The BBC and Master Computer Public Domain Library

A8C3 ATN

Submitted by Steve Fewell

Description:

Call routine &96DA to obtain the Floating-point value (at the BASIC text pointer location) and convert iy to a Float
value (if it was an Integer) and store it in the FWA.

If the FWA Number is zero then exit with FWA = 0.0 and A = #&FF (to indicate a Floating-Point result).
If the FWA value is negative then clear the FWA sign byte (to make the value positive), call the ATN calculation routine,
set the results (FWA) sign byte to #&FF (to make the value negative) [as after the ATN calculation, A = #&FF]
and exit.
Otherwise (FWA value is positive), jump to the ATN calculation routine and exit.

[&A8D5] The ATN Calculation routine:
If the FWA exponent is less than #&81 (i.e. the FWA value is less than 1) then jump to &A8EA to calculate ATN result
for an argument in the range 0.0 - 1.0) and exit.

Otherwise, the FWA value is too large to calculate ATN with, so reduce the FWA value as follows:

* Call &A5E9 to calculate the reciple of the FWA value (FWA = 1 / FWA). The result will be in the range 0.0 - 1.0

* As the FWA value is now in the required range, call &A8EA to calculate the ATN result from the FWA value.

* Set the argp pointer to point to the Floating-Point constant value at location &BF2E, which is the value 1.57079683

* Subtract the FWA (ATN result) value from 1.57079683 [Pi / 2] (FWA = argp - FWA).

* Now, the FWA contains the correct ATN result, so exit with A = #&FF (to indicate that the result is Floating-Point).

[&A8EA] The ATN restricted range (0.0 - 1.0) Calculation routine:
IF the FWA exponent is less than #&73 then the FWA value is too small to calculate (i.e. it is practically 0.0), so
exit with the FWA value unchanged, and with A = #&FF (to indicate a Floating-Point result).

Store the FWA value to temporary Floating-point location &0476 - &047A.
Set the FWB to contain the value -0.5 (using &A576 (Clear FWB) routine, and setting FWB exponent = #&80, FWB Mantissa
byte 1 to #&80 and FWB Sign byte to #&80).
Add the FWB value (-0.5) to the FWA [FWA = FWB + FWA].
Set X to #&97 -> the LSB of the first fraction to apply [i.e. this points to the constant at location &BF97]
Set A to #&C9 -> the LSB of the default value (if the FWA value is too small to evaluate) [i.e. this points to the
constant at location &BFC9]
Set Y to #&04 (the number of cycles to evaluate).

Call routine &A861 which will evaluate the continued-fraction expansion series, as follows:

* If the FWA exponent is less than #&40 thern the FWA is too small to be calculated with, so exit with

   the FWA = the default value at &BFC9, which is 0.927295218.

* Calculate the reciple of the FWA value [FWA = 1 / FWA], store the reciple in &046C. Set argp to &BF97.

* Add argp (&BF97 --> -0.0800532048 (in the Floating-point constant table)) to the FWA value.

* Cycle 1:

* Argp = &BF9C (0.155279656). FWA = argp / FWA.

* Argp = &BFA1 (0.440292008). FWA = argp + FWA.

* FWA = &046C (the reciple value) + FWA.

* Cycle 2:

* Argp = &BFA6 (0.215397413). FWA = argp / FWA.

* Argp = &BFAB (0.240938382). FWA = argp + FWA.

* FWA = &046C (the reciple value) + FWA.

* Cycle 3:

* Argp = &BFB0 (0.0767796872). FWA = argp / FWA.

* Argp = &BFB5 (1.07231162). FWA = argp + FWA.

* FWA = &046C (the reciple value) + FWA.

* Cycle 4:

* Argp = &BFBA (0.956538983). FWA = argp / FWA.

* Argp = &BFBF (-0.513841612). FWA = argp + FWA.

* FWA = &046C (the reciple value) + FWA.

* Argp = &BFC4 (-0.254590442). FWA = argp / FWA.

* Argp = &BFC9 (0.927295218). FWA = argp + FWA.

Jump to &A99F to set the argp to the temporary Floating-Point value at location &0476,
multiply the FWA by argp [FWA = argp * FWA] and exit with A = #&FF (to indicate a Floating-point result).


Example 1: ATN(0.0)

FWA is zero, so exit with FWA = 0.0!

Example 2: ATN(-0.75)

Make FWA sign positive (FWA = 0.75)

[&A8EA] Store FWA (0.75) to &0476

FWA = FWA + -0.5 = 0.25

[&A861] FWA = 1 / FWA = 4

Store FWA to &046C

FWA = -0.0800532 + FWA = 3.9199468

FWA = 0.1552796 / FWA = 0.0396126

FWA = 0.4402920 + FWA = 0.4799046

FWA = &046C + FWA = 4.4799046

FWA = 0.2153974 / FWA = 0.0480808

FWA = 0.2409384 + FWA = 0.2890192

FWA = &046C + FWA = 4.2890192

FWA = 0.0767797 / FWA = 0.0179014

FWA = 1.0723116 + FWA = 1.0902131

FWA = &046C + FWA = 5.0902131

FWA = 0.9565390 / FWA = 0.1879172

FWA = -0.5138416 + FWA = -0.3259243

FWA = &046C + FWA = 3.6740757

FWA = -0.2545904 / FWA = -0.0692937

FWA = 0.9272952 + FWA = 0.8580014

FWA = &0476 (0.75) * FWA = 0.6435011

Make FWA sign negative (FWA = -0.6435011)

Example 3: ATN(2.41)

FWA Exponent >= #&81, so: [A8DB] FWA = 1 / FWA = 0.4149377

[&A8EA] Store FWA (2.41) to &0476

FWA = FWA + -0.5 = -0.0850622

[&A861] FWA = 1 / FWA = -11.756098

Store FWA to &046C

FWA = -0.0800532 + FWA = -11.836151

FWA = 0.1552796 / FWA = -0.013119

FWA = 0.4402920 + FWA = 0.4271729

FWA = &046C + FWA = -11.328925

FWA = 0.2153974 / FWA = -0.019013

FWA = 0.2409384 + FWA = 0.2219253

FWA = &046C + FWA = -11.534173

FWA = 0.0767797 / FWA = -0.0066567

FWA = 1.0723116 + FWA = 1.0656549

FWA = &046C + FWA = -10.690443

FWA = 0.9565390 / FWA = -0.089476

FWA = -0.5138416 + FWA = -0.6033176

FWA = &046C + FWA = -12.359416

FWA = -0.2545904 / FWA = 0.0205989

FWA = 0.9272952 + FWA = 0.9478941

FWA = &0476 (0.4149377) * FWA = 0.393317

As the original FWA value has an exponent >= #'&81, FWA = 1.5707963 - FWA = 1.1774793

Example 4: ATN(0.25)

[&A8EA] Store FWA (0.25) to &0476

FWA = FWA + -0.5 = -0.25

[&A861] FWA = 1 / FWA = -4

Store FWA to &046C

FWA = -0.0800532 + FWA = -4.0800532

FWA = 0.1552796 / FWA = -0.0380582

FWA = 0.4402920 + FWA = 0.4022337

FWA = &046C + FWA = -3.5977662

FWA = 0.2153974 / FWA = -0.0598697

FWA = 0.2409384 + FWA = 0.1810686

FWA = &046C + FWA = -3.8189314

FWA = 0.0767797 / FWA = -0.020105

FWA = 1.0723116 + FWA = 1.0522066

FWA = &046C + FWA = -2.9477934

FWA = 0.9565390 / FWA = -0.3244932

FWA = -0.5138416 + FWA = -0.8383348

FWA = &046C + FWA = -4.8383348

FWA = -0.2545904 / FWA = 0.0526194

FWA = 0.9272952 + FWA = 0.9799146

FWA = &0476 (0.25) * FWA = 0.2449786

Example 5: ATN(5.63)

FWA Exponent >= #&81, so: [A8DB] FWA = 1 / FWA = 0.1776198

[&A8EA] Store FWA (0.1776198) to &0476

FWA = FWA + -0.5 = -0.3223802

[&A861] FWA = 1 / FWA = -3.1019275

Store FWA to &046C

FWA = -0.0800532 + FWA = -3.1819807

FWA = 0.1552796 / FWA = -0.0487996

FWA = 0.4402920 + FWA = 0.3914923

FWA = &046C + FWA = -2.7104352

FWA = 0.2153974 / FWA = -0.0794696

FWA = 0.2409384 + FWA = 0.1614686

FWA = &046C + FWA = -2.9404589

FWA = 0.0767797 / FWA = -0.0261114

FWA = 1.0723116 + FWA = 1.0462001

FWA = &046C + FWA = -2.0557274

FWA = 0.9565390 / FWA = -0.4653043

FWA = -0.5138416 + FWA = -0.9791459

FWA = &046C + FWA = -4.0810735

FWA = -0.2545904 / FWA = 0.0623831

FWA = 0.9272952 + FWA = 0.9896783

FWA = &0476 (0.1776198) * FWA = 0.1757864

As the original FWA value has an exponent >= #'&81, FWA = 1.5707963 - FWA = 1.3950098

Example 6: ATN(1.5)

FWA Exponent >= #&81, so: [A8DB] FWA = 1 / FWA = 0.6666666

[&A8EA] Store FWA (0.6666666) to &0476

FWA = FWA + -0.5 = -0.1666666

[&A861] FWA = 1 / FWA = 6

Store FWA to &046C

FWA = -0.0800532 + FWA = 5.9199468

FWA = 0.1552796 / FWA = 0.0262298

FWA = 0.4402920 + FWA = 0.4665219

FWA = &046C + FWA = 6.4665219

FWA = 0.2153974 / FWA = 0.0333096

FWA = 0.2409384 + FWA = 0.274248

FWA = &046C + FWA = 6.274248

FWA = 0.0767797 / FWA = 0.0122372

FWA = 1.0723116 + FWA = 1.089017

FWA = &046C + FWA = 7.089017

FWA = 0.9565390 / FWA = 0.1349325

FWA = -0.5138416 + FWA = -0.378909

FWA = &046C + FWA = 5.6210909

FWA = -0.2545904 / FWA = -0.0452919

FWA = 0.9272952 + FWA = 0.8820032

FWA = &0476 (0.6666666) * FWA = 0.588002

As the original FWA value has an exponent >= #'&81, FWA = 1.5707963 - FWA = 0.9827942

Example 7: ATN(90)

FWA Exponent >= #&81, so: [A8DB] FWA = 1 / FWA = 0.0111111

[&A8EA] Store FWA (0.0111111) to &0476

FWA = FWA + -0.5 = -0.4888888

[&A861] FWA = 1 / FWA = -2.0454545

Store FWA to &046C

FWA = -0.0800532 + FWA = -2.1255077

FWA = 0.1552796 / FWA = -0.0730552

FWA = 0.4402920 + FWA = 0.3672367

FWA = &046C + FWA = -1.6782178

FWA = 0.2153974 / FWA = -0.1283489

FWA = 0.2409384 + FWA = 0.1125895

FWA = &046C + FWA = -1.932865

FWA = 0.0767797 / FWA = -0.0397232

FWA = 1.0723116 + FWA = 1.0325883

FWA = &046C + FWA = -1.0128662

FWA = 0.9565390 / FWA = -0.9443883

FWA = -0.5138416 + FWA = -1.4582299

FWA = &046C + FWA = -3.5036844

FWA = -0.2545904 / FWA = 0.0726636

FWA = 0.9272952 + FWA = 0.9999588

FWA = &0476 (0.0111111) * FWA = 0.0111106

As the original FWA value has an exponent >= #'&81, FWA = 1.5707963 - FWA = 1.5596857

This diagram shows the relationship between the trig functions.

Disassembly for the ATN routine

A8C3		032 218 150	20 DA 96	JSR &96DA Get and Check Float (convert if Int)
A8C6		032 242 163	20 F2 A3	JSR &A3F2 Obtain Sign of the FWA Floating-Point value
A8C9	[	240 091	F0 5B	BEQ 91 --> &A926 Set A to #&FF and exit
A8CB		016 008	10 08	BPL 8 --> &A8D5
A8CD	d.	100 046	64 2E	STZ &2E
A8CF		032 213 168	20 D5 A8	JSR &A8D5
A8D2	.	133 046	85 2E	STA &2E
A8D4	`	096	60	RTS
A8D5	0	165 048	A5 30	LDA &30
A8D7		201 129	C9 81	CMP#&81
A8D9		144 015	90 0F	BCC 15 --> &A8EA
A8DB		032 233 165	20 E9 A5	JSR &A5E9 Floating-Point Reciple (FWA=1/FWA)
A8DE		032 234 168	20 EA A8	JSR &A8EA
A8E1		032 137 165	20 89 A5	JSR &A589 Set argp to &BF2E
A8E4		032 138 166	20 8A A6	JSR &A68A Floating-Point Subtraction [FWA=argp-FWA]
A8E7		169 255	A9 FF	LDA#&FF
A8E9	`	096	60	RTS
A8EA	0	165 048	A5 30	LDA &30
A8EC	s	201 115	C9 73	CMP#&73
A8EE	6	144 054	90 36	BCC 54 --> &A926 Set A to #&FF and exit
A8F0		032 013 165	20 0D A5	JSR &A50D Store FWA to &0476 and set argp to &0476
A8F3	v	032 118 165	20 76 A5	JSR &A576 Clear FWB Mantissa bytes 2 to 5
A8F6		169 128	A9 80	LDA#&80
A8F8	<	133 060	85 3C	STA &3C
A8FA	=	133 061	85 3D	STA &3D
A8FC	;	133 059	85 3B	STA &3B
A8FE		032 146 166	20 92 A6	JSR &A692 FWA = FWA + FWB
A901		162 151	A2 97	LDX#&97
A903		169 201	A9 C9	LDA#&C9
A905		160 004	A0 04	LDY#&04
A907	a	032 097 168	20 61 A8	JSR &A861 Evaluate continued-fraction expansion series
A90A	L	076 159 169	4C 9F A9	JMP &A99F Multiply FWA by variable at &0476

 

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