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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra float.spad}
\author{Michael Monagan}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{domain FLOAT Float}
As reported in bug number 4733 (rounding of negative numbers)
errors were observed in operations such as
\begin{verbatim}
-> round(-3.9)
-> truncate(-3.9)
\end{verbatim}
The problem is the unexpected behaviour of the shift
with negative integer arguments.
\begin{verbatim}
-> shift(-7,-1)
\end{verbatim}
returns -4 while the code here in float expects the
value to be -3. shift uses the lisp function ASH
'arithmetic shift left' but the spad code expects
an unsigned 'logical' shift. See
\begin{verbatim}
http://www.lispworks.com/reference/HyperSpec/Body/f_ash.htm#ash
\end{verbatim}
A new internal function shift2 is defined in terms of
shift to compensate for the use of ASH and provide the
required function.
It is currently unknown whether the unexpected behaviour
of shift for negative arguments will cause bugs in other
parts of Axiom.
<<domain FLOAT Float>>=
)abbrev domain FLOAT Float
B ==> Boolean
I ==> Integer
S ==> String
PI ==> PositiveInteger
RN ==> Fraction Integer
SF ==> DoubleFloat
N ==> NonNegativeInteger
++ Author: Michael Monagan
++ Date Created:
++ December 1987
++ Change History:
++ 19 Jun 1990
++ Basic Operations: outputFloating, outputFixed, outputGeneral, outputSpacing,
++ atan, convert, exp1, log2, log10, normalize, rationalApproximation,
++ relerror, shift, / , **
++ Keywords: float, floating point, number
++ Description: \spadtype{Float} implements arbitrary precision floating
++ point arithmetic.
++ The number of significant digits of each operation can be set
++ to an arbitrary value (the default is 20 decimal digits).
++ The operation \spad{float(mantissa,exponent,\spadfunFrom{base}{FloatingPointSystem})} for integer
++ \spad{mantissa}, \spad{exponent} specifies the number
++ \spad{mantissa * \spadfunFrom{base}{FloatingPointSystem} ** exponent}
++ The underlying representation for floats is binary
++ not decimal. The implications of this are described below.
++
++ The model adopted is that arithmetic operations are rounded to
++ to nearest unit in the last place, that is, accurate to within
++ \spad{2**(-\spadfunFrom{bits}{FloatingPointSystem})}.
++ Also, the elementary functions and constants are
++ accurate to one unit in the last place.
++ A float is represented as a record of two integers, the mantissa
++ and the exponent. The \spadfunFrom{base}{FloatingPointSystem}
++ of the representation is binary, hence
++ a \spad{Record(m:mantissa,e:exponent)} represents the number \spad{m * 2 ** e}.
++ Though it is not assumed that the underlying integers are represented
++ with a binary \spadfunFrom{base}{FloatingPointSystem},
++ the code will be most efficient when this is the
++ the case (this is true in most implementations of Lisp).
++ The decision to choose the \spadfunFrom{base}{FloatingPointSystem} to be
++ binary has some unfortunate
++ consequences. First, decimal numbers like 0.3 cannot be represented
++ exactly. Second, there is a further loss of accuracy during
++ conversion to decimal for output. To compensate for this, if d
++ digits of precision are specified, \spad{1 + ceiling(log2 d)} bits are used.
++ Two numbers that are displayed identically may therefore be
++ not equal. On the other hand, a significant efficiency loss would
++ be incurred if we chose to use a decimal \spadfunFrom{base}{FloatingPointSystem} when the underlying
++ integer base is binary.
++
++ Algorithms used:
++ For the elementary functions, the general approach is to apply
++ identities so that the taylor series can be used, and, so
++ that it will converge within \spad{O( sqrt n )} steps. For example,
++ using the identity \spad{exp(x) = exp(x/2)**2}, we can compute
++ \spad{exp(1/3)} to n digits of precision as follows. We have
++ \spad{exp(1/3) = exp(2 ** (-sqrt s) / 3) ** (2 ** sqrt s)}.
++ The taylor series will converge in less than sqrt n steps and the
++ exponentiation requires sqrt n multiplications for a total of
++ \spad{2 sqrt n} multiplications. Assuming integer multiplication costs
++ \spad{O( n**2 )} the overall running time is \spad{O( sqrt(n) n**2 )}.
++ This approach is the best known approach for precisions up to
++ about 10,000 digits at which point the methods of Brent
++ which are \spad{O( log(n) n**2 )} become competitive. Note also that
++ summing the terms of the taylor series for the elementary
++ functions is done using integer operations. This avoids the
++ overhead of floating point operations and results in efficient
++ code at low precisions. This implementation makes no attempt
++ to reuse storage, relying on the underlying system to do
++ \spadgloss{garbage collection}. I estimate that the efficiency of this
++ package at low precisions could be improved by a factor of 2
++ if in-place operations were available.
++
++ Running times: in the following, n is the number of bits of precision
++ \spad{*}, \spad{/}, \spad{sqrt}, \spad{pi}, \spad{exp1}, \spad{log2}, \spad{log10}: \spad{ O( n**2 )}
++ \spad{exp}, \spad{log}, \spad{sin}, \spad{atan}: \spad{ O( sqrt(n) n**2 )}
++ The other elementary functions are coded in terms of the ones above.
Float():
Join(FloatingPointSystem, DifferentialRing, ConvertibleTo String,_
CoercibleTo DoubleFloat, TranscendentalFunctionCategory, _
ConvertibleTo InputForm,ConvertibleFrom SF) with
/ : (%, I) -> %
++ x / i computes the division from x by an integer i.
**: (%, %) -> %
++ x ** y computes \spad{exp(y log x)} where \spad{x >= 0}.
normalize: % -> %
++ normalize(x) normalizes x at current precision.
relerror : (%, %) -> I
++ relerror(x,y) computes the absolute value of \spad{x - y} divided by
++ y, when \spad{y \~= 0}.
shift: (%, I) -> %
++ shift(x,n) adds n to the exponent of float x.
rationalApproximation: (%, N) -> RN
++ rationalApproximation(f, n) computes a rational approximation
++ r to f with relative error \spad{< 10**(-n)}.
rationalApproximation: (%, N, N) -> RN
++ rationalApproximation(f, n, b) computes a rational
++ approximation r to f with relative error \spad{< b**(-n)}, that is
++ \spad{|(r-f)/f| < b**(-n)}.
log2 : () -> %
++ log2() returns \spad{ln 2}, i.e. \spad{0.6931471805...}.
log10: () -> %
++ log10() returns \spad{ln 10}: \spad{2.3025809299...}.
exp1 : () -> %
++ exp1() returns exp 1: \spad{2.7182818284...}.
atan : (%,%) -> %
++ atan(x,y) computes the arc tangent from x with phase y.
log2 : % -> %
++ log2(x) computes the logarithm for x to base 2.
log10: % -> %
++ log10(x) computes the logarithm for x to base 10.
outputFloating: () -> Void
++ outputFloating() sets the output mode to floating (scientific) notation, i.e.
++ \spad{mantissa * 10 exponent} is displayed as \spad{0.mantissa E exponent}.
outputFloating: N -> Void
++ outputFloating(n) sets the output mode to floating (scientific) notation
++ with n significant digits displayed after the decimal point.
outputFixed: () -> Void
++ outputFixed() sets the output mode to fixed point notation;
++ the output will contain a decimal point.
outputFixed: N -> Void
++ outputFixed(n) sets the output mode to fixed point notation,
++ with n digits displayed after the decimal point.
outputGeneral: () -> Void
++ outputGeneral() sets the output mode (default mode) to general
++ notation; numbers will be displayed in either fixed or floating
++ (scientific) notation depending on the magnitude.
outputGeneral: N -> Void
++ outputGeneral(n) sets the output mode to general notation
++ with n significant digits displayed.
outputSpacing: N -> Void
++ outputSpacing(n) inserts a space after n (default 10) digits on output;
++ outputSpacing(0) means no spaces are inserted.
arbitraryPrecision
arbitraryExponent
== add
BASE ==> 2
BITS:Reference(PI) := ref 68 -- 20 digits
LENGTH ==> %ilength$Foreign(Builtin)
ISQRT ==> approxSqrt$IntegerRoots(I)
Rep := Record( mantissa:I, exponent:I )
StoredConstant ==> Record( precision:PI, value:% )
UCA ==> Record( unit:%, coef:%, associate:% )
inc ==> increasePrecision
dec ==> decreasePrecision
-- local utility operations
shift2 : (I,I) -> I -- WSP: fix bug in shift
times : (%,%) -> % -- multiply x and y with no rounding
itimes: (I,%) -> % -- multiply by a small integer
chop: (%,PI) -> % -- chop x at p bits of precision
dvide: (%,%) -> % -- divide x by y with no rounding
square: (%,I) -> % -- repeated squaring with chopping
power: (%,I) -> % -- x ** n with chopping
plus: (%,%) -> % -- addition with no rounding
sub: (%,%) -> % -- subtraction with no rounding
negate: % -> % -- negation with no rounding
ceillog10base2: PI -> PI -- rational approximation
floorln2: PI -> PI -- rational approximation
atanSeries: % -> % -- atan(x) by taylor series |x| < 1/2
atanInverse: I -> % -- atan(1/n) for n an integer > 1
expInverse: I -> % -- exp(1/n) for n an integer
expSeries: % -> % -- exp(x) by taylor series |x| < 1/2
logSeries: % -> % -- log(x) by taylor series 1/2 < x < 2
sinSeries: % -> % -- sin(x) by taylor series |x| < 1/2
cosSeries: % -> % -- cos(x) by taylor series |x| < 1/2
piRamanujan: () -> % -- pi using Ramanujans series
shift2(x,y) == sign(x)*shift(sign(x)*x,y)
asin x ==
zero? x => 0
negative? x => -asin(-x)
one? x => pi()/2
x > 1 => error "asin: argument > 1 in magnitude"
inc 5; r := atan(x/sqrt(sub(1,times(x,x)))); dec 5
normalize r
acos x ==
zero? x => pi()/2
negative? x => (inc 3; r := pi()-acos(-x); dec 3; normalize r)
one? x => 0
x > 1 => error "acos: argument > 1 in magnitude"
inc 5; r := atan(sqrt(sub(1,times(x,x)))/x); dec 5
normalize r
atan(x,y) ==
x = 0 =>
positive? y => pi()/2
negative? y => -pi()/2
0
-- Only count on first quadrant being on principal branch.
theta := atan abs(y/x)
if negative? x then theta := pi() - theta
if negative? y then theta := - theta
theta
atan x ==
zero? x => 0
negative? x => -atan(-x)
if x > 1 then
inc 4
r := if zero? fractionPart x and x < [bits(),0] then atanInverse wholePart x
else atan(1/x)
r := pi()/2 - r
dec 4
return normalize r
-- make |x| < O( 2**(-sqrt p) ) < 1/2 to speed series convergence
-- by using the formula atan(x) = 2*atan(x/(1+sqrt(1+x**2)))
k := ISQRT (bits()-100)::I quo 5
k := max(0,2 + k + order x)
inc(2*k)
for i in 1..k repeat x := x/(1+sqrt(1+x*x))
t := atanSeries x
dec(2*k)
t := shift(t,k)
normalize t
atanSeries x ==
-- atan(x) = x (1 - x**2/3 + x**4/5 - x**6/7 + ...) |x| < 1
p := bits() + LENGTH bits() + 2
s:I := d:I := shift(1,p)
y := times(x,x)
t := m := - shift2(y.mantissa,y.exponent+p)
for i in 3.. by 2 while t ~= 0 repeat
s := s + t quo i
t := (m * t) quo d
x * [s,-p]
atanInverse n ==
-- compute atan(1/n) for an integer n > 1
-- atan n = 1/n - 1/n**3/3 + 1/n**5/4 - ...
-- pi = 16 atan(1/5) - 4 atan(1/239)
n2 := -n*n
e:I := bits() + LENGTH bits() + LENGTH n + 1
s:I := shift(1,e) quo n
t:I := s quo n2
for k in 3.. by 2 while t ~= 0 repeat
s := s + t quo k
t := t quo n2
normalize [s,-e]
sin x ==
s := sign x; x := abs x; p := bits(); inc 4
if x > [6,0] then (inc p; x := 2*pi()*fractionPart(x/pi()/2); bits p)
if x > [3,0] then (inc p; s := -s; x := x - pi(); bits p)
if x > [3,-1] then (inc p; x := pi() - x; dec p)
-- make |x| < O( 2**(-sqrt p) ) < 1/2 to speed series convergence
-- by using the formula sin(3*x/3) = 3 sin(x/3) - 4 sin(x/3)**3
-- the running time is O( sqrt p M(p) ) assuming |x| < 1
k := ISQRT (bits()-100)::I quo 4
k := max(0,2 + k + order x)
if positive? k then (inc k; x := x / 3**k::N)
r := sinSeries x
for i in 1..k repeat r := itimes(3,r)-shift(r**3,2)
bits p
s * r
sinSeries x ==
-- sin(x) = x (1 - x**2/3! + x**4/5! - x**6/7! + ... |x| < 1/2
p := bits() + LENGTH bits() + 2
y := times(x,x)
s:I := d:I := shift(1,p)
m:I := - shift2(y.mantissa,y.exponent+p)
t:I := m quo 6
for i in 4.. by 2 while t ~= 0 repeat
s := s + t
t := (m * t) quo (i*(i+1))
t := t quo d
x * [s,-p]
cos x ==
s:I := 1; x := abs x; p := bits(); inc 4
if x > [6,0] then (inc p; x := 2*pi()*fractionPart(x/pi()/2); dec p)
if x > [3,0] then (inc p; s := -s; x := x-pi(); dec p)
if x > [1,0] then
-- take care of the accuracy problem near pi/2
inc p; x := pi()/2-x; bits p; x := normalize x
return (s * sin x)
-- make |x| < O( 2**(-sqrt p) ) < 1/2 to speed series convergence
-- by using the formula cos(2*x/2) = 2 cos(x/2)**2 - 1
-- the running time is O( sqrt p M(p) ) assuming |x| < 1
k := ISQRT (bits()-100)::I quo 3
k := max(0,2 + k + order x)
-- need to increase precision by more than k, otherwise recursion
-- causes loss of accuracy.
-- Michael Monagan suggests adding a factor of log(k)
if positive? k then (inc(k+length(k)**2); x := shift(x,-k))
r := cosSeries x
for i in 1..k repeat r := shift(r*r,1)-1
bits p
s * r
cosSeries x ==
-- cos(x) = 1 - x**2/2! + x**4/4! - x**6/6! + ... |x| < 1/2
p := bits() + LENGTH bits() + 1
y := times(x,x)
s:I := d:I := shift(1,p)
m:I := - shift2(y.mantissa,y.exponent+p)
t:I := m quo 2
for i in 3.. by 2 while t ~= 0 repeat
s := s + t
t := (m * t) quo (i*(i+1))
t := t quo d
normalize [s,-p]
tan x ==
s := sign x; x := abs x; p := bits(); inc 6
if x > [3,0] then (inc p; x := pi()*fractionPart(x/pi()); dec p)
if x > [3,-1] then (inc p; x := pi()-x; s := -s; dec p)
if x > 1 then (c := cos x; t := sqrt(1-c*c)/c)
else (c := sin x; t := c/sqrt(1-c*c))
bits p
s * t
P:StoredConstant := [1,[1,2]]
pi() ==
-- We use Ramanujan's identity to compute pi.
-- The running time is quadratic in the precision.
-- This is about twice as fast as Machin's identity on Lisp/VM
-- pi = 16 atan(1/5) - 4 atan(1/239)
bits() <= P.precision => normalize P.value
(P := [bits(), piRamanujan()]) value
piRamanujan() ==
-- Ramanujans identity for 1/pi
-- Reference: Shanks and Wrench, Math Comp, 1962
-- "Calculation of pi to 100,000 Decimals".
n := bits() + LENGTH bits() + 11
t:I := shift(1,n) quo 882
d:I := 4*882**2
s:I := 0
for i in 2.. by 2 for j in 1123.. by 21460 while t ~= 0 repeat
s := s + j*t
m := -(i-1)*(2*i-1)*(2*i-3)
t := (m*t) quo (d*i**3)
1 / [s,-n-2]
sinh x ==
zero? x => 0
lost:I := max(- order x,0)
2*lost > bits() => x
inc(5+lost); e := exp x; s := (e-1/e)/2; dec(5+lost)
normalize s
cosh x ==
(inc 5; e := exp x; c := (e+1/e)/2; dec 5; normalize c)
tanh x ==
zero? x => 0
lost:I := max(- order x,0)
2*lost > bits() => x
inc(6+lost); e := exp x; e := e*e; t := (e-1)/(e+1); dec(6+lost)
normalize t
asinh x ==
p := min(0,order x)
if zero? x or 2*p < -bits() then return x
inc(5-p); r := log(x+sqrt(1+x*x)); dec(5-p)
normalize r
acosh x ==
if x < 1 then error "invalid argument to acosh"
inc 5; r := log(x+sqrt(sub(times(x,x),1))); dec 5
normalize r
atanh x ==
if x > 1 or x < -1 then error "invalid argument to atanh"
p := min(0,order x)
if zero? x or 2*p < -bits() then return x
inc(5-p); r := log((x+1)/(1-x))/2; dec(5-p)
normalize r
log x ==
negative? x => error "negative log"
zero? x => error "log 0 generated"
p := bits(); inc 5
-- apply log(x) = n log 2 + log(x/2**n) so that 1/2 < x < 2
if negative?(n := order x) then n := n+1
l := if n = 0 then 0 else (x := shift(x,-n); n * log2())
-- speed the series convergence by finding m and k such that
-- | exp(m/2**k) x - 1 | < 1 / 2 ** O(sqrt p)
-- write log(exp(m/2**k) x) as m/2**k + log x
k := ISQRT (p-100)::I quo 3
if k > 1 then
k := max(1,k+order(x-1))
inc k
ek := expInverse (2**k::N)
dec(p quo 2); m := order square(x,k); inc(p quo 2)
m := (6847196937 * m) quo 9878417065 -- m := m log 2
x := x * ek ** (-m)
l := l + [m,-k]
l := l + logSeries x
bits p
normalize l
logSeries x ==
-- log(x) = 2 y (1 + y**2/3 + y**4/5 ...) for y = (x-1) / (x+1)
-- given 1/2 < x < 2 on input we have -1/3 < y < 1/3
p := bits() + (g := LENGTH bits() + 3)
inc g; y := (x-1)/(x+1); dec g
s:I := d:I := shift(1,p)
z := times(y,y)
t := m := shift2(z.mantissa,z.exponent+p)
for i in 3.. by 2 while t ~= 0 repeat
s := s + t quo i
t := m * t quo d
y * [s,1-p]
L2:StoredConstant := [1,1]
log2() ==
-- log x = 2 * sum( ((x-1)/(x+1))**(2*k+1)/(2*k+1), k=1.. )
-- log 2 = 2 * sum( 1/9**k / (2*k+1), k=0..n ) / 3
n := bits() :: N
n <= L2.precision => normalize L2.value
n := n + LENGTH n + 3 -- guard bits
s:I := shift(1,n+1) quo 3
t:I := s quo 9
for k in 3.. by 2 while t ~= 0 repeat
s := s + t quo k
t := t quo 9
L2 := [bits(),[s,-n]]
normalize L2.value
L10:StoredConstant := [1,[1,1]]
log10() ==
-- log x = 2 * sum( ((x-1)/(x+1))**(2*k+1)/(2*k+1), k=0.. )
-- log 5/4 = 2 * sum( 1/81**k / (2*k+1), k=0.. ) / 9
n := bits() :: N
n <= L10.precision => normalize L10.value
n := n + LENGTH n + 5 -- guard bits
s:I := shift(1,n+1) quo 9
t:I := s quo 81
for k in 3.. by 2 while t ~= 0 repeat
s := s + t quo k
t := t quo 81
-- We have log 10 = log 5 + log 2 and log 5/4 = log 5 - 2 log 2
inc 2; L10 := [bits(),[s,-n] + 3*log2()]; dec 2
normalize L10.value
log2(x) == (inc 2; r := log(x)/log2(); dec 2; normalize r)
log10(x) == (inc 2; r := log(x)/log10(); dec 2; normalize r)
exp(x) ==
-- exp(n+x) = exp(1)**n exp(x) for n such that |x| < 1
p := bits(); inc 5; e1:% := 1
if (n := wholePart x) ~= 0 then
inc LENGTH n; e1 := exp1() ** n; dec LENGTH n
x := fractionPart x
if zero? x then (bits p; return normalize e1)
-- make |x| < O( 2**(-sqrt p) ) < 1/2 to speed series convergence
-- by repeated use of the formula exp(2*x/2) = exp(x/2)**2
-- results in an overall running time of O( sqrt p M(p) )
k := ISQRT (p-100)::I quo 3
k := max(0,2 + k + order x)
if positive? k then (inc k; x := shift(x,-k))
e := expSeries x
if positive? k then e := square(e,k)
bits p
e * e1
expSeries x ==
-- exp(x) = 1 + x + x**2/2 + ... + x**i/i! valid for all x
p := bits() + LENGTH bits() + 1
s:I := d:I := shift(1,p)
t:I := n:I := shift2(x.mantissa,x.exponent+p)
for i in 2.. while t ~= 0 repeat
s := s + t
t := (n * t) quo i
t := t quo d
normalize [s,-p]
expInverse k ==
-- computes exp(1/k) via continued fraction
p0:I := 2*k+1; p1:I := 6*k*p0+1
q0:I := 2*k-1; q1:I := 6*k*q0+1
for i in 10*k.. by 4*k while 2 * LENGTH p0 < bits() repeat
(p0,p1) := (p1,i*p1+p0)
(q0,q1) := (q1,i*q1+q0)
dvide([p1,0],[q1,0])
E:StoredConstant := [1,[1,1]]
exp1() ==
if bits() > E.precision then E := [bits(),expInverse 1]
normalize E.value
sqrt x ==
negative? x => error "negative sqrt"
m := x.mantissa; e := x.exponent
l := LENGTH m
p := 2 * bits() - l + 2
if odd?(e-l) then p := p - 1
i := shift2(x.mantissa,p)
-- ISQRT uses a variable precision newton iteration
i := ISQRT i
normalize [i,(e-p) quo 2]
bits() == BITS()
bits(n) == (t := bits(); BITS() := n; t)
precision() == bits()
precision(n) == bits(n)
increasePrecision n == (b := bits(); bits((b + n)::PI); b)
decreasePrecision n == (b := bits(); bits((b - n)::PI); b)
ceillog10base2 n == ((13301 * n + 4003) quo 4004) :: PI
digits() == max(1,4004 * (bits()-1) quo 13301)::PI
digits(n) == (t := digits(); bits (1 + ceillog10base2 n); t)
order(a) == LENGTH a.mantissa + a.exponent - 1
relerror(a,b) == order((a-b)/b)
0 == [0,0]
1 == [1,0]
base() == BASE
mantissa x == x.mantissa
exponent x == x.exponent
one? a == a = 1
zero? a == zero?(a.mantissa)
negative? a == negative?(a.mantissa)
positive? a == positive?(a.mantissa)
chop(x,p) ==
e : I := LENGTH x.mantissa - p
if positive? e then x := [shift2(x.mantissa,-e),x.exponent+e]
x
float(m,e) == normalize [m,e]
float(m,e,b) ==
m = 0 => 0
inc 4; r := m * [b,0] ** e; dec 4
normalize r
normalize x ==
m := x.mantissa
m = 0 => 0
e : I := LENGTH m - bits()
if positive? e then
y := shift2(m,1-e)
if odd? y then
y := (if positive? y then y+1 else y-1) quo 2
if LENGTH y > bits() then
y := y quo 2
e := e+1
else y := y quo 2
x := [y,x.exponent+e]
x
shift(x:%,n:I) == [x.mantissa,x.exponent+n]
x = y ==
order x = order y and sign x = sign y and zero? (x - y)
x < y ==
y.mantissa = 0 => negative? x.mantissa
x.mantissa = 0 => positive? y.mantissa
negative? x and positive? y => true
negative? y and positive? x => false
order x < order y => positive? x
order x > order y => negative? x
negative? (x-y)
abs x == if negative? x then -x else normalize x
ceiling x ==
if negative? x then return (-floor(-x))
if zero? fractionPart x then x else truncate x + 1
wholePart x == shift2(x.mantissa,x.exponent)
floor x == if negative? x then -ceiling(-x) else truncate x
round x == (half := [sign x,-1]; truncate(x + half))
sign x == if negative? x.mantissa then -1 else 1
truncate x ==
if x.exponent >= 0 then return x
normalize [shift2(x.mantissa,x.exponent),0]
recip(x) == if x=0 then "failed" else 1/x
differentiate x == 0
- x == normalize negate x
negate x == [-x.mantissa,x.exponent]
x + y == normalize plus(x,y)
x - y == normalize plus(x,negate y)
sub(x,y) == plus(x,negate y)
plus(x,y) ==
mx := x.mantissa; my := y.mantissa
mx = 0 => y
my = 0 => x
ex := x.exponent; ey := y.exponent
ex = ey => [mx+my,ex]
de := ex + LENGTH mx - ey - LENGTH my
de > bits()+1 => x
de < -(bits()+1) => y
if ex < ey then (mx,my,ex,ey) := (my,mx,ey,ex)
mw := my + shift2(mx,ex-ey)
[mw,ey]
x:% * y:% == normalize times (x,y)
x:I * y:% ==
if LENGTH x > bits() then normalize [x,0] * y
else normalize [x * y.mantissa,y.exponent]
x:% / y:% == normalize dvide(x,y)
x:% / y:I ==
if LENGTH y > bits() then x / normalize [y,0] else x / [y,0]
inv x == 1 / x
times(x:%,y:%) == [x.mantissa * y.mantissa, x.exponent + y.exponent]
itimes(n:I,y:%) == [n * y.mantissa,y.exponent]
dvide(x,y) ==
ew := LENGTH y.mantissa - LENGTH x.mantissa + bits() + 1
mw := shift2(x.mantissa,ew) quo y.mantissa
ew := x.exponent - y.exponent - ew
[mw,ew]
square(x,n) ==
ma := x.mantissa; ex := x.exponent
for k in 1..n repeat
ma := ma * ma; ex := ex + ex
l:I := bits()::I - LENGTH ma
ma := shift2(ma,l); ex := ex - l
[ma,ex]
power(x,n) ==
y:% := 1; z:% := x
repeat
if odd? n then y := chop( times(y,z), bits() )
if (n := n quo 2) = 0 then return y
z := chop( times(z,z), bits() )
x:% ** y:% ==
x = 0 =>
y = 0 => error "0**0 is undefined"
negative? y => error "division by 0"
0
y = 0 => 1
y = 1 => x
x = 1 => 1
p := abs order y + 5
inc p; r := exp(y*log(x)); dec p
normalize r
x:% ** r:RN ==
x = 0 =>
r = 0 => error "0**0 is undefined"
negative? r => error "division by 0"
0
r = 0 => 1
r = 1 => x
x = 1 => 1
n := numer r
d := denom r
negative? x =>
odd? d =>
odd? n => return -((-x)**r)
return ((-x)**r)
error "negative root"
if d = 2 then
inc LENGTH n; y := sqrt(x); y := y**n; dec LENGTH n
return normalize y
y := [n,0]/[d,0]
x ** y
x:% ** n:I ==
x = 0 =>
n = 0 => error "0**0 is undefined"
negative? n => error "division by 0"
0
n = 0 => 1
n = 1 => x
x = 1 => 1
p := bits()
bits(p + LENGTH n + 2)
y := power(x,abs n)
if negative? n then y := dvide(1,y)
bits p
normalize y
-- Utility routines for conversion to decimal
ceilLength10: I -> I
chop10: (%,I) -> %
convert10:(%,I) -> %
floorLength10: I -> I
length10: I -> I
normalize10: (%,I) -> %
quotient10: (%,%,I) -> %
power10: (%,I,I) -> %
times10: (%,%,I) -> %
convert10(x,d) ==
m := x.mantissa; e := x.exponent
--!! deal with bits here
b := bits(); (q,r) := divide(abs e, b)
b := 2**b::N; r := 2**r::N
-- compute 2**e = b**q * r
h := power10([b,0],q,d+5)
h := chop10([r*h.mantissa,h.exponent],d+5)
if negative? e then h := quotient10([m,0],h,d)
else times10([m,0],h,d)
ceilLength10 n == 146 * LENGTH n quo 485 + 1
floorLength10 n == 643 * LENGTH n quo 2136
-- length10 n == DECIMAL_-LENGTH(n)$Lisp
length10 n ==
ln := LENGTH(n:=abs n)
upper := 76573 * ln quo 254370
lower := 21306 * (ln-1) quo 70777
upper = lower => upper + 1
n := n quo (10**lower::N)
while n >= 10 repeat
n:= n quo 10
lower := lower + 1
lower + 1
chop10(x,p) ==
e : I := floorLength10 x.mantissa - p
if positive? e then x := [x.mantissa quo 10**e::N,x.exponent+e]
x
normalize10(x,p) ==
ma := x.mantissa
ex := x.exponent
e : I := length10 ma - p
if positive? e then
ma := ma quo 10**(e-1)::N
ex := ex + e
(ma,r) := divide(ma, 10)
if r > 4 then
ma := ma + 1
if ma = 10**p::N then (ma := 1; ex := ex + p)
[ma,ex]
times10(x,y,p) == normalize10(times(x,y),p)
quotient10(x,y,p) ==
ew := floorLength10 y.mantissa - ceilLength10 x.mantissa + p + 2
if negative? ew then ew := 0
mw := (x.mantissa * 10**ew::N) quo y.mantissa
ew := x.exponent - y.exponent - ew
normalize10([mw,ew],p)
power10(x,n,d) ==
x = 0 => 0
n = 0 => 1
n = 1 => x
x = 1 => 1
p:I := d + LENGTH n + 1
e:I := n
y:% := 1
z:% := x
repeat
if odd? e then y := chop10(times(y,z),p)
if (e := e quo 2) = 0 then return y
z := chop10(times(z,z),p)
--------------------------------
-- Output routines for Floats --
--------------------------------
zero ==> char("0")
separator ==> space()$Character
SPACING : Reference(N) := ref 10
OUTMODE : Reference(S) := ref "general"
OUTPREC : Reference(I) := ref(-1)
fixed : % -> S
floating : % -> S
general : % -> S
padFromLeft(s:S):S ==
zero? SPACING() => s
n:I := #s - 1
t := new( (n + 1 + n quo SPACING()) :: N , separator )
for i in 0..n for j in minIndex t .. repeat
t.j := s.(i + minIndex s)
if (i+1) rem SPACING() = 0 then j := j+1
t
padFromRight(s:S):S ==
SPACING() = 0 => s
n:I := #s - 1
t := new( (n + 1 + n quo SPACING()) :: N , separator )
for i in n..0 by -1 for j in maxIndex t .. by -1 repeat
t.j := s.(i + minIndex s)
if (n-i+1) rem SPACING() = 0 then j := j-1
t
fixed f ==
zero? f => "0.0"
zero? exponent f =>
padFromRight concat(convert(mantissa f)@S, ".0")
negative? f => concat("-", fixed abs f)
d := if OUTPREC() = -1 then digits()::I else OUTPREC()
-- g := convert10(abs f,digits); m := g.mantissa; e := g.exponent
g := convert10(abs f,d); m := g.mantissa; e := g.exponent
if OUTPREC() ~= -1 then
-- round g to OUTPREC digits after the decimal point
l := length10 m
if -e > OUTPREC() and -e < 2*digits()::I then
g := normalize10(g,l+e+OUTPREC())
m := g.mantissa; e := g.exponent
s := convert(m)@S; n := #s; o := e+n
p := if OUTPREC() = -1 then n::I else OUTPREC()
t:S
if e >= 0 then
s := concat(s, new(e::N, zero))
t := ""
else if o <= 0 then
t := concat(new((-o)::N,zero), s)
s := "0"
else
t := s(o + minIndex s .. n + minIndex s - 1)
s := s(minIndex s .. o + minIndex s - 1)
n := #t
if OUTPREC() = -1 then
t := rightTrim(t,zero)
if t = "" then t := "0"
else if n > p then t := t(minIndex t .. p + minIndex t- 1)
else t := concat(t, new((p-n)::N,zero))
concat(padFromRight s, concat(".", padFromLeft t))
floating f ==
zero? f => "0.0"
negative? f => concat("-", floating abs f)
t:S := if zero? SPACING() then "E" else " E "
zero? exponent f =>
s := convert(mantissa f)@S
concat ["0.", padFromLeft s, t, convert(#s)@S]
-- base conversion to decimal rounded to the requested precision
d := if OUTPREC() = -1 then digits()::I else OUTPREC()
g := convert10(f,d); m := g.mantissa; e := g.exponent
-- I'm assuming that length10 m = # s given n > 0
s := convert(m)@S; n := #s; o := e+n
s := padFromLeft s
concat ["0.", s, t, convert(o)@S]
general(f) ==
zero? f => "0.0"
negative? f => concat("-", general abs f)
d := if OUTPREC() = -1 then digits()::I else OUTPREC()
zero? exponent f =>
d := d + 1
s := convert(mantissa f)@S
OUTPREC() ~= -1 and (e := #s) > d =>
t:S := if zero? SPACING() then "E" else " E "
concat ["0.", padFromLeft s, t, convert(e)@S]
padFromRight concat(s, ".0")
-- base conversion to decimal rounded to the requested precision
g := convert10(f,d); m := g.mantissa; e := g.exponent
-- I'm assuming that length10 m = # s given n > 0
s := convert(m)@S; n := #s; o := n + e
-- Note: at least one digit is displayed after the decimal point
-- and trailing zeroes after the decimal point are dropped
if positive? o and o <= max(n,d) then
-- fixed format: add trailing zeroes before the decimal point
if o > n then s := concat(s, new((o-n)::N,zero))
t := rightTrim(s(o + minIndex s .. n + minIndex s - 1), zero)
if t = "" then t := "0" else t := padFromLeft t
s := padFromRight s(minIndex s .. o + minIndex s - 1)
concat(s, concat(".", t))
else if o <= 0 and o >= -5 then
-- fixed format: up to 5 leading zeroes after the decimal point
concat("0.",padFromLeft concat(new((-o)::N,zero),rightTrim(s,zero)))
else
-- print using E format written 0.mantissa E exponent
t := padFromLeft rightTrim(s,zero)
s := if zero? SPACING() then "E" else " E "
concat ["0.", t, s, convert(e+n)@S]
outputSpacing n == SPACING() := n
outputFixed() == (OUTMODE() := "fixed"; OUTPREC() := -1)
outputFixed n == (OUTMODE() := "fixed"; OUTPREC() := n::I)
outputGeneral() == (OUTMODE() := "general"; OUTPREC() := -1)
outputGeneral n == (OUTMODE() := "general"; OUTPREC() := n::I)
outputFloating() == (OUTMODE() := "floating"; OUTPREC() := -1)
outputFloating n == (OUTMODE() := "floating"; OUTPREC() := n::I)
convert(f):S ==
b:Integer :=
OUTPREC() = -1 and not zero? f =>
bits(length(abs mantissa f)::PositiveInteger)
0
s :=
OUTMODE() = "fixed" => fixed f
OUTMODE() = "floating" => floating f
OUTMODE() = "general" => general f
empty()$String
if positive? b then bits(b::PositiveInteger)
s = empty()$String => error "bad output mode"
s
coerce(f):OutputForm ==
f >= 0 => message(convert(f)@S)
- (coerce(-f)@OutputForm)
convert(f):InputForm ==
convert [convert('float), convert mantissa f,
convert exponent f, convert base()]$List(InputForm)
-- Conversion routines
convert(x:%):Float == x pretend Float
convert(x:%):SF == makeSF(x.mantissa,x.exponent)$Lisp
coerce(x:%):SF == convert(x)@SF
convert(sf:SF):% == float(mantissa sf,exponent sf,base()$SF)
retract(f:%):RN == rationalApproximation(f,(bits()-1)::N,BASE)
retractIfCan(f:%):Union(RN, "failed") ==
rationalApproximation(f,(bits()-1)::N,BASE)
retract(f:%):I ==
(f = (n := wholePart f)::%) => n
error "Not an integer"
retractIfCan(f:%):Union(I, "failed") ==
(f = (n := wholePart f)::%) => n
"failed"
rationalApproximation(f,d) == rationalApproximation(f,d,10)
rationalApproximation(f,d,b) ==
t: Integer
nu := f.mantissa; ex := f.exponent
if ex >= 0 then return ((nu*BASE**(ex::N))/1)
de := BASE**((-ex)::N)
if b < 2 then error "base must be > 1"
tol := b**d
s := nu; t := de
p0,p1,q0,q1 : Integer
p0 := 0; p1 := 1; q0 := 1; q1 := 0
repeat
(q,r) := divide(s, t)
p2 := q*p1+p0
q2 := q*q1+q0
if r = 0 or tol*abs(nu*q2-de*p2) < de*abs(p2) then return (p2/q2)
(p0,p1) := (p1,p2)
(q0,q1) := (q1,q2)
(s,t) := (t,r)
@
--% Float: arbitrary precision floating point arithmetic domain
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--Copyright (C) 2007-2010, Gabriel Dos Reis.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
-- - Redistributions of source code must retain the above copyright
-- notice, this list of conditions and the following disclaimer.
--
-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
-- the documentation and/or other materials provided with the
-- distribution.
--
-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
-- names of its contributors may be used to endorse or promote products
-- derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>
<<domain FLOAT Float>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|