You can always double check the spline equations by plugging in the boundary conditions (they should give you the exact values for x and y). Doing a quick check (I may have typed some numbers incorrectly), it seems like you last spline equation does not compute correctly at the boundaries y(750) = .2235? Double check this.

Also, since the spline is piecewise, make sure you are evaluating with the correct piece for a particular value of x when using Newton-Raphson. Also, since it is cubic, there may be multiple solutions for a single y. If you start with an initial guess closer to the expected solution, then you are more likely to get that solution.