//===----------------------------------------------------------------------===//
//
// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
// See https://llvm.org/LICENSE.txt for license information.
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
//
//===----------------------------------------------------------------------===//
// UNSUPPORTED: c++03, c++11, c++14
// <cmath>
// double hermite(unsigned n, double x);
// float hermite(unsigned n, float x);
// long double hermite(unsigned n, long double x);
// float hermitef(unsigned n, float x);
// long double hermitel(unsigned n, long double x);
// template <class Integer>
// double hermite(unsigned n, Integer x);
#include <array>
#include <cassert>
#include <cmath>
#include <limits>
#include <vector>
#include "type_algorithms.h"
template <class Real>
constexpr unsigned get_maximal_order() {
if constexpr (std::numeric_limits<Real>::is_iec559)
return 128;
else { // Workaround for z/OS HexFloat.
// Note |H_n(x)| < 10^75 for n < 39 and x in sample_points().
static_assert(std::numeric_limits<Real>::max_exponent10 == 75);
return 39;
}
}
template <class T>
std::array<T, 11> sample_points() {
return {-12.34, -7.42, -1.0, -0.5, -0.1, 0.0, 0.1, 0.5, 1.0, 5.67, 15.67};
}
template <class Real>
class CompareFloatingValues {
private:
Real abs_tol;
Real rel_tol;
public:
CompareFloatingValues() {
abs_tol = []() -> Real {
if (std::is_same_v<Real, float>)
return 1e-5f;
else if (std::is_same_v<Real, double>)
return 1e-11;
else
return 1e-12l;
}();
rel_tol = abs_tol;
}
bool operator()(Real result, Real expected) const {
if (std::isinf(expected) && std::isinf(result))
return result == expected;
if (std::isnan(expected) || std::isnan(result))
return false;
Real tol = abs_tol + std::abs(expected) * rel_tol;
return std::abs(result - expected) < tol;
}
};
// Roots are taken from
// Salzer, Herbert E., Ruth Zucker, and Ruth Capuano.
// Table of the zeros and weight factors of the first twenty Hermite
// polynomials. US Government Printing Office, 1952.
template <class T>
std::vector<T> get_roots(unsigned n) {
switch (n) {
case 0:
return {};
case 1:
return {T(0)};
case 2:
return {T(0.707106781186548)};
case 3:
return {T(0), T(1.224744871391589)};
case 4:
return {T(0.524647623275290), T(1.650680123885785)};
case 5:
return {T(0), T(0.958572464613819), T(2.020182870456086)};
case 6:
return {T(0.436077411927617), T(1.335849074013697), T(2.350604973674492)};
case 7:
return {T(0), T(0.816287882858965), T(1.673551628767471), T(2.651961356835233)};
case 8:
return {T(0.381186990207322), T(1.157193712446780), T(1.981656756695843), T(2.930637420257244)};
case 9:
return {T(0), T(0.723551018752838), T(1.468553289216668), T(2.266580584531843), T(3.190993201781528)};
case 10:
return {
T(0.342901327223705), T(1.036610829789514), T(1.756683649299882), T(2.532731674232790), T(3.436159118837738)};
case 11:
return {T(0),
T(0.65680956682100),
T(1.326557084494933),
T(2.025948015825755),
T(2.783290099781652),
T(3.668470846559583)};
case 12:
return {T(0.314240376254359),
T(0.947788391240164),
T(1.597682635152605),
T(2.279507080501060),
T(3.020637025120890),
T(3.889724897869782)};
case 13:
return {T(0),
T(0.605763879171060),
T(1.220055036590748),
T(1.853107651601512),
T(2.519735685678238),
T(3.246608978372410),
T(4.101337596178640)};
case 14:
return {T(0.29174551067256),
T(0.87871378732940),
T(1.47668273114114),
T(2.09518325850772),
T(2.74847072498540),
T(3.46265693360227),
T(4.30444857047363)};
case 15:
return {T(0.00000000000000),
T(0.56506958325558),
T(1.13611558521092),
T(1.71999257518649),
T(2.32573248617386),
T(2.96716692790560),
T(3.66995037340445),
T(4.49999070730939)};
case 16:
return {T(0.27348104613815),
T(0.82295144914466),
T(1.38025853919888),
T(1.95178799091625),
T(2.54620215784748),
T(3.17699916197996),
T(3.86944790486012),
T(4.68873893930582)};
case 17:
return {T(0),
T(0.5316330013427),
T(1.0676487257435),
T(1.6129243142212),
T(2.1735028266666),
T(2.7577629157039),
T(3.3789320911415),
T(4.0619466758755),
T(4.8713451936744)};
case 18:
return {T(0.2582677505191),
T(0.7766829192674),
T(1.3009208583896),
T(1.8355316042616),
T(2.3862990891667),
T(2.9613775055316),
T(3.5737690684863),
T(4.2481178735681),
T(5.0483640088745)};
case 19:
return {T(0),
T(0.5035201634239),
T(1.0103683871343),
T(1.5241706193935),
T(2.0492317098506),
T(2.5911337897945),
T(3.1578488183476),
T(3.7621873519640),
T(4.4285328066038),
T(5.2202716905375)};
case 20:
return {T(0.2453407083009),
T(0.7374737285454),
T(1.2340762153953),
T(1.7385377121166),
T(2.2549740020893),
T(2.7888060584281),
T(3.347854567332),
T(3.9447640401156),
T(4.6036824495507),
T(5.3874808900112)};
default: // polynom degree n>20 is unsupported
assert(false);
return {T(-42)};
}
}
template <class Real>
void test() {
if constexpr (
std::numeric_limits<Real>::has_quiet_NaN &&
std::numeric_limits<
Real>::has_signaling_NaN) { // checks if NaNs are reported correctly (i.e. output == input for input == NaN)
using nl = std::numeric_limits<Real>;
for (Real NaN : {nl::quiet_NaN(), nl::signaling_NaN()})
for (unsigned n = 0; n < get_maximal_order<Real>(); ++n)
assert(std::isnan(std::hermite(n, NaN)));
}
if constexpr (std::numeric_limits<Real>::has_quiet_NaN &&
std::numeric_limits<
Real>::has_signaling_NaN) { // simple sample points for n=0..127 should not produce NaNs.
for (Real x : sample_points<Real>())
for (unsigned n = 0; n < get_maximal_order<Real>(); ++n)
assert(!std::isnan(std::hermite(n, x)));
}
{ // checks std::hermite(n, x) for n=0..5 against analytic polynoms
const auto h0 = [](Real) -> Real { return 1; };
const auto h1 = [](Real y) -> Real { return 2 * y; };
const auto h2 = [](Real y) -> Real { return 4 * y * y - 2; };
const auto h3 = [](Real y) -> Real { return y * (8 * y * y - 12); };
const auto h4 = [](Real y) -> Real { return (16 * std::pow(y, 4) - 48 * y * y + 12); };
const auto h5 = [](Real y) -> Real { return y * (32 * std::pow(y, 4) - 160 * y * y + 120); };
for (Real x : sample_points<Real>()) {
const CompareFloatingValues<Real> compare;
assert(compare(std::hermite(0, x), h0(x)));
assert(compare(std::hermite(1, x), h1(x)));
assert(compare(std::hermite(2, x), h2(x)));
assert(compare(std::hermite(3, x), h3(x)));
assert(compare(std::hermite(4, x), h4(x)));
assert(compare(std::hermite(5, x), h5(x)));
}
}
{ // checks std::hermitef for bitwise equality with std::hermite(unsigned, float)
if constexpr (std::is_same_v<Real, float>)
for (unsigned n = 0; n < get_maximal_order<Real>(); ++n)
for (float x : sample_points<float>())
assert(std::hermite(n, x) == std::hermitef(n, x));
}
{ // checks std::hermitel for bitwise equality with std::hermite(unsigned, long double)
if constexpr (std::is_same_v<Real, long double>)
for (unsigned n = 0; n < get_maximal_order<Real>(); ++n)
for (long double x : sample_points<long double>())
assert(std::hermite(n, x) == std::hermitel(n, x));
}
{ // Checks if the characteristic recurrence relation holds: H_{n+1}(x) = 2x H_n(x) - 2n H_{n-1}(x)
for (Real x : sample_points<Real>()) {
for (unsigned n = 1; n < get_maximal_order<Real>() - 1; ++n) {
Real H_next = std::hermite(n + 1, x);
Real H_next_recurrence = 2 * (x * std::hermite(n, x) - n * std::hermite(n - 1, x));
if (std::isinf(H_next))
break;
const CompareFloatingValues<Real> compare;
assert(compare(H_next, H_next_recurrence));
}
}
}
{ // sanity checks: hermite polynoms need to change signs at (simple) roots. checked upto order n<=20.
// root tolerance: must be smaller than the smallest difference between adjacent roots
Real tol = []() -> Real {
if (std::is_same_v<Real, float>)
return 1e-5f;
else if (std::is_same_v<Real, double>)
return 1e-9;
else
return 1e-10l;
}();
const auto is_sign_change = [tol](unsigned n, Real x) -> bool {
return std::hermite(n, x - tol) * std::hermite(n, x + tol) < 0;
};
for (unsigned n = 0; n <= 20u; ++n) {
for (Real x : get_roots<Real>(n)) {
// the roots are symmetric: if x is a root, so is -x
if (x > 0)
assert(is_sign_change(n, -x));
assert(is_sign_change(n, x));
}
}
}
if constexpr (std::numeric_limits<Real>::has_infinity) { // check input infinity is handled correctly
Real inf = std::numeric_limits<Real>::infinity();
for (unsigned n = 1; n < get_maximal_order<Real>(); ++n) {
assert(std::hermite(n, +inf) == inf);
assert(std::hermite(n, -inf) == ((n & 1) ? -inf : inf));
}
}
if constexpr (std::numeric_limits<
Real>::has_infinity) { // check: if overflow occurs that it is mapped to the correct infinity
if constexpr (std::is_same_v<Real, double>) {
// Q: Why only double?
// A: The numeric values (e.g. overflow threshold `n`) below are different for other types.
static_assert(sizeof(double) == 8);
for (unsigned n = 0; n < get_maximal_order<Real>(); ++n) {
// Q: Why n=111 and x=300?
// A: Both are chosen s.t. the first overlow occurs for some `n<get_maximal_order<Real>()`.
if (n < 111) {
assert(std::isfinite(std::hermite(n, +300.0)));
assert(std::isfinite(std::hermite(n, -300.0)));
} else {
double inf = std::numeric_limits<double>::infinity();
assert(std::hermite(n, +300.0) == inf);
assert(std::hermite(n, -300.0) == ((n & 1) ? -inf : inf));
}
}
}
}
}
struct TestFloat {
template <class Real>
void operator()() {
test<Real>();
}
};
struct TestInt {
template <class Integer>
void operator()() {
// checks that std::hermite(unsigned, Integer) actually wraps std::hermite(unsigned, double)
for (unsigned n = 0; n < get_maximal_order<double>(); ++n)
for (Integer x : {-42, -7, -5, -1, 0, 1, 5, 7, 42})
assert(std::hermite(n, x) == std::hermite(n, static_cast<double>(x)));
}
};
int main() {
types::for_each(types::floating_point_types(), TestFloat());
types::for_each(types::type_list<short, int, long, long long>(), TestInt());
}
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