17.6.5.2 The Sine Function

A somewhat limited sine function could be defined as follows, using the well-known Taylor series expansion for ‘sin(x)’:

(defmath mysin ((float (anglep x)))
  (interactive 1 "mysn")
  (setq x (to-radians x))    ; Convert from current angular mode.
  (let ((sum x)              ; Initial term of Taylor expansion of sin.
        newsum
        (nfact 1)            ; "nfact" equals "n" factorial at all times.
        (xnegsqr :"-(x^2)")) ; "xnegsqr" equals -x^2.
    (for ((n 3 100 2))       ; Upper limit of 100 is a good precaution.
      (working "mysin" sum)  ; Display "Working" message, if enabled.
      (setq nfact (* nfact (1- n) n)
            x (* x xnegsqr)
            newsum (+ sum (/ x nfact)))
      (if (~= newsum sum)    ; If newsum is "nearly equal to" sum,
          (break))           ;  then we are done.
      (setq sum newsum))
    sum))

The actual sin function in Calc works by first reducing the problem to a sine or cosine of a nonnegative number less than ‘pi/4’. This ensures that the Taylor series will converge quickly. Also, the calculation is carried out with two extra digits of precision to guard against cumulative round-off in ‘sum’. Finally, complex arguments are allowed and handled by a separate algorithm.

(defmath mysin ((float (scalarp x)))
  (interactive 1 "mysn")
  (setq x (to-radians x))    ; Convert from current angular mode.
  (with-extra-prec 2         ; Evaluate with extra precision.
    (cond ((complexp x)
           (mysin-complex x))
          ((< x 0)
           (- (mysin-raw (- x)))    ; Always call mysin-raw with x >= 0.
          (t (mysin-raw x))))))

(defmath mysin-raw (x)
  (cond ((>= x 7)
         (mysin-raw (% x (two-pi))))     ; Now x < 7.
        ((> x (pi-over-2))
         (- (mysin-raw (- x (pi)))))     ; Now -pi/2 <= x <= pi/2.
        ((> x (pi-over-4))
         (mycos-raw (- x (pi-over-2))))  ; Now -pi/2 <= x <= pi/4.
        ((< x (- (pi-over-4)))
         (- (mycos-raw (+ x (pi-over-2)))))  ; Now -pi/4 <= x <= pi/4,
        (t (mysin-series x))))           ; so the series will be efficient.

where mysin-complex is an appropriate function to handle complex numbers, mysin-series is the routine to compute the sine Taylor series as before, and mycos-raw is a function analogous to mysin-raw for cosines.

The strategy is to ensure that ‘x’ is nonnegative before calling mysin-raw. This function then recursively reduces its argument to a suitable range, namely, plus-or-minus ‘pi/4’. Note that each test, and particularly the first comparison against 7, is designed so that small roundoff errors cannot produce an infinite loop. (Suppose we compared with ‘(two-pi)’ instead; if due to roundoff problems the modulo operator ever returned ‘(two-pi)’ exactly, an infinite recursion could result!) We use modulo only for arguments that will clearly get reduced, knowing that the next rule will catch any reductions that this rule misses.

If a program is being written for general use, it is important to code it carefully as shown in this second example. For quick-and-dirty programs, when you know that your own use of the sine function will never encounter a large argument, a simpler program like the first one shown is fine.