float cosh(float val) {
  float tmp = exp(val);
  float cosH = (tmp + 1.0 / tmp) / 2.0;
  return cosH;
}

float sinh(float val) {
  float tmp = exp(val);
  float sinH = (tmp - 1.0 / tmp) / 2.0;
  return sinH;
}

vec2 cmul(vec2 a, vec2 b) {
  return vec2(a.x * b.x - a.y * b.y, a.x * b.y + a.y * b.x);
}

vec2 cdiv(vec2 a, vec2 b) {
  float denominator = b.x * b.x + b.y * b.y;
  if (denominator == 0.) return a;
  return cmul(a, vec2(b.x, -b.y)) / denominator;
}

vec2 csin(vec2 a) {
  return vec2(sin(a.x) * cosh(a.y), cos(a.x) * sinh(a.y));
}

vec2 ccos(vec2 a) {
  return vec2(cos(a.x) * cosh(a.y), -sin(a.x) * sinh(a.y));
}

vec2 ctan(vec2 a) {
  return cdiv(csin(a), ccos(a));
}

vec2 cpow(vec2 z, int n) {
  vec2 res = z;
  for (int i = 0; i < 20; i++) {
    if (i >= n - 1) break;
    res = cmul(z, res);
  }
  return res;
}

vec2 cexp(vec2 z) {
  return exp(z.x) * vec2(cos(z.y), sin(z.y));
}