| Routines

37E2: THE 'ARCTAN' FUNCTION (offset +24)

The address of this routine is found in the table of addresses. It is called via the calculator literal +24 by the routine at asn. It is also called indirectly via fp_calc_2. This subroutine handles the function ATN X and is the last of the four routines that use the series generator to produce Chebyshev polynomials. It returns a real number between -π/2 and π/2, which is equal to the value in radians of the angle whose tan is X. The approximation to ATN X is found as follows: i. The values W and Y are found for three cases of X, such that: if -1<X<1 then W=0, Y=X (case i) if 1<=X then W=π/2, Y=-1/X (case ii) if X<=-1 then W=-π/2, Y=-1/X (case iii) In each case, -1<=Y<=1, as required for the series to converge. ii. The argument Z is formed, such that: if -1<X<1 then Z=2*Y*Y-1=2*X*X-1 (case i) otherwise Z=2*Y*Y-1=2/(X*X)-1 (cases ii and iii) iii. The series generator is used to produce the required function. iv. Finally a simple multiplication and addition give ATN X.
Input
HL | Address of the first byte of the number (X)
Perform step i.
atn | 37E2 | CALL re_stack | Use the full floating-point form of X.
| 37E5 | LD A,(HL) | Fetch the exponent of X.
| 37E6 | CP $81 | Jump forward for case i: Y=X.
| 37E8 | JR C,SMALL
| 37EA | RST $28 | X
| 37EB | DEFB $A1 | stk_one: X, 1
| 37EC | DEFB $1B | negate: X, -1
| 37ED | DEFB $01 | exchange: -1, X
| 37EE | DEFB $05 | division: -1/X
| 37EF | DEFB $31 | duplicate: -1/X, -1/X
| 37F0 | DEFB $36 | less_0: -1/X, (1/0)
| 37F1 | DEFB $A3 | stk_pi_2: -1/X, (1/0), π/2
| 37F2 | DEFB $01 | exchange: -1/X, π/2, (1/0)
| 37F3 | DEFB $00 | jump_true to CASES for case ii: -1/X, π/2
| 37F4 | DEFB $06
| 37F5 | DEFB $1B | negate: -1/X, -π/2
| 37F6 | DEFB $33 | jump to CASES for case iii: -1/X, -π/2
| 37F7 | DEFB $03
SMALL | 37F8 | RST $28 |
| 37F9 | DEFB $A0 | stk_zero: Y, 0; continue for case i: W=0
Perform step ii.
CASES | 37FA | DEFB $01 | exchange: W, Y
| 37FB | DEFB $31 | duplicate: W, Y, Y
| 37FC | DEFB $31 | duplicate: W, Y, Y, Y
| 37FD | DEFB $04 | multiply: W, Y, Y*Y
| 37FE | DEFB $31 | duplicate: W, Y, Y*Y, Y*Y
| 37FF | DEFB $0F | addition: W, Y, 2*Y*Y
| 3800 | DEFB $A1 | stk_one: W, Y, 2*Y*Y, 1
| 3801 | DEFB $03 | subtract: W, Y, 2*Y*Y-1=Z
Perform step iii, passing to the series generator the parameter '12', and the twelve constants required.
| 3802 | DEFB $8C | series_0C: W, Y, Z
| 3803 | DEFB $10,$B2 |
| 3805 | DEFB $13,$0E |
| 3807 | DEFB $55,$E4,$8D |
| 380A | DEFB $58,$39,$BC |
| 380D | DEFB $5B,$98,$FD |
| 3810 | DEFB $9E,$00,$36,$75 |
| 3814 | DEFB $A0,$DB,$E8,$B4 |
| 3818 | DEFB $63,$42,$C4 |
| 381B | DEFB $E6,$B5,$09,$36,$BE |
| 3820 | DEFB $E9,$36,$73,$1B,$5D |
| 3825 | DEFB $EC,$D8,$DE,$63,$BE |
| 382A | DEFB $F0,$61,$A1,$B3,$0C |
At the end of the last loop the 'last value' is: ATN X/X (case i) ATN (-1/X)/(-1/X) (cases ii and iii) Perform step iv.
| 382F | DEFB $04 | multiply: W, ATN X (case i) or W, ATN (-1/X) (cases ii and iii)
| 3830 | DEFB $0F | addition: ATN X (all cases now)
| 3831 | DEFB $38 | end_calc
| 3832 | RET | Finished: 'last value'=ATN X.