Spectrum ROM: Routine at 3713

The address of this routine is found in the table of addresses. It is called via the calculator literal +25 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 +25 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	3713	RST $28	X
Perform step i.
	3714	DEFB $3D	re_stack: X (in full floating-point form)
	3715	DEFB $31	duplicate: X, X
	3716	DEFB $37	greater_0: X, (1/0)
	3717	DEFB $00	jump_true to VALID: X
	3718	DEFB $04	multiply: X
	3719	DEFB $38	end_calc: X
Report A - Invalid argument.
	371A	RST $08	Call the error handling routine.
	371B	DEFB $09	Call the error handling routine.
Perform step ii.
VALID	371C	DEFB $A0	stk_zero: X, 0 (the deleted 1 is overwritten with zero)
	371D	DEFB $02	delete: X
	371E	DEFB $38	end_calc: X
	371F	LD A,(HL)	The exponent, e, goes into A.
	3720	LD (HL),$80	X is reduced to X'.
	3722	CALL STACK_A	The stack holds: X', e.
	3725	RST $28	X', e
	3726	DEFB $34	stk_data: X', e, 128
	3727	DEFB $38,$00	stk_data: X', e, 128
	3729	DEFB $03	subtract: X', e'
Perform step iii.
	372A	DEFB $01	exchange: e', X'
	372B	DEFB $31	duplicate: e', X', X'
	372C	DEFB $34	stk_data: e', X', X', 0.8
	372D	DEFB $F0,$4C,$CC,$CC,$CD	stk_data: e', X', X', 0.8
	3732	DEFB $03	subtract: e', X', X'-0.8
	3733	DEFB $37	greater_0: e', X', (1/0)
	3734	DEFB $00	jump_true to GRE_8: e', X'
	3735	DEFB $08	jump_true to GRE_8: e', X'
	3736	DEFB $01	exchange: X', e'
	3737	DEFB $A1	stk_one: X', e', 1
	3738	DEFB $03	subtract: X', e'-1
	3739	DEFB $01	exchange: e'-1, X'
	373A	DEFB $38	end_calc
	373B	INC (HL)	Double X' to give 2*X'.
	373C	RST $28	e'-1, 2*X'
GRE_8	373D	DEFB $01	exchange: X', e' (X'>0.8) or 2*X', e'-1 (X'<=0.8)
	373E	DEFB $34	stk_data: X', e', LN 2 or 2*X', e'-1, LN 2
	373F	DEFB $F0,$31,$72,$17,$F8	stk_data: X', e', LN 2 or 2*X', e'-1, LN 2
	3744	DEFB $04	multiply: X', e'LN 2=Y1 or 2X', (e'-1)*LN 2=Y2
Perform step iv.
	3745	DEFB $01	exchange: Y1, X' (X'>0.8) or Y2, 2*X' (X'<=0.8)
	3746	DEFB $A2	stk_half: Y1, X', .5 or Y2, 2*X', .5
	3747	DEFB $03	subtract: Y1, X'-.5 or Y2, 2*X'-.5
	3748	DEFB $A2	stk_half: Y1, X'-.5, .5 or Y2, 2*X'-.5, .5
	3749	DEFB $03	subtract: Y1, X'-1 or Y2, 2*X'-1
Perform step v.
	374A	DEFB $31	duplicate: Y, X'-1, X'-1 or Y2, 2X'-1, 2X'-1
	374B	DEFB $34	stk_data: Y1, X'-1, X'-1, 2.5 or Y2, 2X'-1, 2X'-1, 2.5
	374C	DEFB $32,$20	stk_data: Y1, X'-1, X'-1, 2.5 or Y2, 2X'-1, 2X'-1, 2.5
	374E	DEFB $04	multiply: Y1, X'-1, 2.5X'-2.5 or Y2, 2X'-1, 5*X'-2.5
	374F	DEFB $A2	stk_half: Y1, X'-1, 2.5X'-2.5, .5 or Y2, 2X'-1, 5*X'-2.5, .5
	3750	DEFB $03	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.
	3751	DEFB $8C	series_0C: Y1, X'-1, Z or Y2, 2*X'-1, Z
	3752	DEFB $11,$AC
	3754	DEFB $14,$09
	3756	DEFB $56,$DA,$A5
	3759	DEFB $59,$30,$C5
	375C	DEFB $5C,$90,$AA
	375F	DEFB $9E,$70,$6F,$61
	3763	DEFB $A1,$CB,$DA,$96
	3767	DEFB $A4,$31,$9F,$B4
	376B	DEFB $E7,$A0,$FE,$5C,$FC
	3770	DEFB $EA,$1B,$43,$CA,$36
	3775	DEFB $ED,$A7,$9C,$7E,$5E
	377A	DEFB $F0,$6E,$23,$80,$93
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.
	377F	DEFB $04	multiply: Y1=LN (2e'), LN X' or Y2=LN (2(e'-1)), LN (2*X')
	3780	DEFB $0F	addition: LN (2*e')X')=LN X or LN (2*(e'-1)2*X')=LN X
	3781	DEFB $38	end_calc: LN X
	3782	RET	Finished: 'last value' is LN X.