A fast scheme for evaluating a polynomial such as:

 



−
19
+
7
x
−
4

x

2


+
6

x

3





{\displaystyle -19+7x-4x^{2}+6x^{3}\,}


when

 



x
=
3



{\displaystyle x=3\;}

.
is to arrange the computation as follows:

 



(
(
(
(
0
)
x
+
6
)
x
+
(
−
4
)
)
x
+
7
)
x
+
(
−
19
)



{\displaystyle ((((0)x+6)x+(-4))x+7)x+(-19)\;}


And compute the result from the innermost brackets outwards as in this pseudocode:

coefficients := [-19, 7, -4, 6] # list coefficients of all x^0..x^n in order
x := 3
accumulator := 0
for i in length(coefficients) downto 1 do
    # Assumes 1-based indexing for arrays
    accumulator := ( accumulator * x ) + coefficients[i]
done
# accumulator now has the answer

Task Description

Create a routine that takes a list of coefficients of a polynomial in order of increasing powers of x; together with a value of x to compute its value at, and return the value of the polynomial at that value using Horner's rule.
Cf. Formal power series

$ include "seed7_05.s7i";
  include "float.s7i";
 
const type: coeffType is array float;
 
const func float: horner (in coeffType: coeffs, in float: x) is func
  result
    var float: res is 0.0;
  local
    var integer: i is 0;
  begin
    for i range length(coeffs) downto 1 do
      res := res * x + coeffs[i];
    end for;
  end func;
 
const proc: main is func
  local
    const coeffType: coeffs is [] (-19.0, 7.0, -4.0, 6.0);
  begin
    writeln(horner(coeffs, 3.0) digits 1);
  end func;