WARNING: Numbers in this section are *octal* (and occasionally
*binary*) unless followed by a decimal point. 105=69.. (And
105.=69 hexadecimal.)
[PDP-10 Info]

0/ TLCA 1,1(1) 1/ see below 2/ ROT 1,9 3/ JRST 0

CHANGE: GOOD INITIAL CONTENTS OF 1: none 377767,,377767; 757777,,757757; etc. TLC 1,2(1) 373777,,0; 300000,,0 TLC 1,3(1) -2,,-2; -5,,-1; -6,,-1 ROT 1,1 7,,7; A0000B,,A0000B ROTC 1,11 ;Can't use TLCA over data. AOJA 1,0

DATAI 2 ADDB 1,2 ROTC 2,-22 XOR 1,2 JRST .-4

4000,,4 1000,,2002 2000,,4 0,,1002

Also try the FADR after the XOR, switches = 1001,,1.

NEW X = OLD X - *epsilon* * OLD Y

NEW Y = OLD Y + *epsilon* * NEW(!) X

This makes a very round ellipse centered at the origin with its size
determined by the initial point. *epsilon* determines the angular
velocity of the circulating point, and slightly affects the
eccentricity. If *epsilon* is a power of 2, then we don't even need
multiplication, let alone square roots, sines, and cosines! The
"circle" will be perfectly stable because the points soon become
periodic.

The circle algorithm was invented by mistake when I tried to save one register in a display hack! Ben Gurley had an amazing display hack using only about six or seven instructions, and it was a great wonder. But it was basically line-oriented. It occurred to me that it would be exciting to have curves, and I was trying to get a curve display hack with minimal instructions.

X(N+1) = (2 - *epsilon*^2) * X(N) - X(N-1)

Y(N+1) = (2 - *epsilon*) * Y(N) - Y(N-1).

These are just the Chebychev recurrence with cos *theta* (the angular
increment) = 1-*epsilon*^2/2. Thus X(N) and Y(N) are expressible in the
form R cos(N *theta* + *phi*). The *phi*'s and R for X(N) and Y(N) can be
found from N=0,1. The *phi*'s will differ by less than *pi*/2 so that the
curve is not really a circle. The algorithm is useful nevertheless,
because it needs no sine or square root function, even to get
started.

X(N) and Y(N) are also expressible in closed form in the algebra of ordered pairs described under linear recurrences, but they lack the remarkable numerical stability of the "simultaneous" form of the recurrence.

X^2 - *epsilon* X Y + Y^2 = constant,

where the constant is determined by the initial point. This ellipse
has its major axis at 45 degrees (if *epsilon* > 0) or 135 degrees
(if *epsilon* < 0) and has eccentricity

sqrt(*epsilon*/(1 + *epsilon*/2)).

- If the result loops with period = 1 with sign +, you are on a sign-magnitude machine.
- If the result loops with period = 1 at -1, you are on a twos-complement machine.
- If the result loops with period > 1, including the beginning, you are on a ones-complement machine.
- If the result loops with period > 1, not including the beginning, your machine isn't binary -- the pattern should tell you the base.
- If you run out of memory, you are on a string or Bignum system.
- If arithmetic overflow is a fatal error, some fascist pig with a read-only mind is trying to enforce machine independence. But the very ability to trap overflow is machine dependent.

let X = the sum of many powers of two = ...111111

now add X to itself; X + X = ...111110

thus, 2X = X - 1 so X = -1

therefore algebra is run on a machine (the universe) which is twos-complement.

IMUL A,[377777,,1]

HRLOI A,-1(A) EQVI A,ORIGIN

HRLOI A,-ORIGIN(A) EQVI A,ORIGIN

COS: FADR A,[1.57079632679] ;pi/2 SIN: MOVM B,A ;ARGUMENT IN A CAMG B,[.00017] ; <= 3^(1/3) / 2^13 POPJ P, ;sin X = X, within 27. bits FDVRI A,(-3.0) PUSHJ P,SIN ;sin -X/3 FMPR B,B FSC B,2 FADRI B,(-3.0) FMPRB A,B ;sin X = 4(sin -X/3)^3 - 3(sin -X/3) POPJ P, ;sin in A, sin or |sin| in B ;|sin| in B occurs when angle is smaller than end test

sinh X = 3 sinh X/3 + 4 (sinh X/3)^3.

Changing the first -3.0 to a +9.0, then inserting PUSHJ P,.+1 after PUSHJ P,SIN gains about 20% in speed and uses half the pushdown space (< 5 levels in the first 4 quadrants). PUSHJ P,.+1 is a nice way to have something happen twice. Other useful angle multiplying formulas are

tanh X = (2 tanh X/2) / (1 + (tanh X/2)^2)

tan X = (2 tan X/2) / (1 - (tan X/2)^2),

if infinity is handled correctly. For cos and cosh, one can use

cos X = 1 - 2 (sin X/2)^2, cosh X = 1 + 2 (sinh X/2)^2.

In general, to compute functions like *e*^X, cos X, elliptic functions, etc. by
iterated application of double and triple argument formulas, it is necessary to
subtract out the constant in the Taylor series and transform the range
reduction formula accordingly. Thus:

F(X) = cos(X)-1

F(2 X) = 2 F*(F+2)

F(*epsilon*) = -*epsilon*^2/2

G(X) = *e*^X - 1

G(2 X) = G*(G+2)

G(*epsilon*) = *epsilon*

This is to prevent the destruction of the information in the
range-reduced argument by the addition of a quantity near 1 upon the
success of the *epsilon* test. The addition of such a quantity
in the actual recurrences is OK since the information is restored by
the multiply. In fact, a cheap and dirty test for F(*epsilon*)
sufficiently small is to see if the addition step has no effect.
People lucky enough to have a square root instruction can get
*natural log* by iterating

X <- X/(sqrt(1+X) + 1) until 1+X = 1.

Then multiply by 2^(number of iterations). Here, a LSH or FSC would work.

The correct *epsilon* test in such functions as the foregoing SIN are
generally the largest argument for which addition of the second term
has no effect on the first. In SIN, the first term is x and the
second is -x^3/6, so the answer is roughly the x which makes the ratio
of those terms 1/2^27; so x = sqrt(3) / 2^13. But this is not exact,
since the precise cutoff is where the neglected term is the power of 2
whose 1 bit coincides with the first neglected (28th) bit of the
fraction. Thus, x^3/6 = 1/2^27 * 1/2^13, so x = 3^(1/3) / 2^13.

LOG2: LSHC A,-33 MOVSI C,-201(A) TLC C,211000 ;Speciner's bum MOVI A,200 ;exponent and sign sentinel LOGL: LSH B,-9 REPEAT 7, FMPR B,B ;moby flunderflo LSH B,2 LSHC A,7 SOJG A,LOGL ;fails on 4th try LSH A,-1 FADR A,C POPJ P, ;answer in A

EXCH A,LOC1 EXCH A,LOC2 EXCH A,LOC1

TRCE A,BITS TRCE A,BITS TRCE A,BITS

(SETQ X (PROG2 0 Y (SETQ Y X)))

ROTC A,6 CAMG A,B EXCH A,B ROTC A,-6

ROTC A,N ROT B,-N ROTC B,N

ROTC A,N-36. EXCH A,C ROT B,36.-N ROTC A,N-72.

;B gets 7 bit character in A with even parity IMUL A,[2010040201] ;5 adjacent copies AND A,[21042104377] ;every 4th bit of left 4 copies + right copy IDIVI A,17<-7 ;casting out 15.'s in hexadecimal shifted 7 ;odd parity on 7 bits (Schroeppel) IMUL A,[10040201] ;4 adjacent copies IOR A,[7555555400] ;leaves every 3rd bit+offset+right copy IDIVI A,9<-7 ;powers of 2^3 are +-1 mod 9 ;changing 7555555400 to 27555555400 gives even parity ;if A is a 9 bit quantity, B gets number of 1's (Schroeppel) IMUL A,[1001001001] ;4 copies AND A,[42104210421] ;every 4th bit IDIVI A,17 ;casting out 15.'s in hexadecimal ;if A is 6 bit quantity, B gets 6 bits reversed (Schroeppel) IMUL A,[2020202] ;4 copies shifted AND A,[104422010] ;where bits coincide with reverse repeated base 2^8 IDIVI A,377 ;casting out 2^8 - 1's ;reverse 7 bits (Schroeppel) IMUL A,[10004002001] ;4 copies sep by 000's base 2 (may set arith. o'flow) AND A,[210210210010] ;where bits coincide with reverse repeated base 2^8 IDIVI A,377 ;casting out 377's ;reverse 8 bits (Schroeppel) MUL A,[100200401002] ;5 copies in A and B AND B,[20420420020] ;where bits coincide with reverse repeated base 2^10 ANDI A,41 ;" DIVI A,1777 ;casting out 2^10 - 1's

foo, lat /DATAI switches adm a /ADDB and (707070 adm b iot 14 /output AC sign bit to a music flip-flop jmp foo

FOO: DATAI 2 ADDB 1,2 AND 2,[707070707070] ;or 171717171717, 363636363636, 454545454545, ... ADDB 2,3 LDB 0,[360600,,2] JRST FOO

LDB B,[014300,,A] ;or MOVE B,A then LSH B,-1 AND B,[333333,,333333] SUB A,B LSH B,-1 AND B,[333333,,333333] SUBB A,B ;each octal digit is replaced by number of 1's in it LSH B,-3 ADD A,B AND A,[070707,,070707] IDIVI A,77 ;casting out 63.'s

DPT: IDIVI F,12 HRLM G,(P) ;tuck remainder on pushdown list SKIPE F PUSHJ P,DPT LDB G,[220600,,(P)] ;retrieve low 6 bits of remainder TRCE G,"0 ;convert digit to character SETOM CCT ;that was no digit! TYO: .IOT TYOCHN,G ;or DATA0 or IDPB ... AOS G,CCT POPJ P,

CONS: EXCH A,[EXCH A,[...[PUSHJ P,GC]...]] EXCH A,CONS

UNCONS: HRLI A,(EXCH A,) EXCH A,CONS EXCH A,@CONS

MULI A,400 ;exponent to A, fraction to A+1 TSC A,A ;1's complement magnitude of excess 200 exponent ASH A+1,-200-27.-8(A) ;answer in A+1

On the PDP-10

When you know the sign of A, and |A| < 2^26, you canUFA A,[+-233000,,] ;not in PDP-6 repertoire TLC A+1,233000 ;if those bits really bother you

FAD A,[+-233400,,] ;or FADR for rounded fix! TLC A,233400 ;if those bits are relevant

MOVE B,A MOVN C,B AND C,B ADD A,C MOVE D,A XOR D,B LSH D,-2 IDIVM D,C IOR A,C