We have sin(3*x)=3*sin(x)-4*sin(x)^3 and sinh(3*x)=3*sinh(x)+4*sinh(x)^3 These give us recursive approximations to sin(x), sinh(x), thanks to RWG/Hakmem. So we have f(3*x)=3*f(x)+-4*f(x)^3 What are the functions f(3*x)=3*f(x)+b*f(x)^3, for some real b ? They seem to live in the land between sin(x) and sinh(x). Actually, I'm more interested in f(x), s.t. f(2*x) = 2*f(x)+b*f(x)^3, for some real (probably integer?) constant b, because I want a representation in which binary shifting is easy(er). If f(x) = x + ..., then we have f(x) = 8 17 7 15 6 13 5 11 4 9 3 7 3035153449 b x 32516717 b x 151 b x 163 b x 37 b x b x --------------------- + ----------------- + ---------- + ---------- + -------- + ----- 137019759699497875200 29401699409082000 3204726525 97389600 771120 945 2 5 3 b x b x + ----- + ---- + x 60 6 How to easily compute these coefficients? If b=4, the coefficient of x^3 becomes 2/3, which matches that in sinh(2x)/2, although f(x) doesn't converge quite as fast. If b=1, f(x) converges somewhat faster, but how fast? Does it converge everywhere? What is its radius of convergence? If f(2*x)=2*x+b*x^3, what is the/a differential equation for f(x)?