One out of Infinity: Interpolating polynomials

Question

For this challenge, when given a list of (x,y) points your submission needs to output a polynomial function that goes through all of them.

For example, if your points were [(0,0), (2,5)], you could return y = 2.5*x or y = x + 1/4x^2.

• All points passed in will consist only of integers
• Two points will never have the same y value
• Your function be of the form y = a*x^n + ... b*x^m. Reducing of terms is allowed.
• Constants (the exponent and coefficient) can either be in decimal or fractional form. Decimals should be accurate to at least 3 decimal places
• You should be able to handle an arbitrary number of points
• Output must be a string starting with "y=". Whitespace anywhere in the string is fine.

Test cases

Note, the given output is not the only valid output, just one of the possibilities.

(1,4)                                                    -> y=x+3
(0,0), (1,8)                                             -> y=8x
(0,6), (-1,90)                                           -> y=-84x+6
(6,4), (70,4), (-1,-6)                                   -> y=−10/497x^2+760/497x−316/71
(4,5), (5,6), (6,7)                                      -> y=x+1
(20,1), (-20,1), (0,5), (10, 4), (11,12), (17,4), (2,4)  -> y=488137/10424165400x^6−643187/473825700x^5−87561839/10424165400x^4+550999039/1042416540x^3−21590713027/5212082700x^2+300110420/52120827x+5

This is a code-golf, so make your answers as short as possible in your favorite language!

Answer 1

MATL, 38 bytes

ynq3$ZQ'y=' '%+fx^%i'btn:qP&vYDh1J3-h)

Inputs are an array with the x values and an array with the y values.

Try it online!

y          % Inplicitly take the two inputs, and push another copy of the first
nq         % Get its length minus 1. This is the required polynomial degree.
3$ZQ       % Coefficients of polynomial fitted to those data (`polyfit`)
'y='       % Push this string
'%+fx^%i'  % Push format string for `sprintf`
b          % Bubble up the polynomial coefficients to the top of the stack
tn:qP      % Duplicate and produce [n n-1  ... 0] where n is the polynomial degree.
           % These are the exponents
&v         % Concatenate vertically. This gives a 2D array with the coefficients in
           % the first row and the exponents in the second
YD         % Apply `sprintf` to this array with previous format string. The array
           % is read in column-major order: down, then across
h          % Concatenate 'y=' and formatted string representing the polynomial
1J3-h)     % The string ends in '...x^0'. Remove last three chars to delete this.
           % Implicitly display

Answer 2

Python2 + sympy, 61 58 bytes

from sympy import*;print interpolate(input("y="),var("x"))

Answer 3

Matlab (with Symbolic Toolbox), 52 bytes

Using builtins for calculating an interpolation polynomial, and then converting it to a symbolc expression for pretty printing and back to a string for the actual output. This function takes two lists of coordinates, e.g. [6,70,-1] and [4,4,-6] for the fourth test case.

 @(x,y)['y=',char(poly2sym(polyfit(x,y,numel(x)-1)))]