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A90D SIN
Submitted by Steve Fewell
Description:
Clear the carry flag  to indicate that the Sine result is required.
Continue to the COS routine (&A90E) to continue the calculation as follows.
Store the 6502 flags to the 6502 stack, so that the carry flag can be checked later.
Call routine &A93A to obtain the FloatingPoint argument and calculate the Sine or Cosine value (the value returned
will be either COS or SIN  depending on the value of the FloatingPoint argument.

Routine &A93A does the following:
 * Call routine &96DA to obtain the value at the Text pointer B location (convert it to Float if it is
 Integer or issue 'Type mismatch' error if it is a String value).
 * If the FWA exponent (location &30) is more than or equal to #&98 then issue 'Accuracy lost' error
 as the FWA value is too large to use to calculate the SIN/COS value.
 * Store the FWA value to temporary Floating point value location &046C&0470.
 * Set the argument pointer (&4A&4B) to &BF2E (which points to the Float value 1.57079633 in the
 FloatingPoint constant table).
 * Call &A4E0 to unpack the FloatingPoint constant at &BF2E (1.57079633, which is PI/2) to the FWB
 * Store the FWA sign (location &2E) in the FWB sign byte (location &3B) so that the PI/2 value has
 the same sign as the argument (the FWA value).
 * Decrement the FWB exponent (location &3C) to divide the FWB value by 2  so the FWB now contains the
 value 0.785398165.
 * Call routine &A692 to add the FWB value (PI/4) to the FWA value.
 * Set A to #&33.
 * Call routine &A9D4 to set the argument pointer (&4A&4B) to &BF00 + A = &BF33 (which points to
 the Float value 0.636619772 (which is 2/PI) in the FloatingPoint constant table)
 and then multiply the FWA by the Float value pointed to by the argument pointer (&4A&4B).
 * Call routine &96C3 to convert the Float value (FWA) to an Integer and place the result in the IWA.
 * Store A (the LSB of the IWA value  from location &2A) in location &49.
 * If the Integer value (&2A&2C) is zero then jump to &A98D (as the FWA value does not need to be
 reduced to a value within the required range (PI/4) to (PI/4) [i.e. 0.785398 to 0.785398]),
 so [&A98D] load the FWA with the original Sine/Cosine argument (from location &046C&0470).
 * Otherwise, the FWA value is out of the Sine/Cosine range, and needs to be reduced, so:
 * Call &8189 to convert the Integer value (IWA) to a Float and store the result in the temporary
 Floatingpoint variable located at &0471&0475.
 * Then set the argp pointer (&4A&4B) to pointer to &BF24 (which is the Floatingpoint constant
 value 1.57080078) and multiply the FWA by this constant.
 * Set the argp pointer (&4A&4B) to point to the temporary Floatingpoint variable location &046C
 (which is the original FWA Sine/Cosine argument value) and add the FloatingPoint
 value at &046C&0470 to the FWA value.
 * Store the result in temporary FloatingPoint value location &046C&0470.
 * Set the FWA to the FloatingPoint value stored at location &0471&0475 (the IWA value).
 * Set the argp pointer (&4A&4B) to point to &BF29 (the address of the value 0.00000445445511
 in the FloatingPoint constant table).
 * Multiply the FWA by the FloatingPoint constant 0.00000445445511 (store the result in the FWA).
 * Set the argp pointer (&4A&4B) to &046C and add the FloatingPoint value at &046C&0470
 to the FWA.
 * Jump to &A990 to perform the Sine/Cosine calculation as the value is now within the required range.
 * [A990] Store the FWA value (the actual Sine/Cosine argument, which has been reduced where necessary) to location
 &0476&047A.
 * Multiply the FWA by the Floatingpoint value at &0476&047A (i.e. square the FWA value).
 * Set X to #&74 (i.e. the LSB of the first fraction to apply)
 * Set A to #&92 (i.e. the LSB of the default value (if the FWA value is too small to evaluate)
 * Set Y to #&02 (the number of continuedfraction expansion cycles to evaluate)
 * Call routine &A861 to evaluate the continuedfraction expansions, as follows:
 * If the FWA exponent is less than #&40 then the FWA value is too small to evaluate, so exit with the
 FWA set to the FloatingPoint constant at &BF92 (i.e. 1.0).
 * Calculate the reciple of the FWA value (FWA = 1/FWA) and store the result in the FWA and location
 &046C&0470
 * Add the FloatingPoint constant at &BF74 (0.0119090311) to the FWA
 * Cycle 1:
 * Divide the FloatingPoint constant at &BF79 (0.000107499459) by the FWA, storing the result in the FWA.
 * Add the FloatingPoint constant at &BF7E (0.0171640246) to the FWA.
 * Add the Reciple value (from location &046C&0470) to the FWA.
 * Cycle 2:
 * Divide the FloatingPoint constant at &BF83 (0.0013095369) by the FWA, storing the result in the FWA.
 * Add the FloatingPoint constant at &BF88 (0.0499999922) to the FWA.
 * Add the Reciple value (from location &046C&0470) to the FWA.
 Last Cycle calculation:
 * Divide the FloatingPoint constant at &BF8D (0.166666666) by the FWA, storing the result in the FWA.
 * Add the FloatingPoint constant at &BF92 (1.0) to the FWA.
 * Multiply the FWA by the Temporary FloatingPoint variable stored at location &0476&047A (the argument
 value).
 * Exit with A = #&FF (as the result is a FloatingPoint value in the FWA)
Now, the FWA contains either the Sine result or the Cosine result (which will have to be converted to the value
requested by the user).
Retrieve the carry flag value from the 6502 stack.
The carry flag is clear, indicating that a Sine result is required (therefore, the value at location &49 is not
incremented.
[&A923] If bit 0 of location &49 is clear (i.e. the original argument was an even value (in the case of Sine)  or
an odd value (in the case of Cosine)) then Set A to #&FF (to indicate that the result is a Floatingpoint value) and
[&A917] check the value in location &49  if bit 1 is not set then complement the FWA value (FWA=FWA) and exit;
otherwise, just exit.

Otherwise, bit 0 of location &49 is set (indicating that the original argument was odd (in the case of Sine) or even (in
the case of Cosine); so, adjust the calculated result as follows:
 * Square the FWA value (FWA = FWA * FWA),
 * Subtract the FWA value from 1 (FWA=1.0FWA) and
 * Set the FWA value to the square root of the FWA (FWA=SQR(FWA)).
[&A917] check the value in location &49  if bit 1 is not set then complement the FWA value (FWA=FWA) and exit;
otherwise, just exit.
Example 1: SIN(1.5)

 Set &046C = 1.5 (the argument)
 Set FWB to 0.785398165 (PI/4)
 FWA = FWA + FWB = 2.285398
 FWA = FWA * 0.636619772 = 1.4549294
 ?&49 = 1 (Integer part of FWA value). FWA = IWA = 1
 Store the FWA in &0471
 FWA = FWA * 1.57080078 = 1.57080078
 FWA = FWA + &046C = 0.07080078
 Store FWA in &046C
 Store &0471 (1) in FWA
 FWA = FWA * 0.0000044544511 = 0.0000044544511
 FWA = FWA + &046C = 0.0707963255
 Store the FWA in &0476
 FWA = FWA * FWA = 0.0050121197
 [&A861]: FWA = 1 / FWA = 199.51638
 Store FWA in &046C
 FWA = FWA + 0.0119090311 = 199.504474778
 FWA = 0.000107499 / FWA = 0.00000005388
 FWA = 0.017164024 + FWA = 0.017163485
 FWA = &046C + FWA = 199.499216515
 FWA = 0.0013095369 / FWA = 0.000006564
 FWA = 0.04999999 + FWA = 0.050006554
 FWA = &046C + FWA = 199.566387
 FWA = 0.16666666 / FWA = 0.00083514
 FWA = 1 + FWA = 0.999164856
 FWA = FWA * &0476 = 0.070737200377

 Location &49 contains the value 1
 Bit 0 of &49 is set, so adjust the calculated value as follows:
 FWA = FWA * FWA = 0.005003751517
 FWA = 1  FWA = 0.9949962484828
 FWA = SQR(FWA) = 0.99749498669558
 Bit 1 of &49 is not set, so exit with FWA unchanged.
Example 2: SIN(0.75)

 Set &046C = 0.75 (the argument)
 Set FWB to 0.785398165 (PI/4)
 FWA = FWA + FWB = 1.535398165
 FWA = FWA * 0.636619772 = 0.9774646
 ?&49 = 0 (Integer part of FWA value). FWA = IWA = 0
 [&A98D] As &49 is 0, store value at &046C in FWA = 0.75
 Store the FWA in &0476
 FWA = FWA * FWA = 0.5625
 [&A861]: FWA = 1 / FWA = 1.777777
 Store FWA in &046C
 FWA = FWA + 0.0119090311 = 1.7896868
 FWA = 0.000107499 / FWA = 0.00006
 FWA = 0.017164024 + FWA = 0.0171039
 FWA = &046C + FWA = 1.7606737
 FWA = 0.0013095369 / FWA = 0.0007437
 FWA = 0.04999999 + FWA = 0.0507436
 FWA = &046C + FWA = 1.8285213
 FWA = 0.16666666 / FWA = 0.0911482
 FWA = 1 + FWA = 0.9088517
 FWA = FWA * &0476 = 0.6816387

 Location &49 contains the value 0
 Bits 0 and 1 of &49 are not set, so exit with FWA unchanged.
Example 3: SIN(0.0)

 Set &046C = 0.0 (the argument)
 Set FWB to 0.785398165 (PI/4)
 FWA = FWA + FWB = 0.785398165
 FWA = FWA * 0.636619772 = 0.50000000
 ?&49 = 0 (Integer part of FWA value). FWA = IWA = 0
 [&A98D] As &49 is 0, store value at &046C in FWA = 0.0
 Store the FWA in &0476
 FWA = FWA * FWA = 0.0
 [&A861]: FWA = 1.0 (as argument is zero)
 FWA = FWA * &0476 = 0.0

 Location &49 contains the value 0
 Bits 0 and 1 of &49 are not set, so exit with FWA unchanged.
Example 4: SIN(0.25)

 Set &046C = 0.25 (the argument)
 Set FWB to 0.785398165 (PI/4)
 FWA = FWA + FWB = 1.0353982
 FWA = FWA * 0.636619772 = 0.6591548
 ?&49 = 0 (Integer part of FWA value). FWA = IWA = 0
 [&A98D] As &49 is 0, store value at &046C in FWA = 0.25
 Store the FWA in &0476
 FWA = FWA * FWA = 0.0625
 [&A861]: FWA = 1 / FWA = 16
 Store FWA in &046C
 FWA = FWA + 0.0119090311 = 15.988091
 FWA = 0.000107499 / FWA = 0.0000067
 FWA = 0.0171640246 + FWA = 0.0171572
 FWA = &046C + FWA = 15.982843
 FWA = 0.0013095369 / FWA = 0.0000819
 FWA = 0.04999999 + FWA = 0.0500818
 FWA = &046C + FWA = 16.050082
 FWA = 0.16666666 / FWA = 0.0103841
 FWA = 1 + FWA = 0.9896158
 FWA = FWA * &0476 = 0.2474039

 Location &49 contains the value 0
 Bits 0 and 1 of &49 are not set, so exit with FWA unchanged.
Example 5: SIN(2.41)

 Set &046C = 2.41 (the argument)
 Set FWB to 0.785398165 (PI/4)
 FWA = FWA + FWB = 3.195398
 FWA = FWA * 0.636619772 = 2.0342533
 ?&49 = 2 (Integer part of FWA value). FWA = IWA = 2
 Store the FWA in &0471
 FWA = FWA * 1.57080078 = 3.1416014
 FWA = FWA + &046C = 0.7316014
 Store FWA in &046C
 Store &0471 (2) in FWA
 FWA = FWA * 0.0000044544511 = 0.0000089
 FWA = FWA + &046C = 0.7315924
 Store the FWA in &0476
 FWA = FWA * FWA = 0.5352275
 [&A861]: FWA = 1 / FWA = 1.8683641
 Store FWA in &046C
 FWA = FWA + 0.0119090311 = 1.8564551
 FWA = 0.000107499 / FWA = 0.0000057
 FWA = 0.017164024 + FWA = 0.0171583
 FWA = &046C + FWA = 1.8512058
 FWA = 0.0013095369 / FWA = 0.0007073
 FWA = 0.04999999 + FWA = 0.0507072
 FWA = &046C + FWA = 1.9190714
 FWA = 0.16666666 / FWA = 0.0868475
 FWA = 1 + FWA = 0.9131524
 FWA = FWA * &0476 = 0.6680554

 Location &49 contains the value 2
 Bit 0 of &49 is not set, so no need to adjust the calculated value
 Bit 1 of &49 is set, so FWA =  FWA = 0.6680554
Example 6: SIN(5.63)

 Set &046C = 5.63 (the argument)
 Set FWB to 0.785398165 (PI/4)
 FWA = FWA + FWB = 6.415398
 FWA = FWA * 0.636619772 = 4.0841688
 ?&49 = 4 (Integer part of FWA value). FWA = IWA = 4
 Store the FWA in &0471
 FWA = FWA * 1.57080078 = 6.2832028
 FWA = FWA + &046C = 0.6532028
 Store FWA in &046C
 Store &0471 (4) in FWA
 FWA = FWA * 0.0000044544511 = 0.0000178
 FWA = FWA + &046C = 0.6531849
 [&A990] Store the FWA in &0476
 FWA = FWA * FWA = 0.4266506
 [&A861]: FWA = 1 / FWA = 2.3438381
 Store FWA in &046C
 FWA = FWA + 0.0119090311 = 2.3319291
 FWA = 0.000107499 / FWA = 0.000046
 FWA = 0.017164024 + FWA = 0.0171179
 FWA = &046C + FWA = 2.3267202
 FWA = 0.0013095369 / FWA = 0.0056282
 FWA = 0.04999999 + FWA = 0.0556281
 FWA = &046C + FWA = 2.3994662
 FWA = 0.16666666 / FWA = 0.0694598
 FWA = 1 + FWA = 0.9305401
 FWA = FWA * &0476 = 0.6078147

 Location &49 contains the value 4
 Bits 0 and 1 of &49 are not set, so exit with FWA unchanged.
Example 7: SIN(90)

 Set &046C = 90 (the argument)
 Set FWB to 0.785398165 (PI/4)
 FWA = FWA + FWB = 90.785398
 FWA = FWA * 0.636619772 = 57.795773
 ?&49 = 57 (Integer part of FWA value). FWA = IWA = 57
 Store the FWA in &0471
 FWA = FWA * 1.57080078 = 89.53564
 FWA = FWA + &046C = 0.4643601
 Store FWA in &046C
 Store &0471 (57) in FWA
 FWA = FWA * 0.0000044544511 = 0.0002539
 FWA = FWA + &046C = 0.464614
 Store the FWA in &0476
 FWA = FWA * FWA = 0.2158661
 [&A861]: FWA = 1 / FWA = 4.6324998
 Store FWA in &046C
 FWA = FWA + 0.0119090311 = 4.6205908
 FWA = 0.000107499 / FWA = 0.0000023
 FWA = 0.017164024 + FWA = 0.0171616
 FWA = &046C + FWA = 4.6153381
 FWA = 0.0013095369 / FWA = 0.0002837
 FWA = 0.04999999 + FWA = 0.0502836
 FWA = &046C + FWA = 4.6827834
 FWA = 0.16666666 / FWA = 0.0355913
 FWA = 1 + FWA = 0.9644086
 FWA = FWA * &0476 = 0.4480777

 Location &49 contains the value 57 (0011 1001)
 Bit 0 of &49 is set, so adjust the calculated value as follows:
 FWA = FWA * FWA = 0.2007736
 FWA = 1  FWA = 0.7992263
 FWA = SQR(FWA) = 0.8939945
 Bit 1 of &49 is not set, so exit with FWA unchanged.
This diagram shows the relationship between the trig functions.
Disassembly for the SIN routine
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