Horner's rule for polynomial evaluation

From Rosetta Code
Task
Horner's rule for polynomial evaluation
You are encouraged to solve this task according to the task description, using any language you may know.

A fast scheme for evaluating a polynomial such as:

when

.

is to arrange the computation as follows:

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

360 Assembly

*        Horner's rule for polynomial evaluation - 07/10/2015
HORNER CSECT
USING HORNER,R15 set base register
SR R5,R5 accumulator=0
LA R2,N i=number_of_coeff
LOOP M R4,X accumulator=accumulator*x
LR R1,R2 i
SLA R1,2 i*4
L R3,COEF-4(R1) coef(i)
AR R5,R3 accumulator=accumulator+coef(i)
BCT R2,LOOP i=i-1; loop n times
XDECO R5,PG edit accumulator
XPRNT PG,12 print buffer
XR R15,R15 set return code
BR R14 return to caller
COEF DC F'-19',F'7',F'-4',F'6' <== input values
X DC F'3' <== input value
N EQU (X-COEF)/4 number of coefficients
PG DS CL12 buffer
YREGS
END HORNER
Output:
         128

ACL2

(defun horner (ps x)
(if (endp ps)
0
(+ (first ps)
(* x (horner (rest ps) x)))))

Ada

with Ada.Float_Text_IO; use Ada.Float_Text_IO;
 
procedure Horners_Rule is
type Coef is array(Positive range <>) of Float;
 
function Horner(Coeffs: Coef; Val: Float) return Float is
Res : Float := 0.0;
begin
for P in reverse Coeffs'Range loop
Res := Res*Val + Coeffs(P);
end loop;
return Res;
end Horner;
 
begin
Put(Horner(Coeffs => (-19.0, 7.0, -4.0, 6.0), Val => 3.0), Aft=>1, Exp=>0);
end Horners_Rule;

Output:

128.0

Aime

real
horner(list coeffs, real x)
{
integer i;
real z;
 
z = 0;
 
i = l_length(coeffs);
while (i) {
i -= 1;
z *= x;
z += l_q_real(coeffs, i);
}
 
return z;
}
 
 
integer
main(void)
{
o_plan(horner(l_effect(-19r, 7.0, -real(4), 6r), 3), "\n");
 
return 0;
}

ALGOL 68

Works with: ALGOL 68G
PROC horner = ([]REAL c, REAL x)REAL :
(
REAL res := 0.0;
FOR i FROM UPB c BY -1 TO LWB c DO
res := res * x + c[i]
OD;
res
);
 
main:(
[4]REAL coeffs := (-19.0, 7.0, -4.0, 6.0);
print( horner(coeffs, 3.0) )
)

ATS

#include
"share/atspre_staload.hats"
 
fun
horner
(
x: int, cs: List int
) : int = let
//
implement
list_foldright$fopr<int><int> (a, b) = a + b * x
//
in
list_foldright<int><int> (cs, 0)
end // end of [horner]
 
implement
main0 () = let
val x = 3
val cs = $list{int}(~19, 7, ~4, 6)
val res = horner (x, cs)
in
println! (res)
end // end of [main0]

AutoHotkey

Coefficients = -19, 7, -4, 6
x := 3
 
MsgBox, % EvalPolynom(Coefficients, x)
 
 
 
;---------------------------------------------------------------------------
EvalPolynom(Coefficients, x) { ; using Horner's rule
;---------------------------------------------------------------------------
StringSplit, Co, coefficients, `,, %A_Space%
Result := 0
Loop, % Co0
i := Co0 - A_Index + 1, Result := Result * x + Co%i%
Return, Result
}

Message box shows:

128

AWK

#!/usr/bin/awk -f
function horner(x, A) {
acc = 0;
for (i = length(A); 0<i; i--) {
acc = acc*x + A[i];
}
return acc;
}
BEGIN {
split(p,P);
print horner(x,P);
}
Output:
   awk  -v X=3 -v p="-19  7 -4  6" -f horner.awk
   128

Batch File

 
@echo off
 
call:horners a:-19 b:7 c:-4 d:6 x:3
call:horners x:3 a:-19 c:-4 d:6 b:7
pause>nul
exit /b
 
:horners
setlocal enabledelayedexpansion
set a=0
set b=0
set c=0
set d=0
set x=0
 
for %%i in (%*) do (
for /f "tokens=1,2 delims=:" %%j in ("%%i") do (
set %%j=%%k
)
)
set /a return=((((0)*%x%+%d%)*%x%+(%c%))*%x%+%b%)*%x%+(%a%)
echo %return%
exit /b
 
Output:
>a:-19 b:7 c:-4 d:6 x:3
128
>x:3 a:-19 c:-4 d:6 b:7
128


BBC BASIC

      DIM coefficients(3)
coefficients() = -19, 7, -4, 6
PRINT FNhorner(coefficients(), 3)
END
 
DEF FNhorner(coeffs(), x)
LOCAL i%, v
FOR i% = DIM(coeffs(), 1) TO 0 STEP -1
v = v * x + coeffs(i%)
NEXT
= v

Bracmat

( ( Horner
= accumulator coefficients x coeff
.  !arg:(?coefficients.?x)
& 0:?accumulator
& whl
' ( !coefficients:?coefficients #%@?coeff
& !accumulator*!x+!coeff:?accumulator
)
& !accumulator
)
& Horner$(-19 7 -4 6.3)
);

Output:

128

C

Translation of: Fortran
#include <stdio.h>
 
double horner(double *coeffs, int s, double x)
{
int i;
double res = 0.0;
 
for(i=s-1; i >= 0; i--)
{
res = res * x + coeffs[i];
}
return res;
}
 
 
int main()
{
double coeffs[] = { -19.0, 7.0, -4.0, 6.0 };
 
printf("%5.1f\n", horner(coeffs, sizeof(coeffs)/sizeof(double), 3.0));
return 0;
}

C#

using System;
using System.Linq;
 
class Program
{
static double Horner(double[] coefficients, double variable)
{
return coefficients.Reverse().Aggregate(
(accumulator, coefficient) => accumulator * variable + coefficient);
}
 
static void Main()
{
Console.WriteLine(Horner(new[] { -19.0, 7.0, -4.0, 6.0 }, 3.0));
}
}

Output:

128

C++

The same C function works too, but another solution could be:

#include <iostream>
#include <vector>
 
using namespace std;
 
double horner(vector<double> v, double x)
{
double s = 0;
 
for( vector<double>::const_reverse_iterator i = v.rbegin(); i != v.rend(); i++ )
s = s*x + *i;
return s;
}
 
int main()
{
double c[] = { -19, 7, -4, 6 };
vector<double> v(c, c + sizeof(c)/sizeof(double));
cout << horner(v, 3.0) << endl;
return 0;
}

Yet another solution, which is more idiomatic in C++ and works on any bidirectional sequence:

 
#include <iostream>
 
template<typename BidirIter>
double horner(BidirIter begin, BidirIter end, double x)
{
double result = 0;
while (end != begin)
result = result*x + *--end;
return result;
}
 
int main()
{
double c[] = { -19, 7, -4, 6 };
std::cout << horner(c, c + 4, 3) << std::endl;
}
 

Clojure

(defn horner [coeffs x]
(reduce #(-> %1 (* x) (+ %2)) (reverse coeffs)))
 
(println (horner [-19 7 -4 6] 3))

CoffeeScript

 
eval_poly = (x, coefficients) ->
# coefficients are for ascending powers
return 0 if coefficients.length == 0
ones_place = coefficients.shift()
x * eval_poly(x, coefficients) + ones_place
 
console.log eval_poly 3, [-19, 7, -4, 6] # 128
console.log eval_poly 10, [4, 3, 2, 1] # 1234
console.log eval_poly 2, [1, 1, 0, 0, 1] # 19
 

Common Lisp

(defun horner (coeffs x)
(reduce #'(lambda (coef acc) (+ (* acc x) coef))
coeffs :from-end t :initial-value 0))

Alternate version using LOOP. Coefficients are passed in a vector.

(defun horner (x a)
(loop :with y = 0
:for i :from (1- (length a)) :downto 0
:do (setf y (+ (aref a i) (* y x)))
:finally (return y)))
 
(horner 1.414 #(-2 0 1))

D

The poly() function of the standard library std.math module uses Horner's rule:

void main() {
void main() {
import std.stdio, std.math;
double x = 3.0;
static real[] pp = [-19,7,-4,6];
 
poly(x,pp).writeln;
}
}

Basic implementation:

import std.stdio, std.traits;
 
CommonType!(U, V) horner(U, V)(U[] p, V x) pure nothrow @nogc {
typeof(return) accumulator = 0;
foreach_reverse (c; p)
accumulator = accumulator * x + c;
return accumulator;
}
 
void main() {
[-19, 7, -4, 6].horner(3.0).writeln;
}

More functional style:

import std.stdio, std.algorithm, std.range;
 
auto horner(T, U)(in T[] p, in U x) pure nothrow @nogc {
return reduce!((a, b) => a * x + b)(U(0), p.retro);
}
 
void main() {
[-19, 7, -4, 6].horner(3.0).writeln;
}

E

def makeHornerPolynomial(coefficients :List) {
def indexing := (0..!coefficients.size()).descending()
return def hornerPolynomial(x) {
var acc := 0
for i in indexing {
acc := acc * x + coefficients[i]
}
return acc
}
}
? makeHornerPolynomial([-19, 7, -4, 6])(3)
# value: 128

EchoLisp

Functional version

 
(define (horner x poly)
(foldr (lambda (coeff acc) (+ coeff (* acc x))) 0 poly))
 
(horner 3 '(-19 7 -4 6))128
 

Library

 
(lib 'math)
Lib: math.lib loaded.
 
(define P '(-19 7 -4 6))
(poly->string 'x P) → 6x^3 -4x^2 +7x -19
(poly 3 P)128
 

Elixir

horner = fn(list, x)-> List.foldr(list, 0, fn(c,acc)-> x*acc+c end) end
 
IO.puts horner.([-19,7,-4,6], 3)
Output:
128

Emacs Lisp

Translation of: Common Lisp
 
(defun horner (coeffs x)
(reduce #'(lambda (coef acc) (+ (* acc x) coef) )
coeffs :from-end t :initial-value 0) )
 
(horner '(-19 7 -4 6) 3)
 

Output:

 
128

Erlang

 
horner(L,X) ->
lists:foldl(fun(C, Acc) -> X*Acc+C end,0, lists:reverse(L)).
t() ->
horner([-19,7,-4,6], 3).
 

ERRE

 
PROGRAM HORNER
 
! 2 3
! polynomial is -19+7x-4x +6x
!
 
DIM C[3]
 
PROCEDURE HORNER(C[],X->RES)
LOCAL I%,V
FOR I%=UBOUND(C,1) TO 0 STEP -1 DO
V=V*X+C[I%]
END FOR
RES=V
END PROCEDURE
 
BEGIN
C[]=(-19,7,-4,6)
HORNER(C[],3->RES)
PRINT(RES)
END PROGRAM
 

Euler Math Toolbox

 
>function horner (x,v) ...
$ n=cols(v); res=v{n};
$ loop 1 to n-1; res=res*x+v{n-#}; end;
$ return res
$endfunction
>v=[-19,7,-4,6]
[ -19 7 -4 6 ]
>horner(2,v) // test Horner
27
>evalpoly(2,v) // built-in Horner
27
>horner(I,v) // complex values
-15+1i
>horner(1±0.05,v) // interval values
~-10.9,-9.11~
>function p(x) &= sum(@v[k]*x^(k-1),k,1,4) // Symbolic Polynomial
3 2
6 x - 4 x + 7 x - 19
 

F#

 
let horner l x =
List.rev l |> List.fold ( fun acc c -> x*acc+c) 0
 
horner [-19;7;-4;6] 3
 

Factor

: horner ( coeff x -- res )
[ <reversed> 0 ] dip '[ [ _ * ] dip + ] reduce ;
( scratchpad ) { -19 7 -4 6 } 3 horner .
128

Forth

: fhorner ( coeffs len F: x -- F: val )
0e
floats bounds ?do
fover f* i f@ f+
1 floats +loop
fswap fdrop ;
 
create coeffs 6e f, -4e f, 7e f, -19e f,
 
coeffs 4 3e fhorner f. \ 128.

Fortran

Works with: Fortran version 90 and later
program test_horner
 
implicit none
 
write (*, '(f5.1)') horner ((/-19.0, 7.0, -4.0, 6.0/), 3.0)
 
contains
 
function horner (coeffs, x) result (res)
 
implicit none
real, dimension (:), intent (in) :: coeffs
real, intent (in) :: x
real :: res
integer :: i
 
res = 0.0
do i = size (coeffs), 1, -1
res = res * x + coeffs (i)
end do
 
end function horner
 
end program test_horner

Output:

128.0

Fortran 77

      FUNCTION HORNER(N,A,X)
IMPLICIT NONE
INTEGER I,N
DOUBLE PRECISION A(N),X,Y,HORNER
Y = A(N)
DO I = N - 1,1,-1
Y = Y*X + A(I)
END DO
HORNER=Y
END

As a matter of fact, computing the derivative is not much more difficult (see Roundoff in Polynomial Evaluation, W. Kahan, 1986). The following subroutine computes both polynomial value and derivative for argument x.

      SUBROUTINE HORNER2(N,A,X,Y,Z)
C COMPUTE POLYNOMIAL VALUE AND DERIVATIVE
C SEE "ROUNDOFF IN POLYNOMIAL EVALUATION", W. KAHAN, 1986
C POLY: A(1) + A(2)*X + ... + A(N)*X**(N-1)
C Y: VALUE, Z: DERIVATIVE
IMPLICIT NONE
INTEGER I,N
DOUBLE PRECISION A(N),X,Y,Z
Z = 0.0D0
Y = A(N)
DO 10 I = N - 1,1,-1
Z = Z*X + Y
10 Y = Y*X + A(I)
END

FunL

Translation of: Haskell
import lists.foldr
 
def horner( poly, x ) = foldr( \a, b -> a + b*x, 0, poly )
 
println( horner([-19, 7, -4, 6], 3) )
Output:
128

GAP

# The idiomatic way to compute with polynomials
 
x := Indeterminate(Rationals, "x");
 
# This is a value in a polynomial ring, not a function
p := 6*x^3 - 4*x^2 + 7*x - 19;
 
Value(p, 3);
# 128
 
u := CoefficientsOfUnivariatePolynomial(p);
# [ -19, 7, -4, 6 ]
 
# One may also create the polynomial from coefficients
q := UnivariatePolynomial(Rationals, [-19, 7, -4, 6], x);
# 6*x^3-4*x^2+7*x-19
 
p = q;
# true
 
# Now a Horner implementation
Horner := function(coef, x)
local v, c;
v := 0;
for c in Reversed(coef) do
v := x*v + c;
od;
return v;
end;
 
Horner(u, 3);
# 128

Go

package main
 
import "fmt"
 
func horner(x int64, c []int64) (acc int64) {
for i := len(c) - 1; i >= 0; i-- {
acc = acc*x + c[i]
}
return
}
 
func main() {
fmt.Println(horner(3, []int64{-19, 7, -4, 6}))
}

Output:

128

Groovy

Solution:

def hornersRule = { coeff, x -> coeff.reverse().inject(0) { accum, c -> (accum * x) + c } }

Test includes demonstration of currying to create polynomial functions of one variable from generic Horner's rule calculation. Also demonstrates constructing the derivative function for the given polynomial. And finally demonstrates in the Newton-Raphson method to find one of the polynomial's roots using the polynomial and derivative functions constructed earlier.

def coefficients = [-19g, 7g, -4g, 6g]
println (["p coefficients":coefficients])
 
def testPoly = hornersRule.curry(coefficients)
println (["p(3)":testPoly(3g)])
println (["p(0)":testPoly(0g)])
 
def derivativeCoefficients = { coeff -> (1..<(coeff.size())).collect { coeff[it] * it } }
println (["p' coefficients":derivativeCoefficients(coefficients)])
 
def testDeriv = hornersRule.curry(derivativeCoefficients(coefficients))
println (["p'(3)":testDeriv(3g)])
println (["p'(0)":testDeriv(0g)])
 
def newtonRaphson = { x, f, fPrime ->
while (f(x).abs() > 0.0001) {
x -= f(x)/fPrime(x)
}
x
}
 
def root = newtonRaphson(3g, testPoly, testDeriv)
println ([root:root.toString()[0..5], "p(root)":testPoly(root).toString()[0..5], "p'(root)":testDeriv(root).toString()[0..5]])

Output:

[p coefficients:[-19, 7, -4, 6]]
[p(3):128]
[p(0):-19]
[p' coefficients:[7, -8, 18]]
[p'(3):145]
[p'(0):7]
[root:1.4183, p(root):0.0000, p'(root):31.862]

Haskell

horner :: (Num a) => a -> [a] -> a
horner x = foldr (\a b -> a + b*x) 0
 
main = print $ horner 3 [-19, 7, -4, 6]

HicEst

REAL :: x=3, coeffs(4)
DATA coeffs/-19.0, 7.0, -4.0, 6.0/
 
WRITE(Messagebox) Horner(coeffs, x) ! shows 128
 
FUNCTION Horner(c, x)
DIMENSION c(1)
Horner = 0
DO i = LEN(c), 1, -1
Horner = x*Horner + c(i)
ENDDO
END

Icon and Unicon

 
procedure poly_eval (x, coeffs)
accumulator := 0
every index := *coeffs to 1 by -1 do
accumulator := accumulator * x + coeffs[index]
return accumulator
end
 
procedure main ()
write (poly_eval (3, [-19, 7, -4, 6]))
end
 

J

Solution:
 
 
horner =: 4 : ' (+ *&y)/x'
 
horner1 =: (#."0 _ |.)~
 
horner2=: [: +`*/ [: }: ,@,. NB. Alternate
 
Example:
   _19 7 _4 6 horner 3
128

Note:
The primitive verb p. would normally be used to evaluate polynomials.

   _19 7 _4 6 p. 3
128

Java

Works with: Java version 1.5+
import java.util.ArrayList;
import java.util.Collections;
import java.util.List;
 
public class Horner {
public static void main(String[] args){
List<Double> coeffs = new ArrayList<Double>();
coeffs.add(-19.0);
coeffs.add(7.0);
coeffs.add(-4.0);
coeffs.add(6.0);
System.out.println(polyEval(coeffs, 3));
}
 
public static double polyEval(List<Double> coefficients, double x) {
Collections.reverse(coefficients);
Double accumulator = coefficients.get(0);
for (int i = 1; i < coefficients.size(); i++) {
accumulator = (accumulator * x) + (Double) coefficients.get(i);
}
return accumulator;
}
}

Output:

128.0

JavaScript

Works with: JavaScript version 1.8
which includes
Works with: Firefox version 3
Translation of: Haskell
function horner(coeffs, x) {
return coeffs.reduceRight( function(acc, coeff) { return(acc * x + coeff) }, 0);
}
console.log(horner([-19,7,-4,6],3)); // ==> 128
 

Julia

Imperative:

 
function horner(coef,x)
sum = coef[end]
for k = length(coef)-1:-1:1
sum = coef[k] + x*sum
end
sum
end

Output:

julia> horner([-19,7,-4,6], 3)
128

Functional:

horner2(coef,x) = foldr((u,v) -> u + x*v, 0, coef)

Output:

julia> horner2([-19,7,-4,6], 3)
128

K

 
horner:{y _sv|x}
horner[-19 7 -4 6;3]
128
 

Kotlin

// version 1.1.2
 
fun horner(coeffs: DoubleArray, x: Double): Double {
var sum = 0.0
for (i in coeffs.size - 1 downTo 0) sum = sum * x + coeffs[i]
return sum
}
 
fun main(args: Array<String>) {
val coeffs = doubleArrayOf(-19.0, 7.0, -4.0, 6.0)
println(horner(coeffs, 3.0))
}
Output:
128.0

Liberty BASIC

src$ = "Hello"
coefficients$ = "-19 7 -4 6" ' list coefficients of all x^0..x^n in order
x = 3
print horner(coefficients$, x) '128
 
print horner("4 3 2 1", 10) '1234
print horner("1 1 0 0 1", 2) '19
end
 
function horner(coefficients$, x)
accumulator = 0
'getting length of a list requires extra pass with WORD$.
'So we just started from high above
for index = 100 to 1 step -1
cft$ = word$(coefficients$, index)
if cft$<>"" then accumulator = ( accumulator * x ) + val(cft$)
next
horner = accumulator
end function
 

to horner :x :coeffs
if empty? :coeffs [output 0]
output (first :coeffs) + (:x * horner :x bf :coeffs)
end
 
show horner 3 [-19 7 -4 6]  ; 128

Lua

function horners_rule( coeff, x )
local res = 0
for i = #coeff, 1, -1 do
res = res * x + coeff[i]
end
return res
end
 
x = 3
coefficients = { -19, 7, -4, 6 }
print( horners_rule( coefficients, x ) )


Maple

 
applyhorner:=(L::list,x)->foldl((s,t)->s*x+t,op(ListTools:-Reverse(L))):
 
applyhorner([-19,7,-4,6],x);
 
applyhorner([-19,7,-4,6],3);
 

Output:

                    ((6 x - 4) x + 7) x - 19

                              128

Mathematica / Wolfram Language

Horner[l_List, x_] := Fold[x #1 + #2 &, 0, l]
Horner[{6, -4, 7, -19}, x]
-> -19 + x (7 + x (-4 + 6 x))
 
-19 + x (7 + x (-4 + 6 x)) /. x -> 3
-> 128

MATLAB

function accumulator = hornersRule(x,coefficients)
 
accumulator = 0;
 
for i = (numel(coefficients):-1:1)
accumulator = (accumulator * x) + coefficients(i);
end
 
end

Output:

>> hornersRule(3,[-19, 7, -4, 6])
 
ans =
 
128

Matlab also has a built-in function "polyval" which uses Horner's Method to evaluate polynomials. The list of coefficients is in descending order of power, where as to task spec specifies ascending order.

>> polyval(fliplr([-19, 7, -4, 6]),3)
 
ans =
 
128

Maxima

/* Function horner already exists in Maxima, though it operates on expressions, not lists of coefficients */
horner(5*x^3+2*x+1);
x*(5*x^2+2)+1
 
/* Here is an implementation */
horner2(p, x) := block([n, y, i],
n: length(p),
y: p[n],
for i: n - 1 step -1 thru 1 do y: y*x + p[i],
y
)$
 
horner2([-19, 7, -4, 6], 3);
128
 
/* Another with rreduce */
horner3(p,x):=rreduce(lambda([a,y],x*y+a),p);
horner3([a,b,c,d,e,f],x);
x*(x*(x*(x*(f*x+e)+d)+c)+b)+a
 
/* Extension to compute also derivatives up to a specified order.
See William Kahan, Roundoff in Polynomial Evaluation, 1986
http://www.cs.berkeley.edu/~wkahan/Math128/Poly.pdf */
 
poleval(a, x, [m]) := block(
[n: length(a), v, k: 1],
if emptyp(m) then m: 1 else m: 1 + first(m),
v: makelist(0, m),
v[1]: a[n],
for i from n - 1 thru 1 step -1 do (
for j from m thru 2 step -1 do v[j]: v[j] * x + v[j - 1],
v[1]: v[1] * x + a[i]
),
for i from 2 thru m do (
v[i]: v[i] * k,
k: k * i
),
if m = 1 then first(v) else v
)$
 
poleval([0, 0, 0, 0, 1], x, 4);
[x^4, 4 * x^3, 12 * x^2, 24 * x, 24]
 
poleval([0, 0, 0, 0, 1], x);
x^4

Mercury

 
:- module horner.
:- interface.
:- import_module io.
:- pred main(io::di, io::uo) is det.
:- implementation.
:- import_module int, list, string.
 
main(!IO) :-
io.format("%i\n", [i(horner(3, [-19, 7, -4, 6]))], !IO).
 
:- func horner(int, list(int)) = int.
 
horner(X, Cs) = list.foldr((func(C, Acc) = Acc * X + C), Cs, 0).
 

МК-61/52

ИП0	1	+	П0
ИПE ИПD * КИП0 + ПE
ИП0 1 - x=0 04
ИПE С/П

Input: Р1:РС - coefficients, Р0 - number of the coefficients, РD - x.

NetRexx

/* NetRexx */
options replace format comments java crossref savelog symbols nobinary
 
c = [-19, 7, -4, 6] -- # list coefficients of all x^0..x^n in order
n=3
x=3
r=0
loop i=n to 0 by -1
r=r*x+c[i]
End
Say r
Say 6*x**3-4*x**2+7*x-19

Output:

128
128

Nim

iterator reversed(x) =
for i in countdown(x.high, x.low):
yield x[i]
 
proc horner(coeffs, x): int =
for c in reversed(coeffs):
result = result * x + c
 
echo horner([-19, 7, -4, 6], 3)

Oberon-2

Works with: oo2c
 
MODULE HornerRule;
IMPORT
Out;
 
TYPE
Coefs = POINTER TO ARRAY OF LONGINT;
VAR
coefs: Coefs;
 
PROCEDURE Eval(coefs: ARRAY OF LONGINT;size,x: LONGINT): LONGINT;
VAR
i,acc: LONGINT;
BEGIN
acc := 0;
FOR i := LEN(coefs) - 1 TO 0 BY -1 DO
acc := acc * x + coefs[i]
END;
RETURN acc
END Eval;
 
BEGIN
NEW(coefs,4);
coefs[0] := -19;
coefs[1] := 7;
coefs[2] := -4;
coefs[3] := 6;
Out.Int(Eval(coefs^,4,3),0);Out.Ln
END HornerRule.
 
Output:
128

Objective-C

Works with: Mac OS X version 10.6+
Using blocks
#import <Foundation/Foundation.h>
 
typedef double (^mfunc)(double, double);
 
@interface NSArray (HornerRule)
- (double)horner: (double)x;
- (NSArray *)reversedArray;
- (double)injectDouble: (double)s with: (mfunc)op;
@end
 
@implementation NSArray (HornerRule)
- (NSArray *)reversedArray
{
return [[self reverseObjectEnumerator] allObjects];
}
 
 
- (double)injectDouble: (double)s with: (mfunc)op
{
double sum = s;
for(NSNumber* el in self) {
sum = op(sum, [el doubleValue]);
}
return sum;
}
 
- (double)horner: (double)x
{
return [[self reversedArray] injectDouble: 0.0 with: ^(double s, double a) { return s * x + a; } ];
}
@end
 
int main()
{
@autoreleasepool {
 
NSArray *coeff = @[@-19.0, @7.0, @-4.0, @6.0];
printf("%f\n", [coeff horner: 3.0]);
 
}
return 0;
}

Objeck

 
class Horner {
function : Main(args : String[]) ~ Nil {
coeffs := Collection.FloatVector->New();
coeffs->AddBack(-19.0);
coeffs->AddBack(7.0);
coeffs->AddBack(-4.0);
coeffs->AddBack(6.0);
PolyEval(coeffs, 3)->PrintLine();
}
 
function : PolyEval(coefficients : Collection.FloatVector , x : Float) ~ Float {
accumulator := coefficients->Get(coefficients->Size() - 1);
for(i := coefficients->Size() - 2; i > -1; i -= 1;) {
accumulator := (accumulator * x) + coefficients->Get(i);
};
 
return accumulator;
}
}
 

OCaml

# let horner coeffs x =
List.fold_left (fun acc coef -> acc * x + coef) 0 (List.rev coeffs) ;;
val horner : int list -> int -> int = <fun>
 
# let coeffs = [-19; 7; -4; 6] in
horner coeffs 3 ;;
- : int = 128

It's also possible to do fold_right instead of reversing and doing fold_left; but fold_right is not tail-recursive.

Octave

function r = horner(a, x)
r = 0.0;
for i = length(a):-1:1
r = r*x + a(i);
endfor
endfunction
 
horner([-19, 7, -4, 6], 3)

ooRexx

/* Rexx ---------------------------------------------------------------
* 04.03.2014 Walter Pachl
*--------------------------------------------------------------------*/

c = .array~of(-19,7,-4,6) -- coefficients of all x^0..x^n in order
n=3
x=3
r=0
loop i=n+1 to 1 by -1
r=r*x+c[i]
End
Say r
Say 6*x**3-4*x**2+7*x-19

Output:

128
128

Oz

declare
fun {Horner Coeffs X}
{FoldL1 {Reverse Coeffs}
fun {$ Acc Coeff}
Acc*X + Coeff
end}
end
 
fun {FoldL1 X|Xr Fun}
{FoldL Xr Fun X}
end
in
{Show {Horner [~19 7 ~4 6] 3}}

PARI/GP

Also note that Pari has a polynomial type. Evaluating these is as simple as subst(P,variable(P),x).

horner(v,x)={
my(s=0);
forstep(i=#v,1,-1,s=s*x+v[i]);
s
};

Pascal

Program HornerDemo(output);
 
function horner(a: array of double; x: double): double;
var
i: integer;
begin
horner := a[high(a)];
for i := high(a) - 1 downto low(a) do
horner := horner * x + a[i];
end;
 
const
poly: array [1..4] of double = (-19.0, 7.0, -4.0, 6.0);
 
begin
write ('Horner calculated polynomial of 6*x^3 - 4*x^2 + 7*x - 19 for x = 3: ');
writeln (horner (poly, 3.0):8:4);
end.

Output:

Horner calculated polynomial of 6*x^3 - 4*x^2 + 7*x - 19 for x = 3: 128.0000

Perl

use 5.10.0;
use strict;
use warnings;
 
sub horner(\@$){
my ($coef, $x) = @_;
my $result = 0;
$result = $result * $x + $_ for reverse @$coef;
return $result;
}
 
my @coeff = (-19.0, 7, -4, 6);
my $x = 3;
say horner @coeff, $x;

Functional version

use strict;
use List::Util qw(reduce);
 
sub horner($$){
my ($coeff_ref, $x) = @_;
reduce { $a * $x + $b } reverse @$coeff_ref;
}
 
my @coeff = (-19.0, 7, -4, 6);
my $x = 3;
print horner(\@coeff, $x), "\n";

Recursive version

sub horner {
my ($coeff, $x) = @_;
@$coeff and
$$coeff[0] + $x * horner( [@$coeff[1 .. $#$coeff]], $x )
}
 
print horner( [ -19, 7, -4, 6 ], 3 );

Perl 6

sub horner ( @coeffs, $x ) {
@coeffs.reverse.reduce: { $^a * $x + $^b };
}
 
say horner( [ -19, 7, -4, 6 ], 3 );

A recursive version would spare us the need for reversing the list of coefficients. However, special care must be taken in order to write it, because the way Perl 6 implements lists is not optimized for this kind of treatment. Lisp-style lists are, and fortunately it is possible to emulate them with Pairs and the reduction meta-operator:

multi horner(Numeric $c, $) { $c }
multi horner(Pair $c, $x) {
$c.key + $x * horner( $c.value, $x )
}
 
say horner( [=>](-19, 7, -4, 6 ), 3 );

We can also use the composition operator:

sub horner ( @coeffs, $x ) {
([o] map { $_ + $x * * }, @coeffs)(0);
}
 
say horner( [ -19, 7, -4, 6 ], 3 );
Output:
128

One advantage of using the composition operator is that it allows for the use of an infinite list of coefficients.

sub horner ( @coeffs, $x ) {
map { .(0) }, [\o] map { $_ + $x * * }, @coeffs;
}
 
say horner( [ 1 X/ (1, |[\*] 1 .. *) ], i*pi )[20];
 
Output:
-0.999999999924349-5.28918515954219e-10i

Phix

function horner(atom x, sequence coeff)
atom res = 0
for i=length(coeff) to 1 by -1 do
res = res*x + coeff[i]
end for
return res
end function
 
?horner(3,{-19, 7, -4, 6})
Output:
128

PHP

<?php
function horner($coeff, $x) {
$result = 0;
foreach (array_reverse($coeff) as $c)
$result = $result * $x + $c;
return $result;
}
 
$coeff = array(-19.0, 7, -4, 6);
$x = 3;
echo horner($coeff, $x), "\n";
?>

Functional version

Works with: PHP version 5.3+
<?php
function horner($coeff, $x) {
return array_reduce(array_reverse($coeff), function ($a, $b) use ($x) { return $a * $x + $b; }, 0);
}
 
$coeff = array(-19.0, 7, -4, 6);
$x = 3;
echo horner($coeff, $x), "\n";
?>

PicoLisp

(de horner (Coeffs X)
(let Res 0
(for C (reverse Coeffs)
(setq Res (+ C (* X Res))) ) ) )
: (horner (-19.0 7.0 -4.0 6.0) 3.0)
-> 128

PL/I

 
declare (i, n) fixed binary, (x, value) float; /* 11 May 2010 */
get (x);
get (n);
begin;
declare a(0:n) float;
get list (a);
value = a(n);
do i = n to 1 by -1;
value = value*x + a(i-1);
end;
put (value);
end;
 

Potion

horner = (x, coef) :
result = 0
coef reverse each (a) :
result = (result * x) + a
.
result
.
 
horner(3, (-19, 7, -4, 6)) print

PowerShell

Works with: PowerShell version 4.0
 
function horner($coefficients, $x) {
$accumulator = 0
foreach($i in ($coefficients.Count-1)..0){
$accumulator = ( $accumulator * $x ) + $coefficients[$i]
}
$accumulator
}
$coefficients = @(-19, 7, -4, 6)
$x = 3
horner $coefficients $x
 

Output:

 
128

Prolog

Tested with SWI-Prolog. Works with other dialects.

horner([], _X, 0).
 
horner([H|T], X, V) :-
horner(T, X, V1),
V is V1 * X + H.
 

Output :

 ?- horner([-19, 7, -4, 6], 3, V).
V = 128.

Functional approach

Works with SWI-Prolog and module lambda, written by Ulrich Neumerkel found there http://www.complang.tuwien.ac.at/ulrich/Prolog-inedit/lambda.pl

:- use_module(library(lambda)).
 
 
% foldr(Pred, Init, List, R).
%
foldr(_Pred, Val, [], Val).
foldr(Pred, Val, [H | T], Res) :-
foldr(Pred, Val, T, Res1),
call(Pred, Res1, H, Res).
 
f_horner(L, V, R) :-
foldr(\X^Y^Z^(Z is X * V + Y), 0, L, R).
 

Functional syntax (Ciao)

Works with Ciao (https://github.com/ciao-lang/ciao) and the fsyntax package:

 
:- module(_, [horner/3], [fsyntax, hiord]).
:- use_module(library(hiordlib)).
horner(L, X) := ~foldr((''(H,V0,V) :- V is V0*X + H), L, 0).
 

PureBasic

Procedure Horner(List Coefficients(), b)
Define result
ForEach Coefficients()
result*b+Coefficients()
Next
ProcedureReturn result
EndProcedure

Implemented as

NewList a()
AddElement(a()): a()= 6
AddElement(a()): a()= -4
AddElement(a()): a()= 7
AddElement(a()): a()=-19
Debug Horner(a(),3)

Outputs

128

Python

>>> def horner(coeffs, x):
acc = 0
for c in reversed(coeffs):
acc = acc * x + c
return acc
 
>>> horner( (-19, 7, -4, 6), 3)
128

Functional version

>>> try: from functools import reduce
except: pass
 
>>> def horner(coeffs, x):
return reduce(lambda acc, c: acc * x + c, reversed(coeffs), 0)
 
>>> horner( (-19, 7, -4, 6), 3)
128

Library: numpy

>>> import numpy
>>> numpy.polynomial.polynomial.polyval(3, (-19, 7, -4, 6))
128.0

R

Procedural style:

horner <- function(a, x) {
y <- 0
for(c in rev(a)) {
y <- y * x + c
}
y
}
 
cat(horner(c(-19, 7, -4, 6), 3), "\n")

Functional style:

horner <- function(x, v) {
Reduce(v, right=T, f=function(a, b) {
b * x + a
})
}
Output:
> v <- c(-19, 7, -4, 6)
> horner(3, v)
[1] 128

Racket

Translated from Haskell

 
#lang racket
(define (horner x l)
(foldr (lambda (a b) (+ a (* b x))) 0 l))
 
(horner 3 '(-19 7 -4 6))
 
 

Rascal

import List;
 
public int horners_rule(list[int] coefficients, int x){
acc = 0;
for( i <- reverse(coefficients)){
acc = acc * x + i;}
return acc;
}

A neater and shorter solution using a reducer:

public int horners_rule2(list[int] coefficients, int x) = (0 | it * x + c | c <- reverse(coefficients));

Output:

rascal>horners_rule([-19, 7, -4, 6], 3)
int: 128
 
rascal>horners_rule2([-19, 7, -4, 6], 3)
int: 128

REBOL

rebol []
 
horner: func [coeffs x] [
result: 0
foreach i reverse coeffs [
result: (result * x) + i
]
return result
]
 
print horner [-19 7 -4 6] 3

REXX

version 1

/*REXX program  demonstrates using    Horner's rule    for   polynomial evaluation.     */
numeric digits 30 /*use extra numeric precision. */
parse arg x poly /*get value of X and the coefficients. */
$= /*start with a clean slate equation. */
do deg=0 until poly=='' /*get the equation's coefficients. */
parse var poly c.deg poly; c.deg=c.deg/1 /*get equation coefficient & normalize.*/
if c.deg>=0 then c.deg= '+'c.deg /*if ¬ negative, then prefix with a + */
$=$ c.deg /*concatenate it to the equation. */
if deg\==0 & c.deg\=0 then $=$'∙x^'deg /*¬1st coefficient & ¬0? Append X pow.*/
$=$ ' ' /*insert some blanks, make it look nice*/
end /*deg*/
say ' x = ' x
say ' degree = ' deg
say ' equation = ' $
a=c.deg /*A: is the accumulator (or answer). */
do j=deg-1 by -1 for deg; a=a*x+c.j /*apply Horner's rule to the equations.*/
end /*j*/
say /*display a blank line for readability.*/
say ' answer = ' a /*stick a fork in it, we're all done. */

output   when the following is used for input:   3   -19   7   -4   6

         x =  3
    degree =  3
  equation =   -19    +7∙x^1    -4∙x^2    +6∙x^3

    answer =  128

version 2

/* REXX ---------------------------------------------------------------
* 27.07.2012 Walter Pachl
* coefficients reversed to descending order of power
* I'm used to x**2+x-3
* equation formatting prettified (coefficients 1 and 0)
*--------------------------------------------------------------------*/

Numeric Digits 30 /* use extra numeric precision. */
Parse Arg x poly /* get value of x and coefficients*/
rpoly=''
Do p=0 To words(poly)-1
rpoly=rpoly word(poly,words(poly)-p)
End
poly=rpoly
equ='' /* start with equation clean slate*/
deg=words(poly)-1
pdeg=deg
Do Until deg<0 /* get the equation's coefficients*/
Parse Var poly c.deg poly /* in descending order of powers */
c.deg=c.deg+0 /* normalize it */
If c.deg>0 & deg<pdeg Then /* positive and not first term */
prefix='+' /* prefix a + sign. */
Else prefix=''
Select
When deg=0 Then term=c.deg
When deg=1 Then
If c.deg=1 Then term='x'
Else term=c.deg'*x'
Otherwise
If c.deg=1 Then term='x^'deg
Else term=c.deg'*x^'deg
End
If c.deg<>0 Then /* build up the equation */
equ=equ||prefix||term
deg=deg-1
End
a=c.pdeg
Do p=pdeg To 1 By -1 /* apply Horner's rule. */
pm1=p-1
a=a*x+c.pm1
End
Say ' x = ' x
Say ' degree = ' pdeg
Say ' equation = ' equ
Say ' '
Say ' result = ' a
Output:
        x =  3
   degree =  3
 equation =  6*x^3-4*x^2+7*x-19

   result =  128

Ring

 
coefficients = [-19, 7, -4, 6]
see "x = 3" + nl +
"degree = 3" + nl +
"equation = 6*x^3-4*x^2+7*x-19" + nl +
"result = " + horner(coefficients, 3) + nl
 
func horner coeffs, x
w = 0
for n = len(coeffs) to 1 step -1
w = w * x + coeffs[n]
next
return w
 

Output:

x =  3
degree =  3
equation =  6*x^3-4*x^2+7*x-19
result = 128

RLaB

RLaB implements horner's scheme for polynomial evaluation in its built-in function polyval. What is important is that RLaB stores the polynomials as row vectors starting from the highest power just as matlab and octave do.

This said, solution to the problem is

 
>> a = [6, -4, 7, -19]
6 -4 7 -19
>> x=3
3
>> polyval(x, a)
128
 
 

Ruby

def horner(coeffs, x)
coeffs.reverse.inject(0) {|acc, coeff| acc * x + coeff}
end
p horner([-19, 7, -4, 6], 3) # ==> 128

Rust

fn horner(v: &[f64], x: f64) -> f64 {
v.iter().rev().fold(0.0, |acc, coeff| acc*x + coeff)
}
 
fn main() {
let v = [-19., 7., -4., 6.];
println!("result: {}", horner(&v, 3.0));
}

A generic version that works with any number type and much more. So much more, it's hard to imagine what that may be useful for.
Uses a gated feature (the Zero trait), only works in unstable Rust

#![feature(zero_one)
use std::num::Zero;
use std::ops::{Mul, Add};
 
fn horner<Arr,Arg, Out>(v: &[Arr], x: Arg) -> Out
where Arr: Clone,
Arg: Clone,
Out: Zero + Mul<Arg, Output=Out> + Add<Arr, Output=Out>,
{
v.iter().rev().fold(Zero::zero(), |acc, coeff| acc*x.clone() + coeff.clone())
}
 
fn main() {
let v = [-19., 7., -4., 6.];
let output: f64 = horner(&v, 3.0);
println!("result: {}", output);
}

Run BASIC

coef$ = "-19 7 -4 6" ' list coefficients of all x^0..x^n in order
x = 3
print horner(coef$,x) '128
print horner("1.2 2.3 3.4 4.5 5.6", 8) '25478.8
print horner("5 4 3 2 1", 10) '12345
print horner("1 0 1 1 1 0 0 1", 2) '157
end
 
function horner(coef$,x)
while word$(coef$, i + 1) <> ""
i = i + 1 ' count the num of values
wend
for j = i to 1 step -1
accum = ( accum * x ) + val(word$(coef$, j))
next
horner = accum
end function

Sather

class MAIN is
 
action(s, e, x:FLT):FLT is
return s*x + e;
end;
 
horner(v:ARRAY{FLT}, x:FLT):FLT is
rv ::= v.reverse;
return rv.reduce(bind(action(_, _, x)));
end;
 
main is
#OUT + horner(|-19.0, 7.0, -4.0, 6.0|, 3.0) + "\n";
end;
end;

Scala

def horner(coeffs:List[Double], x:Double)=
coeffs.reverse.foldLeft(0.0){(a,c)=> a*x+c}
 
val coeffs=List(-19.0, 7.0, -4.0, 6.0)
println(horner(coeffs, 3))
-> 128.0
 

Scheme

Works with: Scheme version RRS
(define (horner lst x)
(define (*horner lst x acc)
(if (null? lst)
acc
(*horner (cdr lst) x (+ (* acc x) (car lst)))))
(*horner (reverse lst) x 0))
 
(display (horner (list -19 7 -4 6) 3))
(newline)

Output:

128

Seed7

$ 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;

Output:

128.0

Sidef

Functional:

func horner(coeff, x) {
coeff.reverse.reduce { |a,b| a*x + b };
}
 
say horner([-19, 7, -4, 6], 3); # => 128

Recursive:

func horner(coeff, x) {
coeff.len > 0
&& (coeff[0] + x*horner(coeff.ft(1), x));
}
 
say horner([-19, 7, -4, 6], 3); # => 128

Smalltalk

Works with: GNU Smalltalk
OrderedCollection extend [
horner: aValue [
^ self reverse inject: 0 into: [:acc :c | acc * aValue + c].
]
].
 
(#(-19 7 -4 6) asOrderedCollection horner: 3) displayNl.

Standard ML

(* Assuming real type for coefficients and x *)
fun horner coeffList x = foldr (fn (a, b) => a + b * x) (0.0) coeffList

Swift

func horner(coefs: [Double], x: Double) -> Double {
return reduce(lazy(coefs).reverse(), 0) { $0 * x + $1 }
}
 
println(horner([-19, 7, -4, 6], 3))
Output:
128.0

Tcl

package require Tcl 8.5
proc horner {coeffs x} {
set y 0
foreach c [lreverse $coeffs] {
set y [expr { $y*$x+$c }]
}
return $y
}

Demonstrating:

puts [horner {-19 7 -4 6} 3]

Output:

128

VBA

Note: this function, "Horner", gets its coefficients as a ParamArray which has no specified length. This array collect all arguments after the first one(s). This means you must specify x first, then the coefficients.

 
Public Function Horner(x, ParamArray coeff())
Dim result As Double
Dim ncoeff As Integer
 
result = 0
ncoeff = UBound(coeff())
 
For i = ncoeff To 0 Step -1
result = (result * x) + coeff(i)
Next i
Horner = result
End Function
 

Output:

print Horner(3, -19, 7, -4, 6)
 128 

Visual FoxPro

Coefficients in ascending order.

 
LOCAL x As Double
LOCAL ARRAY aCoeffs[1]
CLEAR
CREATE CURSOR coeffs (c1 I, c2 I, c3 I, c4 I)
INSERT INTO coeffs VALUES (-19,7,-4,6)
SCATTER TO aCoeffs
x = VAL(INPUTBOX("Value of x:", "Value"))
? EvalPoly(@aCoeffs, x)
USE IN coeffs
 
FUNCTION EvalPoly(c, x As Double) As Double
LOCAL s As Double, k As Integer, n As Integer
n = ALEN(c)
s = 0
FOR k = n TO 1 STEP -1
s = s*x + c[k]
ENDFOR
RETURN s
ENDFUNC
 

Coefficients in descending order.

 
LOCAL x As Double
LOCAL ARRAY aCoeffs[1]
CLEAR
CREATE CURSOR tmp (c1 I, c2 I, c3 I, c4 I)
INSERT INTO tmp VALUES (6,-4,7,-19)
SCATTER TO aCoeffs
x = VAL(INPUTBOX("Value of x:", "Value"))
? EvalPolyDesc(@aCoeffs, x)
USE IN tmp
 
FUNCTION EvalPolyDesc(c, x As Double) As Double
LOCAL s As Double, e
s = 0
FOR EACH e IN c FOXOBJECT
s = s*x + e
ENDFOR
RETURN s
ENDFUNC
 

VBScript

 
Function horners_rule(coefficients,x)
accumulator = 0
For i = UBound(coefficients) To 0 Step -1
accumulator = (accumulator * x) + coefficients(i)
Next
horners_rule = accumulator
End Function
 
WScript.StdOut.WriteLine horners_rule(Array(-19,7,-4,6),3)
 
Output:
128

XPL0

code IntOut=11;
 
func Horner(X, N, C); \Return value of polynomial in X
int X, N, C; \variable, number of terms, coefficient list
int A, I;
[A:= 0;
for I:= N-1 downto 0 do
A:= A*X + C(I);
return A;
];
 
IntOut(0, Horner(3, 4, [-19, 7, -4, 6]));

Output:

128

zkl

fcn horner(coeffs,x)
{ coeffs.reverse().reduce('wrap(a,coeff){ a*x + coeff },0.0) }
Output:
horner(T(-19,7,-4,6), 3).println();
128