If you want to avoid the double root you could also solve for sin(2t). The derivative of the arcsin is simpler if anything. You may need some special handeling

If you want to avoid the double root you could also solve for sin(2t). The derivative of the arcsin is simpler if anything. You may need some special handeling for sin(2t)=1 still. You could pretend the funtion continues there (just join it together with asin(z - 1) + z - 1 + pi/2). Or maybe some other transformation.