A conic of revolution (around the z axis) can be defined by the equation
s^2 – 2*R*z + (k+1)*z^2 = 0
where s^2=x^2+y^2, R is the vertex radius of curvature, and k is the conic constant: k<-1 for a hyperbola, k=-1 for a parabola, -1<k<0 for a tall ellipse, k=0 for a sphere, and k>0 for a short ellipse.
Write a function z=conic(s,R,k) to calculate height z as a function of radius s for given R and k. Choose the branch of the solution that gives z=s^2/(2*R)+... for small values of s. This defines a concave surface for R>0 and a convex surface for R<0.
The trick is to get full machine precision for all values of s and R. The test suite will require a relative error less than 4*eps, where eps is the machine precision.
Hint (added 2015/09/03): the straightforward solution is
z = (R-sqrt(R^2-(k+1)*s^2))/(k+1),
but this does not work if k=-1, gives the wrong branch of the solution if R<0, and is subject to severe roundoff error if s^2 is small compared to R^2. It is possible, however, to find a mathematically equivalent form of the solution that solves all three problems at once.
good one.
Replace NaNs with the number that appears to its left in the row.
2221 Solvers
176 Solvers
Golomb's self-describing sequence (based on Euler 341)
122 Solvers
45 Solvers
1498 Solvers