_t<A, C>, A, vec<B, M>, C> {
  using type = vec<ret_t<F, A, B, C>, M>;
  enum { size = M, mm = 0 };
  template <int... I>
  static constexpr type impl(seq<I...>, F f, A a, const vec<B, M> &b, C c) {
    return {f(a, getter<I>{}(b), c)...};
  }
};
template <class F, int M, class A, class B, class C>
struct apply<F, scalars_t<A, B>, A, B, vec<C, M>> {
  using type = vec<ret_t<F, A, B, C>, M>;
  enum { size = M, mm = 0 };
  template <int... I>
  static constexpr type impl(seq<I...>, F f, A a, B b, const vec<C, M> &c) {
    return {f(a, b, getter<I>{}(c))...};
  }
};
template <class F, int M, int N, class A>
struct apply<F, scalars_t<>, mat<A, M, N>> {
  using type = mat<ret_t<F, A>, M, N>;
  enum { size = N, mm = 0 };
  template <int... J>
  static constexpr type impl(seq<J...>, F f, const mat<A, M, N> &a) {
    return {apply<F, void, vec<A, M>>::impl(make_seq<0, M>{}, f,
                                            getter<J>{}(a))...};
  }
};
template <class F, int M, int N, class A, class B>
struct apply<F, scalars_t<>, mat<A, M, N>, mat<B, M, N>> {
  using type = mat<ret_t<F, A, B>, M, N>;
  enum { size = N, mm = 1 };
  template <int... J>
  static constexpr type impl(seq<J...>, F f, const mat<A, M, N> &a,
                             const mat<B, M, N> &b) {
    return {apply<F, void, vec<A, M>, vec<B, M>>::impl(
        make_seq<0, M>{}, f, getter<J>{}(a), getter<J>{}(b))...};
  }
};
template <class F, int M, int N, class A, class B>
struct apply<F, scalars_t<B>, mat<A, M, N>, B> {
  using type = mat<ret_t<F, A, B>, M, N>;
  enum { size = N, mm = 0 };
  template <int... J>
  static constexpr type impl(seq<J...>, F f, const mat<A, M, N> &a, B b) {
    return {apply<F, void, vec<A, M>, B>::impl(make_seq<0, M>{}, f,
                                               getter<J>{}(a), b)...};
  }
};
template <class F, int M, int N, class A, class B>
struct apply<F, scalars_t<A>, A, mat<B, M, N>> {
  using type = mat<ret_t<F, A, B>, M, N>;
  enum { size = N, mm = 0 };
  template <int... J>
  static constexpr type impl(seq<J...>, F f, A a, const mat<B, M, N> &b) {
    return {apply<F, void, A, vec<B, M>>::impl(make_seq<0, M>{}, f, a,
                                               getter<J>{}(b))...};
  }
};
template <class F, class... A>
struct apply<F, scalars_t<A...>, A...> {
  using type = ret_t<F, A...>;
  enum { size = 0, mm = 0 };
  static constexpr type impl(seq<>, F f, A... a) { return f(a...); }
};

// Function objects for selecting between alternatives
struct min {
  template <class A, class B>
  constexpr auto operator()(A a, B b) const ->
      typename std::remove_reference<decltype(a < b ? a : b)>::type {
    return a < b ? a : b;
  }
};
struct max {
  template <class A, class B>
  constexpr auto operator()(A a, B b) const ->
      typename std::remove_reference<decltype(a < b ? b : a)>::type {
    return a < b ? b : a;
  }
};
struct clamp {
  template <class A, class B, class C>
  constexpr auto operator()(A a, B b, C c) const ->
      typename std::remove_reference<decltype(a < b   ? b
                                              : a < c ? a
                                                      : c)>::type {
    return a < b ? b : a < c ? a : c;
  }
};
struct select {
  template <class A, class B, class C>
  constexpr auto operator()(A a, B b, C c) const ->
      typename std::remove_reference<decltype(a ? b : c)>::type {
    return a ? b : c;
  }
};
struct lerp {
  template <class A, class B, class C>
  constexpr auto operator()(A a, B b,
                            C c) const -> decltype(a * (1 - c) + b * c) {
    return a * (1 - c) + b * c;
  }
};

// Function objects for applying operators
struct op_pos {
  template <class A>
  constexpr auto operator()(A a) const -> decltype(+a) {
    return +a;
  }
};
struct op_neg {
  template <class A>
  constexpr auto operator()(A a) const -> decltype(-a) {
    return -a;
  }
};
struct op_not {
  template <class A>
  constexpr auto operator()(A a) const -> decltype(!a) {
    return !a;
  }
};
struct op_cmp {
  template <class A>
  constexpr auto operator()(A a) const -> decltype(~(a)) {
    return ~a;
  }
};
struct op_mul {
  template <class A, class B>
  constexpr auto operator()(A a, B b) const -> decltype(a * b) {
    return a * b;
  }
};
struct op_div {
  template <class A, class B>
  constexpr auto operator()(A a, B b) const -> decltype(a / b) {
    return a / b;
  }
};
struct op_mod {
  template <class A, class B>
  constexpr auto operator()(A a, B b) const -> decltype(a % b) {
    return a % b;
  }
};
struct op_add {
  template <class A, class B>
  constexpr auto operator()(A a, B b) const -> decltype(a + b) {
    return a + b;
  }
};
struct op_sub {
  template <class A, class B>
  constexpr auto operator()(A a, B b) const -> decltype(a - b) {
    return a - b;
  }
};
struct op_lsh {
  template <class A, class B>
  constexpr auto operator()(A a, B b) const -> decltype(a << b) {
    return a << b;
  }
};
struct op_rsh {
  template <class A, class B>
  constexpr auto operator()(A a, B b) const -> decltype(a >> b) {
    return a >> b;
  }
};
struct op_lt {
  template <class A, class B>
  constexpr auto operator()(A a, B b) const -> decltype(a < b) {
    return a < b;
  }
};
struct op_gt {
  template <class A, class B>
  constexpr auto operator()(A a, B b) const -> decltype(a > b) {
    return a > b;
  }
};
struct op_le {
  template <class A, class B>
  constexpr auto operator()(A a, B b) const -> decltype(a <= b) {
    return a <= b;
  }
};
struct op_ge {
  template <class A, class B>
  constexpr auto operator()(A a, B b) const -> decltype(a >= b) {
    return a >= b;
  }
};
struct op_eq {
  template <class A, class B>
  constexpr auto operator()(A a, B b) const -> decltype(a == b) {
    return a == b;
  }
};
struct op_ne {
  template <class A, class B>
  constexpr auto operator()(A a, B b) const -> decltype(a != b) {
    return a != b;
  }
};
struct op_int {
  template <class A, class B>
  constexpr auto operator()(A a, B b) const -> decltype(a & b) {
    return a & b;
  }
};
struct op_xor {
  template <class A, class B>
  constexpr auto operator()(A a, B b) const -> decltype(a ^ b) {
    return a ^ b;
  }
};
struct op_un {
  template <class A, class B>
  constexpr auto operator()(A a, B b) const -> decltype(a | b) {
    return a | b;
  }
};
struct op_and {
  template <class A, class B>
  constexpr auto operator()(A a, B b) const -> decltype(a && b) {
    return a && b;
  }
};
struct op_or {
  template <class A, class B>
  constexpr auto operator()(A a, B b) const -> decltype(a || b) {
    return a || b;
  }
};

// Function objects for applying standard library math functions
struct std_isfinite {
  template <class A>
  constexpr auto operator()(A a) const -> decltype(std::isfinite(a)) {
    return std::isfinite(a);
  }
};
struct std_abs {
  template <class A>
  constexpr auto operator()(A a) const -> decltype(std::abs(a)) {
    return std::abs(a);
  }
};
struct std_floor {
  template <class A>
  constexpr auto operator()(A a) const -> decltype(std::floor(a)) {
    return std::floor(a);
  }
};
struct std_ceil {
  template <class A>
  constexpr auto operator()(A a) const -> decltype(std::ceil(a)) {
    return std::ceil(a);
  }
};
struct std_exp {
  template <class A>
  constexpr auto operator()(A a) const -> decltype(std::exp(a)) {
    return std::exp(a);
  }
};
struct std_log {
  template <class A>
  constexpr auto operator()(A a) const -> decltype(std::log(a)) {
    return std::log(a);
  }
};
struct std_log2 {
  template <class A>
  constexpr auto operator()(A a) const -> decltype(std::log2(a)) {
    return std::log2(a);
  }
};
struct std_log10 {
  template <class A>
  constexpr auto operator()(A a) const -> decltype(std::log10(a)) {
    return std::log10(a);
  }
};
struct std_sqrt {
  template <class A>
  constexpr auto operator()(A a) const -> decltype(std::sqrt(a)) {
    return std::sqrt(a);
  }
};
struct std_sin {
  template <class A>
  constexpr auto operator()(A a) const -> decltype(std::sin(a)) {
    return std::sin(a);
  }
};
struct std_cos {
  template <class A>
  constexpr auto operator()(A a) const -> decltype(std::cos(a)) {
    return std::cos(a);
  }
};
struct std_tan {
  template <class A>
  constexpr auto operator()(A a) const -> decltype(std::tan(a)) {
    return std::tan(a);
  }
};
struct std_asin {
  template <class A>
  constexpr auto operator()(A a) const -> decltype(std::asin(a)) {
    return std::asin(a);
  }
};
struct std_acos {
  template <class A>
  constexpr auto operator()(A a) const -> decltype(std::acos(a)) {
    return std::acos(a);
  }
};
struct std_atan {
  template <class A>
  constexpr auto operator()(A a) const -> decltype(std::atan(a)) {
    return std::atan(a);
  }
};
struct std_sinh {
  template <class A>
  constexpr auto operator()(A a) const -> decltype(std::sinh(a)) {
    return std::sinh(a);
  }
};
struct std_cosh {
  template <class A>
  constexpr auto operator()(A a) const -> decltype(std::cosh(a)) {
    return std::cosh(a);
  }
};
struct std_tanh {
  template <class A>
  constexpr auto operator()(A a) const -> decltype(std::tanh(a)) {
    return std::tanh(a);
  }
};
struct std_round {
  template <class A>
  constexpr auto operator()(A a) const -> decltype(std::round(a)) {
    return std::round(a);
  }
};
struct std_fmod {
  template <class A, class B>
  constexpr auto operator()(A a, B b) const -> decltype(std::fmod(a, b)) {
    return std::fmod(a, b);
  }
};
struct std_pow {
  template <class A, class B>
  constexpr auto operator()(A a, B b) const -> decltype(std::pow(a, b)) {
    return std::pow(a, b);
  }
};
struct std_atan2 {
  template <class A, class B>
  constexpr auto operator()(A a, B b) const -> decltype(std::atan2(a, b)) {
    return std::atan2(a, b);
  }
};
struct std_copysign {
  template <class A, class B>
  constexpr auto operator()(A a, B b) const -> decltype(std::copysign(a, b)) {
    return std::copysign(a, b);
  }
};
}  // namespace detail

/** @addtogroup LinAlg
 * @ingroup Math
 */

/** @addtogroup vec
 * @ingroup LinAlg
 * @brief `linalg::vec<T,M>` defines a fixed-length vector containing exactly
   `M` elements of type `T`.

This data structure can be used to store a wide variety of types of data,
including geometric vectors, points, homogeneous coordinates, plane equations,
colors, texture coordinates, or any other situation where you need to manipulate
a small sequence of numbers. As such, `vec<T,M>` is supported by a set of
algebraic and component-wise functions, as well as a set of standard reductions.

`vec<T,M>`:
- is
  [`DefaultConstructible`](https://en.cppreference.com/w/cpp/named_req/DefaultConstructible):
  ```cpp
  float3 v; // v contains 0,0,0
  ```
- is constructible from `M` elements of type `T`:
  ```cpp
  float3 v {1,2,3}; // v contains 1,2,3
  ```
- is
  [`CopyConstructible`](https://en.cppreference.com/w/cpp/named_req/CopyConstructible)
  and
  [`CopyAssignable`](https://en.cppreference.com/w/cpp/named_req/CopyAssignable):
  ```cpp
  float3 v {1,2,3}; // v contains 1,2,3
  float3 u {v};     // u contains 1,2,3
  float3 w;         // w contains 0,0,0
  w = u;            // w contains 1,2,3
  ```
- is
  [`EqualityComparable`](https://en.cppreference.com/w/cpp/named_req/EqualityComparable)
  and
  [`LessThanComparable`](https://en.cppreference.com/w/cpp/named_req/LessThanComparable):
  ```cpp
  if(v == y) cout << "v and u contain equal elements in the same positions" <<
  endl; if(v < u) cout << "v precedes u lexicographically" << endl;
  ```
- is **explicitly** constructible from a single element of type `T`:
  ```cpp
  float3 v = float3{4}; // v contains 4,4,4
  ```
- is **explicitly** constructible from a `vec<U,M>` of some other type `U`:
  ```cpp
  float3 v {1.1f,2.3f,3.5f}; // v contains 1.1,2.3,3.5
  int3 u = int3{v};          // u contains 1,2,3
  ```
- has fields `x,y,z,w`:
  ```cpp
  float y = point.y;    // y contains second element of point
  pixel.w = 0.5;        // fourth element of pixel set to 0.5
  float s = tc.x;       // s contains first element of tc
  ```
- supports indexing:
  ```cpp
  float x = v[0]; // x contains first element of v
  v[2] = 5;       // third element of v set to 5
  ```
- supports unary operators `+`, `-`, `!` and `~` in component-wise fashion:
  ```cpp
  auto v = -float{2,3}; // v is float2{-2,-3}
  ```
- supports binary operators `+`, `-`, `*`, `/`, `%`, `|`, `&`, `^`, `<<` and
  `>>` in component-wise fashion:
  ```cpp
  auto v = float2{1,1} + float2{2,3}; // v is float2{3,4}
  ```
- supports binary operators with a scalar on the left or the right:
  ```cpp
  auto v = 2 * float3{1,2,3}; // v is float3{2,4,6}
  auto u = float3{1,2,3} + 1; // u is float3{2,3,4}
  ```
- supports operators `+=`, `-=`, `*=`, `/=`, `%=`, `|=`, `&=`, `^=`, `<<=` and
  `>>=` with vectors or scalars on the right:
  ```cpp
  float2 v {1,2}; v *= 3; // v is float2{3,6}
  ```
- supports operations on mixed element types:
  ```cpp
  auto v = float3{1,2,3} + int3{4,5,6}; // v is float3{5,7,9}
  ```
- supports [range-based
  for](https://en.cppreference.com/w/cpp/language/range-for):
  ```cpp
  for(auto elem : float3{1,2,3}) cout << elem << ' '; // prints "1 2 3 "
  ```
- has a flat memory layout:
  ```cpp
  float3 v {1,2,3};
  float * p = v.data(); // &v[i] == p+i
  p[1] = 4; // v contains 1,4,3
  ```
 *  @{
 */
template <class T>
struct vec<T, 1> {
  T x;
  constexpr vec() : x() {}
  constexpr vec(const T &x_) : x(x_) {}
  // NOTE: vec<T,1> does NOT have a constructor from pointer, this can conflict
  // with initializing its single element from zero
  template <class U>
  constexpr explicit vec(const vec<U, 1> &v) : vec(static_cast<T>(v.x)) {}
  constexpr const T &operator[](int i) const { return x; }
  LINALG_CONSTEXPR14 T &operator[](int i) { return x; }

  template <class U, class = detail::conv_t<vec, U>>
  constexpr vec(const U &u) : vec(converter<vec, U>{}(u)) {}
  template <class U, class = detail::conv_t<U, vec>>
  constexpr operator U() const {
    return converter<U, vec>{}(*this);
  }
};
template <class T>
struct vec<T, 2> {
  T x, y;
  constexpr vec() : x(), y() {}
  constexpr vec(const T &x_, const T &y_) : x(x_), y(y_) {}
  constexpr explicit vec(const T &s) : vec(s, s) {}
  constexpr explicit vec(const T *p) : vec(p[0], p[1]) {}
  template <class U, int N>
  constexpr explicit vec(const vec<U, N> &v)
      : vec(static_cast<T>(v.x), static_cast<T>(v.y)) {
    static_assert(
        N >= 2,
        "You must give extra arguments if your input vector is shorter.");
  }
  constexpr const T &operator[](int i) const { return i == 0 ? x : y; }
  LINALG_CONSTEXPR14 T &operator[](int i) { return i == 0 ? x : y; }

  template <class U, class = detail::conv_t<vec, U>>
  constexpr vec(const U &u) : vec(converter<vec, U>{}(u)) {}
  template <class U, class = detail::conv_t<U, vec>>
  constexpr operator U() const {
    return converter<U, vec>{}(*this);
  }
};
template <class T>
struct vec<T, 3> {
  T x, y, z;
  constexpr vec() : x(), y(), z() {}
  constexpr vec(const T &x_, const T &y_, const T &z_) : x(x_), y(y_), z(z_) {}
  constexpr vec(const vec<T, 2> &xy, const T &z_) : vec(xy.x, xy.y, z_) {}
  constexpr explicit vec(const T &s) : vec(s, s, s) {}
  constexpr explicit vec(const T *p) : vec(p[0], p[1], p[2]) {}
  template <class U, int N>
  constexpr explicit vec(const vec<U, N> &v)
      : vec(static_cast<T>(v.x), static_cast<T>(v.y), static_cast<T>(v.z)) {
    static_assert(
        N >= 3,
        "You must give extra arguments if your input vector is shorter.");
  }
  constexpr const T &operator[](int i) const {
    return i == 0 ? x : i == 1 ? y : z;
  }
  LINALG_CONSTEXPR14 T &operator[](int i) {
    return i == 0 ? x : i == 1 ? y : z;
  }
  constexpr const vec<T, 2> &xy() const {
    return *reinterpret_cast<const vec<T, 2> *>(this);
  }
  vec<T, 2> &xy() { return *reinterpret_cast<vec<T, 2> *>(this); }

  template <class U, class = detail::conv_t<vec, U>>
  constexpr vec(const U &u) : vec(converter<vec, U>{}(u)) {}
  template <class U, class = detail::conv_t<U, vec>>
  constexpr operator U() const {
    return converter<U, vec>{}(*this);
  }
};
template <class T>
struct vec<T, 4> {
  T x, y, z, w;
  constexpr vec() : x(), y(), z(), w() {}
  constexpr vec(const T &x_, const T &y_, const T &z_, const T &w_)
      : x(x_), y(y_), z(z_), w(w_) {}
  constexpr vec(const vec<T, 2> &xy, const T &z_, const T &w_)
      : vec(xy.x, xy.y, z_, w_) {}
  constexpr vec(const vec<T, 3> &xyz, const T &w_)
      : vec(xyz.x, xyz.y, xyz.z, w_) {}
  constexpr explicit vec(const T &s) : vec(s, s, s, s) {}
  constexpr explicit vec(const T *p) : vec(p[0], p[1], p[2], p[3]) {}
  template <class U, int N>
  constexpr explicit vec(const vec<U, N> &v)
      : vec(static_cast<T>(v.x), static_cast<T>(v.y), static_cast<T>(v.z),
            static_cast<T>(v.w)) {
    static_assert(
        N >= 4,
        "You must give extra arguments if your input vector is shorter.");
  }
  constexpr const T &operator[](int i) const {
    return i == 0 ? x : i == 1 ? y : i == 2 ? z : w;
  }
  LINALG_CONSTEXPR14 T &operator[](int i) {
    return i == 0 ? x : i == 1 ? y : i == 2 ? z : w;
  }
  constexpr const vec<T, 2> &xy() const {
    return *reinterpret_cast<const vec<T, 2> *>(this);
  }
  constexpr const vec<T, 3> &xyz() const {
    return *reinterpret_cast<const vec<T, 3> *>(this);
  }
  vec<T, 2> &xy() { return *reinterpret_cast<vec<T, 2> *>(this); }
  vec<T, 3> &xyz() { return *reinterpret_cast<vec<T, 3> *>(this); }

  template <class U, class = detail::conv_t<vec, U>>
  constexpr vec(const U &u) : vec(converter<vec, U>{}(u)) {}
  template <class U, class = detail::conv_t<U, vec>>
  constexpr operator U() const {
    return converter<U, vec>{}(*this);
  }
};
/** @} */

/** @addtogroup mat
 * @ingroup LinAlg
 * @brief `linalg::mat<T,M,N>` defines a fixed-size matrix containing exactly
   `M` rows and `N` columns of type `T`, in column-major order.

This data structure is supported by a set of algebraic and component-wise
functions, as well as a set of standard reductions.

`mat<T,M,N>`:
- is
  [`DefaultConstructible`](https://en.cppreference.com/w/cpp/named_req/DefaultConstructible):
  ```cpp
  float2x2 m; // m contains columns 0,0; 0,0
  ```
- is constructible from `N` columns of type `vec<T,M>`:
  ```cpp
  float2x2 m {{1,2},{3,4}}; // m contains columns 1,2; 3,4
  ```
- is constructible from `linalg::identity`:
  ```cpp
  float3x3 m = linalg::identity; // m contains columns 1,0,0; 0,1,0; 0,0,1
  ```
- is
  [`CopyConstructible`](https://en.cppreference.com/w/cpp/named_req/CopyConstructible)
  and
  [`CopyAssignable`](https://en.cppreference.com/w/cpp/named_req/CopyAssignable):
  ```cpp
  float2x2 m {{1,2},{3,4}}; // m contains columns 1,2; 3,4
  float2x2 n {m};           // n contains columns 1,2; 3,4
  float2x2 p;               // p contains columns 0,0; 0,0
  p = n;                    // p contains columns 1,2; 3,4
  ```
- is
  [`EqualityComparable`](https://en.cppreference.com/w/cpp/named_req/EqualityComparable)
  and
  [`LessThanComparable`](https://en.cppreference.com/w/cpp/named_req/LessThanComparable):
  ```cpp
  if(m == n) cout << "m and n contain equal elements in the same positions" <<
  endl; if(m < n) cout << "m precedes n lexicographically when compared in
  column-major order" << endl;
  ```
- is **explicitly** constructible from a single element of type `T`:
  ```cpp
  float2x2 m {5}; // m contains columns 5,5; 5,5
  ```
- is **explicitly** constructible from a `mat<U,M,N>` of some other type `U`:
  ```cpp
  float2x2 m {int2x2{{5,6},{7,8}}}; // m contains columns 5,6; 7,8
  ```
- supports indexing into *columns*:
  ```cpp
  float2x3 m {{1,2},{3,4},{5,6}}; // m contains columns 1,2; 3,4; 5,6
  float2 c = m[0];                // c contains 1,2
  m[1]     = {7,8};               // m contains columns 1,2; 7,8; 5,6
  ```
- supports retrieval (but not assignment) of rows:
  ```cpp
  float2x3 m {{1,2},{3,4},{5,6}}; // m contains columns 1,2; 3,4; 5,6
  float3 r = m.row(1);            // r contains 2,4,6
  ```

- supports unary operators `+`, `-`, `!` and `~` in component-wise fashion:
  ```cpp
  float2x2 m {{1,2},{3,4}}; // m contains columns 1,2; 3,4
  float2x2 n = -m;          // n contains columns -1,-2; -3,-4
  ```
- supports binary operators `+`, `-`, `*`, `/`, `%`, `|`, `&`, `^`, `<<` and
  `>>` in component-wise fashion:
  ```cpp
  float2x2 a {{0,0},{2,2}}; // a contains columns 0,0; 2,2
  float2x2 b {{1,2},{1,2}}; // b contains columns 1,2; 1,2
  float2x2 c = a + b;       // c contains columns 1,2; 3,4
  ```

- supports binary operators with a scalar on the left or the right:
  ```cpp
  auto m = 2 * float2x2{{1,2},{3,4}}; // m is float2x2{{2,4},{6,8}}
  ```

- supports operators `+=`, `-=`, `*=`, `/=`, `%=`, `|=`, `&=`, `^=`, `<<=` and
  `>>=` with matrices or scalars on the right:
  ```cpp
  float2x2 v {{5,4},{3,2}};
  v *= 3; // v is float2x2{{15,12},{9,6}}
  ```

- supports operations on mixed element types:

- supports [range-based
  for](https://en.cppreference.com/w/cpp/language/range-for) over columns

- has a flat memory layout
 *  @{
 */
template <class T, int M>
struct mat<T, M, 1> {
  using V = vec<T, M>;
  V x;
  constexpr mat() : x() {}
  constexpr mat(const V &x_) : x(x_) {}
  constexpr explicit mat(const T &s) : x(s) {}
  constexpr explicit mat(const T *p) : x(p + M * 0) {}
  template <class U>
  constexpr explicit mat(const mat<U, M, 1> &m) : mat(V(m.x)) {}
  constexpr vec<T, 1> row(int i) const { return {x[i]}; }
  constexpr const V &operator[](int) const { return x; }
  LINALG_CONSTEXPR14 V &operator[](int) { return x; }

  template <class U, class = detail::conv_t<mat, U>>
  constexpr mat(const U &u) : mat(converter<mat, U>{}(u)) {}
  template <class U, class = detail::conv_t<U, mat>>
  constexpr operator U() const {
    return converter<U, mat>{}(*this);
  }
};
template <class T, int M>
struct mat<T, M, 2> {
  using V = vec<T, M>;
  V x, y;
  constexpr mat() : x(), y() {}
  constexpr mat(const V &x_, const V &y_) : x(x_), y(y_) {}
  constexpr explicit mat(const T &s) : x(s), y(s) {}
  constexpr explicit mat(const T *p) : x(p + M * 0), y(p + M * 1) {}
  template <class U, int N, int P>
  constexpr explicit mat(const mat<U, N, P> &m) : mat(V(m.x), V(m.y)) {
    static_assert(P >= 2, "Input matrix dimensions must be at least as big.");
  }
  constexpr vec<T, 2> row(int i) const { return {x[i], y[i]}; }
  constexpr const V &operator[](int j) const { return j == 0 ? x : y; }
  LINALG_CONSTEXPR14 V &operator[](int j) { return j == 0 ? x : y; }

  template <class U, class = detail::conv_t<mat, U>>
  constexpr mat(const U &u) : mat(converter<mat, U>{}(u)) {}
  template <class U, class = detail::conv_t<U, mat>>
  constexpr operator U() const {
    return converter<U, mat>{}(*this);
  }
};
template <class T, int M>
struct mat<T, M, 3> {
  using V = vec<T, M>;
  V x, y, z;
  constexpr mat() : x(), y(), z() {}
  constexpr mat(const V &x_, const V &y_, const V &z_) : x(x_), y(y_), z(z_) {}
  constexpr mat(const mat<T, M, 2> &m_, const V &z_)
      : x(m_.x), y(m_.y), z(z_) {}
  constexpr explicit mat(const T &s) : x(s), y(s), z(s) {}
  constexpr explicit mat(const T *p)
      : x(p + M * 0), y(p + M * 1), z(p + M * 2) {}
  template <class U, int N, int P>
  constexpr explicit mat(const mat<U, N, P> &m) : mat(V(m.x), V(m.y), V(m.z)) {
    static_assert(P >= 3, "Input matrix dimensions must be at least as big.");
  }
  constexpr vec<T, 3> row(int i) const { return {x[i], y[i], z[i]}; }
  constexpr const V &operator[](int j) const {
    return j == 0 ? x : j == 1 ? y : z;
  }
  LINALG_CONSTEXPR14 V &operator[](int j) {
    return j == 0 ? x : j == 1 ? y : z;
  }

  template <class U, class = detail::conv_t<mat, U>>
  constexpr mat(const U &u) : mat(converter<mat, U>{}(u)) {}
  template <class U, class = detail::conv_t<U, mat>>
  constexpr operator U() const {
    return converter<U, mat>{}(*this);
  }
};
template <class T, int M>
struct mat<T, M, 4> {
  using V = vec<T, M>;
  V x, y, z, w;
  constexpr mat() : x(), y(), z(), w() {}
  constexpr mat(const V &x_, const V &y_, const V &z_, const V &w_)
      : x(x_), y(y_), z(z_), w(w_) {}
  constexpr mat(const mat<T, M, 3> &m_, const V &w_)
      : x(m_.x), y(m_.y), z(m_.z), w(w_) {}
  constexpr explicit mat(const T &s) : x(s), y(s), z(s), w(s) {}
  constexpr explicit mat(const T *p)
      : x(p + M * 0), y(p + M * 1), z(p + M * 2), w(p + M * 3) {}
  template <class U, int N, int P>
  constexpr explicit mat(const mat<U, N, P> &m)
      : mat(V(m.x), V(m.y), V(m.z), V(m.w)) {
    static_assert(P >= 4, "Input matrix dimensions must be at least as big.");
  }

  constexpr vec<T, 4> row(int i) const { return {x[i], y[i], z[i], w[i]}; }
  constexpr const V &operator[](int j) const {
    return j == 0 ? x : j == 1 ? y : j == 2 ? z : w;
  }
  LINALG_CONSTEXPR14 V &operator[](int j) {
    return j == 0 ? x : j == 1 ? y : j == 2 ? z : w;
  }

  template <class U, class = detail::conv_t<mat, U>>
  constexpr mat(const U &u) : mat(converter<mat, U>{}(u)) {}
  template <class U, class = detail::conv_t<U, mat>>
  constexpr operator U() const {
    return converter<U, mat>{}(*this);
  }
};
/** @} */

/** @addtogroup identity
 * @ingroup LinAlg
 * @brief Define a type which will convert to the multiplicative identity of any
 * square matrix.
 *  @{
 */
struct identity_t {
  constexpr explicit identity_t(int) {}
};
template <class T>
struct converter<mat<T, 1, 1>, identity_t> {
  constexpr mat<T, 1, 1> operator()(identity_t) const { return {vec<T, 1>{1}}; }
};
template <class T>
struct converter<mat<T, 2, 2>, identity_t> {
  constexpr mat<T, 2, 2> operator()(identity_t) const {
    return {{1, 0}, {0, 1}};
  }
};
template <class T>
struct converter<mat<T, 2, 3>, identity_t> {
  constexpr mat<T, 2, 3> operator()(identity_t) const {
    return {{1, 0}, {0, 1}, {0, 0}};
  }
};
template <class T>
struct converter<mat<T, 3, 3>, identity_t> {
  constexpr mat<T, 3, 3> operator()(identity_t) const {
    return {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}};
  }
};
template <class T>
struct converter<mat<T, 3, 4>, identity_t> {
  constexpr mat<T, 3, 4> operator()(identity_t) const {
    return {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}, {0, 0, 0}};
  }
};
template <class T>
struct converter<mat<T, 4, 4>, identity_t> {
  constexpr mat<T, 4, 4> operator()(identity_t) const {
    return {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}};
  }
};
constexpr identity_t identity{1};
/** @} */

/** @addtogroup fold
 * @ingroup LinAlg
 * @brief Produce a scalar by applying f(A,B) -> A to adjacent pairs of elements
 * from a vec/mat in left-to-right/column-major order (matching the
 * associativity of arithmetic and logical operators).
 *  @{
 */
template <class F, class A, class B>
constexpr A fold(F f, A a, const vec<B, 1> &b) {
  return f(a, b.x);
}
template <class F, class A, class B>
constexpr A fold(F f, A a, const vec<B, 2> &b) {
  return f(f(a, b.x), b.y);
}
template <class F, class A, class B>
constexpr A fold(F f, A a, const vec<B, 3> &b) {
  return f(f(f(a, b.x), b.y), b.z);
}
template <class F, class A, class B>
constexpr A fold(F f, A a, const vec<B, 4> &b) {
  return f(f(f(f(a, b.x), b.y), b.z), b.w);
}
template <class F, class A, class B, int M>
constexpr A fold(F f, A a, const mat<B, M, 1> &b) {
  return fold(f, a, b.x);
}
template <class F, class A, class B, int M>
constexpr A fold(F f, A a, const mat<B, M, 2> &b) {
  return fold(f, fold(f, a, b.x), b.y);
}
template <class F, class A, class B, int M>
constexpr A fold(F f, A a, const mat<B, M, 3> &b) {
  return fold(f, fold(f, fold(f, a, b.x), b.y), b.z);
}
template <class F, class A, class B, int M>
constexpr A fold(F f, A a, const mat<B, M, 4> &b) {
  return fold(f, fold(f, fold(f, fold(f, a, b.x), b.y), b.z), b.w);
}
/** @} */

/** @addtogroup apply
 * @ingroup LinAlg
 * @brief apply(f,...) applies the provided function in an elementwise fashion
 * to its arguments, producing an object of the same dimensions.
 *  @{
 */

// Type aliases for the result of calling apply(...) with various arguments, can
// be used with return type SFINAE to constrain overload sets
template <class F, class... A>
using apply_t = typename detail::apply<F, void, A...>::type;
template <class F, class... A>
using mm_apply_t = typename std::enable_if<detail::apply<F, void, A...>::mm,
                                           apply_t<F, A...>>::type;
template <class F, class... A>
using no_mm_apply_t = typename std::enable_if<!detail::apply<F, void, A...>::mm,
                                              apply_t<F, A...>>::type;
template <class A>
using scalar_t =
    typename detail::scalar_type<A>::type;  // Underlying scalar type when
                                            // performing elementwise operations

// apply(f,...) applies the provided function in an elementwise fashion to its
// arguments, producing an object of the same dimensions
template <class F, class... A>
constexpr apply_t<F, A...> apply(F func, const A &...args) {
  return detail::apply<F, void, A...>::impl(
      detail::make_seq<0, detail::apply<F, void, A...>::size>{}, func, args...);
}

// map(a,f) is equivalent to apply(f,a)
template <class A, class F>
constexpr apply_t<F, A> map(const A &a, F func) {
  return apply(func, a);
}

// zip(a,b,f) is equivalent to apply(f,a,b)
template <class A, class B, class F>
constexpr apply_t<F, A, B> zip(const A &a, const B &b, F func) {
  return apply(func, a, b);
}
/** @} */

/** @addtogroup comparison_ops
 * @ingroup LinAlg
 * @brief Relational operators are defined to compare the elements of two
 * vectors or matrices lexicographically, in column-major order.
 *  @{
 */
template <class A, class B>
constexpr typename detail::any_compare<A, B>::type compare(const A &a,
                                                           const B &b) {
  return detail::any_compare<A, B>()(a, b);
}
template <class A, class B>
constexpr auto operator==(const A &a,
                          const B &b) -> decltype(compare(a, b) == 0) {
  return compare(a, b) == 0;
}
template <class A, class B>
constexpr auto operator!=(const A &a,
                          const B &b) -> decltype(compare(a, b) != 0) {
  return compare(a, b) != 0;
}
template <class A, class B>
constexpr auto operator<(const A &a,
                         const B &b) -> decltype(compare(a, b) < 0) {
  return compare(a, b) < 0;
}
template <class A, class B>
constexpr auto operator>(const A &a,
                         const B &b) -> decltype(compare(a, b) > 0) {
  return compare(a, b) > 0;
}
template <class A, class B>
constexpr auto operator<=(const A &a,
                          const B &b) -> decltype(compare(a, b) <= 0) {
  return compare(a, b) <= 0;
}
template <class A, class B>
constexpr auto operator>=(const A &a,
                          const B &b) -> decltype(compare(a, b) >= 0) {
  return compare(a, b) >= 0;
}
/** @} */

/** @addtogroup reductions
 * @ingroup LinAlg
 * @brief Functions for coalescing scalar values.
 *  @{
 */
template <class A>
constexpr bool any(const A &a) {
  return fold(detail::op_or{}, false, a);
}
template <class A>
constexpr bool all(const A &a) {
  return fold(detail::op_and{}, true, a);
}
template <class A>
constexpr scalar_t<A> sum(const A &a) {
  return fold(detail::op_add{}, scalar_t<A>(0), a);
}
template <class A>
constexpr scalar_t<A> product(const A &a) {
  return fold(detail::op_mul{}, scalar_t<A>(1), a);
}
template <class A>
constexpr scalar_t<A> minelem(const A &a) {
  return fold(detail::min{}, a.x, a);
}
template <class A>
constexpr scalar_t<A> maxelem(const A &a) {
  return fold(detail::max{}, a.x, a);
}
template <class T, int M>
int argmin(const vec<T, M> &a) {
  int j = 0;
  for (int i = 1; i < M; ++i)
    if (a[i] < a[j]) j = i;
  return j;
}
template <class T, int M>
int argmax(const vec<T, M> &a) {
  int j = 0;
  for (int i = 1; i < M; ++i)
    if (a[i] > a[j]) j = i;
  return j;
}
/** @} */

/** @addtogroup unary_ops
 * @ingroup LinAlg
 * @brief Unary operators are defined component-wise for linalg types.
 *  @{
 */
template <class A>
constexpr apply_t<detail::op_pos, A> operator+(const A &a) {
  return apply(detail::op_pos{}, a);
}
template <class A>
constexpr apply_t<detail::op_neg, A> operator-(const A &a) {
  return apply(detail::op_neg{}, a);
}
template <class A>
constexpr apply_t<detail::op_cmp, A> operator~(const A &a) {
  return apply(detail::op_cmp{}, a);
}
template <class A>
constexpr apply_t<detail::op_not, A> operator!(const A &a) {
  return apply(detail::op_not{}, a);
}
/** @} */

/** @addtogroup binary_ops
 * @ingroup LinAlg
 * @brief Binary operators are defined component-wise for linalg types, EXCEPT
 * for `operator *`, which does standard matrix multiplication, scalar
 * multiplication, and component-wise multiplication for same-size vectors. Use
 * `cmul` for the matrix Hadamard product.
 *  @{
 */
template <class A, class B>
constexpr apply_t<detail::op_add, A, B> operator+(const A &a, const B &b) {
  return apply(detail::op_add{}, a, b);
}
template <class A, class B>
constexpr apply_t<detail::op_sub, A, B> operator-(const A &a, const B &b) {
  return apply(detail::op_sub{}, a, b);
}
template <class A, class B>
constexpr apply_t<detail::op_mul, A, B> cmul(const A &a, const B &b) {
  return apply(detail::op_mul{}, a, b);
}
template <class A, class B>
constexpr apply_t<detail::op_div, A, B> operator/(const A &a, const B &b) {
  return apply(detail::op_div{}, a, b);
}
template <class A, class B>
constexpr apply_t<detail::op_mod, A, B> operator%(const A &a, const B &b) {
  return apply(detail::op_mod{}, a, b);
}
template <class A, class B>
constexpr apply_t<detail::op_un, A, B> operator|(const A &a, const B &b) {
  return apply(detail::op_un{}, a, b);
}
template <class A, class B>
constexpr apply_t<detail::op_xor, A, B> operator^(const A &a, const B &b) {
  return apply(detail::op_xor{}, a, b);
}
template <class A, class B>
constexpr apply_t<detail::op_int, A, B> operator&(const A &a, const B &b) {
  return apply(detail::op_int{}, a, b);
}
template <class A, class B>
constexpr apply_t<detail::op_lsh, A, B> operator<<(const A &a, const B &b) {
  return apply(detail::op_lsh{}, a, b);
}
template <class A, class B>
constexpr apply_t<detail::op_rsh, A, B> operator>>(const A &a, const B &b) {
  return apply(detail::op_rsh{}, a, b);
}

// Binary `operator *` represents the algebraic matrix product - use cmul(a, b)
// for the Hadamard (component-wise) product.
template <class A, class B>
constexpr auto operator*(const A &a, const B &b) {
  return mul(a, b);
}

// Binary assignment operators a $= b is always defined as though it were
// explicitly written a = a $ b
template <class A, class B>
constexpr auto operator+=(A &a, const B &b) -> decltype(a = a + b) {
  return a = a + b;
}
template <class A, class B>
constexpr auto operator-=(A &a, const B &b) -> decltype(a = a - b) {
  return a = a - b;
}
template <class A, class B>
constexpr auto operator*=(A &a, const B &b) -> decltype(a = a * b) {
  return a = a * b;
}
template <class A, class B>
constexpr auto operator/=(A &a, const B &b) -> decltype(a = a / b) {
  return a = a / b;
}
template <class A, class B>
constexpr auto operator%=(A &a, const B &b) -> decltype(a = a % b) {
  return a = a % b;
}
template <class A, class B>
constexpr auto operator|=(A &a, const B &b) -> decltype(a = a | b) {
  return a = a | b;
}
template <class A, class B>
constexpr auto operator^=(A &a, const B &b) -> decltype(a = a ^ b) {
  return a = a ^ b;
}
template <class A, class B>
constexpr auto operator&=(A &a, const B &b) -> decltype(a = a & b) {
  return a = a & b;
}
template <class A, class B>
constexpr auto operator<<=(A &a, const B &b) -> decltype(a = a << b) {
  return a = a << b;
}
template <class A, class B>
constexpr auto operator>>=(A &a, const B &b) -> decltype(a = a >> b) {
  return a = a >> b;
}
/** @} */

/** @addtogroup swizzles
 * @ingroup LinAlg
 * @brief Swizzles and subobjects.
 *  @{
 */
/**
 * @brief Returns a vector containing the specified ordered indices, e.g.
 * linalg::swizzle<1, 2, 0>(vec4(4, 5, 6, 7)) == vec3(5, 6, 4)
 */
template <int... I, class T, int M>
constexpr vec<T, sizeof...(I)> swizzle(const vec<T, M> &a) {
  return {detail::getter<I>{}(a)...};
}
/**
 * @brief Returns a vector containing the specified index range, e.g.
 * linalg::subvec<1, 4>(vec4(4, 5, 6, 7)) == vec3(5, 6, 7)
 */
template <int I0, int I1, class T, int M>
constexpr vec<T, I1 - I0> subvec(const vec<T, M> &a) {
  return detail::swizzle(a, detail::make_seq<I0, I1>{});
}
/**
 * @brief Returns a matrix containing the specified row and column range:
 * linalg::submat<rowStart, colStart, rowEnd, colEnd>
 */
template <int I0, int J0, int I1, int J1, class T, int M, int N>
constexpr mat<T, I1 - I0, J1 - J0> submat(const mat<T, M, N> &a) {
  return detail::swizzle(a, detail::make_seq<I0, I1>{},
                         detail::make_seq<J0, J1>{});
}
/** @} */

/** @addtogroup unary_STL
 * @ingroup LinAlg
 * @brief Component-wise standard library math functions.
 *  @{
 */
template <class A>
constexpr apply_t<detail::std_isfinite, A> isfinite(const A &a) {
  return apply(detail::std_isfinite{}, a);
}
template <class A>
constexpr apply_t<detail::std_abs, A> abs(const A &a) {
  return apply(detail::std_abs{}, a);
}
template <class A>
constexpr apply_t<detail::std_floor, A> floor(const A &a) {
  return apply(detail::std_floor{}, a);
}
template <class A>
constexpr apply_t<detail::std_ceil, A> ceil(const A &a) {
  return apply(detail::std_ceil{}, a);
}
template <class A>
constexpr apply_t<detail::std_exp, A> exp(const A &a) {
  return apply(detail::std_exp{}, a);
}
template <class A>
constexpr apply_t<detail::std_log, A> log(const A &a) {
  return apply(detail::std_log{}, a);
}
template <class A>
constexpr apply_t<detail::std_log2, A> log2(const A &a) {
  return apply(detail::std_log2{}, a);
}
template <class A>
constexpr apply_t<detail::std_log10, A> log10(const A &a) {
  return apply(detail::std_log10{}, a);
}
template <class A>
constexpr apply_t<detail::std_sqrt, A> sqrt(const A &a) {
  return apply(detail::std_sqrt{}, a);
}
template <class A>
constexpr apply_t<detail::std_sin, A> sin(const A &a) {
  return apply(detail::std_sin{}, a);
}
template <class A>
constexpr apply_t<detail::std_cos, A> cos(const A &a) {
  return apply(detail::std_cos{}, a);
}
template <class A>
constexpr apply_t<detail::std_tan, A> tan(const A &a) {
  return apply(detail::std_tan{}, a);
}
template <class A>
constexpr apply_t<detail::std_asin, A> asin(const A &a) {
  return apply(detail::std_asin{}, a);
}
template <class A>
constexpr apply_t<detail::std_acos, A> acos(const A &a) {
  return apply(detail::std_acos{}, a);
}
template <class A>
constexpr apply_t<detail::std_atan, A> atan(const A &a) {
  return apply(detail::std_atan{}, a);
}
template <class A>
constexpr apply_t<detail::std_sinh, A> sinh(const A &a) {
  return apply(detail::std_sinh{}, a);
}
template <class A>
constexpr apply_t<detail::std_cosh, A> cosh(const A &a) {
  return apply(detail::std_cosh{}, a);
}
template <class A>
constexpr apply_t<detail::std_tanh, A> tanh(const A &a) {
  return apply(detail::std_tanh{}, a);
}
template <class A>
constexpr apply_t<detail::std_round, A> round(const A &a) {
  return apply(detail::std_round{}, a);
}
/** @} */

/** @addtogroup binary_STL
 * @ingroup LinAlg
 * @brief Component-wise standard library math functions. Either argument can be
 * a vector or a scalar.
 *  @{
 */
template <class A, class B>
constexpr apply_t<detail::std_fmod, A, B> fmod(const A &a, const B &b) {
  return apply(detail::std_fmod{}, a, b);
}
template <class A, class B>
constexpr apply_t<detail::std_pow, A, B> pow(const A &a, const B &b) {
  return apply(detail::std_pow{}, a, b);
}
template <class A, class B>
constexpr apply_t<detail::std_atan2, A, B> atan2(const A &a, const B &b) {
  return apply(detail::std_atan2{}, a, b);
}
template <class A, class B>
constexpr apply_t<detail::std_copysign, A, B> copysign(const A &a, const B &b) {
  return apply(detail::std_copysign{}, a, b);
}
/** @} */

/** @addtogroup relational
 * @ingroup LinAlg
 * @brief Component-wise relational functions on vectors. Either argument can be
 * a vector or a scalar.
 *  @{
 */
template <class A, class B>
constexpr apply_t<detail::op_eq, A, B> equal(const A &a, const B &b) {
  return apply(detail::op_eq{}, a, b);
}
template <class A, class B>
constexpr apply_t<detail::op_ne, A, B> nequal(const A &a, const B &b) {
  return apply(detail::op_ne{}, a, b);
}
template <class A, class B>
constexpr apply_t<detail::op_lt, A, B> less(const A &a, const B &b) {
  return apply(detail::op_lt{}, a, b);
}
template <class A, class B>
constexpr apply_t<detail::op_gt, A, B> greater(const A &a, const B &b) {
  return apply(detail::op_gt{}, a, b);
}
template <class A, class B>
constexpr apply_t<detail::op_le, A, B> lequal(const A &a, const B &b) {
  return apply(detail::op_le{}, a, b);
}
template <class A, class B>
constexpr apply_t<detail::op_ge, A, B> gequal(const A &a, const B &b) {
  return apply(detail::op_ge{}, a, b);
}
/** @} */

/** @addtogroup selection
 * @ingroup LinAlg
 * @brief Component-wise selection functions on vectors. Either argument can be
 * a vector or a scalar.
 *  @{
 */
template <class A, class B>
constexpr apply_t<detail::min, A, B> min(const A &a, const B &b) {
  return apply(detail::min{}, a, b);
}
template <class A, class B>
constexpr apply_t<detail::max, A, B> max(const A &a, const B &b) {
  return apply(detail::max{}, a, b);
}
/**
 * @brief Clamps the components of x between l and h, provided l[i] < h[i].
 */
template <class X, class L, class H>
constexpr apply_t<detail::clamp, X, L, H> clamp(const X &x, const L &l,
                                                const H &h) {
  return apply(detail::clamp{}, x, l, h);
}
/**
 * @brief Returns the component from a if the corresponding component of p is
 * true and from b otherwise.
 */
template <class P, class A, class B>
constexpr apply_t<detail::select, P, A, B> select(const P &p, const A &a,
                                                  const B &b) {
  return apply(detail::select{}, p, a, b);
}
/**
 * @brief Linear interpolation from a to b as t goes from 0 -> 1. Values beyond
 * [a, b] will result if t is outside [0, 1].
 */
template <class A, class B, class T>
constexpr apply_t<detail::lerp, A, B, T> lerp(const A &a, const B &b,
                                              const T &t) {
  return apply(detail::lerp{}, a, b, t);
}
/** @} */

/** @addtogroup vec_algebra
 * @ingroup LinAlg
 * @brief Support for vector algebra.
 *  @{
 */
/**
 * @brief shorthand for `cross({a.x,a.y,0}, {b.x,b.y,0}).z`
 */
template <class T>
constexpr T cross(const vec<T, 2> &a, const vec<T, 2> &b) {
  return a.x * b.y - a.y * b.x;
}
/**
 * @brief shorthand for `cross({0,0,a.z}, {b.x,b.y,0}).xy()`
 */
template <class T>
constexpr vec<T, 2> cross(T a, const vec<T, 2> &b) {
  return {-a * b.y, a * b.x};
}
/**
 * @brief shorthand for `cross({a.x,a.y,0}, {0,0,b.z}).xy()`
 */
template <class T>
constexpr vec<T, 2> cross(const vec<T, 2> &a, T b) {
  return {a.y * b, -a.x * b};
}
/**
 * @brief the [cross or vector
 * product](https://en.wikipedia.org/wiki/Cross_product) of vectors `a` and `b`
 */
template <class T>
constexpr vec<T, 3> cross(const vec<T, 3> &a, const vec<T, 3> &b) {
  return {a.y * b.z - a.z * b.y, a.z * b.x - a.x * b.z, a.x * b.y - a.y * b.x};
}
/**
 * @brief [dot or inner product](https://en.wikipedia.org/wiki/Dot_product) of
 * vectors `a` and `b`
 */
template <class T, int M>
constexpr T dot(const vec<T, M> &a, const vec<T, M> &b) {
  return sum(a * b);
}
/**
 * @brief *square* of the length or magnitude of vector `a`
 */
template <class T, int M>
constexpr T length2(const vec<T, M> &a) {
  return dot(a, a);
}
/**
 * @brief length or magnitude of a vector `a`
 */
template <class T, int M>
T length(const vec<T, M> &a) {
  return std::sqrt(length2(a));
}
/**
 * @brief unit length vector in the same direction as `a` (undefined for
 zero-length vectors)

 */
template <class T, int M>
vec<T, M> normalize(const vec<T, M> &a) {
  return a / length(a);
}
/**
 * @brief *square* of the [Euclidean
 * distance](https://en.wikipedia.org/wiki/Euclidean_distance) between points
 * `a` and `b`
 */
template <class T, int M>
constexpr T distance2(const vec<T, M> &a, const vec<T, M> &b) {
  return length2(b - a);
}
/**
 * @brief [Euclidean distance](https://en.wikipedia.org/wiki/Euclidean_distance)
 * between points `a` and `b`
 */
template <class T, int M>
T distance(const vec<T, M> &a, const vec<T, M> &b) {
  return length(b - a);
}
/**
 * @brief Return the angle in radians between two unit vectors.
 */
template <class T, int M>
T uangle(const vec<T, M> &a, const vec<T, M> &b) {
  T d = dot(a, b);
  return d > 1 ? 0 : std::acos(d < -1 ? -1 : d);
}
/**
 * @brief Return the angle in radians between two non-unit vectors.
 */
template <class T, int M>
T angle(const vec<T, M> &a, const vec<T, M> &b) {
  return uangle(normalize(a), normalize(b));
}
/**
 * @brief vector `v` rotated counter-clockwise by the angle `a` in
 * [radians](https://en.wikipedia.org/wiki/Radian)
 */
template <class T>
vec<T, 2> rot(T a, const vec<T, 2> &v) {
  const T s = std::sin(a), c = std::cos(a);
  return {v.x * c - v.y * s, v.x * s + v.y * c};
}
/**
 * @brief vector `v` rotated counter-clockwise by the angle `a` in
 * [radians](https://en.wikipedia.org/wiki/Radian) around the X axis
 */
template <class T>
vec<T, 3> rotx(T a, const vec<T, 3> &v) {
  const T s = std::sin(a), c = std::cos(a);
  return {v.x, v.y * c - v.z * s, v.y * s + v.z * c};
}
/**
 * @brief vector `v` rotated counter-clockwise by the angle `a` in
 * [radians](https://en.wikipedia.org/wiki/Radian) around the Y axis
 */
template <class T>
vec<T, 3> roty(T a, const vec<T, 3> &v) {
  const T s = std::sin(a), c = std::cos(a);
  return {v.x * c + v.z * s, v.y, -v.x * s + v.z * c};
}
/**
 * @brief vector `v` rotated counter-clockwise by the angle `a` in
 * [radians](https://en.wikipedia.org/wiki/Radian) around the Z axis
 */
template <class T>
vec<T, 3> rotz(T a, const vec<T, 3> &v) {
  const T s = std::sin(a), c = std::cos(a);
  return {v.x * c - v.y * s, v.x * s + v.y * c, v.z};
}
/**
 * @brief shorthand for `normalize(lerp(a,b,t))`
 */
template <class T, int M>
vec<T, M> nlerp(const vec<T, M> &a, const vec<T, M> &b, T t) {
  return normalize(lerp(a, b, t));
}
/**
 * @brief [spherical linear interpolation](https://en.wikipedia.org/wiki/Slerp)
 * between unit vectors `a` and `b` (undefined for non-unit vectors) by
 * parameter `t`
 */
template <class T, int M>
vec<T, M> slerp(const vec<T, M> &a, const vec<T, M> &b, T t) {
  T th = uangle(a, b);
  return th == 0 ? a
                 : a * (std::sin(th * (1 - t)) / std::sin(th)) +
                       b * (std::sin(th * t) / std::sin(th));
}
/** @} */

/** @addtogroup quaternions
 * @ingroup LinAlg
 * @brief Support for quaternion algebra using 4D vectors of arbitrary length,
 * representing xi + yj + zk + w.
 *  @{
 */
/**
 * @brief
 * [conjugate](https://en.wikipedia.org/wiki/Quaternion#Conjugation,_the_norm,_and_reciprocal)
 * of quaternion `q`
 */
template <class T>
constexpr vec<T, 4> qconj(const vec<T, 4> &q) {
  return {-q.x, -q.y, -q.z, q.w};
}
/**
 * @brief [inverse or
 * reciprocal](https://en.wikipedia.org/wiki/Quaternion#Conjugation,_the_norm,_and_reciprocal)
 * of quaternion `q` (undefined for zero-length quaternions)
 */
template <class T>
vec<T, 4> qinv(const vec<T, 4> &q) {
  return qconj(q) / length2(q);
}
/**
 * @brief
 * [exponential](https://en.wikipedia.org/wiki/Quaternion#Exponential,_logarithm,_and_power_functions)
 * of quaternion `q`
 */
template <class T>
vec<T, 4> qexp(const vec<T, 4> &q) {
  const auto v = q.xyz();
  const auto vv = length(v);
  return std::exp(q.w) *
         vec<T, 4>{v * (vv > 0 ? std::sin(vv) / vv : 0), std::cos(vv)};
}
/**
 * @brief
 * [logarithm](https://en.wikipedia.org/wiki/Quaternion#Exponential,_logarithm,_and_power_functions)
 * of quaternion `q`
 */
template <class T>
vec<T, 4> qlog(const vec<T, 4> &q) {
  const auto v = q.xyz();
  const auto vv = length(v), qq = length(q);
  return {v * (vv > 0 ? std::acos(q.w / qq) / vv : 0), std::log(qq)};
}
/**
 * @brief quaternion `q` raised to the exponent `p`
 */
template <class T>
vec<T, 4> qpow(const vec<T, 4> &q, const T &p) {
  const auto v = q.xyz();
  const auto vv = length(v), qq = length(q), th = std::acos(q.w / qq);
  return std::pow(qq, p) *
         vec<T, 4>{v * (vv > 0 ? std::sin(p * th) / vv : 0), std::cos(p * th)};
}
/**
 * @brief [Hamilton
 * product](https://en.wikipedia.org/wiki/Quaternion#Hamilton_product) of
 * quaternions `a` and `b`
 */
template <class T>
constexpr vec<T, 4> qmul(const vec<T, 4> &a, const vec<T, 4> &b) {
  return {a.x * b.w + a.w * b.x + a.y * b.z - a.z * b.y,
          a.y * b.w + a.w * b.y + a.z * b.x - a.x * b.z,
          a.z * b.w + a.w * b.z + a.x * b.y - a.y * b.x,
          a.w * b.w - a.x * b.x - a.y * b.y - a.z * b.z};
}
/**
 * @brief Multiply as many input quaternions together as desired.
 */
template <class T, class... R>
constexpr vec<T, 4> qmul(const vec<T, 4> &a, R... r) {
  return qmul(a, qmul(r...));
}
/** @} */

/** @addtogroup quaternion_rotation
 * @ingroup LinAlg
 * @brief Support for 3D spatial rotations using normalized quaternions.
 *  @{
 */
/**
 * @brief efficient shorthand for `qrot(q, {1,0,0})`
 */
template <class T>
constexpr vec<T, 3> qxdir(const vec<T, 4> &q) {
  return {q.w * q.w + q.x * q.x - q.y * q.y - q.z * q.z,
          (q.x * q.y + q.z * q.w) * 2, (q.z * q.x - q.y * q.w) * 2};
}
/**
 * @brief efficient shorthand for `qrot(q, {0,1,0})`
 */
template <class T>
constexpr vec<T, 3> qydir(const vec<T, 4> &q) {
  return {(q.x * q.y - q.z * q.w) * 2,
          q.w * q.w - q.x * q.x + q.y * q.y - q.z * q.z,
          (q.y * q.z + q.x * q.w) * 2};
}
/**
 * @brief efficient shorthand for `qrot(q, {0,0,1})`
 */
template <class T>
constexpr vec<T, 3> qzdir(const vec<T, 4> &q) {
  return {(q.z * q.x + q.y * q.w) * 2, (q.y * q.z - q.x * q.w) * 2,
          q.w * q.w - q.x * q.x - q.y * q.y + q.z * q.z};
}
/**
 * @brief Create an equivalent mat3 rotation matrix from the input quaternion.
 */
template <class T>
constexpr mat<T, 3, 3> qmat(const vec<T, 4> &q) {
  return {qxdir(q), qydir(q), qzdir(q)};
}
/**
 * @brief Rotate a vector by a quaternion.
 */
template <class T>
constexpr vec<T, 3> qrot(const vec<T, 4> &q, const vec<T, 3> &v) {
  return qxdir(q) * v.x + qydir(q) * v.y + qzdir(q) * v.z;
}
/**
 * @brief Return the angle in radians of the axis-angle representation of the
 * input normalized quaternion.
 */
template <class T>
T qangle(const vec<T, 4> &q) {
  return std::atan2(length(q.xyz()), q.w) * 2;
}
/**
 * @brief Return the normalized axis of the axis-angle representation of the
 * input normalized quaternion.
 */
template <class T>
vec<T, 3> qaxis(const vec<T, 4> &q) {
  return normalize(q.xyz());
}
/**
 * @brief Linear interpolation that takes the shortest path - this is not
 * geometrically sensible, consider qslerp instead.
 */
template <class T>
vec<T, 4> qnlerp(const vec<T, 4> &a, const vec<T, 4> &b, T t) {
  return nlerp(a, dot(a, b) < 0 ? -b : b, t);
}
/**
 * @brief Spherical linear interpolation that takes the shortest path.
 */
template <class T>
vec<T, 4> qslerp(const vec<T, 4> &a, const vec<T, 4> &b, T t) {
  return slerp(a, dot(a, b) < 0 ? -b : b, t);
}
/**
 * @brief Returns a normalized quaternion representing a rotation by angle in
 * radians about the provided axis.
 */
template <class T>
vec<T, 4> constexpr rotation_quat(const vec<T, 3> &axis, T angle) {
  return {axis * std::sin(angle / 2), std::cos(angle / 2)};
}
/**
 * @brief Returns a normalized quaternion representing the shortest rotation
 * from orig vector to dest vector.
 */
template <class T>
vec<T, 4> rotation_quat(const vec<T, 3> &orig, const vec<T, 3> &dest);
/**
 * @brief Returns a normalized quaternion representing the input rotation
 * matrix, which should be orthonormal.
 */
template <class T>
vec<T, 4> rotation_quat(const mat<T, 3, 3> &m);
/** @} */

/** @addtogroup mat_algebra
 * @ingroup LinAlg
 * @brief Support for matrix algebra.
 *  @{
 */
template <class T, int M>
constexpr vec<T, M> mul(const vec<T, M> &a, const T &b) {
  return cmul(a, b);
}
template <class T, int M>
constexpr vec<T, M> mul(const T &b, const vec<T, M> &a) {
  return cmul(b, a);
}
template <class T, int M, int N>
constexpr mat<T, M, N> mul(const mat<T, M, N> &a, const T &b) {
  return cmul(a, b);
}
template <class T, int M, int N>
constexpr mat<T, M, N> mul(const T &b, const mat<T, M, N> &a) {
  return cmul(b, a);
}
template <class T, int M>
constexpr vec<T, M> mul(const vec<T, M> &a, const vec<T, M> &b) {
  return cmul(a, b);
}
template <class T, int M>
constexpr vec<T, M> mul(const mat<T, M, 1> &a, const vec<T, 1> &b) {
  return a.x * b.x;
}
template <class T, int M>
constexpr vec<T, M> mul(const mat<T, M, 2> &a, const vec<T, 2> &b) {
  return a.x * b.x + a.y * b.y;
}
template <class T, int M>
constexpr vec<T, M> mul(const mat<T, M, 3> &a, const vec<T, 3> &b) {
  return a.x * b.x + a.y * b.y + a.z * b.z;
}
template <class T, int M>
constexpr vec<T, M> mul(const mat<T, M, 4> &a, const vec<T, 4> &b) {
  return a.x * b.x + a.y * b.y + a.z * b.z + a.w * b.w;
}
template <class T, int M, int N>
constexpr mat<T, M, 1> mul(const mat<T, M, N> &a, const mat<T, N, 1> &b) {
  return {mul(a, b.x)};
}
template <class T, int M, int N>
constexpr mat<T, M, 2> mul(const mat<T, M, N> &a, const mat<T, N, 2> &b) {
  return {mul(a, b.x), mul(a, b.y)};
}
template <class T, int M, int N>
constexpr mat<T, M, 3> mul(const mat<T, M, N> &a, const mat<T, N, 3> &b) {
  return {mul(a, b.x), mul(a, b.y), mul(a, b.z)};
}
template <class T, int M, int N>
constexpr mat<T, M, 4> mul(const mat<T, M, N> &a, const mat<T, N, 4> &b) {
  return {mul(a, b.x), mul(a, b.y), mul(a, b.z), mul(a, b.w)};
}
template <class T, int M, int N, int P>
constexpr vec<T, M> mul(const mat<T, M, N> &a, const mat<T, N, P> &b,
                        const vec<T, P> &c) {
  return mul(mul(a, b), c);
}
template <class T, int M, int N, int P, int Q>
constexpr mat<T, M, Q> mul(const mat<T, M, N> &a, const mat<T, N, P> &b,
                           const mat<T, P, Q> &c) {
  return mul(mul(a, b), c);
}
template <class T, int M, int N, int P, int Q>
constexpr vec<T, M> mul(const mat<T, M, N> &a, const mat<T, N, P> &b,
                        const mat<T, P, Q> &c, const vec<T, Q> &d) {
  return mul(mul(a, b, c), d);
}
template <class T, int M, int N, int P, int Q, int R>
constexpr mat<T, M, R> mul(const mat<T, M, N> &a, const mat<T, N, P> &b,
                           const mat<T, P, Q> &c, const mat<T, Q, R> &d) {
  return mul(mul(a, b, c), d);
}
template <class T, int M>
constexpr mat<T, M, 1> outerprod(const vec<T, M> &a, const vec<T, 1> &b) {
  return {a * b.x};
}
template <class T, int M>
constexpr mat<T, M, 2> outerprod(const vec<T, M> &a, const vec<T, 2> &b) {
  return {a * b.x, a * b.y};
}
template <class T, int M>
constexpr mat<T, M, 3> outerprod(const vec<T, M> &a, const vec<T, 3> &b) {
  return {a * b.x, a * b.y, a * b.z};
}
template <class T, int M>
constexpr mat<T, M, 4> outerprod(const vec<T, M> &a, const vec<T, 4> &b) {
  return {a * b.x, a * b.y, a * b.z, a * b.w};
}
template <class T>
constexpr vec<T, 1> diagonal(const mat<T, 1, 1> &a) {
  return {a.x.x};
}
template <class T>
constexpr vec<T, 2> diagonal(const mat<T, 2, 2> &a) {
  return {a.x.x, a.y.y};
}
template <class T>
constexpr vec<T, 3> diagonal(const mat<T, 3, 3> &a) {
  return {a.x.x, a.y.y, a.z.z};
}
template <class T>
constexpr vec<T, 4> diagonal(const mat<T, 4, 4> &a) {
  return {a.x.x, a.y.y, a.z.z, a.w.w};
}
template <class T, int N>
constexpr T trace(const mat<T, N, N> &a) {
  return sum(diagonal(a));
}
template <class T, int M>
constexpr mat<T, M, 1> transpose(const mat<T, 1, M> &m) {
  return {m.row(0)};
}
template <class T, int M>
constexpr mat<T, M, 2> transpose(const mat<T, 2, M> &m) {
  return {m.row(0), m.row(1)};
}
template <class T, int M>
constexpr mat<T, M, 3> transpose(const mat<T, 3, M> &m) {
  return {m.row(0), m.row(1), m.row(2)};
}
template <class T, int M>
constexpr mat<T, M, 4> transpose(const mat<T, 4, M> &m) {
  return {m.row(0), m.row(1), m.row(2), m.row(3)};
}
template <class T, int M>
constexpr mat<T, 1, M> transpose(const vec<T, M> &m) {
  return transpose(mat<T, M, 1>(m));
}
template <class T>
constexpr mat<T, 1, 1> adjugate(const mat<T, 1, 1> &a) {
  return {vec<T, 1>{1}};
}
template <class T>
constexpr mat<T, 2, 2> adjugate(const mat<T, 2, 2> &a) {
  return {{a.y.y, -a.x.y}, {-a.y.x, a.x.x}};
}
template <class T>
constexpr mat<T, 3, 3> adjugate(const mat<T, 3, 3> &a);
template <class T>
constexpr mat<T, 4, 4> adjugate(const mat<T, 4, 4> &a);
template <class T, int N>
constexpr mat<T, N, N> comatrix(const mat<T, N, N> &a) {
  return transpose(adjugate(a));
}
template <class T>
constexpr T determinant(const mat<T, 1, 1> &a) {
  return a.x.x;
}
template <class T>
constexpr T determinant(const mat<T, 2, 2> &a) {
  return a.x.x * a.y.y - a.x.y * a.y.x;
}
template <class T>
constexpr T determinant(const mat<T, 3, 3> &a) {
  return a.x.x * (a.y.y * a.z.z - a.z.y * a.y.z) +
         a.x.y * (a.y.z * a.z.x - a.z.z * a.y.x) +
         a.x.z * (a.y.x * a.z.y - a.z.x * a.y.y);
}
template <class T>
constexpr T determinant(const mat<T, 4, 4> &a);
template <class T, int N>
constexpr mat<T, N, N> inverse(const mat<T, N, N> &a) {
  return adjugate(a) / determinant(a);
}
/** @} */

/** @addtogroup iterators
 * @ingroup LinAlg
 * @brief Vectors and matrices can be used as ranges.
 *  @{
 */
template <class T, int M>
T *begin(vec<T, M> &a) {
  return &a.x;
}
template <class T, int M>
const T *begin(const vec<T, M> &a) {
  return &a.x;
}
template <class T, int M>
T *end(vec<T, M> &a) {
  return begin(a) + M;
}
template <class T, int M>
const T *end(const vec<T, M> &a) {
  return begin(a) + M;
}
template <class T, int M, int N>
vec<T, M> *begin(mat<T, M, N> &a) {
  return &a.x;
}
template <class T, int M, int N>
const vec<T, M> *begin(const mat<T, M, N> &a) {
  return &a.x;
}
template <class T, int M, int N>
vec<T, M> *end(mat<T, M, N> &a) {
  return begin(a) + N;
}
template <class T, int M, int N>
const vec<T, M> *end(const mat<T, M, N> &a) {
  return begin(a) + N;
}
/** @} */

/** @addtogroup transforms
 * @ingroup LinAlg
 * @brief Factory functions for 3D spatial transformations.
 *  @{
 */
enum fwd_axis {
  neg_z,
  pos_z
};  // Should projection matrices be generated assuming forward is {0,0,-1} or
    // {0,0,1}
enum z_range {
  neg_one_to_one,
  zero_to_one
};  // Should projection matrices map z into the range of [-1,1] or [0,1]?
template <class T>
mat<T, 4, 4> translation_matrix(const vec<T, 3> &translation) {
  return {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {translation, 1}};
}
template <class T>
mat<T, 4, 4> rotation_matrix(const vec<T, 4> &rotation) {
  return {{qxdir(rotation), 0},
          {qydir(rotation), 0},
          {qzdir(rotation), 0},
          {0, 0, 0, 1}};
}
template <class T>
mat<T, 4, 4> scaling_matrix(const vec<T, 3> &scaling) {
  return {{scaling.x, 0, 0, 0},
          {0, scaling.y, 0, 0},
          {0, 0, scaling.z, 0},
          {0, 0, 0, 1}};
}
template <class T>
mat<T, 4, 4> pose_matrix(const vec<T, 4> &q, const vec<T, 3> &p) {
  return {{qxdir(q), 0}, {qydir(q), 0}, {qzdir(q), 0}, {p, 1}};
}
template <class T>
mat<T, 4, 4> lookat_matrix(const vec<T, 3> &eye, const vec<T, 3> &center,
                           const vec<T, 3> &view_y_dir, fwd_axis fwd = neg_z);
template <class T>
mat<T, 4, 4> frustum_matrix(T x0, T x1, T y0, T y1, T n, T f,
                            fwd_axis a = neg_z, z_range z = neg_one_to_one);
template <class T>
mat<T, 4, 4> perspective_matrix(T fovy, T aspect, T n, T f, fwd_axis a = neg_z,
                                z_range z = neg_one_to_one) {
  T y = n * std::tan(fovy / 2), x = y * aspect;
  return frustum_matrix(-x, x, -y, y, n, f, a, z);
}
/** @} */

/** @addtogroup array
 * @ingroup LinAlg
 * @brief Provide implicit conversion between linalg::vec<T,M> and
 * std::array<T,M>.
 *  @{
 */
template <class T>
struct converter<vec<T, 1>, std::array<T, 1>> {
  vec<T, 1> operator()(const std::array<T, 1> &a) const { return {a[0]}; }
};
template <class T>
struct converter<vec<T, 2>, std::array<T, 2>> {
  vec<T, 2> operator()(const std::array<T, 2> &a) const { return {a[0], a[1]}; }
};
template <class T>
struct converter<vec<T, 3>, std::array<T, 3>> {
  vec<T, 3> operator()(const std::array<T, 3> &a) const {
    return {a[0], a[1], a[2]};
  }
};
template <class T>
struct converter<vec<T, 4>, std::array<T, 4>> {
  vec<T, 4> operator()(const std::array<T, 4> &a) const {
    return {a[0], a[1], a[2], a[3]};
  }
};

template <class T>
struct converter<std::array<T, 1>, vec<T, 1>> {
  std::array<T, 1> operator()(const vec<T, 1> &a) const { return {a[0]}; }
};
template <class T>
struct converter<std::array<T, 2>, vec<T, 2>> {
  std::array<T, 2> operator()(const vec<T, 2> &a) const { return {a[0], a[1]}; }
};
template <class T>
struct converter<std::array<T, 3>, vec<T, 3>> {
  std::array<T, 3> operator()(const vec<T, 3> &a) const {
    return {a[0], a[1], a[2]};
  }
};
template <class T>
struct converter<std::array<T, 4>, vec<T, 4>> {
  std::array<T, 4> operator()(const vec<T, 4> &a) const {
    return {a[0], a[1], a[2], a[3]};
  }
};
/** @} */

#ifdef MANIFOLD_DEBUG
template <class T>
std::ostream &operator<<(std::ostream &out, const vec<T, 1> &v) {
  return out << '{' << v[0] << '}';
}
template <class T>
std::ostream &operator<<(std::ostream &out, const vec<T, 2> &v) {
  return out << '{' << v[0] << ',' << v[1] << '}';
}
template <class T>
std::ostream &operator<<(std::ostream &out, const vec<T, 3> &v) {
  return out << '{' << v[0] << ',' << v[1] << ',' << v[2] << '}';
}
template <class T>
std::ostream &operator<<(std::ostream &out, const vec<T, 4> &v) {
  return out << '{' << v[0] << ',' << v[1] << ',' << v[2] << ',' << v[3] << '}';
}

template <class T, int M>
std::ostream &operator<<(std::ostream &out, const mat<T, M, 1> &m) {
  return out << '{' << m[0] << '}';
}
template <class T, int M>
std::ostream &operator<<(std::ostream &out, const mat<T, M, 2> &m) {
  return out << '{' << m[0] << ',' << m[1] << '}';
}
template <class T, int M>
std::ostream &operator<<(std::ostream &out, const mat<T, M, 3> &m) {
  return out << '{' << m[0] << ',' << m[1] << ',' << m[2] << '}';
}
template <class T, int M>
std::ostream &operator<<(std::ostream &out, const mat<T, M, 4> &m) {
  return out << '{' << m[0] << ',' << m[1] << ',' << m[2] << ',' << m[3] << '}';
}
#endif
}  // namespace linalg

namespace std {
/** @addtogroup hash
 * @ingroup LinAlg
 * @brief Provide specializations for std::hash<...> with linalg types.
 *  @{
 */
template <class T>
struct hash<linalg::vec<T, 1>> {
  std::size_t operator()(const linalg::vec<T, 1> &v) const {
    std::hash<T> h;
    return h(v.x);
  }
};
template <class T>
struct hash<linalg::vec<T, 2>> {
  std::size_t operator()(const linalg::vec<T, 2> &v) const {
    std::hash<T> h;
    return h(v.x) ^ (h(v.y) << 1);
  }
};
template <class T>
struct hash<linalg::vec<T, 3>> {
  std::size_t operator()(const linalg::vec<T, 3> &v) const {
    std::hash<T> h;
    return h(v.x) ^ (h(v.y) << 1) ^ (h(v.z) << 2);
  }
};
template <class T>
struct hash<linalg::vec<T, 4>> {
  std::size_t operator()(const linalg::vec<T, 4> &v) const {
    std::hash<T> h;
    return h(v.x) ^ (h(v.y) << 1) ^ (h(v.z) << 2) ^ (h(v.w) << 3);
  }
};

template <class T, int M>
struct hash<linalg::mat<T, M, 1>> {
  std::size_t operator()(const linalg::mat<T, M, 1> &v) const {
    std::hash<linalg::vec<T, M>> h;
    return h(v.x);
  }
};
template <class T, int M>
struct hash<linalg::mat<T, M, 2>> {
  std::size_t operator()(const linalg::mat<T, M, 2> &v) const {
    std::hash<linalg::vec<T, M>> h;
    return h(v.x) ^ (h(v.y) << M);
  }
};
template <class T, int M>
struct hash<linalg::mat<T, M, 3>> {
  std::size_t operator()(const linalg::mat<T, M, 3> &v) const {
    std::hash<linalg::vec<T, M>> h;
    return h(v.x) ^ (h(v.y) << M) ^ (h(v.z) << (M * 2));
  }
};
template <class T, int M>
struct hash<linalg::mat<T, M, 4>> {
  std::size_t operator()(const linalg::mat<T, M, 4> &v) const {
    std::hash<linalg::vec<T, M>> h;
    return h(v.x) ^ (h(v.y) << M) ^ (h(v.z) << (M * 2)) ^ (h(v.w) << (M * 3));
  }
};
/** @} */
}  // namespace std

// Definitions of functions too long to be defined inline
template <class T>
constexpr linalg::mat<T, 3, 3> linalg::adjugate(const mat<T, 3, 3> &a) {
  return {{a.y.y * a.z.z - a.z.y * a.y.z, a.z.y * a.x.z - a.x.y * a.z.z,
           a.x.y * a.y.z - a.y.y * a.x.z},
          {a.y.z * a.z.x - a.z.z * a.y.x, a.z.z * a.x.x - a.x.z * a.z.x,
           a.x.z * a.y.x - a.y.z * a.x.x},
          {a.y.x * a.z.y - a.z.x * a.y.y, a.z.x * a.x.y - a.x.x * a.z.y,
           a.x.x * a.y.y - a.y.x * a.x.y}};
}

template <class T>
constexpr linalg::mat<T, 4, 4> linalg::adjugate(const mat<T, 4, 4> &a) {
  return {{a.y.y * a.z.z * a.w.w + a.w.y * a.y.z * a.z.w +
               a.z.y * a.w.z * a.y.w - a.y.y * a.w.z * a.z.w -
               a.z.y * a.y.z * a.w.w - a.w.y * a.z.z * a.y.w,
           a.x.y * a.w.z * a.z.w + a.z.y * a.x.z * a.w.w +
               a.w.y * a.z.z * a.x.w - a.w.y * a.x.z * a.z.w -
               a.z.y * a.w.z * a.x.w - a.x.y * a.z.z * a.w.w,
           a.x.y * a.y.z * a.w.w + a.w.y * a.x.z * a.y.w +
               a.y.y * a.w.z * a.x.w - a.x.y * a.w.z * a.y.w -
               a.y.y * a.x.z * a.w.w - a.w.y * a.y.z * a.x.w,
           a.x.y * a.z.z * a.y.w + a.y.y * a.x.z * a.z.w +
               a.z.y * a.y.z * a.x.w - a.x.y * a.y.z * a.z.w -
               a.z.y * a.x.z * a.y.w - a.y.y * a.z.z * a.x.w},
          {a.y.z * a.w.w * a.z.x + a.z.z * a.y.w * a.w.x +
               a.w.z * a.z.w * a.y.x - a.y.z * a.z.w * a.w.x -
               a.w.z * a.y.w * a.z.x - a.z.z * a.w.w * a.y.x,
           a.x.z * a.z.w * a.w.x + a.w.z * a.x.w * a.z.x +
               a.z.z * a.w.w * a.x.x - a.x.z * a.w.w * a.z.x -
               a.z.z * a.x.w * a.w.x - a.w.z * a.z.w * a.x.x,
           a.x.z * a.w.w * a.y.x + a.y.z * a.x.w * a.w.x +
               a.w.z * a.y.w * a.x.x - a.x.z * a.y.w * a.w.x -
               a.w.z * a.x.w * a.y.x - a.y.z * a.w.w * a.x.x,
           a.x.z * a.y.w * a.z.x + a.z.z * a.x.w * a.y.x +
               a.y.z * a.z.w * a.x.x - a.x.z * a.z.w * a.y.x -
               a.y.z * a.x.w * a.z.x - a.z.z * a.y.w * a.x.x},
          {a.y.w * a.z.x * a.w.y + a.w.w * a.y.x * a.z.y +
               a.z.w * a.w.x * a.y.y - a.y.w * a.w.x * a.z.y -
               a.z.w * a.y.x * a.w.y - a.w.w * a.z.x * a.y.y,
           a.x.w * a.w.x * a.z.y + a.z.w * a.x.x * a.w.y +
               a.w.w * a.z.x * a.x.y - a.x.w * a.z.x * a.w.y -
               a.w.w * a.x.x * a.z.y - a.z.w * a.w.x * a.x.y,
           a.x.w * a.y.x * a.w.y + a.w.w * a.x.x * a.y.y +
               a.y.w * a.w.x * a.x.y - a.x.w * a.w.x * a.y.y -
               a.y.w * a.x.x * a.w.y - a.w.w * a.y.x * a.x.y,
           a.x.w * a.z.x * a.y.y + a.y.w * a.x.x * a.z.y +
               a.z.w * a.y.x * a.x.y - a.x.w * a.y.x * a.z.y -
               a.z.w * a.x.x * a.y.y - a.y.w * a.z.x * a.x.y},
          {a.y.x * a.w.y * a.z.z + a.z.x * a.y.y * a.w.z +
               a.w.x * a.z.y * a.y.z - a.y.x * a.z.y * a.w.z -
               a.w.x * a.y.y * a.z.z - a.z.x * a.w.y * a.y.z,
           a.x.x * a.z.y * a.w.z + a.w.x * a.x.y * a.z.z +
               a.z.x * a.w.y * a.x.z - a.x.x * a.w.y * a.z.z -
               a.z.x * a.x.y * a.w.z - a.w.x * a.z.y * a.x.z,
           a.x.x * a.w.y * a.y.z + a.y.x * a.x.y * a.w.z +
               a.w.x * a.y.y * a.x.z - a.x.x * a.y.y * a.w.z -
               a.w.x * a.x.y * a.y.z - a.y.x * a.w.y * a.x.z,
           a.x.x * a.y.y * a.z.z + a.z.x * a.x.y * a.y.z +
               a.y.x * a.z.y * a.x.z - a.x.x * a.z.y * a.y.z -
               a.y.x * a.x.y * a.z.z - a.z.x * a.y.y * a.x.z}};
}

template <class T>
constexpr T linalg::determinant(const mat<T, 4, 4> &a) {
  return a.x.x * (a.y.y * a.z.z * a.w.w + a.w.y * a.y.z * a.z.w +
                  a.z.y * a.w.z * a.y.w - a.y.y * a.w.z * a.z.w -
                  a.z.y * a.y.z * a.w.w - a.w.y * a.z.z * a.y.w) +
         a.x.y * (a.y.z * a.w.w * a.z.x + a.z.z * a.y.w * a.w.x +
                  a.w.z * a.z.w * a.y.x - a.y.z * a.z.w * a.w.x -
                  a.w.z * a.y.w * a.z.x - a.z.z * a.w.w * a.y.x) +
         a.x.z * (a.y.w * a.z.x * a.w.y + a.w.w * a.y.x * a.z.y +
                  a.z.w * a.w.x * a.y.y - a.y.w * a.w.x * a.z.y -
                  a.z.w * a.y.x * a.w.y - a.w.w * a.z.x * a.y.y) +
         a.x.w * (a.y.x * a.w.y * a.z.z + a.z.x * a.y.y * a.w.z +
                  a.w.x * a.z.y * a.y.z - a.y.x * a.z.y * a.w.z -
                  a.w.x * a.y.y * a.z.z - a.z.x * a.w.y * a.y.z);
}

template <class T>
linalg::vec<T, 4> linalg::rotation_quat(const vec<T, 3> &orig,
                                        const vec<T, 3> &dest) {
  T cosTheta = dot(orig, dest);
  if (cosTheta >= 1 - std::numeric_limits<T>::epsilon()) {
    return {0, 0, 0, 1};
  }
  if (cosTheta < -1 + std::numeric_limits<T>::epsilon()) {
    vec<T, 3> axis = cross(vec<T, 3>(0, 0, 1), orig);
    if (length2(axis) < std::numeric_limits<T>::epsilon())
      axis = cross(vec<T, 3>(1, 0, 0), orig);
    return rotation_quat(normalize(axis),
                         3.14159265358979323846264338327950288);
  }
  vec<T, 3> axis = cross(orig, dest);
  T s = std::sqrt((1 + cosTheta) * 2);
  return {axis * (1 / s), s * 0.5};
}

template <class T>
linalg::vec<T, 4> linalg::rotation_quat(const mat<T, 3, 3> &m) {
  const vec<T, 4> q{m.x.x - m.y.y - m.z.z, m.y.y - m.x.x - m.z.z,
                    m.z.z - m.x.x - m.y.y, m.x.x + m.y.y + m.z.z},
      s[]{{1, m.x.y + m.y.x, m.z.x + m.x.z, m.y.z - m.z.y},
          {m.x.y + m.y.x, 1, m.y.z + m.z.y, m.z.x - m.x.z},
          {m.x.z + m.z.x, m.y.z + m.z.y, 1, m.x.y - m.y.x},
          {m.y.z - m.z.y, m.z.x - m.x.z, m.x.y - m.y.x, 1}};
  return copysign(normalize(sqrt(max(T(0), T(1) + q))), s[argmax(q)]);
}

template <class T>
linalg::mat<T, 4, 4> linalg::lookat_matrix(const vec<T, 3> &eye,
                                           const vec<T, 3> &center,
                                           const vec<T, 3> &view_y_dir,
                                           fwd_axis a) {
  const vec<T, 3> f = normalize(center - eye), z = a == pos_z ? f : -f,
                  x = normalize(cross(view_y_dir, z)), y = cross(z, x);
  return inverse(mat<T, 4, 4>{{x, 0}, {y, 0}, {z, 0}, {eye, 1}});
}

template <class T>
linalg::mat<T, 4, 4> linalg::frustum_matrix(T x0, T x1, T y0, T y1, T n, T f,
                                            fwd_axis a, z_range z) {
  const T s = a == pos_z ? T(1) : T(-1), o = z == neg_one_to_one ? n : 0;
  return {{2 * n / (x1 - x0), 0, 0, 0},
          {0, 2 * n / (y1 - y0), 0, 0},
          {-s * (x0 + x1) / (x1 - x0), -s * (y0 + y1) / (y1 - y0),
           s * (f + o) / (f - n), s},
          {0, 0, -(n + o) * f / (f - n), 0}};
}
#endif
