reverse_horner

eve::reverse_horner = functor<reverse_horner_t>

inlineconstexpr

Header file

#include <eve/module/math.hpp>

Callable Signatures

namespace eve
{
   // Regular overloads
   constexpr auto reverse_horner(floating_value auto x, value auto ci...)                       noexcept; // 1
   constexpr auto reverse_horner(floating_value auto x, eve::coefficients tci)                  noexcept; // 2
 
   // Semantic options
   constexpr auto reverse_horner[pedantic](/*any of the above overloads*/)                      noexcept; // 3
   constexpr auto reverse_horner[kahan](/*any of the above overloads*/)                         noexcept; // 4
   constexpr auto reverse_horner[widen](/*any of the above overloads*/)                         noexcept; // 5
}

1. Polynom is evaluated at x the other inputs are the polynomial coefficients.
2. Polynom is evaluated at x the other input is a range or a kumi::tuple containing the coefficients
3. fma[pedantic] instead of fma is used in internal computations.
4. a Kahan like compensated algorithm is used to enhance accuracy.
5. the computation is applied to upgraded types values when available.

Parameters

• x: evaluation point floating value arguments.
• ci...: floating values polynom coefficients in increasing power order, Note that the values of the ci are not necessarily floating but the non floating ones are to be scalar
• tci: eve:coefficients is a specialization of tuple of floating values used to avoid possible ambiguities
• c: Conditional expression masking the operation.
• m: Logical value masking the operation.

Return value

If \((c_i)_{0\le i\le n-1}\) denotes the coefficients of the polynomial by increasing power order, the reverse Horner scheme evaluates the polynom \(p\) at \(x\) using the following formula:

\(\qquad\qquad\displaystyle p(x) = (((c_{n-1}x+c_{n-2})x+ ... )x + c_0)\)

1. The value of the polynom at x is returned.
2. same as the call with the elements of the tuple. 3.fma[pedantic] instead of fma is used in internal computations. This is intended to insure more accurate computations where needed. This has no cost (and is automatically done) if the system has hard wired fma but is very expansive if it is not the case.

Note

If the coefficients are simd values of cardinal N, this means you simultaneously compute the values of N polynomials.

• If x is scalar, the polynomials are all computed at the same point
• If x is simd, the nth polynomial is computed on the nth value of x

External references

• [Wikipedia](https://en.wikipedia.org/wiki/Horner's_method)

Example

// revision 1
#include <eve/module/math.hpp>
#include <iostream>
#include <iomanip>
 
int main()
{
  eve::wide xd = {-0.3, 0.5, 0.0, 2.0};
  eve::wide b  = {-2.0, 10.5, -4.0, 0.1};
  double x(0.2);
  kumi::tuple v {1.0, -2.0, 3.0, -4.0};
  using w_t = decltype(xd);
  kumi::tuple wv{ w_t{1.5, 1, 2, 3}, w_t{4, 5, 6, 7}, w_t{8, 9, 10, 11} };
  auto t = kumi::tuple{1.5,4.0,8.0};
 
  std::cout << "<- xd                                       = " << xd << '\n';
  std::cout << "<- x                                        = " << x  << '\n';
  std::cout << "<- v                                        = " << v  << '\n';
  std::cout << "<- wv                                       = " << wv << '\n';
  std::cout << "-> reverse_horner(xd, 1.0, -2.0, 3.0, -4.0) = " << eve::reverse_horner(xd, 1.0, -2.0, 3.0, -4.0) << '\n';
  std::cout << "-> reverse_horner(0.5, 1, b, 3, -4)         = " << eve::reverse_horner(0.5, 1, b, 3, -4) << '\n';
  std::cout << "-> reverse_horner(x, 1, -2, 3, -4)          = " << eve::reverse_horner(xd, 1, -2, 3, -4)  << '\n';
  std::cout << "-> reverse_horner(xd, v)                    = " << eve::reverse_horner(xd, eve::coefficients(v))  << '\n';
  std::cout << "-> reverse_horner(xd, t)                    = " << eve::reverse_horner(xd, eve::coefficients(t))   << '\n';
  std::cout << "-> reverse_horner(x, t)                     = " << eve::reverse_horner(x, eve::coefficients(t))   << '\n';
  std::cout << "-> reverse_horner(x, wv)                    = " << eve::reverse_horner(x, eve::coefficients(wv)) << '\n';
  std::cout << "-> reverse_horner(0.5f, wv)                 = " << eve::reverse_horner(0.5, eve::coefficients(wv)) << '\n';
  std::cout << "-> reverse_horner(xd, wv)                   = " << eve::reverse_horner(xd, eve::coefficients(wv)) << '\n';
  std::cout << "-> reverse_horner(2.0, t)                   = " << eve::reverse_horner(2.0, eve::coefficients(t)) << '\n';
  std::cout << "-> reverse_horner(2.0, 1.5,4.0,8.0)         = " << eve::reverse_horner(2.0, 1.5,4.0,8.0) << '\n';
  std::cout << "-> reverse_horner                           = " << eve::reverse_horner(w_t(2.0),  1.5,4.0,8.0) << '\n';
}