DIRECTORY
Basics,
FloatingPointCommon,
Real,
RealSupport USING [Fix],
RealFns;
RealFnsImpl: CEDAR PROGRAM
IMPORTS Basics, FloatingPointCommon, Real, RealSupport
EXPORTS RealFns
= BEGIN

SingleReal: TYPE = MACHINE DEPENDENT RECORD [
sign (0: 0..0): BOOL,
exp (0: 1..8): Exponent,
m (0: 9..31): Mantissa];

Exponent: TYPE = CARDINAL [0..256);
Mantissa: TYPE = CARDINAL [0..HiddenBit);
HiddenBit: CARDINAL = 800000h;

ExponentBias: INTEGER = 127;
DenormExponent: INTEGER = Exponent.FIRST;
NaNExponent: INTEGER = Exponent.LAST;

SquareReal: TYPE = MACHINE DEPENDENT RECORD [
sign (0: 0..0): BOOL,
expD2 (0: 1..7): CARDINAL [0..128),
index (0: 8..12): CARDINAL [0..32),
m (0: 13..31): CARDINAL [0..2000000B)];



Bomb: PROC [x: REAL, exCode: FloatingPointCommon.Exception ¬ invalidOperation] = {
ERROR FloatingPointCommon.Error[exCode];
};

Ln2: REAL = 0.693147181;
LogBase2ofE: REAL ¬ 1/Ln2;
smallnumber: REAL = 0.00000000005;
PI: REAL = 3.1415926535;
twoPI: REAL ¬ PI*2;
piHalves: REAL ¬ PI/2;
pi3ovr2: REAL ¬ 3*piHalves;
pi3Eighths: REAL = pi3ovr2/4;
piFourths: REAL = PI/4;
piEighths: REAL ¬ PI/8;
radiansPerDegree: REAL ¬ PI/180;
degreesPerRadian: REAL ¬ 180/PI;
rec2pi: REAL ¬ 1.0/twoPI;
rec360: REAL ¬ 1.0/360.0;
fourOverPI: REAL ¬ 4.0/PI;
twoOverPI: REAL ¬ 2.0/PI;
invRoot2: REAL ¬ 0.70710678119;
lnRoot2pi: REAL ¬ 0.9189385332;
sqrtEps: REAL ¬ Real.FScale[1.0, -12];	-- 2­-12 = square root of machine epsilon

Exp: PUBLIC PROC [x: REAL] RETURNS [REAL] = {
r, f: REAL;
k: INTEGER;
fl: SingleReal ¬ LOOPHOLE[1.0, SingleReal];

SELECT ABS[x] FROM
< 3e-4 => RETURN [1+x];
< 3e-3 => RETURN [1+x+x*x*0.5];
ENDCASE;

x ¬ x*LogBase2ofE; --exponent of 2 (instead of e)
IF x < -ExponentBias THEN RETURN [0.0];
IF x > ExponentBias THEN Bomb[x];

k ¬ RealSupport.Fix[x]; --integer part (was InlineFixI)
x ¬ x - k; --fractional part

x ¬ x*Ln2; --now result = 2­k * exp(x) and 0<=x<Ln2
r ¬ x*x;

{
LoopCount: NAT = 3;
f ¬ 4*LoopCount + 2;
f ¬ (4*(LoopCount-0) - 2) + r/f;
f ¬ (4*(LoopCount-1) - 2) + r/f;
f ¬ (4*(LoopCount-2) - 2) + r/f;

};

x ¬ (f + x)/(f - x);
fl.exp ¬ k + ExponentBias;
RETURN[x*LOOPHOLE[fl, REAL]];
};

Log: PUBLIC PROC [base, arg: REAL] RETURNS [REAL] = {
RETURN[Ln[arg]/Ln[base]];
};

Ln: PUBLIC PROC [x: REAL] RETURNS [REAL] = {
fl: SingleReal;
m: INTEGER;
xm1: REAL ¬ x-1;
OneMinusX, OneMinusXPowerRep: REAL;
IF x <= 0 THEN Bomb[x];

SELECT ABS[xm1] FROM
> 3e-3 => {};
> 3e-4 => RETURN [xm1 - 0.5*xm1*xm1];
ENDCASE => RETURN [xm1];

fl ¬ LOOPHOLE[x, SingleReal];
m ¬ fl.exp - (ExponentBias - 1);
fl.exp ¬ ExponentBias - 1; --now 1/2 <= z < 1
x ¬ LOOPHOLE[fl, REAL];

IF x = 0.5 THEN RETURN[Ln2*(m-1)];

OneMinusX ¬ 1 - x;
x ¬ m*Ln2;
OneMinusXPowerRep ¬ 1;
FOR i: CARDINAL ¬ 1, i+1 DO
PreviousVal: REAL ¬ x;
OneMinusXPowerRep ¬ OneMinusXPowerRep*OneMinusX;
x ¬ x - OneMinusXPowerRep/i;
IF x = PreviousVal THEN EXIT;
ENDLOOP;

RETURN[x];
};

SqRt: PUBLIC PROC [a: REAL] RETURNS [b: REAL ¬ 0.0] = {
SELECT TRUE FROM
a <= 0.0 => {
IF a < 0.0 THEN Bomb[a];
b ¬ 0.0;
};
LOOPHOLE[a, SingleReal].exp = 0 =>
b ¬ SqRt[a * (LONG[20000B] * LONG[20000B])] / 20000B;
ENDCASE => {
aFmt: SquareReal ¬ LOOPHOLE[a, SquareReal];
bFmt: SingleReal ¬ LOOPHOLE[sqGuesses[aFmt.index], SingleReal];
aExp: CARDINAL ¬ aFmt.expD2;
bFmt.exp ¬ bFmt.exp + aExp - 63;
b ¬ LOOPHOLE[bFmt, REAL];
b ¬ LOOPHOLE[LOOPHOLE[b + a/b, LONG CARDINAL] - 40000000B, REAL];
b ¬ LOOPHOLE[LOOPHOLE[b + a/b, LONG CARDINAL] - 40000000B, REAL];
};
};

sqGuesses: ARRAY[0..32) OF CARD = [
7715741440B, 7717240700B, 7720514140B, 7721745140B,  -- 0.71782 .. 0.78043
7723155200B, 7724346000B, 7725517500B, 7726654000B,  -- 0.80021 .. 0.85681
7727773400B, 7731077100B, 7732167300B, 7733245000B,  -- 0.87486 .. 0.92691
7734310400B, 7735342500B, 7736363400B, 7737373700B,  -- 0.94362 .. 0.99206
7740370240B, 7741351440B, 7742314540B, 7743243000B,  -- 1.01516 .. 1.10370
7744155200B, 7745054400B, 7745741300B, 7746614600B,  -- 1.13167 .. 1.21172
7747457040B, 7750310640B, 7751132500B, 7751745000B,  -- 1.23725 .. 1.31085
7752550100B, 7753344400B, 7754132600B, 7754712500B   -- 1.33448 .. 1.40299
];

Root: PUBLIC PROC [index, arg: REAL] RETURNS [REAL] = {
RETURN[Exp[Ln[arg]/index]];
};

Power: PUBLIC PROC [base, exponent: REAL] RETURNS [REAL] = {
IF base = 0 THEN IF exponent = 0 THEN RETURN[1] ELSE RETURN[0];
RETURN[Exp[exponent*Ln[base]]];
};

Sin: PUBLIC PROC [radians: REAL] RETURNS [sin: REAL] = {
rem: REAL ¬ ABS[radians];
IF rem > twoPI THEN rem ¬ GetWithinTwoPI[rem];
{
SELECT RealSupport.Fix[rem * fourOverPI] FROM -- was InlineFixC
0 => {GO TO posSin};
1 => {rem ¬ piHalves - rem; GO TO posCos};
2 => {rem ¬ rem - piHalves; GO TO posCos};
3 => {rem ¬ PI - rem; GO TO posSin};
4 => {rem ¬ rem - PI; GO TO negSin};
5 => {rem ¬ pi3ovr2 - rem; GO TO negCos};
6 => {rem ¬ rem - pi3ovr2; GO TO negCos};
7 => {rem ¬ twoPI - rem; GO TO negSin};
ENDCASE => RETURN [0.0];
EXITS
posSin => {sin ¬ SinFirstOctant[rem]; IF radians < 0.0 THEN GO TO neg};
negSin => {sin ¬ SinFirstOctant[rem]; IF radians > 0.0 THEN GO TO neg};
posCos => {sin ¬ CosFirstOctant[rem]; IF radians < 0.0 THEN GO TO neg};
negCos => {sin ¬ CosFirstOctant[rem]; IF radians > 0.0 THEN GO TO neg};
};
EXITS neg => sin ¬ - sin;
};

Cos: PUBLIC PROC [radians: REAL] RETURNS [cos: REAL] = {
rem: REAL ¬ ABS[radians];
IF rem > twoPI THEN rem ¬ GetWithinTwoPI[rem];
{
SELECT RealSupport.Fix[rem * fourOverPI] FROM -- was InlineFixC
0 => {GO TO posCos};
1 => {rem ¬ piHalves - rem; GO TO posSin};
2 => {rem ¬ rem - piHalves; GO TO negSin};
3 => {rem ¬ PI - rem; GO TO negCos};
4 => {rem ¬ rem - PI; GO TO negCos};
5 => {rem ¬ pi3ovr2 - rem; GO TO negSin};
6 => {rem ¬ rem - pi3ovr2; GO TO posSin};
7 => {rem ¬ twoPI - rem; GO TO posCos};
ENDCASE => cos ¬ 1.0;
EXITS
posSin => {cos ¬ SinFirstOctant[rem]};
negSin => {cos ¬ SinFirstOctant[rem]; cos ¬ -cos};
posCos => {cos ¬ CosFirstOctant[rem]};
negCos => {cos ¬ CosFirstOctant[rem]; cos ¬ -cos};
};
};

SinDeg: PUBLIC PROC [degrees: REAL] RETURNS [sin: REAL] = {
IF ABS[degrees] > 360 THEN degrees ¬ GetWithin360[degrees];
sin ¬ Sin[degrees*radiansPerDegree];
};

CosDeg: PUBLIC PROC [degrees: REAL] RETURNS [cos: REAL] = {
degrees ¬ ABS[degrees];
IF degrees > 360 THEN degrees ¬ GetWithin360[degrees];
cos ¬ Cos[degrees*radiansPerDegree];
};

Tan: PUBLIC PROC [radians: REAL] RETURNS [tan: REAL] = {
RETURN[TanCot[radians, FALSE]];
};

CoTan: PUBLIC PROC [radians: REAL] RETURNS [cotan: REAL] = {
RETURN[TanCot[radians, TRUE]];
};

TanCot: PUBLIC PROC [x: REAL, cot: BOOLEAN] RETURNS [tan: REAL] = {
c1: REAL = 201.0/128.0;		-- exactly representable
c2: REAL = .000483826794897;	-- c1 + c2 = pi/2
xn, f, g: REAL;
num, den: REAL;
n: INT;
p1: REAL = -.0958017723;
q1: REAL = -0.429135777;
q2: REAL = .00971685835;

n ¬ Real.Round[twoOverPI*x];
xn ¬ REAL[n];
f ¬ (x - xn*c1) - xn*c2;
IF ABS[f] < sqrtEps THEN {
num ¬ f;
den ¬ 1.0;
}
ELSE {
g ¬ f*f;
num ¬ f + f*g*p1;
den ¬ (q2*g + q1)*g + 1;
};
IF cot THEN
{IF Basics.OddInt[n] THEN RETURN[-num/den] ELSE RETURN[den/num]}
ELSE
{IF Basics.OddInt[n] THEN RETURN[-den/num] ELSE RETURN[num/den]};
};

TanDeg: PUBLIC PROC [degrees: REAL] RETURNS [tan: REAL] = {
IF ABS[degrees] > 360 THEN degrees ¬ GetWithin360[degrees];
tan ¬ SinDeg[degrees] / CosDeg[degrees];
};

CoTanDeg: PUBLIC PROC [degrees: REAL] RETURNS [cotan: REAL] = {
IF ABS[degrees] > 360 THEN degrees ¬ GetWithin360[degrees];
cotan ¬ CoTan[degrees*radiansPerDegree];
};

ArcSin: PUBLIC PROC [x: REAL] RETURNS [ REAL] = {
z: REAL;

SELECT (z ¬ ABS[x]) FROM
< sqrtEps => RETURN[x];	-- so small that arcsin(x) = x + x­3/6 + x­5/40 + ... S x
< 0.5 => {						
RETURN[ArcTan[x, SqRt[1 - z*z]]];
};
ENDCASE => {
RETURN[ArcTan[x, SqRt[2*(1-z) - (1-z)*(1-z)]]];
};

};

ArcCos: PUBLIC PROC [x: REAL] RETURNS [REAL] = {
RETURN[IF x = -1.0
THEN PI
ELSE 2.0*ArcTan[SqRt[(1.0-x)/(1.0+x)], 1.0]];
};

ArcSinDeg: PUBLIC PROC [x: REAL] RETURNS [degrees: REAL] = {
degrees ¬ ArcSin[x]*degreesPerRadian;
};

ArcCosDeg: PUBLIC PROC [x: REAL] RETURNS [degrees: REAL] = {
degrees ¬ ArcCos[x]*degreesPerRadian;
};
tanPI16: REAL = .1989123673;
x2: REAL ¬ 1/(.414213562);
x22: REAL ¬ x2*x2 + 1;
tan3PI16: REAL = .668178638;
x4: REAL = 1.0;
x44: REAL ¬ x4*x4 + 1;
tan5PI16: REAL = 1.496605763;
x6: REAL ¬ 1/(2.41421356);
x66: REAL ¬ x6*x6 + 1;
tan7PI16: REAL = 5.02733949;

p0: REAL = .999999998;
p1: REAL = -.333331726;
p2: REAL = .1997952738;
p3: REAL = -.134450639;

ArcTan: PUBLIC PROC [y, x: REAL] RETURNS [radians: REAL] = {
s, v, t, t2, c, q: REAL;
IF ABS[x] <= ABS[smallnumber] THEN
IF ABS[y] <= ABS[smallnumber]
THEN RETURN[0]
ELSE IF y < 0 THEN RETURN[-piHalves] ELSE RETURN[piHalves];
IF x < 0
THEN IF y < 0 THEN {q ¬ -PI; s ¬ 1} ELSE {q ¬ PI; s ¬ -1}
ELSE IF y < 0 THEN {q ¬ 0; s ¬ -1} ELSE {q ¬ 0; s ¬ 1};
v ¬ ABS[y/x];
SELECT v FROM
< tanPI16 =>  {t ¬ v;  c ¬ 0};
< tan3PI16 => {t ¬ x2 - x22/(x2 + v); c ¬ piEighths};
< tan5PI16 => {t ¬ x4 - x44/(x4 + v); c ¬ piFourths};
< tan7PI16 => {t ¬ x6 - x66/(x6 + v); c ¬ pi3Eighths};
ENDCASE =>  {t ¬ -1/v;  c ¬ piHalves};
t2 ¬ t*t;
radians ¬ s*(t*(p0 + t2*(p1 + t2*(p2 + t2*p3))) + c) + q;
};

ArcTanDeg: PUBLIC PROC [y, x: REAL] RETURNS [degrees: REAL] = {
degrees ¬ ArcTan[y, x]*degreesPerRadian;
};


SinH: PUBLIC PROC [x: REAL] RETURNS [f: REAL] = {
ex: REAL = Exp[x];
emx: REAL = 1/ex;
f ¬ (ex-emx)/2.0
};

CosH: PUBLIC PROC [x: REAL] RETURNS [f: REAL] = {
ex: REAL = Exp[x];
emx: REAL = 1/ex;
f ¬ (ex+emx)/2.0
};

TanH: PUBLIC PROC [x: REAL] RETURNS [f: REAL] = {
IF ABS[x] > 20. THEN f ¬ (IF x<0. THEN -1. ELSE 1.)
ELSE {
ex: REAL = Exp[x];
emx: REAL = 1/ex;
f ¬ (ex-emx)/(ex+emx);
};
};

CotH: PUBLIC PROC [x: REAL] RETURNS [f: REAL] = {
IF ABS[x] > 20. THEN f ¬ (IF x<0. THEN -1. ELSE 1.)
ELSE {
ex: REAL = Exp[x];
emx: REAL = 1/ex;
f ¬ (ex+emx)/(ex-emx);
};
};

InvSinH: PUBLIC PROC [x: REAL] RETURNS [f: REAL] = {
f ¬ Ln[x+SqRt[x*x+1.0]];
};

InvCosH: PUBLIC PROC [x: REAL] RETURNS [f: REAL] = {
IF x < 1.0 THEN Bomb[x];
f ¬ Ln[x+SqRt[x*x-1.0]];
};

InvTanH: PUBLIC PROC [x: REAL] RETURNS [f: REAL] = {
IF ABS[x] >= 1.0 THEN Bomb[x];
f ¬ 0.5*Ln[(1.0+x)/(1.0-x)];
};

InvCotH: PUBLIC PROC [x: REAL] RETURNS [f: REAL] = {
IF ABS[x] <= 1.0 THEN Bomb[x];
f ¬ 0.5*Ln[(x+1.0)/(x-1.0)];
};

LnGamma: PUBLIC PROC [x: REAL] RETURNS [f: REAL] = {
phi: REAL;
lnAdjFact: REAL ¬ 0;
IF x <= 0.0 THEN Bomb[x];
IF x < 8 THEN { -- Gamma[x+1] = x! = x*(x-1)! = x*Gamma[x]
adjFact: REAL ¬ 1.0;
WHILE x < 8 DO
adjFact ¬ adjFact*x;
x ¬ x+1.0;
ENDLOOP;
lnAdjFact ¬ Ln[adjFact];
};
phi ¬ (-0.2762472E-2/(x*x)+0.8333327385E-1)/x; -- LGAM 5400 - 8.82 digits
f ¬ (x-0.5)*Ln[x]-x+lnRoot2pi+phi-lnAdjFact; -- Stirling's approx.
};

Gamma: PUBLIC PROC [x: REAL] RETURNS [REAL] = {
RETURN[Exp[LnGamma[x]]];
};

J0: PUBLIC PROC [x: REAL] RETURNS [f: REAL] = {
IF x < 0.0 THEN x ¬ -x;
IF x < 8.0 THEN {
x2: REAL = x*x;
P: REAL = ((((
(-0.1849052456E3)*x2+
(0.7739233017E5))*x2+
(-0.1121442418E8))*x2+
(0.6516196407E9))*x2+
(-0.1336259035E11))*x2+
(0.5756849057E11);
Q: REAL = ((((
x2+
(0.2678532712E3))*x2+
(0.5927264853E5))*x2+
(0.9494680718E7))*x2+
(0.1029532985E10))*x2+
(0.5756849041E11);
f ¬ P/Q;
}
ELSE {
X0: REAL = x-piFourths;
p0, q0: REAL;
[p0, q0] ¬ PQ0[x];
f ¬ SqRt[2.0/(PI*x)]*(p0*Cos[radians: X0]-q0*Sin[radians: X0]);
};
};
PQ0: PROC [x: REAL] RETURNS [P0, Q0: REAL] = {
z: REAL = 8.0/x;
z2: REAL = z*z;
P0 ¬ (( -- PZERO 6502 - 8.78 digits
(-0.165711693E-5)*z2+
(0.270871648E-4))*z2+
(-0.1098577711E-2))*z2+
(0.9999999983);
Q0 ¬ ((( -- QZERO 6901 - 9.13 digits
(0.576476505E-6)*z2+
(-0.6796319617E-5))*z2+
(0.1430262718E-3))*z2+
(-0.1562499927E-1))*z;
};

pi34: REAL ¬ 3*piFourths;

J1: PUBLIC PROC [x: REAL] RETURNS [f: REAL] = {
sign: REAL ¬ 1.0;
IF x < 0.0 THEN {sign ¬ -1.0; x ¬ -x};
IF x < 8.0 THEN {
x2: REAL = x*x;
P: REAL = (((
(0.6543001487E1)*x2+
(-0.1988185996E4))*x2+
(0.1978615055E6))*x2+
(-0.7064812917E7))*x2+
(0.6706259146E8);
Q: REAL = (((
x2+
(0.1766217266E3))*x2+
(0.2668508067E5))*x2+
(0.2635953855E7))*x2+
(0.1341252501E9);
f ¬ sign*x*P/Q;
}
ELSE {
X1: REAL = x-pi34;
p1, q1: REAL;
[p1, q1] ¬ PQ1[x];
f ¬ sign*SqRt[2.0/(PI*x)]*(p1*Cos[radians: X1]-q1*Sin[radians: X1]);
};
};
PQ1: PROC [x: REAL] RETURNS [P1, Q1: REAL] = {
z: REAL = 8.0/x;
z2: REAL = z*z;
P1 ¬ (( -- PONE 6702 - 8.72 digits
(0.19796752E-5)*z2+
(-0.348677998E-4))*z2+
(0.1830991520E-2))*z2+
(0.1000000001E1);
Q1 ¬ ((( -- QONE 7102 - 9.08 digits
(-0.67222066E-6)*z2+
(0.831923005E-5))*z2+
(-0.2002434943E-3))*z2+
(0.4687499917E-1))*z;
};
OldJn: PUBLIC PROC [n: INT, x: REAL] RETURNS [f: REAL] = {
sign: REAL ¬ 1.0;
IF n<0 THEN {n ¬ -n; sign ¬ IF n MOD 2 # 0 THEN -1.0 ELSE 1.0};
SELECT n FROM
0 => f ¬ J0[x];
1 => f ¬ sign*J1[x];
ENDCASE => {
v0: REAL ¬ J0[x];
v1: REAL ¬ J1[x];
FOR m: INT IN [2..n] DO
v2: REAL ¬ 2*(m-1)*v1/x-v0;
v0 ¬ v1;
v1 ¬ v2;
ENDLOOP;
f ¬ sign*v1;
};
};

lagestNumber: REAL ¬ Real.LargestNumber;
factLimit: NAT ¬ 8;

Jn: PUBLIC PROC [n: INT, x: REAL] RETURNS [f: REAL] = {
sign: REAL ¬ 1.0;
IF n<0 THEN {n ¬ -n; sign ¬ IF n MOD 2 # 0 THEN -1.0 ELSE 1.0};
SELECT TRUE FROM
n=0 => RETURN [J0[x]];
n=1 => f ¬ sign*J1[x];
x=0.0 => RETURN [0.0];
ENDCASE => {
halfX: REAL ~ x*0.5;
halfX2: REAL ~ halfX * halfX;
term, sum: REAL;
IF n <= factLimit THEN {
term ¬ halfX;
FOR j: NAT IN [2..n] DO term ¬ term * (halfX / j) ENDLOOP;
}
ELSE term ¬ Exp[n*Ln[halfX] - LnGamma[n+1] ];
sum ¬ term;
FOR k: REAL ¬ 1, k+1 DO
newSum: REAL;
term ¬ -term * (halfX2 / (k*(n+k)));
newSum ¬ sum + term;
IF newSum=sum THEN EXIT ELSE sum ¬ newSum;
ENDLOOP;
f ¬ sum*sign;
};
};

RationalFromReal: PUBLIC PROC [x: REAL, limit: INT ¬ LAST[INT]]
RETURNS [RealFns.Rational]
~ {

tooLargeToFix: REAL ~ Real.FScale[1.0, BITS[INT] - 1];			-- 2147483648.0
tooSmallToFix: REAL ~ Real.FScale[1.0, 1 - BITS[INT]];			-- 4.65661280e-10
halfTooSmallToFix: REAL ~ Real.FScale[tooSmallToFix, -1];	-- 4.65661280e-10 / 2
expForInt: INT ~ 150;		-- This exponent in SingleReal makes the significand an INT
ratZero: RealFns.Rational ~ [0, 1];

sign: INT ¬ 1;
flip: BOOL ¬ FALSE;									-- If TRUE, nk and dk are swapped
scale: INT;											-- Power of 2 to get x into integer domain
ak2, ak1, ak: CARD;									-- GCD arguments, initially 1 and x
ck, climit: CARD;										-- ck is GCD quotient and c.f. term k
nk, dk: INT;											-- Result num. and den., recursively found
nk1, dk2: INT ¬ 0; 									-- History terms for recursion
nk2, dk1: INT ¬ 1; 

IF limit <= 0 THEN RETURN[ratZero];				-- Insist limit > 0
IF x < 0.0 THEN { x ¬ -x; sign ¬ -1; };
IF x >= 1.0
THEN {											-- First continued fraction term is non-zero
rest: REAL;
IF x >= tooLargeToFix OR (ck ¬ Real.Fix[x]) >= CARD[limit]
THEN RETURN [[sign*limit, 1]];
flip ¬ TRUE;									-- Keep denominator larger, for fast loop test
nk ¬ 1;  dk ¬ ck;							-- Make new numerator and denominator
rest ¬ x - ck;
scale ¬ expForInt - LOOPHOLE[1.0, SingleReal].exp;
ak ¬ Real.Fix[Real.FScale[rest, scale]];
ak1 ¬ Real.Fix[Real.FScale[1.0, scale]];
}
ELSE {											-- First continued fraction term is zero
n, num: CARD;
IF x <= tooSmallToFix						-- Is x too tiny to be 1/INT ?
THEN {
IF x <= halfTooSmallToFix THEN RETURN[ratZero];
IF CARD[limit] > CARD[Real.Fix[0.5/x]]
THEN RETURN [[sign, limit]]
ELSE RETURN[ratZero];
};
scale ¬ expForInt - LOOPHOLE[x, SingleReal].exp;
ak1 ¬ Real.Fix[Real.FScale[x, scale]];
n ¬ MIN[BITS[CARD]-1, scale];			-- Stay within CARD arithmetic
num ¬ Basics.BITLSHIFT[1, n];
ck ¬ num/ak1;								-- First attempt at 1/x
ak ¬ num MOD ak1;							-- First attempt at remainder
WHILE (scale ¬ scale-n) > 0 DO			-- Shift quotient and remainder until done
n ¬ MIN[8, scale];						-- That 8 is from 24 bits of x in a 32 bit CARD
num ¬ Basics.BITLSHIFT[ak, n];
ck ¬ Basics.BITLSHIFT[ck, n] + num/ak1;
ak ¬ num MOD ak1;						-- Reduce remainder
ENDLOOP;
IF ck >= CARD[limit]					-- Is x too tiny to be 1/limit ?
THEN {
IF CARD[2*limit] > ck
THEN RETURN [[sign, limit]]
ELSE RETURN[ratZero];
};
nk ¬ 1;  dk ¬ ck;						-- Make new numerator and denominator
};

WHILE ak # 0 DO								-- If possible, quit when have exact result
ak2 ¬ ak1;  ak1 ¬ ak;						-- Prepare for next term
nk2 ¬ nk1;  nk1 ¬ nk;						-- (This loop does essentially all the work)
dk2 ¬ dk1;  dk1 ¬ dk;
ck ¬ ak2/ak1;								-- Get next term of continued fraction
ak ¬ ak2 - ck*ak1;							-- Get remainder (GCD step)
climit ¬ (limit - dk2)/dk1;					-- Anticipate result of recursion on denominator
IF climit <= ck THEN GOTO Interpolate;	-- Do not let denominator exceed limit
nk ¬ ck*nk1 + nk2;							-- Make new result numerator and denominator
dk ¬ ck*dk1 + dk2;
REPEAT
Interpolate => {
twoClimit: CARD ¬ 2*climit;
IF twoClimit >= ck THEN {				-- If climit < ck/2 no improvement possible
nk ¬ climit*nk1 + nk2;				-- Make limited numerator and denominator
dk ¬ climit*dk1 + dk2;
IF twoClimit = ck THEN TRUSTED {-- If climit = ck improvement not sure
-- Using climit is better only when dk2/dk1 > ak/ak1
dk2ak1Hi, dk2ak1Lo, dk1akHi, dk1akLo: CARD32;
[dk2ak1Hi, dk2ak1Lo] ¬ Mul32[IF flip THEN nk2 ELSE dk2, ak1];
[dk1akHi, dk1akLo] ¬ Mul32[IF flip THEN nk1 ELSE dk1, ak];
IF dk2ak1Hi < dk1akHi
OR (dk2ak1Hi = dk1akHi AND dk2ak1Lo <= dk1akLo)
THEN { nk ¬ nk1;  dk ¬ dk1; };	-- Not an improvement, so undo step
}
};
};
ENDLOOP;

IF flip
THEN RETURN [[sign*dk, nk]]
ELSE RETURN [[sign*nk, dk]];
};

OldRationalFromReal: PROC [x: REAL, limit: INT] RETURNS [RealFns.Rational] = {

oneOverX: REAL;
sign: INT ¬ 1;
inverse: BOOL ¬ FALSE;
p0, q0, lambda: INT;
q1, p2: INT ¬ 0;
p1, q2: INT ¬ 1;

IF x < 0.0 THEN { x ¬ -x; sign ¬ -1 };
IF x = 1.0 THEN RETURN [[numerator: sign, denominator: 1]];
IF x > 1.0 THEN { x ¬ 1.0/x; inverse ¬ TRUE };

lambda ¬ 0; -- = FLOOR[x]
WHILE (oneOverX ¬ (x-lambda))*REAL[limit-q1] > REAL[q2] DO

x ¬ 1.0/oneOverX;
lambda ¬ RealSupport.Fix[x]; -- was InlineFix
q0 ¬ q1;
q1 ¬ q2;
q2 ¬ lambda*q1+q0;
p0 ¬ p1;
p1 ¬ p2;
p2 ¬ lambda*p1+p0;
ENDLOOP;

IF NOT inverse THEN RETURN [[numerator: sign*p2, denominator: q2]]
ELSE RETURN [[numerator: sign*q2, denominator: p2]];
};

AlmostZero: PUBLIC PROC [x: REAL, distance: INTEGER [-126..127]] RETURNS [BOOL] = {
RETURN[LOOPHOLE[x, SingleReal].exp < (distance + ExponentBias)];
};

AlmostEqual: PUBLIC PROC [y, x: REAL, distance: INTEGER [-126..0]] RETURNS [BOOL] = {
xe: SingleReal ¬ LOOPHOLE[x];
ye: SingleReal ¬ LOOPHOLE[y];
SELECT TRUE FROM
xe.exp = NaNExponent => RETURN [FALSE];
ye.exp = NaNExponent => RETURN [FALSE];
x = y => RETURN [TRUE];
xe.sign # ye.sign => RETURN [FALSE];
distance = 0 => RETURN [TRUE];
xe.exp > ye.exp+1, ye.exp > xe.exp+1 => RETURN [FALSE];
ENDCASE => {
IF xe.exp = 0 OR ye.exp = 0 THEN {
reNorm: REAL ¬ 1.0;
LOOPHOLE[reNorm, SingleReal].exp ¬ LOOPHOLE[reNorm, SingleReal].exp + 24;
xe ¬ LOOPHOLE[x ¬ x * reNorm];
ye ¬ LOOPHOLE[y ¬ y * reNorm];
};
{
delta: SingleReal ¬ LOOPHOLE[(x - y)];
keyExp: INTEGER ¬ MAX[xe.exp, ye.exp] + distance;
SELECT INTEGER[delta.exp] FROM
< keyExp => RETURN [TRUE];
> keyExp => RETURN [FALSE];
ENDCASE => RETURN [delta.m = 0];
};
};
};


fact9recip: REAL ¬ 1.0/(INT[1]*2*3*4*5*6*7*8*9);
fact8recip: REAL ¬ 1.0/(INT[1]*2*3*4*5*6*7*8);
fact7recip: REAL ¬ 1.0/(INT[1]*2*3*4*5*6*7);
fact6recip: REAL ¬ 1.0/(INT[1]*2*3*4*5*6);
fact5recip: REAL ¬ 1.0/(INT[1]*2*3*4*5);
fact4recip: REAL ¬ 1.0/(INT[1]*2*3*4);
fact3recip: REAL ¬ 1.0/(INT[1]*2*3);
fact2recip: REAL ¬ 1.0/(INT[1]*2);

minEps: REAL ¬ 1.0 - LOOPHOLE[LOOPHOLE[1.0, LONG CARDINAL]-1, REAL];
sinCosEps: REAL ¬ SqRt[minEps];

SinFirstOctant: PROC [radians: REAL] RETURNS [REAL] = {
IF radians <= sinCosEps THEN RETURN [radians] ELSE {
r2: REAL ~ radians*radians;
RETURN [(((fact9recip*r2-fact7recip)*r2+fact5recip)*r2-fact3recip)*r2*radians+radians];
};
};

CosFirstOctant: PROC [radians: REAL] RETURNS [REAL] = {
IF radians <= sinCosEps THEN RETURN [1.0] ELSE {
r2: REAL ~ radians*radians;
RETURN [((((fact8recip*r2-fact6recip)*r2+fact4recip)*r2-fact2recip)*r2+1)];
};
};

PIc1: REAL ¬ 6.28125;	-- 402.0/64 (exact representation near twoPI)
PIc2: REAL ¬ twoPI-PIc1;	-- 2*PI - c1

GetWithinTwoPI: PROC [radians: REAL] RETURNS [REAL] = {
xn: REAL ¬ RealSupport.Fix[radians * rec2pi]; -- was InlineFix
RETURN [(radians - xn*PIc1) - xn*PIc2];
};

GetWithin360: PROC [degrees: REAL] RETURNS [REAL] = {
xn: REAL ¬ RealSupport.Fix[degrees * rec360]; -- was InlineFix
RETURN [degrees - xn*360];
};

Mul32: PROC [x, y: CARD32] RETURNS [CARD32, CARD32]
~ {
a: Basics.LongNumber ¬ [card[x]];
b: Basics.LongNumber ¬ [card[y]];
hi, lo, t1, t2: Basics.LongNumber;
cy: CARDINAL;
lo.card ¬ CARD32[a.lo]*CARD32[b.lo];
hi.card ¬ CARD32[a.hi]*CARD32[b.hi];
t1.card ¬ CARD32[a.hi]*CARD32[b.lo];
t2.card ¬ CARD32[a.lo]*CARD32[b.hi];
[cy, lo.hi] ¬ ADC3[lo.hi, t1.lo, t2.lo];
hi.card ¬ hi.card + t1.hi + t2.hi + cy;
RETURN[hi.card, lo.card];
};


ADC3: PROC [a, b, c: CARD16] RETURNS [CARD16, CARD16] = INLINE {
s: Basics.LongNumber;
s.card ¬ CARD32[a] + CARD32[b] + CARD32[c];
RETURN[s.hi, s.lo];
};


END.

September 16, 1980  3:13 PM, Stewart; Bug in Exp
September 28, 1980  8:28 PM, Stewart; Add AlmostEqual and AlmostZero, format
November 7, 1980  3:33 PM, Stewart; fix AlmostZero
August 27, 1982 1:07 pm, Stewart; CEDAR
October 24, 1984 4:37:17 pm PDT, Plass; New impls for Sin, Cos, SinDeg, CosDeg
November 2, 1984, Atkinson; Faster & more accurate SqRt, Sin, Cos, SinDeg, CosDeg
!ê
RealFnsImpl.mesa
Copyright Ó 1984, 1985, 1986, 1987, 1988, 1989, 1991 by Xerox Corporation.  All rights reserved.
Stewart on August 27, 1982 1:07 pm
Paul Rovner on May 4, 1983 10:13 am
Levin, August 8, 1983 4:39 pm
Plass, August 5, 1991 11:28 am PDT
E. McCreight, November 16, 1984 11:48:22 am PST (RealFnsExtrasImpl)
Russ Atkinson (RRA), July 5, 1989 11:36:56 pm PDT
Doug Wyatt, January 15, 1987 1:59:52 pm PST
Chauser, March 16, 1990 4:06 pm PST
Loner, February 26, 1991 8:33 am PST
Willie-s, August 6, 1991 1:06 pm PDT
the bias, amount to subtract to make the exponent "correct"
biased, just as it appears in SingleReal
biased, just as it appears in SingleReal
This declaration obviously depends on IEEE format 32-bit numbers (SingleReal).  The idea is that the index field has the low bit of the exponent and the high 4 bits of the mantissa, which is used to index a table of initial guesses at the square root, each guess is roughly halfway between the lowest and highest root with the given index, which allows convergence with only two iterations.  I don't know where the table originated.
Some useful constants (must be variables)
compute e^x using continued fractions
ACM Algorithm 229, Morelock, James C., "Elementary functions by continued fractions"
To keep the error small near 0.0, use this Taylor's series
Expand the continued fraction loop a few times
Note, if LoopCount > 3, then use the following comments
f _ (4*(LoopCount-3) - 2) + r/f;
f _ (4*(LoopCount-4) - 2) + r/f;
unscaled result is (f+x)/(f-x)
scale: multipy by 2^k by adding k to the exponent
logxy = Ln(y)/Ln(x)
To keep the error small near 1.0, use this Taylor's series
if x = z * 2^m then Ln(x) = m * Ln2 + Ln(z).  Here, 1/2 <= z < 1
special case for limit condition (x a power of 2)
RRA: The idea is to lookup a square root guess from the table, adjusting for the exponent, then perform a few iterations to get within a reasonable error.  HIGHLY IEEE DEPENDENT!
Denormalized exponent.  The guess is only valid when we have a normalized number, so we force it to be normalized, then correct for the shift.
Get the initial guess based on the top 4 bits of the mantissa and the bottom bit of the exponent.
Adjust the guess exponent based on adding half of the initial exponent (adjusted for the offset inherent in IEEE format).
The result is initially the guess
This is really b _ (b + a/b)/2, but much faster
This is really b _ (b + a/b)/2, but much faster
yth root of x = e^(Ln(x)/y)
x^y = e^(y*Ln(x))
We need to have rem IN [0..twoPI]
Now select the appropriate formula given the octant.
We need to have rem IN [0..twoPI]
Now select the appropriate formula given the octant.
We get better results if we compute in the degrees domain for a while.
We get better results if we compute in the degrees domain for a while.
if f is small, then tan(f) S f, otherwise use ratl approximation from Cody & Waite
RETURN[2*ArcTan[SqRt[(1-x)/(1+x)], 1]];

David Goldberg notes regarding the accuracy of two methods for computing ArcCos, where Method 1 is ArcTan[SqRt[1.0-x*x], x], and Method 2 is the one above:
When x ~ y, then x-y is dangerous if x and/or y have suffered roundoff error, because all the significant digits may be cancelled, leaving only roundoff error.  That's the trouble with 1-x*x; there will be roundoff error in x*x, and if x ~ 1, then 1-x*x can have a large relative error.  However, if x ~ 1, (in fact, if 1/2 < x < 2), then 1-x is exact because there is no roundoff error.  Thus 1/(1-x) is very accurate.  That's the theory according to floating point experts.
Method 1 is worse when x = 1-n*2^(-24), n = sqrt(2^(23)), i.e., x = 0.9998274.  The correct value is 1.857983e-02; Method 1 gives 1.858144e-2 (error .000161e-2) while Method 2 gives 1.858064e-2 (error .000081e-2).  So, in this case, Method 2 has half the error of Method 1.
 [Artwork node; type 'Artwork on' to command tool] 
The ArcCos Geometry for Method 2

Constants for ArcTan
This function is good to 8.7 decimal places and is taken from:
Computer Approximations, John F. Hart et.al. pp129(top 2nd col).
The following approximations are taken from the book "Computer Approximations", by Hart, Cheney, Lawson, Maehly, Mesztenyi, Rice, Thacher, and Witzgall, published by Wiley.  Function names and indexes refer to that book.

Bessel function of first kind of order 0
JZERO 5838 - 8.54 digits

Bessel function of first kind of order 1
JONE 6036 - 7.08 digits
..actually, JONE 6038 - 8.19 digits would be better, but that page is missing from my copy.


Bessel function of first kind of order n
Bessel function of first kind of order n
Jn(x) = Sum[k>=0] (-1)k(x/2)n+2k / k! (n+k)!
-- All best rational approximations of a real x may be obtained from x's continued fraction representation, x = c0 + 1/(c1 + 1/(c2 + 1/(...))) = [c0, c1, c2, ... ], by truncation to k terms and possibly "interpolation" of the last term.  The continued fraction expansion itself is obtained by a variant of the standard GCD algorithm, which is folded into the recursions generating successive numerators and denominators.  These recursions both have the same form: fk _ ck*fk-1 + fk-2.  For further information, see CSL-Notebook entry 88CSLN-0045, Nov. 28, 1988; cited there is Fundamentals of Mathematics, Volume I, MIT Press, 1983.  On return, |numer| <= limit, 0 < denom <= limit.
-- Handle first non-zero term of continued fraction, rest setup for integer GCD, sure to fit.
-- Treating 1.0 and x as integers, divide 1/x in a peculiar way to get accurate remainder
num _ Basics.BITLSHIFT[[lc[1]], n].lc;
num _ Basics.BITLSHIFT[[li[ak]], n].lc;
ck _ Basics.BITLSHIFT[[lc[ck]], n].lc + num/ak1;
All done with divide
[dk2ak1Hi, dk2ak1Lo] _ Mul32[[lc[IF flip THEN nk2 ELSE dk2]],[lc[ak1]]];
[dk1akHi, dk1akLo] _ Mul32[[lc[IF flip THEN nk1 ELSE dk1]],[lc[ak]]];
The best rational approximation to a real x is obtained using continued fractions.  The following algorithm is adapted from Wm. J. LeVeque's book, Fundamentals of Number Theory, Addison Wesley, 1977, page 229.  |num| <= limit, 0 < denom <= limit.
Here x lies in [0..1)
This test ensures that q2 never exceeds limit.
Because the initial x was <1, p2<=q2.
The best approximation so far is p2/q2, and q2 > p2.
The range of distance has been carefully chosen to
1.  make zero and all denormalized values of x return TRUE
2.  make all NaN values of x return FALSE
AlmostEqual[y, x, distance] ==
NotNaN[x] AND NotNaN[y] AND
ABS[x-y] <= MAX[ABS[x], ABS[y]]*2^distance
for valid x & y:
x = y => TRUE
Sign[x] # Sign[y] => FALSE
Sign[x] = Sign[y] AND distance = 0 => TRUE
Rewritten by RRA on November 11, 1986, previous implementation returned FALSE when x = y = 0.0, which was counterintuitive, to say the least.
NotNaN[x] => FALSE
NotNaN[y] => FALSE
exactly equal valid numbers are always AlmostEqual
(use REAL equality to check with -0.0)
xe.sign # ye.sign => ABS[x-y] > MAX[ABS[x],ABS[y]]
xe.sign = ye.sign AND x # y => ABS[x-y] < MAX[ABS[x],ABS[y]]
ABS[x-y] > MAX[ABS[x],ABS[y]]*0.5 
Sigh, one of them is denormalized!
Helpful internal functions
Coefficients for the taylor series for sin and cos.
roughly 5.96e-8, the minimum relative error between two different reals
RRA: IF radians <= sinEps THEN SinFirstOctant[radians] = radians AND CosFirstOctant[radians] = 1.0 to the precision of our number system.  The contribution of the smaller term of two consecutive terms in the Sin or Cos polynomials is down by the square from the larger (and a factor of at least two).  So sinCosEps roughly equals 1.726e-4.  Testing for this case improves accuracy at small values.
This version of Sin is only guaranteed to be accurate within the first octant: [0.0..PI/4], which is approximately [0.0..0.785+].  When radians = PI/4 we have the maximum contribution of the term after the fact9recip term, for a truncation error of roughly 1.76e-9, which is within minEps*Sin[radians] (= 4.2e-8).
This version of Cos is only guaranteed to be accurate within the first octant: [0.0..PI/4].  When radians = PI/4 we have the maximum contribution of the terms after the fact8recip term, for a truncation error of roughly 2.46e-8, which is within minEps*Cos[PI/4] (= 4.2e-8).
Two constants used to add N*PI with more than normal precision.
Let the returned value be R.  Then
radians <= 0 J radians S xn*twoPI + R & -twoPI < R < 0
radians >= 0 J radians S xn*twoPI + R & 0 < R < twoPI
Let the returned value be R.  Then
degrees <= 0 J degrees S xn*360 + R & -360 < R < 0
degrees >= 0 J degrees S xn*360 + R & 0 < R < 360
Add with Carry
Carl Hauser, January 21, 1988 2:08:18 pm PST
For PortableCedar
Removed instances of Real.InlineFix* , replacing with RealSupport.Fix and labelling what was replaced.

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