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14306: THE 'ARCTAN' FUNCTION (offset 36)
The address of this routine is found in the table of addresses. It is called via the calculator literal 36 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 14306 CALL re_stack Use the full floating-point form of X.
14309 LD A,(HL) Fetch the exponent of X.
14310 CP 129 Jump forward for case i: Y=X.
14312 JR C,SMALL
14314 RST 40 X
14315 DEFB 161 stk_one: X, 1
14316 DEFB 27 negate: X, -1
14317 DEFB 1 exchange: -1, X
14318 DEFB 5 division: -1/X
14319 DEFB 49 duplicate: -1/X, -1/X
14320 DEFB 54 less_0: -1/X, (1/0)
14321 DEFB 163 stk_pi_2: -1/X, (1/0), π/2
14322 DEFB 1 exchange: -1/X, π/2, (1/0)
14323 DEFB 0 jump_true to CASES for case ii: -1/X, π/2
14324 DEFB 6
14325 DEFB 27 negate: -1/X, -π/2
14327 DEFB 3
SMALL 14328 RST 40
14329 DEFB 160 stk_zero: Y, 0; continue for case i: W=0
Perform step ii.
CASES 14330 DEFB 1 exchange: W, Y
14331 DEFB 49 duplicate: W, Y, Y
14332 DEFB 49 duplicate: W, Y, Y, Y
14333 DEFB 4 multiply: W, Y, Y*Y
14334 DEFB 49 duplicate: W, Y, Y*Y, Y*Y
14335 DEFB 15 addition: W, Y, 2*Y*Y
14336 DEFB 161 stk_one: W, Y, 2*Y*Y, 1
14337 DEFB 3 subtract: W, Y, 2*Y*Y-1=Z
Perform step iii, passing to the series generator the parameter '12', and the twelve constants required.
14338 DEFB 140 series_0C: W, Y, Z
14339 DEFB 16,178
14341 DEFB 19,14
14343 DEFB 85,228,141
14346 DEFB 88,57,188
14349 DEFB 91,152,253
14352 DEFB 158,0,54,117
14356 DEFB 160,219,232,180
14360 DEFB 99,66,196
14363 DEFB 230,181,9,54,190
14368 DEFB 233,54,115,27,93
14373 DEFB 236,216,222,99,190
14378 DEFB 240,97,161,179,12
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.
14383 DEFB 4 multiply: W, ATN X (case i) or W, ATN (-1/X) (cases ii and iii)
14384 DEFB 15 addition: ATN X (all cases now)
14385 DEFB 56 end_calc
14386 RET Finished: 'last value'=ATN X.
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