### cos =====================================================================

Tags
----
scripting, JS, internal, treenode, general concept

Parameters
----------
x

General description
-------------------
Returns the cosine of x.

###


Eval_cos = ->
  push(cadr(p1))
  Eval()
  cosine()

cosine = ->
  save()
  p1 = pop()
  if (car(p1) == symbol(ADD))
    cosine_of_angle_sum()
  else
    cosine_of_angle()
  restore()

# Use angle sum formula for special angles.

#define A p3
#define B p4

cosine_of_angle_sum = ->
  p2 = cdr(p1)
  while (iscons(p2))
    p4 = car(p2); # p4 is B
    if (isnpi(p4)) # p4 is B
      push(p1)
      push(p4); # p4 is B
      subtract()
      p3 = pop(); # p3 is A
      push(p3);  # p3 is A
      cosine()
      push(p4); # p4 is B
      cosine()
      multiply()
      push(p3); # p3 is A
      sine()
      push(p4); # p4 is B
      sine()
      multiply()
      subtract()
      return
    p2 = cdr(p2)
  cosine_of_angle()

cosine_of_angle = ->

  if (car(p1) == symbol(ARCCOS))
    push(cadr(p1))
    return

  if (isdouble(p1))
    d = Math.cos(p1.d)
    if (Math.abs(d) < 1e-10)
      d = 0.0
    push_double(d)
    return

  # cosine function is symmetric, cos(-x) = cos(x)

  if (isnegative(p1))
    push(p1)
    negate()
    p1 = pop()

  # cos(arctan(x)) = 1 / sqrt(1 + x^2)

  # see p. 173 of the CRC Handbook of Mathematical Sciences

  if (car(p1) == symbol(ARCTAN))
    push_integer(1)
    push(cadr(p1))
    push_integer(2)
    power()
    add()
    push_rational(-1, 2)
    power()
    return

  # multiply by 180/pi to go from radians to degrees.
  # we go from radians to degrees because it's much
  # easier to calculate symbolic results of most (not all) "classic"
  # angles (e.g. 30,45,60...) if we calculate the degrees
  # and the we do a switch on that.
  # Alternatively, we could look at the fraction of pi
  # (e.g. 60 degrees is 1/3 pi) but that's more
  # convoluted as we'd need to look at both numerator and
  # denominator.

  push(p1)
  push_integer(180)
  multiply()

  if evaluatingAsFloats
    push_double(Math.PI)
  else
    push_symbol(PI)

  divide()

  n = pop_integer()

  # most "good" (i.e. compact) trigonometric results
  # happen for a round number of degrees. There are some exceptions
  # though, e.g. 22.5 degrees, which we don't capture here.
  if (n < 0 || isNaN(n))
    push(symbol(COS))
    push(p1)
    list(2)
    return

  switch (n % 360)
    when 90, 270
      push_integer(0)
    when 60, 300
      push_rational(1, 2)
    when 120, 240
      push_rational(-1, 2)
    when 45, 315
      push_rational(1, 2)
      push_integer(2)
      push_rational(1, 2)
      power()
      multiply()
    when 135, 225
      push_rational(-1, 2)
      push_integer(2)
      push_rational(1, 2)
      power()
      multiply()
    when 30, 330
      push_rational(1, 2)
      push_integer(3)
      push_rational(1, 2)
      power()
      multiply()
    when 150, 210
      push_rational(-1, 2)
      push_integer(3)
      push_rational(1, 2)
      power()
      multiply()
    when 0
      push_integer(1)
    when 180
      push_integer(-1)
    else
      push(symbol(COS))
      push(p1)
      list(2)