Make tan function match sbcl

[Izaakwltn]

Trigonometric:tan doesn't match SBCL:

coalton-user> (coalton (tan (Complex 1.0d0 -1.0d0)))
#C(0.2717525853195118 -1.0839233273386946)
NEW-TRIG> (cl:tan #C (1.0d0 -1.0d0))
#C(0.27175258531951174 -1.0839233273386943)

I made an attempt as seen below, though it still doesn't match exactly, and causes NaN errors during identity tests.

;; unused first attempt

(defpackage #:new-trig
  (:use #:coalton
        #:coalton-prelude)
  (:local-nicknames (#:math #:coalton-library/math)))

(in-package #:new-trig)

(coalton-toplevel

  (define (old-tan z)
    (let x = (real-part z))
    (let y = (imag-part z))
    (let c = (exp (complex y (negate x))))
    (let recip-c = (/ 1 c))
    ;; (i (e^(y - i x) - e^(-y + i x)))
    ;; / (e^(-y + i x) + e^(y - i x))
    (/ (* math:ii (- c recip-c)) (+ c recip-c))))

  (define (new-tan z)
    (let x = (real-part z))
    (let y = (imag-part z))
    (let e+y = (exp y))
    (let e-y = (exp (negate y)))
    (let a = (/ (* (+ e+y e-y) (math:sin x)) 2))
    (let b = (/ (* (- e+y e-y) (math:cos x)) 2))
    (let c = (/ (* (+ e+y e-y) (cos x)) 2))
    (let d = (/ (* (- e+y e-y) (sin x)) -2))
    (let denom = (+ (^ c 2) (^ d 2)))
    (Complex (/ (+ (* c a) (* b d))
                denom)
             (/ (- (* b c) (* a d))
                denom)))

;; NEW-TRIG> (coalton (old-tan (Complex 1.0d0 -1.0d0)))
;; #C(0.2717525853195118 -1.0839233273386946)
;; NEW-TRIG> (coalton (new-tan (Complex 1.0d0 -1.0d0)))
;; #C(0.2717525853195117 -1.0839233273386946)
;; NEW-TRIG> (cl:tan #C (1.0d0 -1.0d0))
;; #C(0.27175258531951174 -1.0839233273386943)