Spectrum ROM: Routine at 14099

The address of this routine is found in the table of addresses. It is called via the calculator literal 37 by the routine at to_power. It is also called indirectly via fp_calc_2.

This subroutine handles the function LN X and is the second of the four routines that use the series generator to produce Chebyshev polynomials.

The approximation to LN X is found as follows:

i. X is tested and report A is given if X is not positive.
ii. X is then split into its true exponent, e', and its mantissa X'=X/(2**e'), where 0.5<=X'<1.
iii. The required value Y1 or Y2 is formed: if X'>0.8 then Y1=e'*LN 2, otherwise Y2=(e'-1)*LN 2.
iv. If X'>0.8 then the quantity X'-1 is stacked; otherwise 2*X'-1 is stacked.
v. Now the argument Z is formed, being 2.5*X'-3 if X'>0.8, otherwise 5*X'-3. In each case, -1<=Z<=1, as required for the series to converge.
vi. The series generator is used to produce the required function.
vii. Finally a simple multiplication and addition leads to LN X being returned as the 'last value'.

The address of this routine is found in the table of addresses. It is called via the calculator literal 37 by the routine at to_power. It is also called indirectly via fp_calc_2. This subroutine handles the function LN X and is the second of the four routines that use the series generator to produce Chebyshev polynomials. The approximation to LN X is found as follows: i. X is tested and report A is given if X is not positive. ii. X is then split into its true exponent, e', and its mantissa X'=X/(2*e'), where 0.5<=X'<1. iii. The required value Y1 or Y2 is formed: if X'>0.8 then Y1=e'LN 2, otherwise Y2=(e'-1)LN 2. iv. If X'>0.8 then the quantity X'-1 is stacked; otherwise 2X'-1 is stacked. v. Now the argument Z is formed, being 2.5X'-3 if X'>0.8, otherwise 5X'-3. In each case, -1<=Z<=1, as required for the series to converge. vi. The series generator is used to produce the required function. vii. Finally a simple multiplication and addition leads to LN X being returned as the 'last value'.
ln	14099	RST 40	X
Perform step i.
	14100	DEFB 61	re_stack: X (in full floating-point form)
	14101	DEFB 49	duplicate: X, X
	14102	DEFB 55	greater_0: X, (1/0)
	14103	DEFB 0	jump_true to VALID: X
	14104	DEFB 4	multiply: X
	14105	DEFB 56	end_calc: X
Report A - Invalid argument.
	14106	RST 8	Call the error handling routine.
	14107	DEFB 9	Call the error handling routine.
Perform step ii.
VALID	14108	DEFB 160	stk_zero: X, 0 (the deleted 1 is overwritten with zero)
	14109	DEFB 2	delete: X
	14110	DEFB 56	end_calc: X
	14111	LD A,(HL)	The exponent, e, goes into A.
	14112	LD (HL),128	X is reduced to X'.
	14114	CALL STACK_A	The stack holds: X', e.
	14117	RST 40	X', e
	14118	DEFB 52	stk_data: X', e, 128
	14119	DEFB 56,0	stk_data: X', e, 128
	14121	DEFB 3	subtract: X', e'
Perform step iii.
	14122	DEFB 1	exchange: e', X'
	14123	DEFB 49	duplicate: e', X', X'
	14124	DEFB 52	stk_data: e', X', X', 0.8
	14125	DEFB 240,76,204,204,205	stk_data: e', X', X', 0.8
	14130	DEFB 3	subtract: e', X', X'-0.8
	14131	DEFB 55	greater_0: e', X', (1/0)
	14132	DEFB 0	jump_true to GRE_8: e', X'
	14133	DEFB 8	jump_true to GRE_8: e', X'
	14134	DEFB 1	exchange: X', e'
	14135	DEFB 161	stk_one: X', e', 1
	14136	DEFB 3	subtract: X', e'-1
	14137	DEFB 1	exchange: e'-1, X'
	14138	DEFB 56	end_calc
	14139	INC (HL)	Double X' to give 2*X'.
	14140	RST 40	e'-1, 2*X'
GRE_8	14141	DEFB 1	exchange: X', e' (X'>0.8) or 2*X', e'-1 (X'<=0.8)
	14142	DEFB 52	stk_data: X', e', LN 2 or 2*X', e'-1, LN 2
	14143	DEFB 240,49,114,23,248	stk_data: X', e', LN 2 or 2*X', e'-1, LN 2
	14148	DEFB 4	multiply: X', e'LN 2=Y1 or 2X', (e'-1)*LN 2=Y2
Perform step iv.
	14149	DEFB 1	exchange: Y1, X' (X'>0.8) or Y2, 2*X' (X'<=0.8)
	14150	DEFB 162	stk_half: Y1, X', .5 or Y2, 2*X', .5
	14151	DEFB 3	subtract: Y1, X'-.5 or Y2, 2*X'-.5
	14152	DEFB 162	stk_half: Y1, X'-.5, .5 or Y2, 2*X'-.5, .5
	14153	DEFB 3	subtract: Y1, X'-1 or Y2, 2*X'-1
Perform step v.
	14154	DEFB 49	duplicate: Y, X'-1, X'-1 or Y2, 2X'-1, 2X'-1
	14155	DEFB 52	stk_data: Y1, X'-1, X'-1, 2.5 or Y2, 2X'-1, 2X'-1, 2.5
	14156	DEFB 50,32	stk_data: Y1, X'-1, X'-1, 2.5 or Y2, 2X'-1, 2X'-1, 2.5
	14158	DEFB 4	multiply: Y1, X'-1, 2.5X'-2.5 or Y2, 2X'-1, 5*X'-2.5
	14159	DEFB 162	stk_half: Y1, X'-1, 2.5X'-2.5, .5 or Y2, 2X'-1, 5*X'-2.5, .5
	14160	DEFB 3	subtract: Y1, X'-1, 2.5X'-3=Z or Y2, 2X'-1, 5*X'-3=Z
Perform step vi, passing to the series generator the parameter '12', and the twelve constants required.
	14161	DEFB 140	series_0C: Y1, X'-1, Z or Y2, 2*X'-1, Z
	14162	DEFB 17,172
	14164	DEFB 20,9
	14166	DEFB 86,218,165
	14169	DEFB 89,48,197
	14172	DEFB 92,144,170
	14175	DEFB 158,112,111,97
	14179	DEFB 161,203,218,150
	14183	DEFB 164,49,159,180
	14187	DEFB 231,160,254,92,252
	14192	DEFB 234,27,67,202,54
	14197	DEFB 237,167,156,126,94
	14202	DEFB 240,110,35,128,147
At the end of the last loop the 'last value' is: LN X'/(X'-1) if X'>0.8 LN (2X')/(2X'-1) if X'<=0.8 Perform step vii.
	14207	DEFB 4	multiply: Y1=LN (2e'), LN X' or Y2=LN (2(e'-1)), LN (2*X')
	14208	DEFB 15	addition: LN (2*e')X')=LN X or LN (2*(e'-1)2*X')=LN X
	14209	DEFB 56	end_calc: LN X
	14210	RET	Finished: 'last value' is LN X.