fpprec:100$
/* Compute the circumference of an ellipse with semi-axes a and b
with a relative accuracy of fpprec decimal digits.  Code is adapted
from Carlson (1995), eqs 2.36 - 2.39. */
C(a,b) := block( [x : bfloat(max(a, b)), y : bfloat(min(a, b)),
  s : 0, m : 1, tol : 0.1b0^fpprec],
  if y < tol * x then return( 4 * x ),
  tol:sqrt(tol),
  while x - y > tol * y do
  block( [t : (x + y) / 2],
    y : sqrt(x * y),
    x : t,
    m : 2 * m,
    s : s + m * (x - y)^2 ),
  return( bfloat(%pi) * ((a + b)^2 - s) / (x + y) ) )$